BAYESIAN OBJECT RECOGNITION FOR THE ANALYSIS OF COMPLEX FOREST SCENES IN AIRBORNE LASER SCANNER DATA

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1 BAYESIAN OBJECT RECOGNITION FOR THE ANALYSIS OF COMPLEX FOREST SCENES IN AIRBORNE LASER SCANNER DATA Hans-Erik Andersen a, *, Sephen E. Reuebuch b, Gerard F. Schreuder a a Universiy of Washingon, College of Fores Resources, Seale, WA, USA - (hanserik, gsch)@u.washingon.edu b USDA Fores Service, Pacific Norhwes Research Saion, Seale, WA, USA - sreuebuch@fs.fed.us Commission III, WG III/3 KEY WORDS: Foresry, laser scanning, LIDAR, objec, recogniion, remoe sensing, saisics ABSTRACT: Bayesian objec recogniion is applied o he analysis of complex fores objec configuraions measured in high-densiy airborne laser scanning (LIDAR) daa. Wih he emergence of high-resoluion acive remoe sensing echnologies, highly deailed, spaially explici fores measuremen informaion can be exraced hrough he applicaion of saisical objec recogniion algorihms. A Bayesian approach o objec recogniion incorporaes a probabilisic model of he acive sensing process and places a prior probabiliy model on objec configuraions. LIDAR sensing geomery is explicily modelled in he domain of scan space, a hreedimensional analogue o wo-dimensional image space. Prior models for objec configuraions ake he form of Markov marked poin processes, where pair-wise objec ineracions depend upon objec aribues. Inferences are based upon he poserior disribuion of he objec configuraion given he observed LIDAR. Given he complexiy of he poserior disribuion, inferences are based upon dependen samples generaed via Markov chain Mone Carlo simulaion. This algorihm was applied o a 0.21 ha area wihin Capiol Sae Fores, WA, USA. Algorihm-based esimaes are compared o phoogrammeric crown measuremens and field invenory daa. 1. INTRODUCTION 1.1 Auomaed fores invenory While naional and local invenories ofen uilize remoely sensed daa for sraified sampling and classificaion of general fores ype, mos of hese programs remain heavily relian upon expensive field daa for individual ree-level informaion. A he naional level in he Unied Saes, individual ree invenory informaion is colleced a considerable expense. I is significan ha senior researchers wihin he USDA Fores Service Fores Invenory and Analysis (FIA) program have recognized he need for he developmen of auomaed fores inerpreaion and measuremen algorihms o reduce human inervenion and labor coss (Gulden, 2000). 1.2 The LIDAR echnology LIDAR (LIgh Deecion And Ranging) is an operaionally maure remoe sensing echnology ha can provide highly accurae measuremens of boh fores canopy and ground surface. While specificaions vary among sysems, LIDAR sysems emi from 5, ,000 pulses per second. In foresed areas, individual LIDAR pulses can penerae he fores canopy hrough gaps, and can herefore acquire informaion relaing o hree-dimensional fores srucure as well he underlying errain surface. 1.3 LIDAR analysis for fores measuremen applicaions In recen years, here has been increasing ineres in he use of LIDAR for auomaed deecion and measuremen of fores feaures. Research effors in he las fifeen years were focused on he use of small fooprin (0-1 m pulse diameer) LIDAR sysems o esimae fores sand level parameers (Nelson e al., 1988; Means e al., 2000). Researchers in Canada used a model-based approach o recover ree heighs from LIDAR canopy heigh measuremens (Magnussen e al., 1999). Threedimensional mahemaical morphology has been applied o a high-resoluion LIDAR-based canopy surface model o exrac individual ree measuremens (Andersen e al., 2001). 1.4 Auomaed individual ree crown recogniion hrough emplae maching Wih he recogniion ha high-resoluion remoely sensed spaial daa can suppor more inensive fores managemen pracices, here has been increasing ineres in recen years in he developmen of algorihms for auomaed idenificaion and measuremen of individual rees using high-resoluion, wo dimensional digial imagery. Several sudies have used a modelbased approach o locae individual rees using ree crown emplae models (Pollock, 1996; Larsen, 1998; Sheng e al., 2001). Researchers in Scandinavia have aemped o model he relaionship beween he spaial disribuion of individual rees and he posiion of specral maxima in a digial image (Dralle and Rudemo, 1997). Anoher Scandinavian sudy has used deerminisic parameer search mehods for maximum likelihood esimaion on a spaial poin process model o infer he parameers of a disurbance model ha relaes he rue posiion of ree-ops o hose observed on an aerial phoograph (Lund and Rudemo, 2000). * Corresponding auhor.

