Using fractal analysis of crown images to measure the structural condition of trees

Using fractal analysis of crown images to measure the structural condition of trees Abstract Observations of tree canopy structure are routinely used as an indicator of tree condition for the purposes of monitoring tree health, assessing habitat characteristics or evaluating the potential risk of tree failure. Trees are assigned to broad categories of structural condition using largely subjective methods based upon ground-based, visual observations by a surveyor. Such approaches can suffer from a lack of consistency between surveyors; are qualitative in nature and have low precision. In this study, a technique is developed for acquiring, processing and analysing hemispherical images of sessile oak (Quercus petraea (Matt.) Liebl.) tree crowns. We demonstrate that by calculating the fractal dimensions of tree crown images it is possible to define a continuous measurement scale of structural condition and to be able to quantify intra-category variance of tree crown structure. This approach corresponds with traditional categorical methods; however, we recognize that further work is required to precisely define interspecies thresholds. Our study demonstrates that this approach has the potential to form the basis of a new, transferable and objective methodology that can support a wide range of uses in arboriculture, ecology and forest science. Introduction Traditionally, the assessment of tree structural condition, as used in general tree surveys, relies upon simple methodologies and ground-based observations due to the physical complexities of directly measuring tree crowns. However, these traditional techniques are time consuming, manual and largely subjective. Subjectivity has been shown to prevent the same conclusions being reached during independent tree surveys, including surveys of the same trees by different, experienced tree surveyors (Norris, 2007). Predominantly, these assessments rely on a tree surveyor’s knowledge of ideal tree form, tree health, their ability to identify pests and disease, and the consideration of potential hazards and targets that are at risk of harm. Blennow et al. (2013) state that when managing trees or woodlands the use of subjective tree condition observations are not ideal, particularly where objective tree assessments would provide greater insights in the tree management decision process. Ultimately, traditional tree assessment procedures can result in subjective and potentially biased, field observations of tree condition, irrespective of how knowledgeable and experienced the surveyor is (Norris, 2007; Britt and Johnston, 2008). Trees are self-optimizing organisms that respond to a range of recurrent environmental demands and employ strategies to alter their form to minimize potential negative effects or optimize their structure for the greatest physiological benefit (Zimmerman and Brown, 1971; Mattheck and Breloer, 1994; Fourcaud and Dupuy et al., 2004; Pollardy, 2008). In most angiosperms, the lateral branches grow almost as fast, or in some instances faster, than the terminal leader. This process results in the characteristic broad crown structure common in this tree type (Pollardy, 2008; Burkhart and Tome, 2012). Tree form is typically the result of various influences combining the genetic potential, the demands of physiological processes, spatial competition in the crown and the effects of other environmental conditions, such as thigmomorphogenic change caused by repeated wind force effects. The shedding of branches through responsive self-pruning driven by abscission is a characteristic found in many tree species which has a direct effect on the shape of the crown (Pollardy, 2008). There are many additional reasons for trees to shed branches, or parts thereof; which are accelerated by the effects of colonizing pathogens e.g. fungal infestation, or external forces such as gravity or wind force. Indeed, the tree’s own physiology also increases the potential for crown dieback as trees age (King, 2011). Despite many potential stimuli affecting overall tree structure, the growth habits of trees are fundamentally controlled by the genetic predisposition of individual species throughout different tree growth stages. Therefore, the characteristic structure and form of differing tree species remain visually recognizable even after the external impacts are considered (Zimmerman and Brown, 1971). When trees reach late-maturity, there is a combined slowing down of both the stem diameter increment and extension growth in the crown, as a response of the influence of the tree species, genotype or its local environment (King, 2011). It is the recognition of these types of biotic and abiotic structural changes that tree surveyors use to aid the classifying of trees into discrete categories, ultimately aiming to gain insights into the tree’s condition. There have been many studies of tree crown structure in recent years, many of which utilize high-end technology such as light detecting and ranging (LiDAR) as the main method of data capture (Ørka et al., 2009; Ferraz et al., 2016). Specifically with LiDAR data investigations, it is understood that the success of tree investigation algorithms for location detection or height estimation is strongly correlated to the type of tree structure under analysis (Vauhkonen et al., 2012). Through analysis of aerial LiDAR data, boreal tree species have been identified at a species level due to differences in their tree structure signatures (Lina and Hyyppä, 2016), or through LiDAR waveform analysis which identifies structural features within the LiDAR wave (Hovi and Korhonen et al., 2016). Aerial LiDAR investigations are often supported with aerial imagery which is captured simultaneously as image based investigations also provide opportunities for tree canopy structure analysis (Dash and Watt et al., 2016). Furthermore, photogrammetric techniques such as digital stereo imagery and radar imagery have been used in tree canopy structure investigations (Holopainen and Vastaranta et al., 2014). For many researchers or environmental managers, a restrictive element of these types of investigations is the requirement for expensive, specialized research equipment that is often mounted on an aerial platform, such as an unmanned aerial vehicle, aeroplane or satellite. The use of hemispherical photography to undertake proximal tree crown assessments has a field history of more than 50 years, with forest ecologists, Evans and Coombe (1959) using the technique to investigate the available light climate under woodland canopies with an early prototype ‘Hill’ (fisheye) camera. This has remained a readily used, accessible and repeatable method for the investigation of tree canopy structure (Hale, 2004; Chianucci, 2016). Researchers have also previously used hemispherical imagery to assess canopy gap fraction or provide leaf area index assessments (Weiss et al., 2004; Beckschäfer et al., 2013), as it is understood that images captured by hemispherical, or fisheye, lenses provide opportunities for photogrammetric measurement (Schwalbe et al., 2009). Conducting photogrammetric analysis on hemispherical imagery falls within the remote, or indirect, methods of measurement which enable rapid, non-destructive determination of crown properties (Chason et al., 1991; Weiss et al., 2004). Modern advancements in digital cameras, coupled with readily available hemispherical lenses or lens adaptors, provide the opportunity for an off-the-shelf approach to photogrammetric research (Leblanc et al., 2005). When tree crowns are viewed from directly beneath, looking upwards towards the zenith viewing point (90° from the horizontal elevation), holes can be observed within the crown structure. The tree crown area is a complex arrangement of tree branches, combined with observable unoccupied areas between the different parts of the tree crown. This upward looking view provides a visual separation between the tree structure and the sky, which when photographed can be converted into a binary image with the occupied and background regions of the image coded ‘1’ and ‘0’, respectively (Beckschäfer et al., 2013; Sossa-Azuela et al., 2013). Image analysis techniques for pattern recognition in tree structures have identified features of lacunarity (the size and distribution of holes), complex spatial distributions or other morphologic features (Zheng et al., 1995; Frazer et al., 2005). Due to the unique geometry found in nature, the dimensions of natural, physical forms cannot readily be described in simple, integral terms (Mandelbrot, 1982; Dimri, 2000). Mandelbrot (1982) argues that more insightful measurements are required to measure pattern complexity, such as quantifying the degree of complexity in a structure. As trees exhibit natural structural variance, Mandelbrot (1982), also notes that it is the frequently anomalous nature of tree structure whose form is sculpted by, ‘chance, irregularities and non-uniformity’, that provides the opportunity for statistical investigation. Rian and Sassone (2014) demonstrate that the crown structures of trees are unique in their self-affine and highly irregular branching patterns. It has been stated that fractal dimensions (Df) can be used to quantify structural complexity in a continuous measure, theoretically ranging from 0 to infinity, which can be expressed as a single value (Mandelbrot, 1967; Kaye, 2008). Although tree crown structures are complex shapes, there are various examples of Df being used as a predictor variable for the classification of forest canopies (Zeide and Pfeifer, 1991; Zeide, 1998; Jonckheere et al., 2006; Zhang et al., 2007). The aim of this study was to develop an objective methodology to assess the structural condition of broadleaved tree crowns (Quercus sp.) by quantifying the complexity of the tree crowns through hemispherical images taken under leaf-off conditions. This approach was designed to overcome the limitations of current subjective field methodologies. The first objective was to develop an in-field data capture technique that was suitable for a range of subject trees across a variety of structural conditions. The second objective was to develop image processing methods for the assessment of crown structural condition. The third objective was to propose a new and objective means of evaluating tree structural condition on a continuous scale. Methodology Throughout three study areas across northwest Lancashire, England, 64 Sessile Oak trees (Quercus petraea (Matt.) Liebl.) were individually photographed using hemispherical imagery obtained from beneath subject tree canopies, looking towards the zenith viewpoint (Figure 1). The trees used in the study were either individual maiden trees, or trees that were located in closed canopy, woodland conditions. The trees were photographed over a single winter season in leaf-off condition, thereby allowing an unobscured view of the tree crown structure. To minimize potentially confounding variables, this method was applied to trees of the same species that were in the mature phases of tree development, specifically: early-mature (28 per cent), mature (25 per cent), late-mature (25 per cent), veteran and senescent (22 per cent) (Fay and de Berker, 1997). To achieve a suitable sample size, a locally prolific species was used in this study. Figure 1 View largeDownload slide A schematic of the field method for taking a hemispherical picture from beneath a tree canopy. The camera is situated on a standard tripod, and is levelled and pointing towards the zenith viewing point (90° from the horizontal elevation). In this example, the full extent of the crown is 4 m along the southern axis, and the image is taken at the 2 m mid-point. Figure 1 View largeDownload slide A schematic of the field method for taking a hemispherical picture from beneath a tree canopy. The camera is situated on a standard tripod, and is levelled and pointing towards the zenith viewing point (90° from the horizontal elevation). In this example, the full extent of the crown is 4 m along the southern axis, and the image is taken at the 2 m mid-point. Field methodology development Reference data on the trees structural condition were collected using a four-point categorical system, as is common in arboricultural assessments using traditional field techniques. The four-point method used in this research is not based upon a single specific method, but broadly upon several arboricultural tree survey methods (e.g. BS5837:2012 surveys which use a four level condition hierarchy, the ISA tree hazard evaluation, which uses four classification categories to generate an accumulative hazard score (Matheny and Clark, 1994; BSI, 2012), and is also comparable with a qualitative tree condition category assignment as described in Swetnam et al. 2016). Consequently, this approach is representative of similar tree survey methods where the assessment of trees leads to an empirical categorization of tree condition. Box 1 provides an overview of the classification descriptors. Once identified, the tree’s cardinal orientation was determined by the use of a field compass. The part of the crown that extended towards the southernmost point (i.e. the tree crown’s southern axis), was marked out along the ground with a standard surveyors tape and used as the linear axis upon which the crown images were taken at specific intervals. Camera set-up A high-resolution digital single-lens reflex (dSLR) camera (Canon EOS 550D DS126271) was used with an 18 mm lens and a hemispherical lens adaptor (Opteka Super Wide Fisheye Lens 0.20×). The lens adaptor permits focal length conversion into a 3.6 mm circular lens. The wide angle of the hemispherical lens enabled as much of each tree crown to be captured within each image as possible. The