A REDUCED RANK REGRESSION MIXTURE MODEL FOR CHANGE VALIDATION IN AERIAL IMAGES

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1 A EDUCED AK EGEIO MIXTUE MODEL FO CHAGE VALIDATIO I AEIAL IMAGE FEADO PÉEZ AVA Departameto de Estadística Ivestigació Operativa y Computació. Uiversidad de la Lagua. Islas Caarias pai. fdoperez@ull.es JOE MAUEL GÁLVEZ LAMOLDA Departameto de Física Fudametal y Experimetal Electróica y istemas Uiversidad de La Lagua. Islas Caarias pai jmgalvez@ull.es Chage detectio is a importat part of image iterpretatio ad automated geographical data collectio. I this paper we show a reduced rak regressio mixture model for the verificatio of image chages detected by a huma operator. Maximum likelihood estimators are used to lear the operator behaviour. The the operator uses the traied system to validate the image chages foud. Computatioal results are give with real image data that show the performace of the system.. Itroductio Chage detectio o the terrestrial surface from aerial images is a importat problem i the update ad maiteace of digital cartographic maps. Oe way to detect these chages is the use of pairs of images of the same area at differet times. A huma operator the detects chages by meas of the visual aalysis of images. Huma visual aalysis is favoured by the implicit icorporatio of itelliget processig associated to the operator. However ad also due to the explicit itervetio of the operator this aalysis is ot free of disadvatages due to operator fatigue operator icosistecies or operator costs. To achieve a automatizatio of this process several techiques have bee developed: Image differecig [] Image regressio [2] or Pricipal compoet Aalysis (PCA) [3] [4]. I this paper we preset a quality cotrol system for the verificatio of the chages detected by a huma operator i pairs of images. The systems checks if each chage ca be explaied by the operator global behavior ad evetually select a subset of the chages for revisio. The learig system is

2 2 based o a reduced rak regressio mixture model [5] whose parameters are estimated from the chages detected by the operator. This model is simpler ad computatioally more efficiet tha the full regressio model. 2. A Probabilistic Model of Image Chages I this sectio we propose a geerative model of operator's respose for two patches x from image I captured at time t ad y from image I 2 captured at time t 2. These patches will be represeted as vectors ad are small eough so that they approximately belog to the Chage (CH) class or o Chage (C) class. The proposed model show i Figure states that itesity levels o patch y deped o the itesity levels patch x some ucotrolled factors (illumiatio chages or others) ad whether there is chage or ot as expressed i the True Class () variable. The model also states that oce the true class is iferred there may be a radom disturbace (due to operator's fatigue or other causes) that gives the actual respose of the operator (Operator's Class or ). Patch x Ucotrolled Factors (u) True Class () Patch y Operator's Class () Figure. Probabilistic Model for image chages To validate the operator's behaviour we are iterested i the evaluatig the probability P( x y). Followig the idepedece assumptios i the Bayesia etwork i Figure we have: P( x y) p(y x u ) P() P( ). () The to complete the model we eed to defie the three terms i the right side of equatio (). The proposed model for p(y x u ) is: y µ t B + A x t + C u + ε u ( 0 I) ε ( 0 σ 2 I) CH C (2)

