Báo cáo khoa học: "Polyhedral representation of crown shape. A geometric tool for growth modelling"

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Original article Polyhedral representation of crown shape. A geometric tool for growth modelling C Cluzeau B Courbaud JL Dupouey 1 INRA, unité croissance, production et qualité des bois, 54280 Champenoux; 2 INRA, unité d’écophysiologie forestière, 54280 Champenoux; CEMAGREF, division protection contre les érosions, BP 76, 3 38402 Saint-Martin-d’Hères cedex, France 14 March (Received 1994; accepted 17 January 1995) Summary — Tree or stand growth modelling often needs an explicit representation of crown shape. This is necessary for crown volume or external surface calculations, or light penetration modelling. Many different representations have been used for this purpose. In this paper, we explore the use of the polyhedral convex hull of the crown as a type of boundary representation. We present an application of this representation for the calculation of geometrical characteristics of common ash trees (Fraxinus excelsior L). Crown projection area calculated with the convex hull is closely related to the measured value. Moreover, a strong relation exists between the basal area of the most external tree rings and the convex hull volume or surface area which indirectly validates the polyhedral representations. This relation, however, is no stronger than exists with the simpler crown projection surface area measure- ment. The convex hull is intermediate in terms of computation costs and efficiency between classical geometric shapes and more elaborate computer graphic representations. It is a simple and versatile tool for modelling purposes. shape representation / computational geometry / crown volume / convex hull / pipe crown model / Fraxinus excelsior L Résumé — Représentation polyédrique de la forme du houppier. Un outil géométrique pour la modélisation de la croissance. La modélisation de la croissance des arbres ou des peuplements fait souvent appel à une représentation géométrique du houppier. Cela est nécessaire pour, par exemple, les calculs de volume et de surface externe du houppier, ou la modélisation de la pénétration de la lumière dans les peuplements. De nombreuses représentations ont été utilisées jusqu’à maintenant, le plus souvent une combinaison de différents solides de révolution. Nous explorons dans cet article les possibilités offertes par la représentation polyédrique convexe où la frontière du houppier est représentée par son enveloppe convexe. Cette représentation est appliquée au calcul des caracté- ristiques géométriques de houppiers de frênes (Fraxinus excelsior L). La projection au sol du houppier, calculée à partir de son enveloppe convexe, est très proche de celle mesurée sur le terrain. De plus,
une forte relation d’allométrie apparaît entre l’accroissement en surface terrière des cernes les plus externes du tronc et le volume ou la surface de l’enveloppe convexe du houppier. Cette liaison est maxi- male lorsque l’accroissement est cumulé sur les 3 derniers cernes annuels. Cependant, elle n’est pas meilleure que celle observée avec la surface de projection au sol du houppier, plus facilement mesu- rable. Ces relations valident indirectement la représentation polyédrique convexe. Cette représenta- tion est un compromis intéressant, en termes de complexité et de précision, entre les solides de révo- lution classiques et les représentations volumiques plus élaborées. / volume du houppier /enveloppe convexe / géométrie informatique / Fraxi- forme du houppier excelsior L nus Geometric characteristics are most often INTRODUCTION obtained from the position of a few distin- guishing points of the crown, such as the Crown shape is a key factor in architectural top, base and maximum horizontal exten- and functional tree modelling. The crown is sion of branches. Length (vertical exten- at the interface between the tree and the sion) is readily calculated. Horizontal exten- atmosphere and as such controls the inter- sion is sometimes approximated by the ception of water, light and pollutants. It inter- maximum or mean width of the crown, but acts directly with other trees by mechanical more often ground crown projection is used. contact or indirectly by shading. Crown Its area is calculated from the position of shape both conditions and reflects tree eco- intersections between radii centered on the physiological functioning. bole and the crown edge projection. Calcu- Many geometric characteristics of crown lation of volume of external surface area of shape are used in modelling tree or stand the crown needs reference to an explicit growth. Crown length or horizontal exten- representation of the crown boundary shape. often used for the calculation of sion are Classical forms used are cylinders, various competition indices. Crown volume and sur- conics and vertical or radial combinations face area have been shown to be closely of these. Koop (1989) presents an extended related to foliar biomass (Zeide and Pfeifer, review of these different forms. His own 1991; Jack and Long, 1992; Makela and description of a crown, one of the most elab- Albrekton, 1992) or bole increment (Mitchell, orate, uses the measured position of 8 1975; Seymour and Smith, 1987; Sprinz points on the crown boundary to fit 4 slices and Burkhart, 1987; Ottorini, 1991). Eco- of ellipsoids. physiological parameters such as leaf con- However, these