dSLR was placed on a standard photographic tripod, adjusted at each image capture location ensuring that the dSLR was positioned and levelled with the camera lens pointing vertically upward at ~0.5 m from the ground level. To account for variability in solar illumination, the images were taken during uniform sky conditions. These conditions occur predominantly when the sky is overcast, although this technique can also be used just before sunrise or just after sunset, should bright daytime conditions be expected (Song et al., 2014). Image acquisition and spatial sampling strategy Initially, the number of images captured per subject tree was influenced by the overall length of the crown along the southern axis. Early trials with image capture involved taking images at 1 m intervals along the southern axis, to the full extent of the crown. However, this produced a high number of replicates with large amounts of image content overlap. Inspection of these images identified two problems with this approach. Firstly, that there was ~90 per cent replication of content between the overlapping images (Figure 2a), and secondly, that additional tree features that were not required for the analysis were also captured. For example, additional stem wood was photographed in the images closest to the base of the tree (e.g. at 1 m and 2 m intervals), while large amounts of ‘sky’ was captured towards the canopy edge. Neither of these image components was required in the analysis. It followed that many of the repeated images was not within the optimal range for representing the fullest area of tree crown within an image. Repeated testing indicated that the optimal location for image capture was around the mid-point of the crown axis (Figure 2). Where there was no mid-point location on an exact 1 m interval of the southern axis mid-point, the distance was rounded up to the next whole metre. The southern axis was used for standardization purposes as the subject trees are located in the Northern hemisphere and our preference was to capture images on the non-shaded, south facing side of the trees. Figure 2 View largeDownload slide A schematic showing the optimized range for image capture (a), and the area of tree canopy structure analysed within this study (b). The area of interest is specifically the structural elements of the canopy. Too much ‘sky’ within the image reduces the amount of structure that can be analysed (a). Stem wood and other elements not required, are removed from the image by only analysing the structure inside a user selected bounding box area (b). The use of a bounding box allows images of both individual trees and trees within closed canopies to be analysed. Figure 2 View largeDownload slide A schematic showing the optimized range for image capture (a), and the area of tree canopy structure analysed within this study (b). The area of interest is specifically the structural elements of the canopy. Too much ‘sky’ within the image reduces the amount of structure that can be analysed (a). Stem wood and other elements not required, are removed from the image by only analysing the structure inside a user selected bounding box area (b). The use of a bounding box allows images of both individual trees and trees within closed canopies to be analysed. Immediately after acquisition, the quality of each image was visually assessed. This step was taken to ensure the images were suitable for later analysis and to allow additional images to be captured should the original image be unusable. The process of identifying the southern axis, setting-up the camera and completing image acquisition took between ~45 sec and ~1.5 min, depending on the complexity of the local topographic environment. Image preparation Upon return from the field, the images were re-examined on a desktop computer to check for image clarity, suitability in showing the area of interest, and for the presence of key features (Jones and Vaughn, 2010). A limitation of the in-field image proofing was that this was completed on the dSLR camera’s 2.7-inch screen; therefore it was conducted at a very coarse resolution. Of the original 247 images, 87 were removed for blurring or distortion errors, 96 images were removed as duplicates, leaving the sample size reduced to 64 images of individual trees, with a single image representing each tree. Box 1 Classification descriptors for the subjective arboricultural assessment of trees. Estimated remaining contribution (ERC) refers to a methodology used to consider the health, condition and structure of the tree and aids in classifying the tree in to the different categories adapted from (Barrell, 1993, 2001; Lonsdale, 1999; NTSG, 2011; BSI, 2012). Note: The images show trees in leaf-on condition to enable ease of comparison for the condition types. 4. Good    Dominant trees. Full crown, good extension growth and form. Typical for age and species. High number of buds. Healthy reaction growth to any injuries. Acceptable levels of colonization. ERC: > 40+ years  3. Moderate    Some signs of stress, crown dieback or retrenchment. Deadwood. Other signs of stress likely to be present. Remedial work may have previously taken place. Cavities rot or early disease may be present. ERC: ≥10–40 years  2. Poor    Obvious signs of dieback. Frequent deadwood. Clear signs of disease and decay. Overwhelming of the trees natural defences. Colonization by fungi, wood boring insects and other decay biota highly likely. ERC: ≤10 years  1. Dead    Physiological processes have ceased. Lack of active photosynthetic area. Colonization of fungi, wood boring insects and other decay biota highly likely. Extensive crown retrenchment, bark slough, brittle or collapsing structure. ERC: ≤ 0 years.  4. Good    Dominant trees. Full crown, good extension growth and form. Typical for age and species. High number of buds. Healthy reaction growth to any injuries. Acceptable levels of colonization. ERC: > 40+ years  3. Moderate    Some signs of stress, crown dieback or retrenchment. Deadwood. Other signs of stress likely to be present. Remedial work may have previously taken place. Cavities rot or early disease may be present. ERC: ≥10–40 years  2. Poor    Obvious signs of dieback. Frequent deadwood. Clear signs of disease and decay. Overwhelming of the trees natural defences. Colonization by fungi, wood boring insects and other decay biota highly likely. ERC: ≤10 years  1. Dead    Physiological processes have ceased. Lack of active photosynthetic area. Colonization of fungi, wood boring insects and other decay biota highly likely. Extensive crown retrenchment, bark slough, brittle or collapsing structure. ERC: ≤ 0 years.  4. Good    Dominant trees. Full crown, good extension growth and form. Typical for age and species. High number of buds. Healthy reaction growth to any injuries. Acceptable levels of colonization. ERC: > 40+ years  3. Moderate    Some signs of stress, crown dieback or retrenchment. Deadwood. Other signs of stress likely to be present. Remedial work may have previously taken place. Cavities rot or early disease may be present. ERC: ≥10–40 years  2. Poor    Obvious signs of dieback. Frequent deadwood. Clear signs of disease and decay. Overwhelming of the trees natural defences. Colonization by fungi, wood boring insects and other decay biota highly likely. ERC: ≤10 years  1. Dead    Physiological processes have ceased. Lack of active photosynthetic area. Colonization of fungi, wood boring insects and other decay biota highly likely. Extensive crown retrenchment, bark slough, brittle or collapsing structure. ERC: ≤ 0 years.  4. Good    Dominant trees. Full crown, good extension growth and form. Typical for age and species. High number of buds. Healthy reaction growth to any injuries. Acceptable levels of colonization. ERC: > 40+ years  3. Moderate    Some signs of stress, crown dieback or retrenchment. Deadwood. Other signs of stress likely to be present. Remedial work may have previously taken place. Cavities rot or early disease may be present. ERC: ≥10–40 years  2. Poor    Obvious signs of dieback. Frequent deadwood. Clear signs of disease and decay. Overwhelming of the trees natural defences. Colonization by fungi, wood boring insects and other decay biota highly likely. ERC: ≤10 years  1. Dead    Physiological processes have ceased. Lack of active photosynthetic area. Colonization of fungi, wood boring insects and other decay biota highly likely. Extensive crown retrenchment, bark slough, brittle or collapsing structure. ERC: ≤ 0 years.  Pre-processing interventions removed errors from the images that could affect the measurement of image metrics. Chromatic aberration (CA) is the misregistration of RGB channels causing interference with the dSLR Bayer-pattern sensor, leading to image deterioration and interference with pixel-based classification techniques (Schwalbe et al., 2009). In this study, CA was corrected by removing the red and blue channels, and converting the image to the green element of the RGB channels only. Quadratic or ‘barrel’ distortion is also associated with images captured using hemispherical lenses. A distortion correction algorithm (Vries, 2012) transformed the images from the distorted barrel extension to replicate an image captured at a normal focal length. This perspective distortion effect is influenced by the relative distances between the lens and subject canopy at which the image is captured, therefore, it is important that the relative distance was maintained during image capture. In order to reduce the effects of blurred images caused by contrast errors between colour ranges, an image sharpening algorithm was used. This algorithm was based upon un-sharp masking, where the image is sharpened by removing a blurred negative copy of the same image. The copied mask was laid over the original, resulting in a combined image that is visually sharper. Where there were instances of unsuccessful pre-processing, the affected images were not used in the investigation. The images were analysed in Matlab (version 2015a), where each image pixel was indexed and converted into binary form. This was achieved through applying uniform quantization where limited intensity resolution breaks the image colour space into individual pixels, which are indexed, and the pixel locations are mapped. A process of dithering corrects any potential quantization errors and limits the greyscale range of the image. This binarization procedure allows differentiation between the tree structure and other parts of the image, as optimum image analysis conditions are best achieved where there is high contrast between tree structure and the sky (Chen, Black et al. 1991). Defining the image analysis area Chianucci and Cutini (2012) describe that it is beneficial in image processing to reduce the field of view by masking some elements of the full hemisphere, thereby achieving greater spatial representation of heterogeneous tree crowns i.e. the inclusion of both dense and sparse crown regions in the analysis. At Figure 2b, image analysis is restricted to the part of tree canopy contained within the black bounding box, created on a per image basis. The analysis extent is influenced by standard forestry measurement conventions (West, 2009), with the lower bounding box edge originating at the point of estimated timber height. In decurrent trees, this is where the main stem bifurcates to such a degree that the main stem is no longer discernible. From here, the analysis area is bordered by the upper bounding box at the edge of the tree crown and avoids the image’s vignette region caused by the visible inner walls of the camera lens. The left and right boundaries of the image analysis area are demarked by adjoining lines between the upper and lower bounding box extents maximizing the crown analysis area, while again, also avoiding the vignette region at the edges of the image. Predictor variable creation Multiple indices were generated from the tree images that were developed into image metrics which were tested, both individually and in combination, for their suitability in describing the tree structural character. A description of the metrics is shown at Table 1. Table 1 Descriptions of analytical metrics used in an investigation to quantify tree structural condition. Name  Description  Convex hull area  An area value of the smallest potential convex polygon used to envelop the indexed region in a p-by-2 matrix  Equivalent diameter  A scalar value for a computed circle with the same area as the indexed image  Euler number (32)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 32-bit imagery  Euler number (48)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 48-bit imagery  Euler number (64)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 64-bit imagery  Filled area  A scalar count identifying the number of pixels used to ‘fill-in’ the indexed image (removal of image/crown porosity), with the count extending to the full perimeter of the structure using a logical test of the region index  Fractal dimension  A continuous, scaled measurement of self-affinity, where repeating x and y curves are magnified by different factors and a logarithmic mean is calculated  Name  Description  Convex hull area  An area value of the smallest potential convex polygon used to envelop the indexed region in a p-by-2 matrix  Equivalent diameter  A scalar value for a computed circle with the same area as the indexed image  Euler number (32)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 32-bit imagery  Euler number (48)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 48-bit imagery  Euler number (64)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 64-bit imagery  Filled area  A scalar count identifying the number of pixels used to ‘fill-in’ the indexed image (removal of image/crown porosity), with the count extending to the