2 2.1 LIDAR daa 2. STUDY AREA AND DATA LIDAR daa were acquired wih a Saab TopEye sysem over a 5 km 2 area wihin Capiol Sae Fores, WA in he spring of The sensor seings and fligh parameers are shown in Table 1. Daa were provided in he form of an ASCII ex file, wih GPS ime, aircraf posiion, and coordinae posiion for he firs laser reflecion included. Flying heigh 200 m Flying speed 25 m/s Swah widh 70 m Forward il 8 degrees Laser pulse densiy 3.5 pulses/m 2 Laser pulse rae 7000 pulses/sec Table 1. Fligh parameers and LIDAR sysem seings. The LIDAR vendor also provided a LIDAR-derived digial errain model (DTM) for he sudy area wih a 4.57-meer (15- f) resoluion. 2.2 Aerial phoography Large-scale (1:7000) normal-color aerial phoography was acquired over he sudy area in This phoography was oriened in an analyical sereoploer. 3.1 Bayesian image analysis 3. METHODS In general, Bayesian image analysis provides a means o incorporae prior knowledge or beliefs ino he analysis of remoely sensed daa (Besag, 1993). These a priori beliefs are represened in he form of a prior disribuion, or prior model, ha is placed over he image and is updaed upon observaion of he daa. Formally, if his prior descripion of he image is denoed as p(x), hen he condiional spaial disribuion of his descripion, given he observed image y, is given by: px ( y ). ly ( xpx ) ( ) (1) In Bayesian parlance, his condiional disribuion p(x y) is referred o as he poserior disribuion, on which all inferences are based. In Bayesian inference his poserior disribuion is always represened as he produc of he likelihood l(y x) and he prior p(x). Typically, he goal in Bayesian inference is o calculae expecaions or credible inervals (explici probabiliy saemens made regarding he range of a parameer given he observed daa). 3.2 Bayesian objec recogniion More recenly, he mehods of Bayesian image analysis have been applied o he problem of objec recogniion (Baddeley and van Lieshou, 1993; van Lieshou, 1995; Rue and Syversveen, 1998; Rue and Hurn, 1999). The objecive of his ype of analysis is ypically o locae and characerize various objecs of ineres in space, incorporaing prior knowledge of he spaial disribuion of hese objecs. Therefore, prior models based upon discree grid-based neighborhood srucures end o be less appropriae. The descripion of Bayesian objec recogniion presened here generally follows van Lieshou (1995). In Bayesian objec recogniion, he observed daa consis of an image, y = ( y ;. T ), where T (he image space) is an arbirary finie se. The class of possible objecs U, is an arbirary se, ermed objec space. Objecs can be seen as poins u in U, and each deermine a subse R( u ). T of image space ha is occupied by he objec. Any paricular configuraion is a finie se of disinc objecs x = x, x,..., x. The objecive in { 1 2 n } objec recogniion is o esimae he (unobserved) rue underlying paern x given he observed image y. This rue configuraion x is relaed o he observed image y hrough he likelihood funcion l(y x). As van Lieshou (1995) describes, he likelihood l(y x) represens boh he deerminisic influence of he rue configuraion x, and he sochasic effecs wihin he remoe sensing process ha produces he image, y. In a Bayesian analysis, he prior models will represen our prior beliefs regarding he spaial disribuion of objecs, and can be formulaed o assign low probabiliy o configuraions ha we do no expec o occur frequenly, such as a large number of overlapping objecs. The maximum a poseriori (MAP) esimaor of x is he configuraion xû ha maximizes he funcion l(y x)p(x), and he prior essenially is a penaly assigned o his maximizaion. Therefore MAP esimaion is also called penalized maximum likelihood esimaion. 3.3 Bayesian objec recogniion for he analysis of hreedimensional LIDAR daa in foresed areas While Bayesian objec recogniion has previously been applied o he analysis of wo-dimensional images, his approach can also be applied o analyze srucure wihin hree-dimensional LIDAR daa. In his case, he observed daa, y, are no defined in erms of a raser image space, T. Insead, he scan space becomes a collecion of vecors, T, deermined by he LIDAR scanning process. Therefore, an individual pulse vecor,, represens he hree-dimensional direcion of each LIDAR pulse, from he aircraf o he errain surface. The observed daa, y, hen represen range measuremens along hese vecors a which poin he reurning signal inensiy exceeded a predeermined hreshold (see Figure 1). Bayesian image analysis has radiionally been carried ou using digial images consising of a discree grid of picure elemens (or pixels). Ofen he objecive is o reconsruc an "underlying" image ha has been disored hrough a noise process.

