USING LINEAR REGRESSION FOR THE AUTOMATION OF SUPERVISED CLASSIFICATION IN MULTITEMPORAL IMAGES 1 Fetosa, R.Q., 2 Merelles, M.S.P., 3 Blos, P. A. 1,3 Dept. of Electrcal Engneerng ; Catholc Unversty of Ro de Janero Rua Marquês de São Vcente, 225 22453-900, Ro de Janero, Brazl Tel.: +55 21 5299438 1,2 Dept. of Computer Engneerng State Unversty of Ro de Janero Rua São Francsco Xaver, 524 20550-013, Ro de Janero, Brazl Tel.: +55 21 587 7442 e-mal: raul@ele.puc-ro.br, magge@eng.uerj.br, prblos@dog.com A method for the automatc estmaton of spectral sgnatures n multtemporal remotely sensed mages s proposed. The method resembles relatve atmospherc correctons technques where a lnear relatonshp between mage bands across tme s assumed. Before computng the regresson coeffcents an outler flterng procedure s appled resultng n an mproved accuracy n the estmated spectral sgnatures. The work compares also band-toband regresson and multple regresson. Moreover the use of sngle model for all classes and one model per class s nvestgated. Scenes of two areas undergong a rapd envronmental degradaton process were used as database to evaluate the proposed approach. The combnaton of outler flterng wth one regresson model per class produced the best results for moderate to hgh percentage of change pxels n the multtemporal mages. 1 Introducton Supervsed classfcaton of remotely sensed mages nvolves three basc steps: a) the photo nterpreter selects some representatve tranng data for each classe n the mage, b) an automatc tool estmates the spectral sgnatures based on the tranng data and generates the frst verson of a thematc map usng some classfcaton model, and c) the photo nterpreter refnes the ntal thematc map by ncorporatng some a pror knowledge not represented n the classfer used n step b. A great effort has been done worldwde to buld mage nterpretaton systems able to carry out all these tasks automatcally [1] [2]. Ths work presents an approach to generate the tranng data (step a) n an automatc fashon. Although no explct radometrc correcton s performed, there s a close analogy between the methods consdered here and relatve atmospherc correcton technques [3] [4] [5]. In fact smple ntutve extensons to the procedure dscussed n ths work would suffce to convert ths approach nto a relatve correcton technque. The proposed methods, as well as relatve atmospherc correcton, are based on the assumpton of a lnear relatonshp between mage bands across tme [6]. Pxels that change from one class n the earler mage to a dfferent class n the later mage (change pxels) behave lke outlers and affect the accuracy of these technques.
Ths work proposes and nvestgates methods to flter the tranng data from outlers before estmatng the coeffcents of the lnear relatonshp between mage bands. The procedure dscrmnates between change and stable pxels by evaluatng how well the lnear model fts to them. The next secton descrbes the outler flterng procedures consdered n ths study. The experments performed to evaluate the performance assocated to ths method are explaned n the secton 3. Secton 4 dscusses the experment results. The paper ends summarzng the man conclusons of ths work. I A TM A CLASSIFER TM B I B Fgure 1: Structure of the Automatc Classfcaton System 2 Flterng Outlers 2.1 Problem Formulaton The problem consdered n ths work s llustrated n Fgure 1. An automatc classfcaton system has three nputs: ) the multspectral mage I A of some area acqured at date t A, ) a relable thematc map TM A relatve to the mage I A, and ) a second multspectral mage I B of the same area acqured at later date t B. The task of the system conssts of generatng a thematc map (TM B ) for mage I B wthout any user nterventon. Ths work focuses on the step of automatc selecton of tranng data. So the task for whch an automatc soluton s requred conssts of developng an automatc procedure to collect data examples from mage I B for each land cover type, on the bass of the three system nputs. 2.2 Frst Basc Assumpton The basc assumpton underlyng the methods proposed n ths work s that the dfferences n the land cover between I A and I B are moderate. Ths s n most cases a realstc assumpton f the tme elapsed between t A and t B s not long. Accordng to ths assumpton the thematc map TM A s a good ntal guess for the thematc map TM B. Snce most pxels are assumed to reman n the same class n both mages (stable pxels), a representatve tranng set can be bult by pckng up pxels from mage I B assumng the class assgnment defned n TM A.