full perimeter of the structure using a logical test of the region index  Fractal dimension  A continuous, scaled measurement of self-affinity, where repeating x and y curves are magnified by different factors and a logarithmic mean is calculated  Table 1 Descriptions of analytical metrics used in an investigation to quantify tree structural condition. Name  Description  Convex hull area  An area value of the smallest potential convex polygon used to envelop the indexed region in a p-by-2 matrix  Equivalent diameter  A scalar value for a computed circle with the same area as the indexed image  Euler number (32)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 32-bit imagery  Euler number (48)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 48-bit imagery  Euler number (64)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 64-bit imagery  Filled area  A scalar count identifying the number of pixels used to ‘fill-in’ the indexed image (removal of image/crown porosity), with the count extending to the full perimeter of the structure using a logical test of the region index  Fractal dimension  A continuous, scaled measurement of self-affinity, where repeating x and y curves are magnified by different factors and a logarithmic mean is calculated  Name  Description  Convex hull area  An area value of the smallest potential convex polygon used to envelop the indexed region in a p-by-2 matrix  Equivalent diameter  A scalar value for a computed circle with the same area as the indexed image  Euler number (32)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 32-bit imagery  Euler number (48)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 48-bit imagery  Euler number (64)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 64-bit imagery  Filled area  A scalar count identifying the number of pixels used to ‘fill-in’ the indexed image (removal of image/crown porosity), with the count extending to the full perimeter of the structure using a logical test of the region index  Fractal dimension  A continuous, scaled measurement of self-affinity, where repeating x and y curves are magnified by different factors and a logarithmic mean is calculated  Euler numbers represent the amount of tree crown occupied by solid tree structure through quantifying connected pixel components, holes and vertices within the image. Initially an RGB image is indexed and an inverse colour map algorithm restricts the number of possible RGB colour values to a predetermined range, e.g. 32, 48 or 64 colours, to refine the image resolution. Each pixel is then matched to the closest colour in the colour map, and the image is subsequently binarized for analysis purposes. Euler numbers are then used to measure image topology through the frequency and area occupancy of ‘holes’ within the binarized image. These holes are subtracted from the total number of objects that occupy the image region, therefore the Euler value represents pixel occupation in the image (Chen and Yan, 1988). The creation of the Euler number is defined as:   E=N−H (1)where N is the number of connected image components (region) and H is the number of image holes identified as separate from the region (Sossa-Azuela, Santiago-Montero et al. 2013). Convex hulls are used to delineate a computed shape edge; therefore in this application, region convex hulls are considered representative of the tree crown edge extent and provide the opportunity to quantify the area covered by the hull shape. Region convex hulls were created demarking a polyhedron boundary in the Euclidean plane around a known distribution of data points (X). This process defines a measurable boundary where the polygon is considered convex if all of the dataset X lie within the boundary, and any two points in X can be joined using a straight-line segment that also remains within the boundary. A limitation of convex hulls is that the outer bounds of the polygon may extend beyond the data range in order to maintain convexity, thereby potentially adding additional area to the generated polygon. Successful convex hull algorithms, however, provide the smallest convex contour area within a given region (Gargano, Bellotti et al. 2007). A similar method used in photogrammetric analysis is the calculation of equivalent diameters. The projections of equivalent diameters are frequently used in RS investigations to model the spatial distribution of tree crowns. Within this study, the equivalent diameter metric represents the area occupied by the tree crown structure in each image, while also providing a potentially continuous index of equivalent circle areas. A scalar value is defined that is the equivalent area of the irregular shape within the image (Kara, Sayinci et al. 2013), and is compared to the area of a known shape, e.g. a circle, using the equation;   de=4a/π (2)where a is the area of the irregular shape, and de is the equivalent diameter. Finally, in order to quantify the complexity of the tree crown structure, a fractal geometric analysis approach was used to assess each image for self-affinity by calculating the logarithmic mean for the Df of each image. Df is used as a measure of complexity as Mandelbrot (1967) recognized the merits of using Df to quantify complex change in pattern detail relative to scale. Fractal dimensions should be considered an approximation of the Kolmogorov capacity, driven by a recursive process where small elements of the image are analysed individually, before the overall Kolmogorov capacity for the image is calculated. Equation (3) describes the Df calculation:   Df=limR→∞lnN(R)/ln(R) (3)where N is the number of boxes needed to cover the fractal shape where it is present, R represents the unit size of the boxes, and N(R) is the number of boxes required to fulfil the fractal element for the image region. Lim refers to the limit of R, as R approaches infinity (Bonnet et al., 2001; Moisy, 2008). In order to generate an individual Df model, a box-counting function (Moisy, 2008) is applied that derives a local Df at each box size, integrated with the power law:   N(R)=N0*R−Df (4)where N0 is the expected value when R equals one. As this approach is dependent on both R and Df the result is a logarithmic mean of all the Df values generated for the fractal region of the image, and is interpreted as a quantification of the structural complexity of tree crowns. The steps required to process the tree images and compute individual tree metrics are summarized at Figure 3. Figure 3 View largeDownload slide A procedural workflow showing how tree structure images are processed for the computation of image metrics. Figure 3 View largeDownload slide A procedural workflow showing how tree structure images are processed for the computation of image metrics. Calculating statistical probabilities The suitability of the predictor variables in quantifying tree structure was tested via multinomial regression, where the observed tree conditions are categorical responses, given as:   log(πi(j)πi(0))=α(j)+β1(j)X1i+⋯+βk(j)Xki (5)where Xki is the kth predictor variable for i, the imaginary unit. 0 is the reference standard, j is the non-reference standard, and α(j) and β1(j),...,βk(j) are the various unknown population parameters. The predictor variables are used to discern where a response, i.e. the tree structure, relates to the same tree characteristics that are indicative of an observed condition. Multinomial regression, therefore, creates a proportional odds model where a single category of trees is specified as the reference standard and is used as a comparative measure against which all other tree categories are compared. Probability (P) estimates are calculated for all trees, to quantify the likelihood that they share the same structural characteristics as the reference standard trees. For the purposes of this study, ‘Good’ category trees (Box 1), are used as the reference standard. The probability that the non-reference standard trees share the same structural characteristics of the reference standard is expressed as a P estimate percentage. Outlining classification thresholds To allow the comparison of continuous and categorical data, several predictor variables were used to create quantified indices to represent the structural character of the individual trees (Table 1). These variables were analysed to discriminate between the structural characteristics of individual trees and to determine how well the indices represented the field-observed classification. The predictor variable indices were grouped and analysed as individual indices, i.e. all Df values grouped as one dataset, all Euler (64) values as another dataset, etc. An empirical data mapping test was undertaken where homogeneity traits were observed in the predictor variable indices. Data mapping is achieved where the categorical data are plotted over the ordinal data using the two available values for each tree image e.g. categorical: Good, ordinal/predictor value: Df 1.875. The tree images were grouped by their field-observed classifications; Good, Moderate, Poor and Dead. For each of these four groups, the minimum and maximum predictor indices values showed the threshold value extent for each classification. Results At Figure 4a, the Df predictor variable quantifies the structural characteristics of all the assessed trees with individual Df values on a continuous scale, and displays homogenous clustering of the field-observed condition types. The group threshold extents are demarked as horizontal classification lines for the Df predictor variable in Figure 4a, where there are four separate groups of Df values consistent with their given field classifications; Good, Moderate, Poor and Dead (Table 2). In instances where heterogeneity was observed in the predictor variable indices, the data mapping could not be applied and it was not possible to define threshold extents (Figure 4b–d). Figure 4 View largeDownload slide Sample subset of predictor variables used to define the characteristics of different tree structures (n64). The annotations Good, Moderate, Poor and Dead refer to the field-observed condition of the individual trees. Only with the measure of fractal dimension (a.), provides homogeneous clustering of field-observed conditions as identified by the threshold lines. Not all predictor variables used in this study are visualized in this plot. Figure 4 View largeDownload slide Sample subset of predictor variables used to define the characteristics of different tree structures (n64). The annotations Good, Moderate, Poor and Dead refer to the field-observed condition of the individual trees. Only with the measure of fractal dimension (a.), provides homogeneous clustering of field-observed conditions as identified by the threshold lines. Not all predictor variables used in this study are visualized in this plot. Table 2 Threshold limits of tree condition categories, expressed in fractal dimensions (Df). Field categories  Df threshold  Good  ≥1.6021  Moderate  ≤1.6020 to >1.4815  Poor  ≤1.4814 to >1.3423  Dead  ≤1.3422  Field categories  Df threshold  Good  ≥1.6021  Moderate  ≤1.6020 to >1.4815  Poor  ≤1.4814 to >1.3423  Dead  ≤1.3422  View Large Table 2 Threshold limits of tree condition categories, expressed in fractal dimensions (Df). Field categories  Df threshold  Good  ≥1.6021  Moderate  ≤1.6020 to >1.4815  Poor  ≤1.4814 to >1.3423  Dead  ≤1.3422  Field categories  Df threshold  Good  ≥1.6021  Moderate  ≤1.6020 to >1.4815  Poor  ≤1.4814 to >1.3423  Dead  ≤1.3422  View Large At Figure 4b–d, there are heterogeneous clusters of field classifications as denoted by the mixed colouring and absence of threshold lines. All sub-plots in Figure 4 show similarities with generally decreasing indices, suggesting a continuous nature to the data, and implying that the trees included in the study possessed a varying range of structural conditions. In Figure 4b–d, all field-observed conditions are shown in heterogeneous grouping for the different predictor variable indices, therefore demonstrating inconsistency with the field-observed classification for each predictor variable (Table 1). It follows that the remaining predictor variables (Table 1 and Figure 4 b–d) do not provide a suitable mechanism to discriminate between different structural characteristics. Euler (64) (Figure 4b.) is the only variable to output negatively skewed data, and repeatedly quantified a number of individual trees with a Euler value of ‘1’, thereby also providing limited information on potential structural differences in these trees. Figure 4a shows the validity of Df as a continuous measure of tree structure complexity. We further demonstrate the relationship between the categorical classifications and the probability that Df values are representative of these categories in Figure 5. Within the good category, there is a ~99 per cent probability that the trees share the same structural characteristics as the trees in the reference standard. Within the moderate category, the probability that the trees show the same structural characteristics of a good tree structure has fallen to ~89 per cent at the median, thereby identifying a probability shift between good and moderate structural characteristics. There is a further, large median shift between the moderate and poor categories, as the median reduces to ~29 per cent for poor category trees when compared with the reference standard. Where trees were field observed as belonging in the dead category, there is a decrease in probability to <1 per cent that these trees show the same structural characteristics as the reference standard. Figure 5 View largeDownload slide A proportional odds model to indicate the probability (P) that tree structure images, quantified in fractal dimensions (Df), are indicative of an observable tree structure condition and known reference standard (n64). Tree