3 T a foresed area. The spaial disribuion of foliage is a funcion of individual ree locaions, sizes, crown forms, and an average leaf area densiy (LAD). Crown forms were represened as generalized ellipsoids following Sheng e al.. (2001), where he space occupied by he foliage wihin an individual ree crown is deermined by four parameers: crown widh (cw), crown heigh (ch), crown curvaure (cc), ree heigh (h), and he 2-D coordinae of he crown op (X op,y op,) (see Figure 2). * * * * * * * y y Figure 1. LIDAR sensing geomery (red sars represen LIDAR measuremens; blue lines represen pulse vecors composing he scan space T; y represens a single range measuremen along pulse vecor ). The disribuion of ree crowns over he enire scene is hen modeled as an objec configuraion, x. If individual plans were acually solid objecs (e.g. ellipsoids, spheres, ec.) in objec space U, hese LIDAR measuremens, y, would represen he locaion where each vecor inerceped he surface of he objec. In he erminology inroduced above, hese measuremens would represen he signal, or he deerminisic influence of he acual configuraion of objecs x on he series of LIDAR range measuremens ha are observed. A more realisic approach, however, would need o accoun for he fac ha plans are no solid geomeric objecs, and LIDAR pulses acually penerae a cerain disance ino he canopy hrough foliage gaps. This would incorporae a sochasic elemen o he LIDAR measuremens, y, due o he irregular spaial disribuion of foliage elemens (leaves, branches, ec.) in he pahway of a laser pulse as i inersecs a ree crown. Therefore, again using he noaion inroduced above, he condiional disribuion of individual LIDAR measuremens, given he signal, is given by a family of densiies g( y x). In he conex of LIDAR measuremen of ree crowns, hese probabiliy densiies will be relaed o he laser aenuaion funcion, which in he case of discree LIDAR sysems is direcly relaed o he probabiliy of reflecion. If he values of individual LIDAR measuremens along a pulse vecor can be considered condiionally independen, given he rue configuraion of objecs, x, he likelihood funcion, represening he join probabiliy of he daa, is given by: l( y x) =. g( y x ). T Modelling he disribuion of foliage in complex fores scenes: Previous sudies of laser ransmission hrough he fores canopy have uilized hree-dimensional grid models populaed wih generalized geomeric forms ha represen individual plans (Sun and Ranson, 2000). In his sudy, a hree-dimensional array, wih 0.91-m cell size, was used o model he disribuion of foliage densiy hroughou (2) a) h:47, cw:11, ch:14, cc:1.45 b) h:30, cw:8, ch:18, cc:0.75 Figure 2. Generalized ellipsoid crown models. The surface of a ree crown is hen given by he following mahemaical expression: 2 2 cc /2 cc (Z + ch + Z ) (( X - X ) + ( Y -Y op op ) op ) + = 1 (3) cc cc ch where he elevaion of he crown op, Z op, is deermined by adding he ree heigh o he elevaion of he base as deermined from he DTM. Values for LAD were obained from previous research findings (Webb and Ungs, 1993) Modelling laser-canopy foliage ineracion: The analysis of daa acquired from acive remoe sensing echnologies requires an undersanding of he ineracions beween he emied radiaion and he physical properies of he arge. In our model, where he laser fooprin (0.4 m) is significanly less han he cell size (0.91 m), he probabiliy ha a direc ligh beam ha eners a cell exis from he cell wihou being inerceped is calculaed as a funcion of he leaf area densiy and he off-nadir angle (θ) of he laser pulse. The model hen calculaes he probabiliy of peneraion o he cener of any cell by direc laser energy originaing from ouside he cell. Specifically, he probabiliy of a laser beam ha eners a canopy cell, z i, a a specific off-nadir angle, θ, reflecs from his cell wih foliage densiy LAD i is given by he following funcion (Vanderbil, 1990): p ( reflecion ) = 1 -exp(- 1/ cos. LAD i G(., z i ) dz ) (4) i where LAD i is he leaf area densiy (m 2 / m 3 ) wihin cell z i, G(θ, z i ) is he projecion of he vegeaion wihin cell z i in he direcion of θ, and dz i is he deph of cell z i. Ofen, a spherical leaf angle disribuion can be assumed, in which case G(θ, z i ) is cr