axisymmetrical shapes ductance, internal CO concentration 2 or impose heavy constraints on the repre- efficiency are significantly corre- water use sentation of the crown boundary. A more lated with crown volume (Samuelson et al, relaxed representation can be obtained 1992) and crown surface area has been using a set of points selected on the bound- used for the study of pollution impacts (see ary, and a graph of proximity on this set of eg, Dong et al, 1989). Geometrical and topo- points. Various geometric structures can logical information about the shape of the be used for this purpose (Boissonnat, crown is needed to model mechanical inter- 1984). In this paper, we explore the use of actions between trees or light interception the polyhedral convex hull of the crown, within stands. Finally, computer graphics which is one of the simplest structures and also need the use of such data for the syn- has not yet been tested for crown repre- thesis of realistic tree or stand pictures sentation. Such a representation can be (Reffye et al, 1988).
framework, the azimuth from the north and the used as a by-product of architectural mod- length of the leafy part were measured. Second els of crown development. These models order branches, directly attached to the bole, were are based on the quantitative analysis of distinguished from tertiary branches attached to tree organization at the branch or growth the secondary branches. From these data, we unit level (see eg, Mitchell, 1975; Ottorini, calculated the Cartesian coordinates (x,y,z) of the 1991; Reffye et al, 1991; Prusinkiewicz et al, origin and tip of each growth unit. A stem analysis gave basal area (at breast height) and bole vol- 1993). They imply the precise spatial posi- increments for each year. Disks were anal- ume tioning of phytoelements inside the crown. at 1 year intervals along the bole. For each ysed Thus, they provide the set of data neces- disk, annual radial increments were measured sary for the polyhedral representation of along 4 radii. A more detailed presentation of boundary. crown sampling and measurements is given in Cluzeau et al (1994). We an application of this poly- present hedral representation in the calculation of crown shape parameters of common ash Calculation of the polyhedral (Fraxinus excelsior L). To verify the relia- representation and crown bility of this representation, we also study shape parameters the classical allometric relationships between crown dimensions calculated with the polyhedral representation and radial tree All the measured points delimiting each growth unit are included in the crown. From this set of growth (see eg, Coyea and Margolis, 1992). points, we calculated the crown’s polyhedral hull. This work makes use of data initially col- There is no unique solution for this problem, but, lected for the modelling of common ash among all possible solutions, the convex hull is growth development (Cluzeau et al, 1994). the simplest and also has some properties that make it easy to manipulate. By definition, the poly- hedral convex hull is the smallest convex set con- MATERIALS AND METHODS taining all the above points. For any pair of points inside a convex set, the segment joining these 2 points is entirely inside the convex set. The hull of the calculated Tree sampling and measurements convex crown was using the gift wrapping algorithm (Preparata and Shamos, 1985) which gives a triangulation of the common ashes were sampled in Twenty-seven set of points belonging to the convex hull. Each various forests of north-eastern France. Trees facet of this convex polyhedron is, by construction, were chosen in order to represent different ages a triangle. We developed an application software and crown forms, including free growing trees for the calculation of convex hulls and image syn- with a large crown as well as crowded trees with thesis. This representation allows the calculation a thinner crown. Before cutting, each tree was of various form parameters. measured for diameter at breast height, total The position of the center of gravity of this height, crown length and crown projection sur- polyhedron was calculated. This gives informa- face area. This latter surface was delimited with tion on the asymmetry of the crown. Crown length a plumbline, from the branches which had the (CL) is the difference between the highest and longest horizontal projection all around the tree. lowest point ordinates. Total area of the hull (CS) Common ash has only a few second order is the sum of the elementary triangular facet areas. branches, thus 8 to 12 branches were sufficient. Volume (CV) is calculated as the sum of the vol- umes of elementrary tetrahedra based on each After harvest, annual length increments (growth units) of the branches and the stem were facet and with the centrer of gravity as the summit. determined. Boundaries between growth units Crown projection surface area is the area, on a were localized using bud scars. Length, diameter horizontal plane, of the convex hull of the verti- and age of each growth unit were determined. cal projections of all the points of the crown con- For each main branch, making up the basic crown vex hull. The surface area of the top part of the