images were measured for structural complexity in Df. The box plot extents identify the P that the structures show characteristics of the reference standard. Figure 5 View largeDownload slide A proportional odds model to indicate the probability (P) that tree structure images, quantified in fractal dimensions (Df), are indicative of an observable tree structure condition and known reference standard (n64). Tree images were measured for structural complexity in Df. The box plot extents identify the P that the structures show characteristics of the reference standard. Also in Figure 5, it is noticeable that there is no overlap between the overall visible spread (OVS) in the good field-observed population and any of the other potential categories, due to the OVS separation between all other field-observed categories. Similarly, this trend of OVS separation continues for each field-observed category when compared with any other category. Trees quantified as having structural characteristics of either the moderate or poor groups have a larger interquartile range then trees observed to be in either good or dead condition. This indicates that there is a greater degree of uncertainty in characterizing the moderate or poor groups of trees, particularly as the trees with the good or dead characteristics, are assigned to their relative categories with a high degree of precision. In order to identify potential subgrouping effects, where similar classification probabilities may be clustered around specific probability values, a linear regression model was calculated which identified that there was no evidence of subgrouping and that the probability data range is randomly spread (r2 = 0.86, P-value 0.01). Discussion This study presents a methodology for the objective assessment of tree crown structure, through analysing tree crown structure in hemispherical images. The underlying aim of this study is to reduce the degree of subjectivity currently accepted within tree surveying and assessment and to provide opportunities for high-resolution intra-category assessment of tree structure. Mandelbrot (1967) states that the question of how to accurately measure tree crowns, with the inherent complexity of objectively assessing various shapes, forms, structural porosity, all of varying sizes, is not a simple task that can be solved with classical geometry. Following the findings of this study, it is possible to quantify tree structural complexity using Df as an objective predictor variable using a relatively proximal photogrammetric method and computational analysis (Figure 4), thereby increasing the objectivity and repeatability of structural assessment, whilst also reducing the potential for bias from field measurements. Through quantifying tree structure in Df and creating a proportional odds model, the probabilities that field-observed, ‘good’ classified trees displayed the structural characteristics of structurally sound trees, was found to be statistically very high at P ~99 per cent. Due to the way the proportional odds model functions, achieving this high level of probability is essential for the reliable characterization of the remaining structural condition types. It is suggested that this method of analysis could be transferred to many other investigations of tree structure where the model is trained on a species-specific basis across differing structural architectures. Following the creation of the model, the probabilities of trees with moderate, poor or dead observed classes reduce at the median to P ~89 per cent, P ~29 per cent and P < 1 per cent, respectively, when compared with the reference standard images (Figure 5). These changes in median levels reflect a measured reduction of the tree crown structure complexity. The continuous nature of the Df scale provides a unique measurement of individual tree structure characteristics, as opposed to individual trees being arbitrarily grouped into coarse-resolution, homogenous categories where intra-category differences cannot be easily identified. This insight provides the researcher or practitioner with the opportunity to further sub-divide each classification group, and to monitor intra-category variance over time. This methodology has the potential for the long-term monitoring of pest, disease or pathogen progression, or for the quantification of structural decline, particularly with trees of high conservation, landscape or heritage value. This could include the monitoring of naturally occurring veteran trees, to quantify their rate of structural decline, particularly in areas where there is potential conflict with the public. Furthermore, this method could also be used to guide and inform the process of tree veteranisation, where pre-veteran, mature trees are intentionally injured and receive structural alterations to mimic the structure of naturally occurring veterans with the aim of providing valuable habitats that would otherwise only be found on the most mature trees (Bengtsson et al., 2012). As shown in Figure 4a, there is a wide range of Df values, homogenous grouping of field observations, and no clustering of the P ranges for each potential category. Therefore, it can be stated that tree structure is more accurately quantified in a structural condition continuum than with traditional categorical classification methods. Tree structure measurably degenerates the more trees senesce; tree crown structures change as branch death and limb shedding occur, which ultimately leads to a general decrease in the fractal nature of tree crowns (Mäkelä and Valentine, 2006). Through understanding phenotypic tree structures and the biological response of trees to environmental stress, there is the potential to relate tree structure complexity to an overall indication of tree health or general condition. Tree crown structures are indicative of the amount of photosynthetically active area in the tree required for homoeostatic equilibrium, and therefore is considered to act as a reliable indicator of tree health (Burkhart and Tome, 2012). An advantage of this method is the potential to measure intra-category differences in tree structure complexity and with the computerized storage and easy retrieval of this data, the same analysis can be repeated over time, allowing the accurate tracking of tree structure change. Sudden catastrophic damage to a tree crown is readily recognizable, such as when following a strong wind event. However, more subtle or prolonged tree crown degeneration as a result of biotic or abiotic stress; such as pathogen ingress, or sudden death as a result of heavy, late frosts, could be measured and identified over repeat iterations of surveying. It is recognized that in the immediate period after the sudden death of a tree via these more subtle means, that the structure will likely not have changed significantly, and although potentially dead, a tree could still be classified as good due to the immediate retention of its ‘good’ structure, further reinforcing the requirement for temporal studies to monitor the subtle changes of the tree crown. Further developments of this method should include a refinement of the methodology to accurately measure more subtle structural change in the finer structures of the crown edge. The traditional coarse categorical classification methods do not provide a clear mechanism for measuring subtle structural degeneration as the thresholds for the each potential category are poorly defined and only provide generalized categories for the tree classification. For tree-risk managers such as local government tree officers or utility company infrastructure managers, a structural condition continuum can be used to objectively quantify the probabilities that their tree stock is in a suitable condition. Through quantifying tree structure in a continuous Df scale, specific, measurable thresholds for remedial intervention may be defined. With a categorical approach, tree-risk managers have the limitation of allocating broad categories such as ‘poor’ or ‘dead’ as the triggers for remedial intervention. This limitation greatly increases the number of trees that will be designated as requiring remedial work, compounded by the additional costs and labour requirements. As a higher resolution method, our new approach has the potential to limit unnecessary remedial works, lowering tree management expenditure, and would facilitate limited resources being used in more focussed interventions. We acknowledge that additional work is required to quantify the extent of these improvements, particularly in respect to health and safety related tree management This investigation used a single broadleaved tree species, and we recognize that further work is required to determine where categorical thresholds exist for other tree species. This would follow the work of Morse et al. (1985), who observed that there are differences in the structural complexity of varying vegetation species when they are measured in Df. During a pilot study phase, we identified that there are different thresholds for condition categories in different tree species. The other broadleaved species photographed in various quantities prior to this investigation, were; Acer pseudoplatanus (L.), Fraxinus excelsior (L.), Quercus rubra (L.), Fagus sylvatica (L.), Betula pubescens (Ehrh.), Crataegus monogyna (Jacq.) and Pinus sylvestris (L.). Initial observations suggest that there are likely to be interspecies differences from the small sample numbers used, therefore, this research could also be extended to consider other tree species. In training, the reference category for the proportional odds model, trees that are observed as being in a sound structural condition and are representative of trees in good condition for that species, are identified as the reference category trees. These become the standard against which the remaining trees of the same species are compared. In the process of developing the model, a small degree of user intervention is required to define the parameters of the model and to interpret the model efficacy. Similarly, a user defined bounding box is created to identify the area of interest for the image analysis. This method ensures the procedure can be applied across the full range of tree crown images. The creation of the bounding box is governed by the user following a set of standards that are influenced by standard forestry conventions (West, 2009), and the simple requirement to only identify the tree crown of interest and no other elements, such as the image vignette region. An important distinction to highlight is that the procedure remains a dependable and independent methodology, despite the user intervention as the image analysis, statistical querying and computation of the Df value are all autonomous and therefore, remain objective. This methodology does not purport to entirely remove the requirement for practitioner intervention. We also recognize a potential limitation of this methodology is the reliance on the southern axis for capturing crown images. During methodology development, the southern axis was used to standardize fieldwork when capturing tree crown images. It is recommended that additional field trials should be undertaken to determine the sensitivity of capturing images from differing cardinal points or multiple locations per tree. Conclusion The methodology described in this study for assessing the structural condition of trees is commensurate with traditional techniques. The development of a proximal, hemispherical image field methodology enabled the data capture of many trees in a range of different physical conditions and locations, and satisfies the first objective of this study. The second objective was met with the analysis and objective measurement of hemispherical tree structure images. Finally the ranking of individual trees by the automated calculation of the continuous Df values, satisfies the third objective. It can be stated that the traditional techniques which identify broad categories of structural condition are very coarse, as they do not account for intra-category structural variability and are highly subjective. Our approach enables the assessment of tree condition to be completed with a greater level of precision than was previously possible due to the continuous nature of the Df measurement. Fundamentally, this concept provides a repeatable and objective way to characterize tree crown structure, which can be used to improve the objectivity of tree surveying and inform the specific management of trees with high amenity value. We recognize that further work is required to define the sensitivity of the image acquisition protocol, and to gain further understanding of the full extent of intra-species differences. Nonetheless, it is envisaged that this methodology could form the basis for a new range of analytical measures that will enable tree, environmental or ecological managers to gain greater insights and make more informed decisions about the tree stock under their management. Supplementary data Supplementary data are available at Forestry online. Funding This research is supported by an Engineering and Physical Sciences Research Council (EPSRC) studentship for the lead author [EP/L504804/1]. Acknowledgements The lead author would like to thank Dr. V. Karloukovski, Lancaster Environment Centre, Lancaster University, for discussions on fractal dimensions, and, Dr. E. Eastoe, Department of Mathematics and Statistics, Lancaster University, for providing statistical advice. The authors would also like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of this paper. Conflict of interest statement None declared. 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Using fractal analysis of crown images to measure the structural condition of trees

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Abstract

