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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 1, JANUARY 2001 49

Multiscale MAP Filtering of SAR ImagesSamuel Foucher, Student Member, IEEE, Goze Bertin Bénié, and Jean-Marc Boucher

Abstract—Synthetic aperture radar (SAR) images are disturbedby a multiplicative noise depending on the signal (the ground re-flectivity) due to the radar wave coherence. Images have a strongvariability from one pixel to another reducing essentially the ef-ficiency of the algorithms of detection and classification. In thisstudy, we propose to filter this noise with a multiresolution anal-ysis of the image. The wavelet coefficient of the reflectivity is esti-mated with a Bayesian model, maximizing thea posterioriproba-bility density function. The different probability density functionare modeled with the Pearson system of distributions. The resultingfilter combines the classical adaptive approach with wavelet de-composition where the local variance of high-frequency images isused in order to segment and filter wavelet coefficients.

Index Terms—Adaptive filtering, synthetic aperture radar(SAR), speckle, wavelet.

I. INTRODUCTION

W ITH the advent of the synthetic aperture radar (SAR),resolution performances keep improving reaching nom-

inal resolutions of about 10 m for RADARSAT. Unfortunately,in radar imagery, the high-spatial resolution implies a poorradiometric resolution. The radar wave coherence produces arandom aspect on the extended homogeneous targets of theimage. Consequently, a value from a single pixel is not relevantand the radar backscattering coefficient cannot be derived fromonly one pixel information. In order to overcome this seriousdrawback, we can apply either the multilook technique duringthe radar signal processing, or image filtering methods. Thelatters are critical for monolook images when assigned to tasksof classification or segmentation.

Up to now, speckle reduction remains a major issue in SARimagery processing. Usual filtering techniques designed for anadditive noise as the Wiener filter, fail due to the multiplicativenature of speckle. The first filtering techniques were heuristicas the median or Crimmins filters. Then, statistical adaptiveapproaches appeared using optimization criteria as the LMMSE(local minimum mean square error) for the Kuan filter [1] andthe Lee filter [2]. The adaptive approach takes into account thenatural nonstationarity of the image by adapting the filter to thelocal image information content. Later, Nezry introduced the

Manuscript received October 27, 1997; revised August 9, 2000. This workwas supported by the Natural Sciences and Engineering Research Council ofCanada, Fonds de Délveloppement, Technologique du Québec, and ViasatGéotechnologie, Inc., Montréal, QC, Canada. The associate editor coordinatingthe review of this manuscript and approving it for publication was Dr. SridharLakshmanan.

S. Foucher and G. B. Bénié are with the Centre d’applications et de Rechercheen Télédétection, Université de Sherbrooke, J1K 2R1, Sherbrooke, QC, Canada(e-mail: [email protected]).

J.-M. Boucher is with the École Nationale Supérieure des Télécommunica-tions de Bretagne, Bretagne, France (e-mail: [email protected]).

Publisher Item Identifier S 1057-7149(01)00104-X.

Gamma-MAP filter ([3], [4]), both considering well establishedspeckle and reflectivity Probability Density Functions (pdf).The ground reflectivity value is thereby estimated through aMaximumA Posterioricriteria (MAP). Most of these adaptivefilters are based on a preliminary segmentation dividing theimage into homogeneous areas and heterogeneous areas. Forhomogeneous surfaces, the reflectivity can be assessed with asimple local mean, whereas for heterogeneous surfaces, pixelvalues must be weighted from the local statistics or remainunchanged in order to preserve texture or edges. Usually, thissegmentation is achieved with the help of the local estimationof the normalized standard deviation of the image which onlyneeds the knowledge of the number of looks. Improvementshave been proposed, using more efficient edge detectorssuch as the Touzi’s detector [4], [5]. However, the detectionperformance is strongly related to the accurate determinationof several thresholds.