4 0.5 (Goudriaan, 1988). Therefore, he probabiliy of an individual LIDAR pulse reflecing from a specific cell z i in he grid (and no reflecing from he cells i-1, i-2,, 0 ha i has already passed hrough) will be given by: g( y x exp( 1/ cos G(., z k ) dz ) =. i-1 [ -. LAD k k =0 [1- exp( - 1/ cos. LAD i G(., z i ) dz i )] k )] This funcion defines a probabiliy densiy for LIDAR reflecion y, anywhere along a hree-dimensional pulse vecor. In addiion, in our model i is assumed ha he probabiliy of a laser pulse reflecing if i peneraes o wihin 6 meers of he errain (DTM) elevaion is 1. In addiion, foliage reflecance is assumed o be consan. The likelihood funcion is hen given by he following expression, which represens he join probabiliy of he LIDAR daa: i-1. exp( 1/ cos k. k k l( y x ) =. k =0. T [ 1 - exp( - 1/ cos. LAD i G(., z i ) dz i )] (5) [ -. LAD G(, z ) dz )] (6) where n(x) is he number of poins in x and g(x i,x j ) is an ineracion funcion (Ripley, 1981). This model herefore places a consan muliplicaive penaly on each pair of ineracing poins. This ype of model can be used o represen varying radii of inhibiion surrounding biological phenomenon, and herefore can provide a useful model for fores objec processes where rees exhibi pair-wise ineracions. A marked poin process is a poin process wih a characerisic (mark) aached o each poin in he process. Therefore a marked poin process on R d is a random sequence x = [ sn; m n ] where he poins s consiue an (unmarked) poin process in R d and he m are he marks corresponding o each locaion s. In his model, s denoes he locaion of a ree, while m represens a vecor of objec aribues including heigh, crown widh, crown heigh, and crown curvaure. Given a probabiliy disribuion for he marks, ν(m), he prior model, represening he Markov objec process, akes he following form, where fores objec ineracions depend upon he individual ree aribues (marks): p ( ) ( x n x ) = aj (8).. (m.i ) g ( x, x ) i j i i< j The objec configuraion xû ha maximizes his funcion will represen he maximum likelihood esimae (MLE) of he rue objec configuraion x. However, given ha MLE does no penalize large numbers of overlapping objecs, i is likely ha he MLE will be overly sensiive o he daa and herefore will no represen a realisic fores objec configuraion. Through a Bayesian approach, prior knowledge relaing o ree disribuions and ineracions can be incorporaed ino he model hrough he specificaion of he prior model, leading o more accurae esimaes of he rue objec configuraion Fores objec processes: In he Bayesian objec recogniion approach, he underlying prior disribuion, and he resuling poserior probabiliy disribuion of he rue objec configuraion given he observed image daa, usually akes he form of a spaial poin process, a sochasic geomeric model for an irregular, random paern of poins. These models allow for inference o be carried ou relaing o he spaial posiion of individual objecs as well as he aribues of hese individuals. These models also allow iner-objec ineracion, as well as possible global properies of a disribuion of objecs o be incorporaed ino he spaial model (Ripley, 1991). If we define he environmen E(A) of a se A o be he se of neighbors of poins in A, a poin process is a Markov process if he condiional disribuion on A given he res of he process depends only on he process in E(A). One of he mos common Markov poin process models is he pair-wise ineracion model, which has he form: In our model, wo crowns were considered o be overlapping if he raio of he disance beween he cener of he crowns and he sum of he crown radii was less han The mark disribuion was a mulivariae normal disribuion, wih parameers deermined from sand observaions Simulaion-based poserior inference