Abstract Observations of tree canopy structure are routinely used as an indicator of tree condition for the purposes of monitoring tree health, assessing habitat characteristics or evaluating the potential risk of tree failure. Trees are assigned to broad categories of structural condition using largely subjective methods based upon ground-based, visual observations by a surveyor. Such approaches can suffer from a lack of consistency between surveyors; are qualitative in nature and have low precision. In this study, a technique is developed for acquiring, processing and analysing hemispherical images of sessile oak (Quercus petraea (Matt.) Liebl.) tree crowns. We demonstrate that by calculating the fractal dimensions of tree crown images it is possible to define a continuous measurement scale of structural condition and to be able to quantify intra-category variance of tree crown structure. This approach corresponds with traditional categorical methods; however, we recognize that further work is required to precisely define interspecies thresholds. Our study demonstrates that this approach has the potential to form the basis of a new, transferable and objective methodology that can support a wide range of uses in arboriculture, ecology and forest science. Introduction Traditionally, the assessment of tree structural condition, as used in general tree surveys, relies upon simple methodologies and ground-based observations due to the physical complexities of directly measuring tree crowns. However, these traditional techniques are time consuming, manual and largely subjective. Subjectivity has been shown to prevent the same conclusions being reached during independent tree surveys, including surveys of the same trees by different, experienced tree surveyors (Norris, 2007). Predominantly, these assessments rely on a tree surveyor’s knowledge of ideal tree form, tree health, their ability to identify pests and disease, and the consideration of potential hazards and targets that are at risk of harm. Blennow et al. (2013) state that when managing trees or woodlands the use of subjective tree condition observations are not ideal, particularly where objective tree assessments would provide greater insights in the tree management decision process. Ultimately, traditional tree assessment procedures can result in subjective and potentially biased, field observations of tree condition, irrespective of how knowledgeable and experienced the surveyor is (Norris, 2007; Britt and Johnston, 2008). Trees are self-optimizing organisms that respond to a range of recurrent environmental demands and employ strategies to alter their form to minimize potential negative effects or optimize their structure for the greatest physiological benefit (Zimmerman and Brown, 1971; Mattheck and Breloer, 1994; Fourcaud and Dupuy et al., 2004; Pollardy, 2008). In most angiosperms, the lateral branches grow almost as fast, or in some instances faster, than the terminal leader. This process results in the characteristic broad crown structure common in this tree type (Pollardy, 2008; Burkhart and Tome, 2012). Tree form is typically the result of various influences combining the genetic potential, the demands of physiological processes, spatial competition in the crown and the effects of other environmental conditions, such as thigmomorphogenic change caused by repeated wind force effects. The shedding of branches through responsive self-pruning driven by abscission is a characteristic found in many tree species which has a direct effect on the shape of the crown (Pollardy, 2008). There are many additional reasons for trees to shed branches, or parts thereof; which are accelerated by the effects of colonizing pathogens e.g. fungal infestation, or external forces such as gravity or wind force. Indeed, the tree’s own physiology also increases the potential for crown dieback as trees age (King, 2011). Despite many potential stimuli affecting overall tree structure, the growth habits of trees are fundamentally controlled by the genetic predisposition of individual species throughout different tree growth stages. Therefore, the characteristic structure and form of differing tree species remain visually recognizable even after the external impacts are considered (Zimmerman and Brown, 1971). When trees reach late-maturity, there is a combined slowing down of both the stem diameter increment and extension growth in the crown, as a response of the influence of the tree species, genotype or its local environment (King, 2011). It is the recognition of these types of biotic and abiotic structural changes that tree surveyors use to aid the classifying of trees into discrete categories, ultimately aiming to gain insights into the tree’s condition. There have been many studies of tree crown structure in recent years, many of which utilize high-end technology such as light detecting and ranging (LiDAR) as the main method of data capture (Ørka et al., 2009; Ferraz et al., 2016). Specifically with LiDAR data investigations, it is understood that the success of tree investigation algorithms for location detection or height estimation is strongly correlated to the type of tree structure under analysis (Vauhkonen et al., 2012). Through analysis of aerial LiDAR data, boreal tree species have been identified at a species level due to differences in their tree structure signatures (Lina and Hyyppä, 2016), or through LiDAR waveform analysis which identifies structural features within the LiDAR wave (Hovi and Korhonen et al., 2016). Aerial LiDAR investigations are often supported with aerial imagery which is captured simultaneously as image based investigations also provide opportunities for tree canopy structure analysis (Dash and Watt et al., 2016). Furthermore, photogrammetric techniques such as digital stereo imagery and radar imagery have been used in tree canopy structure investigations (Holopainen and Vastaranta et al., 2014). For many researchers or environmental managers, a restrictive element of these types of investigations is the requirement for expensive, specialized research equipment that is often mounted on an aerial platform, such as an unmanned aerial vehicle, aeroplane or satellite. The use of hemispherical photography to undertake proximal tree crown assessments has a field history of more than 50 years, with forest ecologists, Evans and Coombe (1959) using the technique to investigate the available light climate under woodland canopies with an early prototype ‘Hill’ (fisheye) camera. This has remained a readily used, accessible and repeatable method for the investigation of tree canopy structure (Hale, 2004; Chianucci, 2016). Researchers have also previously used hemispherical imagery to assess canopy gap fraction or provide leaf area index assessments (Weiss et al., 2004; Beckschäfer et al., 2013), as it is understood that images captured by hemispherical, or fisheye, lenses provide opportunities for photogrammetric measurement (Schwalbe et al., 2009). Conducting photogrammetric analysis on hemispherical imagery falls within the remote, or indirect, methods of measurement which enable rapid, non-destructive determination of crown properties (Chason et al., 1991; Weiss et al., 2004). Modern advancements in digital cameras, coupled with readily available hemispherical lenses or lens adaptors, provide the opportunity for an off-the-shelf approach to photogrammetric research (Leblanc et al., 2005). When tree crowns are viewed from directly beneath, looking upwards towards the zenith viewing point (90° from the horizontal elevation), holes can be observed within the crown structure. The tree crown area is a complex arrangement of tree branches, combined with observable unoccupied areas between the different parts of the tree crown. This upward looking view provides a visual separation between the tree structure and the sky, which when photographed can be converted into a binary image with the occupied and background regions of the image coded ‘1’ and ‘0’, respectively (Beckschäfer et al., 2013; Sossa-Azuela et al., 2013). Image analysis techniques for pattern recognition in tree structures have identified features of lacunarity (the size and distribution of holes), complex spatial distributions or other morphologic features (Zheng et al., 1995; Frazer et al., 2005). Due to the unique geometry found in nature, the dimensions of natural, physical forms cannot readily be described in simple, integral terms (Mandelbrot, 1982; Dimri, 2000). Mandelbrot (1982) argues that more insightful measurements are required to measure pattern complexity, such as quantifying the degree of complexity in a structure. As trees exhibit natural structural variance, Mandelbrot (1982), also notes that it is the frequently anomalous nature of tree structure whose form is sculpted by, ‘chance, irregularities and non-uniformity’, that provides the opportunity for statistical investigation. Rian and Sassone (2014) demonstrate that the crown structures of trees are unique in their self-affine and highly irregular branching patterns. It has been stated that fractal dimensions (Df) can be used to quantify structural complexity in a continuous measure, theoretically ranging from 0 to infinity, which can be expressed as a single value (Mandelbrot, 1967; Kaye, 2008). Although tree crown structures are complex shapes, there are various examples of Df being used as a predictor variable for the classification of forest canopies (Zeide and Pfeifer, 1991; Zeide, 1998; Jonckheere et al., 2006; Zhang et al., 2007). The aim of this study was to develop an objective methodology to assess the structural condition of broadleaved tree crowns (Quercus sp.) by quantifying the complexity of the tree crowns through hemispherical images taken under leaf-off conditions. This approach was designed to overcome the limitations of current subjective field methodologies. The first objective was to develop an in-field data capture technique that was suitable for a range of subject trees across a variety of structural conditions. The second objective was to develop image processing methods for the assessment of crown structural condition. The third objective was to propose a new and objective means of evaluating tree structural condition on a continuous scale. Methodology Throughout three study areas across northwest Lancashire, England, 64 Sessile Oak trees (Quercus petraea (Matt.) Liebl.) were individually photographed using hemispherical imagery obtained from beneath subject tree canopies, looking towards the zenith viewpoint (Figure 1). The trees used in the study were either individual maiden trees, or trees that were located in closed canopy, woodland conditions. The trees were photographed over a single winter season in leaf-off condition, thereby allowing an unobscured view of the tree crown structure. To minimize potentially confounding variables, this method was applied to trees of the same species that were in the mature phases of tree development, specifically: early-mature (28 per cent), mature (25 per cent), late-mature (25 per cent), veteran and senescent (22 per cent) (Fay and de Berker, 1997). To achieve a suitable sample size, a locally prolific species was used in this study. Figure 1 View largeDownload slide A schematic of the field method for taking a hemispherical picture from beneath a tree canopy. The camera is situated on a standard tripod, and is levelled and pointing towards the zenith viewing point (90° from the horizontal elevation). In this example, the full extent of the crown is 4 m along the southern axis, and the image is taken at the 2 m mid-point. Figure 1 View largeDownload slide A schematic of the field method for taking a hemispherical picture from beneath a tree canopy. The camera is situated on a standard tripod, and is levelled and pointing towards the zenith viewing point (90° from the horizontal elevation). In this example, the full extent of the crown is 4 m along the southern axis, and the image is taken at the 2 m mid-point. Field methodology development Reference data on the trees structural condition were collected using a four-point categorical system, as is common in arboricultural assessments using traditional field techniques. The four-point method used in this research is not based upon a single specific method, but broadly upon several arboricultural tree survey methods (e.g. BS5837:2012 surveys which use a four level condition hierarchy, the ISA tree hazard evaluation, which uses four classification categories to generate an accumulative hazard score (Matheny and Clark, 1994; BSI, 2012), and is also comparable with a qualitative tree condition category assignment as described in Swetnam et al. 2016). Consequently, this approach is representative of similar tree survey methods where the assessment of trees leads to an empirical categorization of tree condition. Box 1 provides an overview of the classification descriptors. Once identified, the tree’s cardinal orientation was determined by the use of a field compass. The part of the crown that extended towards the southernmost point (i.e. the tree crown’s southern axis), was marked out along the ground with a standard surveyors tape and used as the linear axis upon which the crown images were taken at specific intervals. Camera set-up A high-resolution digital single-lens reflex (dSLR) camera (Canon EOS 550D DS126271) was used with an 18 mm lens and a hemispherical lens adaptor (Opteka Super Wide Fisheye Lens 0.20×). The lens adaptor permits focal length conversion into a 3.6 mm circular lens. The wide angle of the hemispherical lens enabled as much of each tree crown to be captured within each image as possible. The dSLR was placed on a standard photographic tripod, adjusted at each image capture location ensuring that the dSLR was positioned and levelled with