Independently, wavelet theory provides a new powerfultool for studies relying on the time-frequency signal analysis.The main advantage of wavelet transformation remains in itsability to locally describe signal frequency content. Unlike theshort term spectral analysis, the wavelet transform provideslocalized frequential and spatial information about the signal.Small-scale details corresponding to high-frequency imagesare represented on a wavelet basis of small spatial support,whereas large-scale variations are projected on wavelets withlarge spatial support. The wavelet transform has been appliedby Mallat to image processing ([6], [7]), in the particularcase of dyadic decomposition. The image is decomposed intoseveral high-frequency images containing wavelet coefficientsrepresenting details with increasing scale and different ori-entations. Wavelet filtering methods have been successful inthe additive case using thresholding techniques developedby Donoho ([8], [9]). Attempts at speckle reduction using awavelet decomposition exist, essentially by filtering the imagelogarithm transform. Recently, Gagnonet al. [10] proposed toapply a thresholding of the wavelet coefficients of the imagelogarithm employing a complex form of the Daubechies’wavelet. Nevertheless, the logarithm transformation leads toa biased estimation of the reflectivity [11]. Moreover, mostof these noise reduction methods are based on a pyramidalrepresentation with undersampling, which was originally usedfor a image compression purpose. However, this representationpresents a serious drawback by which the invariance by trans-lation is not preserved. Artifacts like pseudo-Gibbs oscillationsmay appear near discontinuities in the reconstructed signal.Invariance by translation ensures that edges are similarlyrepresented by wavelet coefficients independently of theirposition in the image. In order to allow this stationarity in thewavelet representation, we have to suppress the undersampling

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50 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 1, JANUARY 2001

process. This derived transformation obtained, called “á trous”algorithm [12] or stationary wavelet transform [13], [26],becomes redundant and no longer orthogonal. The stationarywavelet transform has been previously used with success inestimation/detection problems [14], [24].

The purpose of this paper, is an extension of the clas-sical adaptive methods and particularly an extension of theGamma-MAP filter to the image wavelet representation. Theapplication of a bayesian analysis requires models for the dif-ferent probability density functions. Consequently we proposean application of the Pearson system of distributions in order toapproximate wavelet coefficient pdf, assuming gamma distribu-tions for both radar reflectivity and speckle. After a brief reviewof the usual statistical hypothesis for SAR images (Section I)and a presentation of the discrete wavelet decomposition (Sec-tion III), we describe the wavelet coefficients behavior usingthe second-order moments for a speckled image (Section IV).The use of the multiplicative model allow a segmentation ofthe high-frequency images. Before applying a MAP criteria,we also demonstrate in Section IV that the wavelet probabilitydensity function (pdf) of a gamma distributed image is wellapproximated by a Pearson type IV distribution. Therefore, thelocal bayesian estimate of the wavelet coefficient of the groundreflectivity is the solution of a third-degree equation. Results ofthis method are then compared to the Gamma-MAP filter withedge detection (Touzi’s detector).

II. STATISTICAL IMAGE MODEL AND HYPOTHESIS

A. Nature and Origin of the Speckle

Speckle noise is a consequence of radar wave coherence il-luminating the scene [15]. Each ground resolution cell is com-posed of a large number of elementary reflectors backscatteringthe radar wave in the sensor direction. For a rough surface incomparison with the radar wavelength, these elementary reflec-tors are present in a large enough number to ensure the statisticalindependence in phase and amplitude of these backscattered ele-mentary contributions. For this type of target, the speckle is fullydeveloped. Elementary phases are then uniformly distributedrandom variables. The total component backscattered by the res-olution cell is the vectorial sum of these elementary backscat-tered electrical fields. The energy registrated by the sensor froma resolution cell can be either nil or significant according to theconstructive or destructive interferences between the elemen-tary contributions. Consequently, the backscattered energy canrandomly fluctuate from one resolution cell to another indepen-dently of the radar backscattering coefficient. The homogeneousareas within the SAR image present a particular texture calledspeckle. The ground radar reflectivity proportional to the sur-face backscattering coefficient cannot then be derived froma single pixel digital number. The latter is only estimated by anaverage of a pixel set. One way to reduce the radiometric vari-ability due to the speckle is the multilook processing which con-sists of taking an average oflooks of the same scene producedby a signal bandwith subband extraction.

