INr. J. REMoTEsENSrNc,1993,vol,. 14,No. 3,615-619 The wavelet transform for the analysisof remotely sensedimages THIERRY RANCHIN and LUCIEN WALD Centred'Energétique-GroupeTélédétection& Modélisation,EcoledesMines deParis,BP 207,06904SophiaAntipolisCedex,France (Receiued15April 1992;infinalform 13 October1992) Abstract. Thewavelettransformisamathematicaltool allowinganimageto be decomposedin termsof its structuresandcharacteristicscales.This transformis reviewedbriefly and appliedto a remotelysensedimage.Perspectivesfor the analysisandprocessingof remotelysensedimagesarepresented. Wavelet transform and multiresolution analysis Wavelet theory is a powerful mathematical tool recently developed for signal processing(Meyer 1990).It is adapted to the analysisof non-stationary signalsof finite energy for which the classical formalism based on variance and correlation function doesnot hold. Remotely sensedimages are such a signal. Furthermore, the wavelet transform leads to the concept of multiresolution analysis (MRA) (Mallat 1989),where imagesare decomposedinto structures and then analyzedat successive scales(or spatial resolutions). The wavelet transform makes any arbitrary function of finite energy as a summation of elementary functions: the wavelets. The respective weights of the wavelets in the summation are called the wavelet coefficients. Wavelets are well- locatedin both domains:spaceand scale(Meyer et al.1987; Daubechies1990;Rioul and Vetterli 1991).Wavelets are obtained from a singlefunction, the mother wavelet, by dilatations and shifts. Wavelet coefficients are a measure of the intensity of the local variations of the signal for the scale under consideration. The value of a coefficient will be large when the dilation of the wavelet is close to the scaleof the heterogeneity as the signal will be irregular. The value of a coefûcient will be negligible when the local signal is regular (smooth) for this particular scale.Hence the value of a coefficient for a particular location and at any scalecan be understood as a characterization of the structures having this scale and present at this geographical location. The MRA reorganizesthe information content of the original image in terms of structuresor scaleswhich are composing the image (Mallat 1989).Mathematically, the structures (also called details) of an image at the spatial resolution j are defined asthe difference betweenits approximation at the resolution j and its approximation at the resolution (7* l). In the detail image at resolution j appear all the structures having a characteristic length comprised between (7- 1) and j. This image is composed of the wavelet coefficients. The MRA provides a hierarchical pyramid for interpreting the image in terms of structures. In the course of the analysis, the image containing the informations due to the structures which scalesare greater than the current scaleis called 'context image'. If the analysis is pursued, this context image will be in turn decomposedin details and another context image. The details and the 0143-l16ll93$10.00O 1993Taylor& FrancisLtd 616 T. Ranchin and L, Wald contextimagesareobtainedin thediscretecaseby filtering and subsamplingof the original image.In the following example,the wavelettransformused(Daubechies 1988)providesonecontextimageandthreedirectionaldetailsimagesby resolution (horizontal,vertical,diagonal,seetablel). Exampleof multiresolutionanalysis(MRA) The MRA wasappliedto a SPOTHRV panchromaticimageof Ryadh (Saudi Arabia),usingthealgorithmdescribedin Mallat (1989).Ryadhisaverymoderncity for themostpart with largeavenuesformingrectangulararrays,andlargebuildings (figureI (a)).Theoldestpartof thecityislocatedin thelowerpartof thefigure.It is composedof smallhousesand buildingsand of narrow winding streets.In this image,theold town doesnot exhibitregularstructuresasdo theotherparts.This clearlyappearsin theMRA madefor scalesfrom 10mup to 40m anddisplayedin figure1(à).Thecontextimage(upperleft part of theimage)containsonly structures with characteristiclengthsgreaterthan40m. Therectangularpatternof themodern cityisenhanced.Thewidestavenuesarevisible.Becauseitswidthisabout100m,an highwaywith centralseparationis seenrunningNW-SEin theupperright cornerof this contextimage.The CCDs in the panchromaticchannelare affectedby noises which structuresappearin the detailsimagesat 10-20m. The horizontalnoise affectingeachline of imagehasbeennotedby C.N.E.S.