3 3 where t is a r dimesioal radom variable u is q dimesioal radom variable ad t u ε are idepedetly distributed. The dimesio of matrix A is d r the dimesio of matrix B is r d ad both matrices have rak r. The dimesio of matrix C is d q ad its rak is q. Therefore we ca express the d dimesioal variable y i terms of two latet variables t ad u of dimesios r ad q where usually d >> r + q. The proposed model the states that image patch y is related to: Latet variable t that liearly depeds o image patch x. The term A t A B x ca be iterpreted as a reduced rak regressio [5] of y over x. Latet variable u that accouts for those aspects i y ot explaied by t. This term is related to probabilistic pricipal compoet aalysis (PPCA) [6]. A overall mea µ adom variable ε that expresses a isotropic oise model. We ca simplify the model i Figure itegratig out variable u. The: T 2 y x ( µ + A B x M ) M C C + σ I. (3) The probability distributio for P( ) ad P() are defied from Beroulli radom variables. 2. The Learig et Before we show how to estimate the parameters i the model we will describe the learig set. This set is give by the operator's work ad cosist i a set of poit locatios i the image where she observes a chage. The for each detected chage a image patch aroud this locatio is geerated. This provides the learig set for the 'Chage' class. After a operator processes a image there are o examples for the 'o Chage' class. To provide a set of examples for this class we radomly sample locatios from the image except i a eighborhood of the locatios i the 'Chage' class. The image patches geerated from these locatios are the learig set for the 'o Chage' class. I Figure 2 we ca see two examples from the learig set. Figure 2. ample image patches from the learig set of the Chage class (left) ad o Chage class (right).

4 4 2.2 Model parameter estimatio We will estimate the parameters of the model after elimiatig variable u by Maximum Likelihood Estimatio i the joit probability distributio of the observed variables p(y x): p(y x) p(y x )P( )P(). (4) It is simpler however to use the alterative factorisatio: p(y x) p(y x )P( )P(). (5) We will deote P( ) as P() as P( ) as ad P() as. These probability distributios are related by: (6) The give a learig set H{(y x ) (y 2 x 2 2 ) (y x )} the Maximum Likelihood Estimators (MLE) for the model are evaluated iteratively with the Expectatio Maximizatio (EM) algorithm [7] as follows: For each sample i H iitialise its resposibilities for each of the CH C classes. repeat { Evaluate the parameters i the model from: i [ ] [ ] δ C CH [ ] C CH (7)

5 5 µ A yx xx ( µ ABµ ) V y r x ( y µ )( x µ ) ( x µ )( x µ ) B V µ y x T r y yx T x x xx where the r colum vectors i the d r matrix T y µ x x V r are the eigevectors correspodig to the r largest eigevalues of C CH. C yx xx xy U q ( y µ A B x )( y µ A B x ) ( Λ q 2 σ I) / 2 where the q colum vectors i the d q matrix U q are the eigevectors of with correspodig eigevalues i order of decreasig magitude i the q q diagoal matrix defied as: d 2 j d q j q + σ λ T Λ q where λ q +... λ d are the smallest eigevalues of. 2 ad σ is (7) For each sample reevaluate the resposibilities: p( y x ) (8) C CH p( y x ) } util likelihood coverges

6 6 To classify as C or CH a ew pair of image patches x ad y we apply the optimal Bayesia rule [7]: P( C x y ) / P( CH x y ) (9) p( y x C) / p( y x CH ) > ( bias C CH < ) To test a arbitrary patch we ca set the variable as missig obtaiig: P( C x y) / P( CH x y) (0) p( y x C) / p( y x CH ) > ( < bias C CH ) The bias variable is usually set to (to miimize the probability of error) but other values may be used if other criteria are employed. 2.3 Model electio Oce the estimatio problem is solved we have to select the complexity of the model i (). It depeds o the rak parameters: r CH r C q CH q C. To study the optimal values for the parameters we have made 5 replicatios of two fold cross-validatio (5x2 cv) i several images. esults show that for most images the optimal values are: r CH 0 q CH q C ad r C i the rage [2535]. For example we show i Figure 3 the average egative log-likelihood (the smaller the better) of the traiig ad validatio data after 5x2 cv i terms of the r C for the image pair partially reproduced i Figure 4 after fixig r CH 0 q CH q C. Figure 3 Average egative log-likelihood of the validatio set (cotiuous lie) ad learig set (discotiuous lie) after 5 replicatios of 2 fold cross-validatio. The miimum value for the average egative log-likelihood of the validatio set is obtaied for r C 48 ad its value is equal to l(r C ) with a stadard deviatio of.3. To select the optimal value of r C we searched for