B. Statistical Models of SAR Images

In this paper, we only considered an intensity image. Resultsfor a square-root of intensity image can be easily deduced by achange of variable. We note and as random processes ofthe observed intensity and the ground reflectivity, respectively.

1) Speckle Probability Density Function:For a looks in-tensity image, the conditional observed intensity to the under-lying radar reflectivity is distributed [15]

2) Multiplicative Model: Usually, the speckle randomprocess is normalized, which gives a random processofmean and the density is

This normalization leads to the multiplicative model largelyemployed in the literature

(1)

Random variables and are considered to be independent,when the speckle is assumed to be fully developed. The mul-tiplicative model is considered valid within homogeneous andweakly textured areas. The relation (1) leads to the followingrelation between the different normalized standard deviationsof the ground reflectivity, the speckle, and the intensity:

(2)

(3)

with , , .In most adaptive filtering techniques [2], [4], [16], the

local estimate of allows to distinguish homogeneousregions ( ) where from heterogeneousareas ( ) where . In the Section IV-B, theserelations will be extended to the wavelet coefficients, in orderto permit the segmentation of high-frequency images.

3) Reflectivity and Intensity Probability Density Func-tions: The first MAP speckle filter [16] was based on aGaussian hypothesis for the reflectivity

The Nezry’s gamma-MAP filter [4] assumes a more real-istic gamma distribution for the reflectivity which supposed aPoisson distribution for the elementary reflectors

where is the mean reflectivity in the considered zone and thedegree of heterogeneity is measured by .

FOUCHERet al.: MULTISCALE MAP FILTERING OF SAR IMAGES 51

Presuming a gamma pdf for , we obtained a pdf for theobserved intensity

where is the modified Bessel function of the third kind oforder . This distribution proves to be well adapted at describingreality [17].

III. M ULTISCALE ANALYSIS

A. Principles

In the following sections, we note the Hilbert space ofreal square summable functions, with a scalar product

.A multiresolution analysis with levels of a signalof finite energy is a projection of on a basis

[18]. Basis functionsresults from translations and dilations

of a same function called scale function, verifying. The family span a sub-space

in . The projection of on gives an approximationof at the scale .

In the same way, basis functionsare the result of dilations and translations of the same func-

tion called the wavelet, which verifies .The family span a sub-space of . The pro-jection of on gives the wavelet coefficients

of representing the details between two suc-cessive approximations. Consequently, subspace is thecomplement of in

(4)

Subspaces realizes a multiresolution analysis. Theypresent the following properties [18]:

1) , ;2) ;3) , ;4) is dense in and ;5) scale function exists as:

is a basis of .Consequently, a multiresolution analysis withlevels gives

a decomposition of as depicted

(5)

All functions of can be decomposed in the followingway:

(6)

Duals functions and have to be defined in orderto ensure a perfect reconstruction.

B. Filter Bank

The connection between filter banks and wavelets stems fromdilation equations allowing us to pass from a finer scale to acoarser one [18]

(7)

with and .The normalization of the scale function implies .

In the same way, implies . A signalmultiresolution analysis can be performed with a filter bankcomposed of a low-pass analysis filter and a high-passanalysis filter

(8)

As a result, successive coarser approximations ofat scaleare provided by successive low-pass filtering (an undersamplingoperation is applied on each filter output). Wavelet coefficientsat scale are obtained by a high-pass filtering of an approxi-mation of at the scale , followed by an undersampling.

The signal reconstruction is directly derived from relation (4)

(9)

where the coefficients and define the synthesis filters.1) Orthogonal Wavelet:We can construct and in

order to realize an orthogonal decomposition of the signal, thenis the orthogonal complement of in . Quadra-

ture mirror filters (QMF) satisfy all these constraints ([19], [18])with . Despite the mathematical elegance ofthe decomposition, constraints imposed on filters do not allowa symmetric design and establish the bandwith value.