β β β The naccuracy of the spectral sgnatures estmated on the bass of such data wll ncrease wth the proporton γ of change pxels n the tranng set. A more accurate estmate for the spectral sgnatures could be obtaned, f one were able to dentfy the change pxels and refne the frst guess by elmnatng them from the tranng set before estmatng the sgnatures. The second assumpton provdes a way to dentfy potental change pxels. 2.3 Second Basc Assumpton Changes n the atmospherc condtons, llumnaton and vew angles and sensor calbraton, among other factors, nfluence the spectral sgnatures. The proposed method assumes that all these factors affect the spectral response of a sngle land cover type qute n the same way, so that the ntenstes of a stable pxel n the later mage (I B ) can be estmated wth a good accuracy by a lnear functon of the ntenstes of the same pxel n the earler mage (I A ). Such a functon s expected to ft well to stable pxels. For the small proporton of change pxels the functon wll n most cases not ft well. Therefore the dstncton between stable and change pxels can be made on the bass on how well the functon fts to them. 2.4 The Methods The methods studed n ths work start wth the ntal guess and refne t by dentfyng the change pxels and excludng them from the ntal tranng data. Clearly the performance assocated to ths proposal wll depend on the functon used to map pxel ntenstes from one mage to another. Inspred by many prevous approaches for relatve atmospherc calbraton four lnear regresson methods are consdered n ths study: a) Band-to-band regresson, one model for the entre mage, b) Multple regresson from several bands, one model for the entre mage, c) Band-to-band regresson, one model per class, and d) Multple regresson from several bands, one model per class. The band-to-band regresson models (cases a and c) take the form: y β + β + ε j = x j 1 0 1 j where y j denotes the value of pxel for band j n an later mage and x j n an earler mage, β 0 and β 1 are the regresson coeffcents, and ε j s the correspondng error term. Equaton 2 shows the model relatve to cases b and d for mages wth p bands. y = β + x + x + L + x + ε j 2 0 1 1 2 2 p p j
β β β In ths equaton y j denotes de value of pxel for band j n an later mage and x j n an earler mage, β 0 to β p are the regresson coeffcents, and ε j s the error term for mages wth p spectral bands. In the cases a and b the land cover nformaton s not taken nto account, and a sngle model for all land cover types s assumed. In the cases c and d the model s ftted to each land cover type ndvdually. Many statstcal tests have been proposed for detectng and rejectng outlers n lnear regresson models. (see [7] for a dscusson of some approaches). Some of these tests base on the resduals r gven respectvely by equatons 3 (cases a and c) and equaton 4 (cases b and d). = y β + x 3 j r j ( 0 β1 j ) ( + x + x + x ) r = y β L + 4 0 1 1 2 By assumpton, these resduals have ndependent normal dstrbutons wth zero mean and a constant varance. It s a common practce to scale the resduals so that they have untary varance. So the normalzed resduals r wll be gven by 2 ( r ) p p * r 5 r * = var where var(r ) s the estmated error varance. In the equaton above the normalzed resdual has a Student s t dstrbuton wth (n-p-1) degrees of freedom, where n s the total number of data ponts and p s the number of predctor varables. The confdence nterval for the mean of each error wll be gven by 6 c r 1± t 1 1 α / 2 var r [ ] = ( n p )( ) ( ) where t (n-p-1) (1-α/2) s the upper 100(1-α/2)th percentle of a Stutendt s t dstrbuton wth (n-p-1) degrees of freedom. A confdence nterval gven by equaton 6 not ncludng 0 (zero), s a good evdence that the data pont s an outler. Ths work follows ths reasonng to dscrmnate between change and stable pxels n the ntal guess. Accordngly, the followng sequence of actons s proposed to estmate the spectral sgnatures n the later mage. 1. buld a tranng set for the earler mage (TM B ) based on the later thematc map (TM A ) assumng that all pxels are stable, 2. apply the lnear regresson model, 3. elmnate from the tranng set pxels for whch the error confdence nterval does not contan the value 0 (zero), 4. compute the spectral sgnatures usng remanng the pxels. It s worth mentonng that for α = 0 (zero) corresponds to a method n whch no outler flterng takes place.
3 Experment Desgn 3.1 Study Areas The magery avalable for ths nvestgaton conssted of spectral bands 3 to 5 of two pars of Landsat TM-5 scenes mapped to RGB channels. The frst par of scenes was obtaned n 1996 and n 1999 both durng the dry season. They cover part of the Taquar Watershed n western Brazl. In the second par of scenes part of the Atlantc Forest n southeastern Brazl s depcted. They were obtaned n 1993 and 2000 also durng the dry season. Only segments wth 400 by 400 pxels cropped from the scenes were used n the experments. A well traned photo nterpreter performed the regstraton and classfcaton of all mages accordng to 9 dfferent land cover types wth the help of automatc software tools, eventually explorng addtonal nformaton avalable n a GIS and applyng some a pror knowledge about the covered areas. In the experments reported n secton 4 the earler mage (I A ), ts thematc map (TM A ) produced by the photo nterpreter, along wth the later mage (I B ) were used as nput to the system shown n Fgure 1: Structure of the Automatc Classfcaton System. The thematc map correspondng to the later mage produced by the photo nterpreter (TM Bref ) was consdered as the reference tranng data for performance evaluaton. 3.2 Buldng the Dataset The frst basc assumpton (secton 2.2) requres a low percentage (γ) of change pxels n the ntal guess. In order to have control over ths parameter durng the experments not all change pxels were ncluded n the data set. For each land cover type a number of change pxels were randomly selected for tranng. Ths number was gven by the product between γ and the number of stable pxels of the correspondng land cover type n TM A. Beyond the so selected change pxels all stable pxels were ncluded n the tranng data. 3.3 Performance Metrcs The most representatve tranng set of the land cover types for the mage I B conssts of all pxels of I B wth the classes assgned n TM Bref. The spectral sgnatures estmated on the bass of such a tranng set were used n the experments as reference. The performance measure wll be gven by the average percentage of agreement between the outputs of a maxmum lkelhood classfer traned wth the reference tranng set and wth the fltered tranng set accordng to the proposed methods, whereby all classes were assumed to have a multvarate normal dstrbuton.