the camera lens pointing vertically upward at ~0.5 m from the ground level. To account for variability in solar illumination, the images were taken during uniform sky conditions. These conditions occur predominantly when the sky is overcast, although this technique can also be used just before sunrise or just after sunset, should bright daytime conditions be expected (Song et al., 2014). Image acquisition and spatial sampling strategy Initially, the number of images captured per subject tree was influenced by the overall length of the crown along the southern axis. Early trials with image capture involved taking images at 1 m intervals along the southern axis, to the full extent of the crown. However, this produced a high number of replicates with large amounts of image content overlap. Inspection of these images identified two problems with this approach. Firstly, that there was ~90 per cent replication of content between the overlapping images (Figure 2a), and secondly, that additional tree features that were not required for the analysis were also captured. For example, additional stem wood was photographed in the images closest to the base of the tree (e.g. at 1 m and 2 m intervals), while large amounts of ‘sky’ was captured towards the canopy edge. Neither of these image components was required in the analysis. It followed that many of the repeated images was not within the optimal range for representing the fullest area of tree crown within an image. Repeated testing indicated that the optimal location for image capture was around the mid-point of the crown axis (Figure 2). Where there was no mid-point location on an exact 1 m interval of the southern axis mid-point, the distance was rounded up to the next whole metre. The southern axis was used for standardization purposes as the subject trees are located in the Northern hemisphere and our preference was to capture images on the non-shaded, south facing side of the trees. Figure 2 View largeDownload slide A schematic showing the optimized range for image capture (a), and the area of tree canopy structure analysed within this study (b). The area of interest is specifically the structural elements of the canopy. Too much ‘sky’ within the image reduces the amount of structure that can be analysed (a). Stem wood and other elements not required, are removed from the image by only analysing the structure inside a user selected bounding box area (b). The use of a bounding box allows images of both individual trees and trees within closed canopies to be analysed. Figure 2 View largeDownload slide A schematic showing the optimized range for image capture (a), and the area of tree canopy structure analysed within this study (b). The area of interest is specifically the structural elements of the canopy. Too much ‘sky’ within the image reduces the amount of structure that can be analysed (a). Stem wood and other elements not required, are removed from the image by only analysing the structure inside a user selected bounding box area (b). The use of a bounding box allows images of both individual trees and trees within closed canopies to be analysed. Immediately after acquisition, the quality of each image was visually assessed. This step was taken to ensure the images were suitable for later analysis and to allow additional images to be captured should the original image be unusable. The process of identifying the southern axis, setting-up the camera and completing image acquisition took between ~45 sec and ~1.5 min, depending on the complexity of the local topographic environment. Image preparation Upon return from the field, the images were re-examined on a desktop computer to check for image clarity, suitability in showing the area of interest, and for the presence of key features (Jones and Vaughn, 2010). A limitation of the in-field image proofing was that this was completed on the dSLR camera’s 2.7-inch screen; therefore it was conducted at a very coarse resolution. Of the original 247 images, 87 were removed for blurring or distortion errors, 96 images were removed as duplicates, leaving the sample size reduced to 64 images of individual trees, with a single image representing each tree. Box 1 Classification descriptors for the subjective arboricultural assessment of trees. Estimated remaining contribution (ERC) refers to a methodology used to consider the health, condition and structure of the tree and aids in classifying the tree in to the different categories adapted from (Barrell, 1993, 2001; Lonsdale, 1999; NTSG, 2011; BSI, 2012). Note: The images show trees in leaf-on condition to enable ease of comparison for the condition types. 4. Good    Dominant trees. Full crown, good extension growth and form. Typical for age and species. High number of buds. Healthy reaction growth to any injuries. Acceptable levels of colonization. ERC: > 40+ years  3. Moderate    Some signs of stress, crown dieback or retrenchment. Deadwood. Other signs of stress likely to be present. Remedial work may have previously taken place. Cavities rot or early disease may be present. ERC: ≥10–40 years  2. Poor    Obvious signs of dieback. Frequent deadwood. Clear signs of disease and decay. Overwhelming of the trees natural defences. Colonization by fungi, wood boring insects and other decay biota highly likely. ERC: ≤10 years  1. Dead    Physiological processes have ceased. Lack of active photosynthetic area. Colonization of fungi, wood boring insects and other decay biota highly likely. Extensive crown retrenchment, bark slough, brittle or collapsing structure. ERC: ≤ 0 years.  4. Good    Dominant trees. Full crown, good extension growth and form. Typical for age and species. High number of buds. Healthy reaction growth to any injuries. Acceptable levels of colonization. ERC: > 40+ years  3. Moderate    Some signs of stress, crown dieback or retrenchment. Deadwood. Other signs of stress likely to be present. Remedial work may have previously taken place. Cavities rot or early disease may be present. ERC: ≥10–40 years  2. Poor    Obvious signs of dieback. Frequent deadwood. Clear signs of disease and decay. Overwhelming of the trees natural defences. Colonization by fungi, wood boring insects and other decay biota highly likely. ERC: ≤10 years  1. Dead    Physiological processes have ceased. Lack of active photosynthetic area. Colonization of fungi, wood boring insects and other decay biota highly likely. Extensive crown retrenchment, bark slough, brittle or collapsing structure. ERC: ≤ 0 years.  4. Good    Dominant trees. Full crown, good extension growth and form. Typical for age and species. High number of buds. Healthy reaction growth to any injuries. Acceptable levels of colonization. ERC: > 40+ years  3. Moderate    Some signs of stress, crown dieback or retrenchment. Deadwood. Other signs of stress likely to be present. Remedial work may have previously taken place. Cavities rot or early disease may be present. ERC: ≥10–40 years  2. Poor    Obvious signs of dieback. Frequent deadwood. Clear signs of disease and decay. Overwhelming of the trees natural defences. Colonization by fungi, wood boring insects and other decay biota highly likely. ERC: ≤10 years  1. Dead    Physiological processes have ceased. Lack of active photosynthetic area. Colonization of fungi, wood boring insects and other decay biota highly likely. Extensive crown retrenchment, bark slough, brittle or collapsing structure. ERC: ≤ 0 years.  4. Good    Dominant trees. Full crown, good extension growth and form. Typical for age and species. High number of buds. Healthy reaction growth to any injuries. Acceptable levels of colonization. ERC: > 40+ years  3. Moderate    Some signs of stress, crown dieback or retrenchment. Deadwood. Other signs of stress likely to be present. Remedial work may have previously taken place. Cavities rot or early disease may be present. ERC: ≥10–40 years  2. Poor    Obvious signs of dieback. Frequent deadwood. Clear signs of disease and decay. Overwhelming of the trees natural defences. Colonization by fungi, wood boring insects and other decay biota highly likely. ERC: ≤10 years  1. Dead    Physiological processes have ceased. Lack of active photosynthetic area. Colonization of fungi, wood boring insects and other decay biota highly likely. Extensive crown retrenchment, bark slough, brittle or collapsing structure. ERC: ≤ 0 years.  Pre-processing interventions removed errors from the images that could affect the measurement of image metrics. Chromatic aberration (CA) is the misregistration of RGB channels causing interference with the dSLR Bayer-pattern sensor, leading to image deterioration and interference with pixel-based classification techniques (Schwalbe et al., 2009). In this study, CA was corrected by removing the red and blue channels, and converting the image to the green element of the RGB channels only. Quadratic or ‘barrel’ distortion is also associated with images captured using hemispherical lenses. A distortion correction algorithm (Vries, 2012) transformed the images from the distorted barrel extension to replicate an image captured at a normal focal length. This perspective distortion effect is influenced by the relative distances between the lens and subject canopy at which the image is captured, therefore, it is important that the relative distance was maintained during image capture. In order to reduce the effects of blurred images caused by contrast errors between colour ranges, an image sharpening algorithm was used. This algorithm was based upon un-sharp masking, where the image is sharpened by removing a blurred negative copy of the same image. The copied mask was laid over the original, resulting in a combined image that is visually sharper. Where there were instances of unsuccessful pre-processing, the affected images were not used in the investigation. The images were analysed in Matlab (version 2015a), where each image pixel was indexed and converted into binary form. This was achieved through applying uniform quantization where limited intensity resolution breaks the image colour space into individual pixels, which are indexed, and the pixel locations are mapped. A process of dithering corrects any potential quantization errors and limits the greyscale range of the image. This binarization procedure allows differentiation between the tree structure and other parts of the image, as optimum image analysis conditions are best achieved where there is high contrast between tree structure and the sky (Chen, Black et al. 1991). Defining the image analysis area Chianucci and Cutini (2012) describe that it is beneficial in image processing to reduce the field of view by masking some elements of the full hemisphere, thereby achieving greater spatial representation of heterogeneous tree crowns i.e. the inclusion of both dense and sparse crown regions in the analysis. At Figure 2b, image analysis is restricted to the part of tree canopy contained within the black bounding box, created on a per image basis. The analysis extent is influenced by standard forestry measurement conventions (West, 2009), with the lower bounding box edge originating at the point of estimated timber height. In decurrent trees, this is where the main stem bifurcates to such a degree that the main stem is no longer discernible. From here, the analysis area is bordered by the upper bounding box at the edge of the tree crown and avoids the image’s vignette region caused by the visible inner walls of the camera lens. The left and right boundaries of the image analysis area are demarked by adjoining lines between the upper and lower bounding box extents maximizing the crown analysis area, while again, also avoiding the vignette region at the edges of the image. Predictor variable creation Multiple indices were generated from the tree images that were developed into image metrics which were tested, both individually and in combination, for their suitability in describing the tree structural character. A description of the metrics is shown at Table 1. Table 1 Descriptions of analytical metrics used in an investigation to quantify tree structural condition. Name  Description  Convex hull area  An area value of the smallest potential convex polygon used to envelop the indexed region in a p-by-2 matrix  Equivalent diameter  A scalar value for a computed circle with the same area as the indexed image  Euler number (32)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 32-bit imagery  Euler number (48)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 48-bit imagery  Euler number (64)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 64-bit imagery  Filled area  A scalar count identifying the number of pixels used to ‘fill-in’ the indexed image (removal of image/crown porosity), with the count extending to the full perimeter of the structure using a logical test of the region index  Fractal dimension  A continuous, scaled measurement of self-affinity, where repeating x and y curves are magnified by different factors and a logarithmic mean is calculated  Name  Description  Convex hull area  An area value of the smallest potential convex polygon used to envelop the indexed region in a p-by-2 matrix  Equivalent diameter  A scalar value for a computed circle with the same area as the indexed image  Euler number (32)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 32-bit imagery  Euler number (48)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 48-bit imagery  Euler number (64)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 64-bit imagery  Filled area  A scalar count identifying the number of pixels used to ‘fill-in’ the indexed image (removal of image/crown porosity), with the count extending to the full perimeter of the structure using a logical test of the region index  Fractal dimension  A continuous, scaled measurement of self-affinity, where