2) Biorthogonal Wavelets:Leaving out the orthogonalityconstraint, we can have symmetric filters which are suitable forimage processing. Synthesis filters are derived from analysisfilters with the help of the following relations [20]:

C. Stationary Wavelet Transform

The pyramidal algorithm described above, does not preservethe translation invariance. In other words, a translation of theoriginal signal does not necessarily imply a translation of thecorresponding wavelet coefficients. This property is essentialin image processing. On the contrary, wavelet coefficientsgenerated by an image discontinuity could disappeared arbi-trarily. This nonstationarity in the representation is a directconsequence of the undersampling operations following each


filtration. In order to preserve the translation invariance prop-erty, some authors have introduced the concept of stationarywavelet transforms [13], [26] originally called “á trous” algo-rithm [12]. The undersampling operation is then suppressedbut filters are dilated by inserting zeros between low-passand high-pass filter coefficients of a given level. This isperformed, in order to reduce the bandwidth by a factor of twofrom one level to another

, if

else

, if

else.

D. Stationary Wavelet Decomposition of an Image

Image multiresolution analysis was introduced by Mallat [7].The unidimensional filter bank used for the stationary waveletdecomposition can be applied in the two dimensional case.Image rows and columns are then filtered separately. Filteringequations to obtain the level from the level are thefollowing (where is for the pixel position):

(10)

where is the approximation of the original image atthe scale , giving the low-frequency content in the subband

. Image details are contained in three high-frequencyimages , and corresponding to horizontal,vertical and diagonal detail orientations, respectively. Waveletcoefficients of the level give high-frequency information inthe subband . For each decomposition level,images preserve the original size in so far as undersamplingoperations after each filter have been suppressed. We candefine a filtering operator permitting to obtain the levelhigh-frequency image for any orientation. This operator is theresult of successive convolutions

(11)

with , and

(12)

(13)

IV. SPECKLE INFLUENCE ONWAVELET COEFFICIENTS

A. Wavelet Coefficient Behavior in Presence of Speckle

The moment generating function of a distributedrandom variable is by definition the integral [21]

(14)

If is changed to we obtain the characteristic function. Thederivatives of at the origin equal the moments of . Inour case, it is more covenient to use the second (generating)moment function defined by

(15)

Considering the wavelet coefficients at position on a high-frequency image of level and with any orientation; accordingto the relation (11), the random variable is a linearcombination of random variables as-sumed independent and equally distributed. Therefore, using thesecond moment function of the pdf, we obtain the second mo-ment function of

The value of the th order derivative of at the origin,gives the cumulants of order [21]:

(16)

We note

(17)

defined for images et . For the high-frequency im-ages with a diagonal orientation we obtain:

(18)

The condition on the wavelet function im-plies .

The relation (16) will be used in Section IV to establish mo-ments of the wavelet coefficients useful for the computation ofdifferent parameters of the Pearson distribution system.


Assuming a distribution for the reflectivity pdfand a distribution for the speckle pdf, we obtain

(19)

Applying the relation (16), second-order cumulant can bederived from the second-order cumulant

(20)

(21)

In fact, the second-order cumulant of equals its variance,which can be expressed as a function of and

(22)

Consequently, we have a relation equivalent to (3), in defining anormalized standard deviation on wavelet coefficients

(23)

Analogous to the relation (3), we can define a normalized stan-

dard deviation of the speckle wavelet coeffi-cient. Within image homogeneous areas without texture (), we have

Therefore, the multiplicative model on the original image in-volves that the standard deviation of the wavelet coefficients

is proportional to the mean intensity of thehomogeneous region under consideration.

B. High-Frequency Image Segmentation

The relation (23), in a similar way to the SAR image seg-mentation using the grey level normalized standard deviation,allows to segment high-frequency images with the help of thenormalized standard deviation . To do this, we locally es-timate within a neighborhood centered on each pixel

the standard deviation of wavelet coefficientsand the mean on the original SAR image. The neigh-borhood is defined by a window of size enlargingin relation to the level in order to take into account the scaleincrease

The window size on the levelcan follow a similar progressionto wavelet support, i.e., (where isthe original size on the first level).