hull was also calculated since it can be and 35 points, respectively (table I). The convex used to estimate the leaf surface exposed to sun. empty interior volume is small compared to This surface is composed of all the facets which the total volume (10% on average). Calcu- have their normal vector at more than 90° above lated values of volume are rather low in com- the horizontal. Finally, the empty interior volume parison with values observed elsewhere. of the crown ("bare inner core", Jack and Long, Vrestiak (1989) observed average values 1992) and the leafy volume were estimated with of 800 m for free growing common ashes at 3 the same convex hull approach applied to the set 50 years old. of points delimiting the leafy and leafless zones of the branches inside the crown. Allometric relationships between Verifying the reliability of crown dimensions and growth the polyhedral representation Figure 3 gives the relationship between the In order toverify the reliability of the polyhedral measured and calculated crown projection representation, we compared crown projection surface area. The correlation is very high surface area calculated with this representation to (r= 0.98), indicating that our representation that measured in the field. Furthermore, allomet- gives a valid view of the real crown, in 2-D ric relationships between crown volume or sur- space at least. A slight underestimation face area and basal area or bole volume incre- occurs for the largest trees, above 25 m 2 ment were studied. Correlation coefficients (r) of the crown projection surface area. and regression equations were calculated using the SAS package (SAS Institute Inc, 1989). Both Table II gives the linear correlation coef- raw variables and their squares were tested in ficients between various crown shape the regression equations. The quality of fit was parameters calculated with our convex rep- assessed by standard error of the estimates and resentation (surface area and volume of the adjusted coefficient of determination (R as well ), 2 external convex hull, crown projection sur- as visual inspection of the residuals. Although consistent with all the calculated regression lines, face area) and measurements of tree growth 1 large tree was removed from the calculations (annual basal area and bole volume incre- due to extreme and influential values for all vari- ments). A strong allometric relation exists ables. between the measured tree basal area and calculated crown surface area (r= 0.82) or volume (r= 0.81).However, the correlation RESULTS is even better with measured crown projec- tion surface area (r= 0.89). the correlation of surface Interestingly, Calculation and graphical representation area, volume of the convex hull, or the crown of the polyhedral representation projection surface area, with squared basal area increment cumulated from the most Figure 1 shows 1 tree and figure 2 its poly- external ring over the last 10 years, reaches hedral convex hull. For each figure, the tree a maximum for the 3 external rings (r= 0.93 is represented from 2 different directions, with the convex hull surface area, r= 0.92 shifted by 5°. This allows the reconstruction with its volume, and r= 0.93 with measured of a 3-D view of the tree using a classical crown projection surface area). It is inter- stereoscope. The complexity of the convex esting to note that this relationship (fig 4) hull increases with the number of branches holds for all trees in our sample, either free from 16 facets for the smallest tree to 66 growing or suppressed. An analysis of the facets for the biggest, corresponding to 10 residuals of this regression shows that over-
where CS is the convex hull surface area estimation of the crown volume occurs when (m CV is the convex hull volume (m and ), 3 ), 2 very low branches are developed down the BAI is the squared surface increments of the main stem. In this occurrence of outliers, 3 most external rings at breast height (dm ). 2 convexity assumes the presence of a con- tinuous layer of leaves from these low These 2 relations present an efficient branches upwards, whereas these branches way to rapidly calculate crown volume and are isolated at the base of the tree, without surface area based on simple measure- any leaves. ments of external tree-ring increments. The equations of the regression lines are: Neither the top part of the crown surface the leafy volume were better correlated nor with radial growth than total crown surface
and volume (table II). These relations were height and both crown surface at breast improved by removing 3rd order not and volume. This relationship between area branches from the hull calculation. conducting and evaporating surfaces (the convex so-called pipe model) has been extensively documented (see Coyea and Margolis, DISCUSSION 1992). It is a consequence of the fact that hydraulic conductance in stems is the prod- uct of conducting tissue area multiplied by Crown projection surface area is the only the specific conductivity of these tissues. measurement we have to directly verify the For ash trees, the relationship is better with validity of our representation. The agree- a squared value of conducting surface area. ment between measured and calculated val- This is in accordance with the theoretical ues is very high. However, this is only a 2-D hydraulic conductance, given by the Hagen- validity test of our 3-D representation. Valid- Poiseuille equation, which is proportional to ity of the polyhedral representation is also the 4th power of the capillary radius (Ewers strengthened by the very strong correlation we observed between current increments and Zimmermann, 1984).