for he MAP fores objec configuraion: In Bayesian analysis, all inferences are based upon he poserior disribuion: p( x y ). l( y x) p( x ). The ypical objecive of Bayesian objec recogniion is o esimae he rue configuraion of objecs x, given he observed daa y. In paricular, he maximum a poseriori (MAP) esimae, represening he mode of he poserior disribuion, is of primary ineres in he conex of objec recogniion. Wihin our model formulaion, he poserior disribuion is also a Markov objec process. Due o he complex naure of he poserior disribuion in his case, poserior inference was conduced via Markov chain Mone Carlo (MCMC) simulaion. In MCMC, one consrucs a Markov chain wih an equilibrium disribuion converging o he arge disribuion (he poserior disribuion in he case of Bayesian inference). Ideally, his Markov chain should be consruced so as o efficienly move hroughou he se of possible configuraions, while mainaining he correc equilibrium disribuion. nx ( ) p( x ) =aj (7). g( x i, x j ) i< j

5 In his case, proposed moves for he Markov chain include 1) change of objec parameers, 2) birh of an objec, 3) deah of an objec, 4) merging of wo objecs, and 5) spliing a single objec ino wo objecs. Due o he change of dimension when proposing o add, delee, merge, or spli objecs in he configuraion, he Meropolis-Hasings-Green reversible jump MCMC algorihm was used o mainain he correc equilibrium disribuion (Green, 1995; Hurn and Syversveen, 1998). The change of parameers is achieved by drawing a sample from a mulivariae normal disribuion cenered on he parameer vecor for a seleced objec. The birh of an objec is carried ou by drawing a sample from he mulivariae normal mark disribuion for ree objecs. The locaions, heighs, and ree crown dimensions corresponding o he MAP esimae of he rue objec configuraion wihin his area are shown in Figure 4, superimposed on he hree-dimensional scaer plo of he LIDAR daa and errain model. A wo-dimensional represenaion of he MAP esimae and LIDAR daa is shown in Figure MAP esimaion via simulaed annealing: Finding he objec configuraion ha maximizes he poserior probabiliy (i.e. MAP esimae) is essenially a combinaorial opimizaion problem. Simulaed annealing, an opimizaion echnique wih is origins in saisical mechanics, provides a means of arriving a he global opimum for a given sysem hrough a process of firs meling he sysem a a high emperaure, hen gradually lowering he emperaure unil he sysem freezes a he opimal configuraion (Kirkparick e al., 1983). In he conex of Bayesian objec recogniion, i has been shown ha samples obained, via MCMC, from he empered poserior disribuion 1/ H [ p( x y )] will converge o he MAP soluion as H 0 (van Lieshou, 1994). In our algorihm, an annealing schedule of H n+1 =. Hn was used, where H is he emperaure a ieraion n, and λ is he cooling rae. Figure 4. Three-dimensional perspecive view of MAP esimae of ree locaions and crown dimensions superimposed on LIDAR daa. LIDAR pulse fooprins are drawn o scale and are color-coded by elevaion. Figure 5. Planimeric view of MAP esimae of crown configuraion (black circles) superimposed on LIDAR daa. LIDAR pulse fooprins are drawn o scale and are color-coded by elevaion. 4. EXAMPLE The algorihm was run on a 0.21 ha area wihin a lighly hinned uni of he Capiol Sae Fores (see Figure 3). The ineracion parameer used in he prior disribuion was se o e-750, which places a moderaely heavy penaly on severely overlapping ree crowns. The inensiy parameer for he objec process, β, was also se o e-100. For his example, ieraions of he MCMC algorihm were run wih he cooling rae, λ, se o , and iniial emperaure (H) of 20. The MCMC algorihm sared wih zero objecs. Figure 3. Locaion of example area (delineaed in whie) wihin area of Capiol Sae Fores, WA. The esimaes obained from he objec recogniion algorihm were compared o phoogrammeric measuremens of crown locaions made from large-scale (1:7000) aerial phoography wihin a 0.21 ha area (see Figure 6).