Fgure 2: Average percentage of classfcaton agreement between the reference and the proposed tranng set as a functon of the percentage of change pxels (γ) for dfferent confdence levels (α); mages from the Taquar Watershed; curves correspond to a) band-to-band regresson, one model for the entre mage, b) multple regresson from several bands, one model for the entre mage, c) band-to-band regresson, one model per class, and d) multple regresson from several bands, one model per class, e) no outler flterng. 4 Experment Results Fgure 2 and Fgure 3 show the performance results respectvely from the experments on the magery from the Taquar Watershed and from the Atlantc Forest. Each fgure presents three graphs correspondng to dfferent levels of confdence (α= 0.01, 0.05 and 0.09). Each graph presents fve curves correspondng to the four methods outlned n secton 3 (curves a to d), and to a method nvolvng no outler flterng (curve e n sold lne). Ths last curve represents the ntal guess wthout any outler flterng. Each performance curve presents qute the same form across the graphs for both stes. As expected, the results are heavly dependent on the percentage of change pxels (γ) n the mages. For γ below 3% to 5% outler flterng brought no performance mprovement. For γ above 5% the use of one regresson model per class (curves c and d) consstently brought a performance mprovement n relaton to a sngle model for all classes (curves a and b). As a matter of fact the use of outler flterng wth one sngle model for the entre data resulted n no relevant performance mprovement relatve to no flterng for the Atlantc Forest.
The graphs of Fgure 2 and Fgure 3 allow also to compare the performance assocated to the band-to-band regresson (equaton 1) and to the multple regresson from several bands (equaton 2). When just one model s appled to all classes band-to-band regresson (curve a) performs a lttle better than multple regresson (curve b) for γ below 5% to 10%. Above these values multple regresson presented a clear superorty only n the magery from the Taquar Watershed. For one model per class band-to-band regresson (curve c) performed as well as multple regresson (curve d), beng eventually better for some values of γ. In these cases the ncreased complexty of the multple regresson was not worth the trouble. The last mportant observaton s concerned wth the level of confdence. Comparng the three graphs n each fgure one notces that the best performance corresponds to the lowest value assgned to α n these experments (=0.01). Ths happened n both stes for γ as large as 15%. Fgure 3: Average percentage of classfcaton agreement between the reference and the proposed tranng set as a functon of the percentage of change pxels (γ) for dfferent confdence levels (α); mages from the Atlantc Forest; curves correspond to a) band-to-band regresson, one model for the entre mage, b) multple regresson from several bands, one model for the entre mage, c) band-to-band regresson, one model per class, and d) multple regresson from several bands, one model per class e) no outler flterng. 5 Concluson Ths work nvestgates automatc methods to generate tranng data for change detecton n multtemporal mages. The method assumes a lnear relatonshp
between mage bands across tme, and apples an outler flterng procedure to ncrease accuracy. The work also compares the use band-to-band regresson and the multple regresson. Moreover the use of sngle model for all classes and one model per class s nvestgated. Experments performed on scenes from two dfferent stes ndcated that outler flterng combned wth one regresson model per class generally mproves classfcaton performance. Multple regresson dd not shown any mportant superorty over the band-to-band alternatve. Acknowledgement Ths work was supported by CNPq, an agency of the Brazlan government for scentfc and technologc development. References 1 Rchards, J. A., Ja, X., Remote Sensng Dgtal Image Analyss - An Introducton, 3 rd Ed. Sprnger Verlag, 1999. 2 Mather, P., Computer Processng of Remotely-Sensed Images- An Introducton, 2 nd Ed. Wley, 1999. 3 Chaves, P. S., Jr. Macknson, D. J., Automatc detecton of vegetaton changes n the southwestern Unted States usng remotely sensng mages, Photogramm. Eng. Remote Sens. 60(5), 1994, pp. 571-583. 4 Ouadrar, H., Vermote, E. F., Operatonal Atmospherc Correcton of Landsat TM Data, Remote Sensng and Envronment, vol. 70, 1999, pp. 4-15. 5 Song, C., Woodcock, C. E., Seto, K. C., Lenney, M. P., Macomber, S. A., Classfcaton and change Detecton Usng Landsat TM Data: When and How to Correct Atmospherc Effects?, Remote Sensng and Envronment, vol. 75, 2001, pp. 230-244. 6 Tokola, T., Löfman, S., Erkklä, A., Relatve Calbraton of Multtemporal Landsat Data for Forest Cover Change Detecton, Remote Sensng and Envronment, 68, 1999, pp. 1-11. 7 Douglas C. Montgomery, Elzabeth A. Peck, G. Geoffrey Vnng, Introducton to Lnear Regresson Analyss, 3 rd Ed. John Wley and Sons, 2001.