The correlation of both crown volume ilar observation was made for oaks (Rogers and surface area is highest with the area of and Hinckley, 1979) and Norway spruce the last 3 annual rings. This suggests that (Sellin, 1993), and direct measurements water flow in common ash could be confirmed the assumption for oaks (Granier restricted to these external rings rather than et al, 1994). Hence, the functional value of stretching throughout the sapwood. A sim- the morphologically defined sapwood as an
has been shown to strongly control radia- indicator of xylem conductive surface is sus- tion absorption, photosynthesis and tran- pect and the necessity of direct measure- spiration (Wang and Jarvis, 1990; White- ments of water flow area within the trunk is head et al, 1990). Non-boundary made apparent. representations, such as the computer tech- However, the calculated crown external nique of voxel space, can handle the inter- surface area is not a better predictor of radial nal heterogeneity of distribution of phy- growth then the measured crown projection toelements (Green, 1989). surface area. This means that, for a growth However, the need for such more elab- model of common ash using an allometric orated representations of the crown depends relationship between radial increment of on the study scale and final objectives. Poly- bole and crown structure, the simple mea- hedral representation is very efficient in surement of the crown projection surface terms of computation time and memory area gives a sufficient estimation. On the space requirements for computer graphics contrary, Maguire and Hann (1988) and calculations of light penetration at the observed a better correlation of sapwood stand level. It is less demanding than voxel area with crown surface area then with sim- representations. Computation requirements pler variables. and accuracy of representation for the dif- The polyhedral representation is theo- ferent methods are roughly opposed and retically more precise than that of the correlated with the number of points used axisymmetrical solids because the real for the crown representation, from 8 points crown shape is often randomly built by con- in the Koop’s model (1989), to 10-35 points tacts with neighbors, shading or illumina- for our polyhedral representation, to hun- tion and various injuries (insect attacks, dreds of points in voxel spaces. Therefore, snow and ice damage, etc). Developmental the polyhedral representation offers an inter- asymmetry can be very important. Thus, esting alternative to these solutions. Due to the polyhedral representation is closer to the large number of data needed for its cal- the real shape. Whereas no overestimation culation, it is best suited when the position of of surface area, volume nor crown crown growth units within the crown is already projection surface area was apparent in our known, either from previous measurements results, the polyhedral representation is sen- or as a by-product of an architectural model sitive to extreme outliers. This could be of crown development. improved by calculating non-convex hulls, allowing the representation to account thus for depressions on the crown boundary ACKNOWLEDGMENTS (Boissonnat, 1984). Finally, convexity seems to be an acceptable constraint on the rep- resentation of crown shape since concave are indebted to JM Ottorini and N The authors Le Goff who were involved in the initial steps of shapes are seldom observed in nature, this work. We also thank R Canta and L Garros for especially for broadleaves (Koop, 1989). technical assistance during sampling and mea- The intrisic limit of the polyhedral repre- surements. sentation is that it is only a boundary rep- resentation of the crown. Thus, the internal distribution of leaves inside the crown is REFERENCES assumed homogeneous. Rather than homo- geneous, this distribution could be fractal (1984) Geometric structures for three- Boissonnat JD (Zeide and Pfeifer, 1991).The architectural shape representation. ACM Trans Graph dimensional arrangement of foliage in the tree crowns 3, 266-286
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