(cf Anonymous,1986)as well as the diagonalnoisewhich is due to an undesirablecoupling betweenthe multispectraland panchromaticmodesof the sensors.The latternoiseis initially verticalandappearsasdiagonalbecauseof thegeometricalprocessingof theimage up to level lB. In the detailsimagesare visible the skeletonsof the streetsand avenuesof correspondingwidths. Of evidenceis the lack of regularstructuresof typicalscalesgreaterthan 10m in theoldesttownwhichischaracterizedbyverylow absolutevaluesin detailsimages.Thetwo housingareaslocatedin theuppermiddle right of the picture(figureI (a)) immediatelysouthof the airport areof particular interest.Bothexhibitrectangularpatterns,but thesizesof thelotsandbuildingsand thewidthsof the streetsarelargerfor theleftmostareathan for therightmostone. Thelatter areais likely a working-classdistrictwhiletheformerismoreresidential. Thesedifferencesin pattern appearclearly in the vertical and horizontal details imagesat l0-20m and20-40m.Whilethepatternof theresidentialdistrictis still visiblein thedetailsimages20-40m,thepatternof theworking-classdistrictdoes not appearany more, showingthat the typical scalesin this district are lessthan Table l. Schemeof a hierarchical pyramid produced by a multiresolution analysis Contextimage(all scalesgreaterthan (i+ t;; Imageof the 'horizontal' structures at scale(j+ l) Imagesof the 'vertical'structures at scale(j+ l) Imageof the 'diagonal'structures at scale(7+ 1) r4Ëçù ur ruE rlurrzurlletl slruuturEs scale j Imageof the 'vertical'structuresat scale7 Imageof the 'diagonal'structuresat scaleI I (a) (b) Figurel. SPOTHRV imagerecordedon l0 April 1986of Ryadh,SaudiArabia.(a)Original imagein the panchromaticchannel;the spatial resoiutionis lOm and.1024x1024 pixelsareshownwith alevelof processinglB. (ô)Contextimage(theupperleftimage) hasa spatialresolutionof 40m, thoseimagessurroundingit hâvèu rpuiiut resolutËn of 20m andthosethreeimagesto thefar right andbottom havea spaiialresolutionof l0m (table1). 618 T. Ranchin and L. Wald 20m. This exampleillustrateshow MRA enhancesthe discrepanciesin the urban architectureand how the different patternscan be separatedincluding the sensor noise. Perspectives Analysisof the spatial structures'.the wavelettransform provides an effrcient characterizationof thestructures,theknowledgeof themisimportantin manyfields of Earth sciences.RanchinandWald (1992)provideanexampleof MRA appliedto a SPOT HRV imagefor the study of structuresappearingat the surfaceof the ocean. Geometricalmergingof data:thestudyof thenaturalprocessesusuallyrequiresa largeamount of data and makesnecessarytheir geometricalsuperimposition.An important effort hasbeenmadein thisfieldby theresearchteamof RogerManière at Universityof Nice Sophia-Antipolis,France,for Landsat-MSSand SPOTHRV images. Segmentationand classificationof multi-spectralimages:many of the recent methodsfor theclassificationof multi-spectralimagesusethe textureinformations within each spectral image. Since the wavelet transform provides a complete descriptionof the texture of the imageat all availablescales,it is expectedthat soundresultscanbereachedby usingwaveletcoefflcientsin classificationschemes. Changeof the informationwith the scale:sinceclassifiersare currently using spatialstatistics,it is important to studyhow thesestatisticsbehavewhenchanging sensorresolution if imagestaken by various sensorsare to be used.A similar problem arisesif both high and low-resolutionsensorsare usedto monitor an environmentalparametersuchasthenormalizeddifferencevegetationindex(NDVI) or the temperature.The MRA is likely an efficientapproachfor suchstudies. Speckleremoualin SARimagery:theSAR imageryis affectedby thepresenceof a multiplicative noise,called the speckle.An adaptativefiltering of the Fourier coefficientsof theimagesuppressesit (Lopesetal.1990),but impliesafilteringof all the structureswithin the imagewhile theyarenot affectedin the sameway by this noise.Cauneauand Ranchin (1992)overcamethis drawbackby filtering out the waveletcoefficientsonly at the corruptedscales. Datacompression:datacompressionmayhelpreducingthelargevolumeof data producedby remotesensingsystemsandwavelettransformis oneof themanytools that areusedto compressdata(Antonini 1991). Conclusions The wavelettransform and MRA havebeenbriefly presentedand an example provestheir usefulnessin the studyof structurescomposingan image.Indeedboth toolsareverynewandmanyeffortsmuststill bedevotedfor a full understandingof their properties.A number of studiesdealingwith the applicationsof waveletto remotelysensedimagesareunderway,at leastin Franceandparticularlyin various institutesin Nice SophiaAntipolis. The domain of applicationsof this.transform and MRA is ratherwide. Acknowledgments The authorsare grateful to Michael Barlaud, PierreMathieu, Albert Bijaoui, Jean-PierreDjamdji and RogerManière. RemoteSensingLetters 619 References ANoNYtvtous,1986,GuidedesutilisateursdedonnéesSPOT.EditeursCNESandSpOTImage, . ToulouseFrance,3 volumes,revisedJanuary1991. 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