repeating x and y curves are magnified by different factors and a logarithmic mean is calculated  Table 1 Descriptions of analytical metrics used in an investigation to quantify tree structural condition. Name  Description  Convex hull area  An area value of the smallest potential convex polygon used to envelop the indexed region in a p-by-2 matrix  Equivalent diameter  A scalar value for a computed circle with the same area as the indexed image  Euler number (32)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 32-bit imagery  Euler number (48)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 48-bit imagery  Euler number (64)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 64-bit imagery  Filled area  A scalar count identifying the number of pixels used to ‘fill-in’ the indexed image (removal of image/crown porosity), with the count extending to the full perimeter of the structure using a logical test of the region index  Fractal dimension  A continuous, scaled measurement of self-affinity, where repeating x and y curves are magnified by different factors and a logarithmic mean is calculated  Name  Description  Convex hull area  An area value of the smallest potential convex polygon used to envelop the indexed region in a p-by-2 matrix  Equivalent diameter  A scalar value for a computed circle with the same area as the indexed image  Euler number (32)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 32-bit imagery  Euler number (48)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 48-bit imagery  Euler number (64)  A scalar value that specifies the frequency of indexed objects in the image region. The Euler number subtracts porosity values (holes) representative of crown porosity using 64-bit imagery  Filled area  A scalar count identifying the number of pixels used to ‘fill-in’ the indexed image (removal of image/crown porosity), with the count extending to the full perimeter of the structure using a logical test of the region index  Fractal dimension  A continuous, scaled measurement of self-affinity, where repeating x and y curves are magnified by different factors and a logarithmic mean is calculated  Euler numbers represent the amount of tree crown occupied by solid tree structure through quantifying connected pixel components, holes and vertices within the image. Initially an RGB image is indexed and an inverse colour map algorithm restricts the number of possible RGB colour values to a predetermined range, e.g. 32, 48 or 64 colours, to refine the image resolution. Each pixel is then matched to the closest colour in the colour map, and the image is subsequently binarized for analysis purposes. Euler numbers are then used to measure image topology through the frequency and area occupancy of ‘holes’ within the binarized image. These holes are subtracted from the total number of objects that occupy the image region, therefore the Euler value represents pixel occupation in the image (Chen and Yan, 1988). The creation of the Euler number is defined as:   E=N−H (1)where N is the number of connected image components (region) and H is the number of image holes identified as separate from the region (Sossa-Azuela, Santiago-Montero et al. 2013). Convex hulls are used to delineate a computed shape edge; therefore in this application, region convex hulls are considered representative of the tree crown edge extent and provide the opportunity to quantify the area covered by the hull shape. Region convex hulls were created demarking a polyhedron boundary in the Euclidean plane around a known distribution of data points (X). This process defines a measurable boundary where the polygon is considered convex if all of the dataset X lie within the boundary, and any two points in X can be joined using a straight-line segment that also remains within the boundary. A limitation of convex hulls is that the outer bounds of the polygon may extend beyond the data range in order to maintain convexity, thereby potentially adding additional area to the generated polygon. Successful convex hull algorithms, however, provide the smallest convex contour area within a given region (Gargano, Bellotti et al. 2007). A similar method used in photogrammetric analysis is the calculation of equivalent diameters. The projections of equivalent diameters are frequently used in RS investigations to model the spatial distribution of tree crowns. Within this study, the equivalent diameter metric represents the area occupied by the tree crown structure in each image, while also providing a potentially continuous index of equivalent circle areas. A scalar value is defined that is the equivalent area of the irregular shape within the image (Kara, Sayinci et al. 2013), and is compared to the area of a known shape, e.g. a circle, using the equation;   de=4a/π (2)where a is the area of the irregular shape, and de is the equivalent diameter. Finally, in order to quantify the complexity of the tree crown structure, a fractal geometric analysis approach was used to assess each image for self-affinity by calculating the logarithmic mean for the Df of each image. Df is used as a measure of complexity as Mandelbrot (1967) recognized the merits of using Df to quantify complex change in pattern detail relative to scale. Fractal dimensions should be considered an approximation of the Kolmogorov capacity, driven by a recursive process where small elements of the image are analysed individually, before the overall Kolmogorov capacity for the image is calculated. Equation (3) describes the Df calculation:   Df=limR→∞lnN(R)/ln(R) (3)where N is the number of boxes needed to cover the fractal shape where it is present, R represents the unit size of the boxes, and N(R) is the number of boxes required to fulfil the fractal element for the image region. Lim refers to the limit of R, as R approaches infinity (Bonnet et al., 2001; Moisy, 2008). In order to generate an individual Df model, a box-counting function (Moisy, 2008) is applied that derives a local Df at each box size, integrated with the power law:   N(R)=N0*R−Df (4)where N0 is the expected value when R equals one. As this approach is dependent on both R and Df the result is a logarithmic mean of all the Df values generated for the fractal region of the image, and is interpreted as a quantification of the structural complexity of tree crowns. The steps required to process the tree images and compute individual tree metrics are summarized at Figure 3. Figure 3 View largeDownload slide A procedural workflow showing how tree structure images are processed for the computation of image metrics. Figure 3 View largeDownload slide A procedural workflow showing how tree structure images are processed for the computation of image metrics. Calculating statistical probabilities The suitability of the predictor variables in quantifying tree structure was tested via multinomial regression, where the observed tree conditions are categorical responses, given as:   log(πi(j)πi(0))=α(j)+β1(j)X1i+⋯+βk(j)Xki (5)where Xki is the kth predictor variable for i, the imaginary unit. 0 is the reference standard, j is the non-reference standard, and α(j) and β1(j),...,βk(j) are the various unknown population parameters. The predictor variables are used to discern where a response, i.e. the tree structure, relates to the same tree characteristics that are indicative of an observed condition. Multinomial regression, therefore, creates a proportional odds model where a single category of trees is specified as the reference standard and is used as a comparative measure against which all other tree categories are compared. Probability (P) estimates are calculated for all trees, to quantify the likelihood that they share the same structural characteristics as the reference standard trees. For the purposes of this study, ‘Good’ category trees (Box 1), are used as the reference standard. The probability that the non-reference standard trees share the same structural characteristics of the reference standard is expressed as a P estimate percentage. Outlining classification thresholds To allow the comparison of continuous and categorical data, several predictor variables were used to create quantified indices to represent the structural character of the individual trees (Table 1). These variables were analysed to discriminate between the structural characteristics of individual trees and to determine how well the indices represented the field-observed classification. The predictor variable indices were grouped and analysed as individual indices, i.e. all Df values grouped as one dataset, all Euler (64) values as another dataset, etc. An empirical data mapping test was undertaken where homogeneity traits were observed in the predictor variable indices. Data mapping is achieved where the categorical data are plotted over the ordinal data using the two available values for each tree image e.g. categorical: Good, ordinal/predictor value: Df 1.875. The tree images were grouped by their field-observed classifications; Good, Moderate, Poor and Dead. For each of these four groups, the minimum and maximum predictor indices values showed the threshold value extent for each classification. Results At Figure 4a, the Df predictor variable quantifies the structural characteristics of all the assessed trees with individual Df values on a continuous scale, and displays homogenous clustering of the field-observed condition types. The group threshold extents are demarked as horizontal classification lines for the Df predictor variable in Figure 4a, where there are four separate groups of Df values consistent with their given field classifications; Good, Moderate, Poor and Dead (Table 2). In instances where heterogeneity was observed in the predictor variable indices, the data mapping could not be applied and it was not possible to define threshold extents (Figure 4b–d). Figure 4 View largeDownload slide Sample subset of predictor variables used to define the characteristics of different tree structures (n64). The annotations Good, Moderate, Poor and Dead refer to the field-observed condition of the individual trees. Only with the measure of fractal dimension (a.), provides homogeneous clustering of field-observed conditions as identified by the threshold lines. Not all predictor variables used in this study are visualized in this plot. Figure 4 View largeDownload slide Sample subset of predictor variables used to define the characteristics of different tree structures (n64). The annotations Good, Moderate, Poor and Dead refer to the field-observed condition of the individual trees. Only with the measure of fractal dimension (a.), provides homogeneous clustering of field-observed conditions as identified by the threshold lines. Not all predictor variables used in this study are visualized in this plot. Table 2 Threshold limits of tree condition categories, expressed in fractal dimensions (Df). Field categories  Df threshold  Good  ≥1.6021  Moderate  ≤1.6020 to >1.4815  Poor  ≤1.4814 to >1.3423  Dead  ≤1.3422  Field categories  Df threshold  Good  ≥1.6021  Moderate  ≤1.6020 to >1.4815  Poor  ≤1.4814 to >1.3423  Dead  ≤1.3422  View Large Table 2 Threshold limits of tree condition categories, expressed in fractal dimensions (Df). Field categories  Df threshold  Good  ≥1.6021  Moderate  ≤1.6020 to >1.4815  Poor  ≤1.4814 to >1.3423  Dead  ≤1.3422  Field categories  Df threshold  Good  ≥1.6021  Moderate  ≤1.6020 to >1.4815  Poor  ≤1.4814 to >1.3423  Dead  ≤1.3422  View Large At Figure 4b–d, there are heterogeneous clusters of field classifications as denoted by the mixed colouring and absence of threshold lines. All sub-plots in Figure 4 show similarities with generally decreasing indices, suggesting a continuous nature to the data, and implying that the trees included in the study possessed a varying range of structural conditions. In Figure 4b–d, all field-observed conditions are shown in heterogeneous grouping for the different predictor variable indices, therefore demonstrating inconsistency with the field-observed classification for each predictor variable (Table 1). It follows that the remaining predictor variables (Table 1 and Figure 4 b–d) do not provide a suitable mechanism to discriminate between different structural characteristics. Euler (64) (Figure 4b.) is the only variable to output negatively skewed data, and repeatedly quantified a number of individual trees with a Euler value of ‘1’, thereby also providing limited information on potential structural differences in these trees. Figure 4a shows the validity of Df as a continuous measure of tree structure complexity. We further demonstrate the relationship between the categorical classifications and the probability that Df values are representative of these categories in Figure 5. Within the good category, there is a ~99 per cent probability that the trees share the same structural characteristics as the trees in the reference standard. Within the moderate category, the probability that the trees show the same structural characteristics of a good tree structure has fallen to ~89 per cent at the median, thereby identifying a probability shift between good and moderate structural characteristics. There is a further, large median shift between the moderate and poor categories, as the median reduces to ~29 per cent for poor category trees when compared with the reference standard. Where trees were field observed as belonging in the dead category, there is a decrease in probability to <1 per cent that these trees show the same structural characteristics as the reference standard. Figure 5 View largeDownload slide A proportional odds model to indicate the probability (P) that tree structure images, quantified in fractal dimensions (Df), are indicative of an observable tree structure condition and known reference standard (n64). Tree images were measured for