6 Fuure research will focus on comparing algorihm resuls o field-based measuremens and assessing he influence of auomaed measuremen error on sand-level parameer esimaes. In addiion, Bayesian objec recogniion offers a flexible modelling approach ha allows for fusing he informaion conen from muliple sources of daa. Such muliple daa sources are becoming more available as vendors offer simulaneous acquisiion of georeferenced imagery and LIDAR daa. As he daa ener he model only hrough he likelihood funcion in Bayesian objec recogniion, oher ypes of remoely sensed daa (including aerial phoography and high resoluion saellie imagery) can be easily incorporaed ino he model hrough adjusmen of he likelihood funcion. 7. REFERENCES Figure 6. Planimeric view of MAP esimae of crown configuraion (black circles), phoogrammeric crown measuremens (shor dashes) and ha circular invenory plo boundary (long dashes). 5. DISCUSSION Resuls indicae ha he algorihm is generally successful in idenifying srucures associaed wih individual ree crowns wihin his fores area. The MAP esimae of he crown configuraion generaed by he algorihm closely maches he spaial paerns eviden in he LIDAR daa (Figure 5). The algorihm appears o be very sensiive o he daa, and in some areas added spurious small crowns o increase he likelihood of he daa. In general, he MAP esimae of crown locaions corresponds o he phoogrammeric crown measuremens (see Figure 6). I should be noed ha accurae recogniion and delineaion of overlapping ree crowns is difficul even in high-resoluion aerial imagery. In his case, here is a sysemaic discrepancy of 1-4 meers in he norh-souh direcion beween algorihm-based crown locaions and phoo-based crown locaions. This offse is probably due o he effec of crown layover and/or misregisraion of he aerial phoography. Field daa was available for a ha circular invenory plo locaed wihin he sudy area (see Figure 6). Ineresingly, he number of codominan (oversory) rees found wihin he plo in he field (14) maches he number found by he algorihm and measured in he phoographs. 6. CONCLUSIONS Bayesian objec recogniion provides a promising framework for he analysis of complex fores scenes using high-densiy, hree-dimensional LIDAR daa. I is clear ha modelling assumpions will have a srong influence on he resuls; for example, i is apparen ha crowns wih an asymmerical, irregular shape will be difficul o deec given he consrains of he generalized ellipsoidal crown model used here. The use of more complex crown models may improve recogniion of irregularly shaped crowns. Andersen, H., S. Reuebuch, and G. Schreuder, Auomaed individual ree measuremen hrough morphological analysis of a LIDAR-based canopy surface model. In: Proceedings of he Firs Inernaional Precision Foresry Symposium, Seale, WA, USA. Baddeley, A., and M. van Lieshou, Sochasic geomery models in high-level vision. In: Mardia, K.V. and G.K. Kanjii, eds. Advances in Applied Saisics 1. Carfax, Abingdon, Oxfordshire, pp Besag, J., Towards Bayesian image analysis. In: Mardia, K.V. and G.K. Kanjii, eds. Advances in Applied Saisics 1. Carfax, Abingdon, Oxfordshire, pp Dralle, K., and M. Rudemo, Auomaic esimaion of individual ree posiions from aerial phoos. Canadian Journal of Fores Research, 27, pp Goudriaan, J., The bare bones of leaf-angle disribuion in