1) Highly Heterogeneous Region Detection:When theresolution cell has only a few strong dominant reflectors, the

imaging system response is deterministic, consequently thepixel value must be preserved. We choose to use an upperthreshold beyond which the region will be consideredhighly heterogeneous. Previous work provides a thresholdderived from the maximum likelihood detector of Frost ([3],[5])

A similar threshold can be defined for

(24)

Images depicted on Fig. 1 give an example of segmentation be-tween homogeneous, heterogeneous and highly heterogeneousregions.

V. MAP WAVELET COEFFICIENTFILTERING

A. Weighting of the Wavelet Coefficient Using a MAP Criteria

The wavelet decomposition operation can be written as

with andwhich are centered and uncorrelated random processes

Thea posterioriprobability density function conditional to theobservation can be expressed from the Bayes relation

The estimate maximizing theA Posterioripdf, is

(25)

In order to apply this Bayesian estimation, we have to establisha model for the different pdf.

B. PDF of the Wavelet Coefficient

The grey level distribution of the original images, acquiredwith a small number of looks, are highly asymmetric. Theextreme case for monolook images, results in an exponentialdistribution. Consequently, we can still expect asymmetric pdffor the high-frequency images of the first levels. When theconsidered level increases, the pdf becomes symmetric andcan be assumed Gaussian (consequence of the central limittheorem). In first approximation, we assume a Gaussian modelfor and . The MAP criteria is


Fig. 1. Segmentation between homogeneous areas (white), heterogeneous areas (grey), and very heterogeneous areas (black) on high-frequency images with avertical orientation and for different levels: (a)J = 1, (b) J = 2, and (c)J = 3.

then equivalent to applying a MMSE criteria. Then, a morerealistic model, introducing a certain amount of asymmetry inthe wavelet coefficient pdf, will be established in the frameworkof the Pearson distribution system (c.f. Section V-B2).

1) Gaussian Assumption:We assume Gaussian distribu-tions and for both andrespectively. From relation (22), we obtain

(26)

Similarly, for the variance of , we have

(27)

In the Gaussian case, the uncorrelation ( ) issufficient to ensure the independence between and .Consequently, the likelihood term in theBayesian relation is equal to the pdf of . The


MAP equation in the Gaussian assumption can be written asfollows:

(28)

This leads to

(29)

Using the relation (3), we obtain

(30)

This expression is similar to the Kuan’s filter [1] determinedaccording to a local MMSE criteria.

2) Pearson System of Distributions:The distribution familyof Pearson verifies the following differential equation:

(31)

Distributions verifying this relation are unimodal and have atangential contact with the axis on extremities ( )[22]. By integrating this differential equation, Pearson parame-ters can be derived directly from moments of the observation

(32)

The distribution is unimodal with a maximum at . Withan origin translation , the differential equation (31)can take a simple form

(33)

with

(34)

(35)

(36)

The behavior of zeros of can be examinedthrough the values of the index . From this,we can identify three main types for

• roots are reals with opposite signs, ;• roots are reals with same signs, ;• roots are complexes, .

The index is equivalent to from which wecan identify, in the same way, the type of distribution obtained.The advantage of using, is that we can derive its value directlyfrom the Pearson parameters (32), and then from the moments.However, the previous translation origin can affectthe root signs

1) type I distribution if:;

2) type VI distribution if:;

3) type VI distribution if: ;4) type IV distribution if: .

The indices and are often used re-spectively as measures of skewness and kurtosis ( and

for a normal distribution).In order to express this parameter, we have to calculate

wavelet coefficient moments up to the fourth moment. Therelation between second moment functions ofand canbe exploited from assumptions on the image pdf. At first, weare going to establish Pearson parameters for a general gammamodel for the image pdf. Then, results obtainedwill be applied to a gamma pdf for the speckle ( , )and for the reflectivity ( , ).