7 the smallest value of r C whose differece with that miimum is ot sigificatively differet. To obtai this value the combied 5x2 cv test [8] was take with sigificace level The obtaied value is r C 28 with likelihood l(r C ) ad stadard deviatio of.64. Also the reduced rak model is sigificatively better tha the full regressio model. It ca be show that the ucotrolled factor foud by the model (q CH q C ) correspods to local chages of illumiatio ad disappears if all patches are preset to a costat mea of illumiatio. Therefore the model ca be uderstood as a mixture of reduced rak regressios with rak r CH 0 for the CH class (o predictio is possible) ad rak r C << d for the C class Computatioal esults We ow preset some computatioal results from the mixture model. I Figure 4 we show a partial reproductio (350x350 pixels) of a aerial image from the islad of Gra Caaria (pai). The learig set extracted from the full image (250x250 pixels) comprised 493 chages detected by the operator. Image patches were size 5 5 pixels hece d 225. After a 5x2 cv step rak values were r CH 0 q CH r C 28 q CH. I the left image of Figure 5 coicideces betwee the operator ad the system are show with white squares ad differeces with black circles. I the right image of Figure 5 the differece of log-likelihood betwee the CH class ad C is show. Whiter regios deote a greater likelihood of chage. This likelihood is also show i 3D i Figure 6. The chages i Figure 5 (left) ot validated by the system are show i Figure 7. Figure 4 Detail of oe pair of the image from the experimets

8 8 Figure 5 Left: Coicideces of the operator ad the system are deoted with white squares ad differeces with black circles. ight: Log-likelihood of chage. Figure 6 Likelihood of chage represeted i 3D over the right image i Figure 4. (the higher the more likelihood of chage) Figure 7 Chages ot validated by the system i Figure 5 (left) are show from top left to dow right sorted by the differeces of log-likelihood betwee the two classes (that is from less likelihood of chage to more).

9 9 4. Coclusios Chage detectio is a importat part of image iterpretatio ad automated data collectio for Geographical Iformatio ystems. I this paper we have preseted a probabilistic model for chage validatio i aerial images. This model uses a mixtute of reduced rak regressors whose parameters are estimated by Maximum Likelihood usig the Expectatio Maximizatio algorithm. Computatioal results are give that show the validity of the results. efereces. D.L. Williams y M.L. tauffer (978). Moitorig Gypsy Moth Defoliatio by Applyig Chage Detectio Techiques to Ladsat Imagery. Proc. of the ymp. for Vegetatio Damage Am. oc. for Photogrammetry pp K. Igram E. Kapp ad J.W. obiso (98). Chage detectio techique developmet for improved urbaized area delieatio Techical memoradum CC/TM-8/6087 Comp. ci. Corp Marylad UA. 3. G.F. Byre P.F. Crapper ad K.K. Mayo (980). Moitorig Lad Cover Chage by Pricipal Compoet Aalysis of Multitemporal Ladsat Data. emote esig of Eviromet Vol. 0 pp Wiemker A. peck D. Kulbach H. pitzer ad J. Bielei (997).Usupervised obust Chage Detectio o Multispectral Imagery Usig pectral ad patial Features. Proceedigs of the Third Iteratioal Airbore emote esig Coferece ad Exhibitio Copehague 5. G. C. eisel ad.p. Velu. (998) Multivariate educed ak egressio: Theory ad Applicatios. ew- York. priger-verlag. 6. M. Tippig ad Christopher Bishop. (997). Mixtures of probabilistic pricipal compoet aalyzers. Techical eport CG/97/003 eural Computig esearch Group Asto Uiversity. 7. A.P. Dempster.M. Laird ad D.B. ubi. (977) Maximum Likelihood from Icomplete Data via The EM Algorithm Joural of oyal tatistical ociety Vol. 39 pp E. Alpaydi (999) Combied 5x2 cv F Test for Comparig upervised Classificatio Learig Algorithms eural Computatio (8) Ackowledgmets The authors would like to tha all the facilities ad support provided by GrafCa (