structural complexity in Df. The box plot extents identify the P that the structures show characteristics of the reference standard. Figure 5 View largeDownload slide A proportional odds model to indicate the probability (P) that tree structure images, quantified in fractal dimensions (Df), are indicative of an observable tree structure condition and known reference standard (n64). Tree images were measured for structural complexity in Df. The box plot extents identify the P that the structures show characteristics of the reference standard. Also in Figure 5, it is noticeable that there is no overlap between the overall visible spread (OVS) in the good field-observed population and any of the other potential categories, due to the OVS separation between all other field-observed categories. Similarly, this trend of OVS separation continues for each field-observed category when compared with any other category. Trees quantified as having structural characteristics of either the moderate or poor groups have a larger interquartile range then trees observed to be in either good or dead condition. This indicates that there is a greater degree of uncertainty in characterizing the moderate or poor groups of trees, particularly as the trees with the good or dead characteristics, are assigned to their relative categories with a high degree of precision. In order to identify potential subgrouping effects, where similar classification probabilities may be clustered around specific probability values, a linear regression model was calculated which identified that there was no evidence of subgrouping and that the probability data range is randomly spread (r2 = 0.86, P-value 0.01). Discussion This study presents a methodology for the objective assessment of tree crown structure, through analysing tree crown structure in hemispherical images. The underlying aim of this study is to reduce the degree of subjectivity currently accepted within tree surveying and assessment and to provide opportunities for high-resolution intra-category assessment of tree structure. Mandelbrot (1967) states that the question of how to accurately measure tree crowns, with the inherent complexity of objectively assessing various shapes, forms, structural porosity, all of varying sizes, is not a simple task that can be solved with classical geometry. Following the findings of this study, it is possible to quantify tree structural complexity using Df as an objective predictor variable using a relatively proximal photogrammetric method and computational analysis (Figure 4), thereby increasing the objectivity and repeatability of structural assessment, whilst also reducing the potential for bias from field measurements. Through quantifying tree structure in Df and creating a proportional odds model, the probabilities that field-observed, ‘good’ classified trees displayed the structural characteristics of structurally sound trees, was found to be statistically very high at P ~99 per cent. Due to the way the proportional odds model functions, achieving this high level of probability is essential for the reliable characterization of the remaining structural condition types. It is suggested that this method of analysis could be transferred to many other investigations of tree structure where the model is trained on a species-specific basis across differing structural architectures. Following the creation of the model, the probabilities of trees with moderate, poor or dead observed classes reduce at the median to P ~89 per cent, P ~29 per cent and P < 1 per cent, respectively, when compared with the reference standard images (Figure 5). These changes in median levels reflect a measured reduction of the tree crown structure complexity. The continuous nature of the Df scale provides a unique measurement of individual tree structure characteristics, as opposed to individual trees being arbitrarily grouped into coarse-resolution, homogenous categories where intra-category differences cannot be easily identified. This insight provides the researcher or practitioner with the opportunity to further sub-divide each classification group, and to monitor intra-category variance over time. This methodology has the potential for the long-term monitoring of pest, disease or pathogen progression, or for the quantification of structural decline, particularly with trees of high conservation, landscape or heritage value. This could include the monitoring of naturally occurring veteran trees, to quantify their rate of structural decline, particularly in areas where there is potential conflict with the public. Furthermore, this method could also be used to guide and inform the process of tree veteranisation, where pre-veteran, mature trees are intentionally injured and receive structural alterations to mimic the structure of naturally occurring veterans with the aim of providing valuable habitats that would otherwise only be found on the most mature trees (Bengtsson et al., 2012). As shown in Figure 4a, there is a wide range of Df values, homogenous grouping of field observations, and no clustering of the P ranges for each potential category. Therefore, it can be stated that tree structure is more accurately quantified in a structural condition continuum than with traditional categorical classification methods. Tree structure measurably degenerates the more trees senesce; tree crown structures change as branch death and limb shedding occur, which ultimately leads to a general decrease in the fractal nature of tree crowns (Mäkelä and Valentine, 2006). Through understanding phenotypic tree structures and the biological response of trees to environmental stress, there is the potential to relate tree structure complexity to an overall indication of tree health or general condition. Tree crown structures are indicative of the amount of photosynthetically active area in the tree required for homoeostatic equilibrium, and therefore is considered to act as a reliable indicator of tree health (Burkhart and Tome, 2012). An advantage of this method is the potential to measure intra-category differences in tree structure complexity and with the computerized storage and easy retrieval of this data, the same analysis can be repeated over time, allowing the accurate tracking of tree structure change. Sudden catastrophic damage to a tree crown is readily recognizable, such as when following a strong wind event. However, more subtle or prolonged tree crown degeneration as a result of biotic or abiotic stress; such as pathogen ingress, or sudden death as a result of heavy, late frosts, could be measured and identified over repeat iterations of surveying. It is recognized that in the immediate period after the sudden death of a tree via these more subtle means, that the structure will likely not have changed significantly, and although potentially dead, a tree could still be classified as good due to the immediate retention of its ‘good’ structure, further reinforcing the requirement for temporal studies to monitor the subtle changes of the tree crown. Further developments of this method should include a refinement of the methodology to accurately measure more subtle structural change in the finer structures of the crown edge. The traditional coarse categorical classification methods do not provide a clear mechanism for measuring subtle structural degeneration as the thresholds for the each potential category are poorly defined and only provide generalized categories for the tree classification. For tree-risk managers such as local government tree officers or utility company infrastructure managers, a structural condition continuum can be used to objectively quantify the probabilities that their tree stock is in a suitable condition. Through quantifying tree structure in a continuous Df scale, specific, measurable thresholds for remedial intervention may be defined. With a categorical approach, tree-risk managers have the limitation of allocating broad categories such as ‘poor’ or ‘dead’ as the triggers for remedial intervention. This limitation greatly increases the number of trees that will be designated as requiring remedial work, compounded by the additional costs and labour requirements. As a higher resolution method, our new approach has the potential to limit unnecessary remedial works, lowering tree management expenditure, and would facilitate limited resources being used in more focussed interventions. We acknowledge that additional work is required to quantify the extent of these improvements, particularly in respect to health and safety related tree management This investigation used a single broadleaved tree species, and we recognize that further work is required to determine where categorical thresholds exist for other tree species. This would follow the work of Morse et al. (1985), who observed that there are differences in the structural complexity of varying vegetation species when they are measured in Df. During a pilot study phase, we identified that there are different thresholds for condition categories in different tree species. The other broadleaved species photographed in various quantities prior to this investigation, were; Acer pseudoplatanus (L.), Fraxinus excelsior (L.), Quercus rubra (L.), Fagus sylvatica (L.), Betula pubescens (Ehrh.), Crataegus monogyna (Jacq.) and Pinus sylvestris (L.). Initial observations suggest that there are likely to be interspecies differences from the small sample numbers used, therefore, this research could also be extended to consider other tree species. In training, the reference category for the proportional odds model, trees that are observed as being in a sound structural condition and are representative of trees in good condition for that species, are identified as the reference category trees. These become the standard against which the remaining trees of the same species are compared. In the process of developing the model, a small degree of user intervention is required to define the parameters of the model and to interpret the model efficacy. Similarly, a user defined bounding box is created to identify the area of interest for the image analysis. This method ensures the procedure can be applied across the full range of tree crown images. The creation of the bounding box is governed by the user following a set of standards that are influenced by standard forestry conventions (West, 2009), and the simple requirement to only identify the tree crown of interest and no other elements, such as the image vignette region. An important distinction to highlight is that the procedure remains a dependable and independent methodology, despite the user intervention as the image analysis, statistical querying and computation of the Df value are all autonomous and therefore, remain objective. This methodology does not purport to entirely remove the requirement for practitioner intervention. We also recognize a potential limitation of this methodology is the reliance on the southern axis for capturing crown images. During methodology development, the southern axis was used to standardize fieldwork when capturing tree crown images. It is recommended that additional field trials should be undertaken to determine the sensitivity of capturing images from differing cardinal points or multiple locations per tree. Conclusion The methodology described in this study for assessing the structural condition of trees is commensurate with traditional techniques. The development of a proximal, hemispherical image field methodology enabled the data capture of many trees in a range of different physical conditions and locations, and satisfies the first objective of this study. The second objective was met with the analysis and objective measurement of hemispherical tree structure images. Finally the ranking of individual trees by the automated calculation of the continuous Df values, satisfies the third objective. It can be stated that the traditional techniques which identify broad categories of structural condition are very coarse, as they do not account for intra-category structural variability and are highly subjective. Our approach enables the assessment of tree condition to be completed with a greater level of precision than was previously possible due to the continuous nature of the Df measurement. Fundamentally, this concept provides a repeatable and objective way to characterize tree crown structure, which can be used to improve the objectivity of tree surveying and inform the specific management of trees with high amenity value. We recognize that further work is required to define the sensitivity of the image acquisition protocol, and to gain further understanding of the full extent of intra-species differences. Nonetheless, it is envisaged that this methodology could form the basis for a new range of analytical measures that will enable tree, environmental or ecological managers to gain greater insights and make more informed decisions about the tree stock under their management. Supplementary data Supplementary data are available at Forestry online. Funding This research is supported by an Engineering and Physical Sciences Research Council (EPSRC) studentship for the lead author [EP/L504804/1]. Acknowledgements The lead author would like to thank Dr. V. Karloukovski, Lancaster Environment Centre, Lancaster University, for discussions on fractal dimensions, and, Dr. E. Eastoe, Department of Mathematics and Statistics, Lancaster University, for providing statistical advice. The authors would also like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of this paper. Conflict of interest statement None declared. 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Forestry: An International Journal Of Forest ResearchOxford University Press

Published: Mar 28, 2018

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