radiaion models for canopy phoosynhesis and energy exchange. Agriculural and Fores Meeorology 43, pp Green, P., Reversible jump Markov chain Mone Carlo compuaion and Bayesian model deerminaion. Biomerika, 82(4), pp Gulden, R., Remoe sensing a he dawn of a new millennium: A Washingon DC perspecive. In: Proceedings of he Eighh Biennial Remoe Sensing Applicaions Conference, Albuquerque, NM, USA. Kirkparick, S., C.D. Gela, Jr., and M.P. Vecchi, Opimizaion by simulaed annealing. Science, 220(4598), pp Larsen, M., Finding an opimal mach window for spruce op deecion based upon an opical ree model. In: Proceedings of he Inernaional Forum on Auomaed Inerpreaion of High Spaial Resoluion Digial Imagery for Foresry, Vicoria, BC, Canada. Lund, J. and M. Rudemo, Models for poin processes observed wih noise. Biomerika, 87(2), pp Magnussen, S., P. Eggermon, and V.N. LaRiccia, Recovering ree heighs from airborne laser scanner daa. Fores Science, 45(3), pp

7 Means, J., S. Acker, B. Fi, M. Renslow, L. Emerson, and C. Hendrix, Predicing fores sand characerisics wih airborne scanning LIDAR. Phoogrammeric Engineering and Remoe Sensing, 66, pp Nelson, R., W. Krabill and J. Tonelli, Esimaing fores biomass and volume using airborne laser daa. Remoe Sensing of he Environmen, 24, pp Pollock, R., The auomaic recogniion of individual rees in aerial images of foress based upon a synheic ree crown image model. PhD hesis, Deparmen of Compuer Science, Universiy of Briish Columbia, Vancouver, BC, Canada. Ripley, B., Spaial Saisics. Wiley, New York. Ripley, B., The use of spaial models as image priors. In: Anonio Possolo, ed., Spaial Saisics and Imaging, vol. 20 of Lecure Noes Monograph Series, Insiue of Mahemaical Saisics, Hayward, CA, USA. Rue, H., and M. Hurn, Bayesian objec recogniion. Biomerika, 86(3), pp Rue, H., and A. Syversveen, Bayesian objec recogniion wih Baddeley s dela loss. Advances in Applied Probabiliy, 30, pp Sheng, Y., P. Gong, and G. Biging, Model-based conifercrown surface reconsrucion from high-resoluion aerial images. Phoogrammeric Engineering and Remoe Sensing, 67(8), pp Sun, G., and K. Ranson, Modeling lidar reurns from fores canopies. IEEE Transacions on Geoscience and Remoe Sensing 38(6), pp Vanderbil, V.C., L.F. Silva, and M.E. Bauer, Canopy archiecure measured wih a laser. Applied Opics, 29(1), pp van Lieshou, M., Sochasic annealing for nearesneighbour poin processes wih applicaion o objec recogniion. Advances in Applied Probabiliy, 26, van Lieshou, M., Sochasic geomery models in image analysis and spaial saisics. CWI, Amserdam, he Neherlands. Webb, W. and M. Ungs, Three-dimensional disribuion of needle and sem surface area in a Douglas-fir. Tree Physiology 13, pp Acknowledgemens The Precision Foresry Cooperaive a he Universiy of Washingon College of Fores Resources and he USDA Fores Service Pacific Norhwes Research Saion provided financial suppor and daa for his research. The auhors would also like o hank Rober McGaughey (USDA Fores Service PNW Research Saion) for assisance in carrying ou his research. Ciaion for his aricle: Andersen, H.-E., S. Reuebuch, and G. Schreuder Bayesian objec recogniion for he analysis of complex fores scenes in airborne laser scanner daa. Inernaional Archives of Phoogrammery and Remoe Sensing, Graz, Ausria, 2002, Vol. XXXIV, Par 3A, pp