3) Probability Density Function of the Wavelet Coefficientsfor a Gamma Distributed Image:Assuming a distribution

for the grey level distribution, its second momentfunction is

By applying the relation (16), we obtained-order cumulantsof the wavelet coefficients

(37)

The high-pass filtering ( ) leads to centered moments

Pearson parameter expressions (32) lead to the following for-mulation for

(38)

with

(39)

Analogous to the classical Pearson indicesand , we candefine similar indices characterizing the wavelets

(40)

Moreover

(41)


TABLE IVALUES OF ~� , ~� AND � FOR THEFIRST LEVEL OF THE MULTISCALE

DECOMPOSITION(j = 1) AND FOR TWO WAVELETS USED IN THIS STUDY:DAUBECHIES’ WAVELETS WITH FOUR AND EIGHT COEFFICIENTS(D4,D8),

BIORTHOGONALS WITH FIVE AND NINE COEFFICIENTS(B5,B9). WE

OBSERVETHAT 0 < ~� < ~� < 1 and� << 1 WHICH ARE THE

CONDITIONS IN ORDER TOHAVE A PEARSONTYPE IV DISTRIBUTION FOR THE

WAVELET COEFFICIENTSPDF

Table I gives numerical examples of indices for and sometypical wavelets [Daubechies (D4, D8) and biorthogonals (B5,B9, B13)]. For all the tested wavelets, we observe

(see Table I). Consequently,verifies

(42)

which implies from relation (38)

(43)

The minimal value for is the positive solution of, which leads to a second-degree equation. Table I gives some

values for which are all largely inferior to 1. Therefore, wehave when the following conditions apply:

(44)

(45)

When takes large values, the distribution tends to the Gaussiancase [where ]. According to the value, the waveletcoefficient pdf for a gamma distributed image is a Pearson typeIV [22]

(46)

with

4) Probability Density Function of the Speckle Wavelet Co-efficients: When the ground radar reflectivity is constant (pdfof is a dirac), the image pdf conditionally to is the specklepdf. Applying relations (38) with and we obtain

(47)

The lower bound (45) on is largely verified since in practice. So, the type IV model is always valid for the

Fig. 2. Pearson typeIV distributions (solid lines) and histograms for a purespeckle image (L = 1). Three different levels:J = 1 (dot),J = 2 (circle),andJ = 3 (cross).

pdf of the random process . Fig. 2 shows an example ofan image of pure speckle filtered with a bi-orthogonal wavelet(B5). Table II gives Pearson and distribution parameters for dif-ferent values and for the decomposition level. As expected,the type IV distribution leads to a Gaussian whenor in-creases.

5) Probability Density Function for the Wavelet Coefficientsof the Reflectivity: In a gamma assumption for the reflectivityand from relations (38) with and , we obtain thefirst fourth moments

(48)

6) Probability Density Function of : We have the fol-lowing moments for the noisy term:

(49)


TABLE IIPEARSON ANDTYPE IV PDF PARAMETERS FOR THESPECKLE PDFMODEL FOR

DIFFERENTL AND j VALUES. WHEN j OR L INCREASES, PEARSON

PARAMETERSTEND TOWARD THE GAUSSIAN CASE (� = 3; � = 0; � = 0)

where and giving expression are

Conditions for a type IV distribution ( ) are easilyverified for all the and values.

C. Maximum A Posteriori Equation

Assuming a type IV model for the probability function ofand , the MAP equation (25) can be expressed as follows:

derivating according to

The single point MAP estimate is the solution of a third-degree equation

(50)

with

(51)

(52)

VI. FILTER IMPLEMENTATION

First, the image is decomposed inlevels, each level havingthree high-frequency images with a particular orientation. Foreach high-frequency image, the algorithm is as follows:

for

or

We estimate and within a neighborhoodaround each pixel of the high-frequency and original image, re-spectively. The value of measures the degreeof local homogeneity, and therefore, determines the type of es-timator to apply

• If .The considered neighborhood is textured, the normalized stan-dard deviation of can be estimated with

Then, the point estimate of requires the following steps:

1) Calculate the first fourth moments of and [rela-tions (48) and (49)].

2) Calculate the Pearson parameters for the two pdf models,then the MAP equation coefficients [relations (51) et(52)].

3) Solve numerically the third-degree equation

• If .


The local neighborhood is strongly heterogeneous, and maycontain a point target or a strong edge

• If .The local neighborhood is considered homogeneous, conse-quently

The resulting filtered image is obtained by applying thereconstruction algorithm from the previously filtered high-fre-quency images.

VII. RESULTS

A. Weighting Behavior

Fig. 3 represents the weighting of obtained by solving theMAP equation in relation to the normalized standard deviationof the wavelet coefficients for a monolook image ( ) with

. The Gaussian case established in Section V-B1is represented with a dotted line. We observe as expected a dif-ference between the weighting and the Gaussian case on thefirst levels, due to the asymmetry introduced by the type IVmodels. This asymmetry diminishes when the level increases,the weighting becomes similar to the Gaussian case beyond thefourth level.

B. Results on SAR Images

The proposed filter is tested on a monolook RADARSATimage of the town of Sorel (Québec, Canada). The resolutioncell is about m m. The image contains a lot of large ho-mogeneous areas (see Fig. 4). Wavelets used are the orthog-onal Daubechies wavelet [18] (noted ) and the biorthonalwavelet of Cohenet al. [20] (noted ). Best results are ob-tained with the shortest filters. Beyond three levels of decom-position ( ), speckle reduction performances remain un-changed. The size of the windows used for the local statisticsestimation is ( ) on the first level, hence the size in-creases according to the proposed progression mentioned above.Results are compared with the Nezry Gamma-MAP filter [4]implemented with the Touzi’s ratio edge detector [5]. The esti-mation window size decreases from to while thereis an edge detection inside.

In terms of speckle reduction in homogeneous areas, per-formances are measured with an estimation of the equivalentnumber of looks (ENL) within three homogeneous windowsselected in the image Fig. 4. Smoothing increases with thelow-pass filter size, however for short filters, performances areequivalent to the Gamma-MAP filter (Table III).

In terms of edges and meaningful detail preservation, theproposed multiscale filter retains more information in homoge-neous areas and preserves well strong edges (roads, agriculturalboundaries, …). Strong reflectors are slightly smoothed com-pared to the one-scale Gamma-MAP filter but without targetblurring.

Fig. 3. Weighting behavior as a function of the normalized standard deviationof the wavelet coefficientsC for the first fourth levels. The dotted linerepresents the case of Gaussian weighting.

VIII. C ONCLUSION

The proposed filter combines image multiscale analysis andclassical techniques of adaptive filtering. The multiplicativemodel introduced in high-frequency images, permits to retaincoefficients produced by significant structures present in theimage and suppress those produced by the speckle noise.From gamma assumptions for the pdf of the reflectivity andthe speckle, and with the help of second generating momentfunctions, we have expressed wavelet coefficients momentsup to the fourth moment. On these results, we applied thePearson distribution system, leading to a type IV model for thewavelet coefficient pdf. This asymmetric distribution becomesnaturally Gaussian when the level increases. The Pearson typeIV model combined with the explicit calculation of the firstfourth moments of the wavelet coefficients, gives us an entirelyparametric model for the wavelet coefficient pdf. Parametersdepend only on the local estimation of the reflectivity nor-malized standard deviation and the number of looks on theoriginal image. From this statistical modeling, the applicationof a MAP criteria leads to an estimate of the wavelet coefficientof the reflectivity as the solution of a simple third-degreeequation. The resulting multiscale filter shows interestingresults compared to the classical Gamma-MAP filter. The mainadvantage of the multiscale approach is the multiscale detectionof discontinuities along different orientations provided by thewavelet decomposition. High-frequency images of the firstlevel are sensitive to small-scale variations of the image, sogrey level variations due to the speckle are mainly concentratedon the first level. Whereas, meaningful image discontinuitiescan be represented on many scales, and therefore influencehigh-frequency images on all levels. Consequently, the proba-bility of a significant discontinuity detection increases with the


Fig. 4. (a) Original SAR image (monolook RADARSAT image). (b) Proposed filter with biorthogonal wavelets (B5) and three levels of decomposition (J = 3).(c) Proposed filter with Daubechies’ wavelets (D4) and three levels of decomposition (J = 3). (d) Result of the Gamma-MAP filter applied on image (a) withsize adapted window and edge detection.

TABLE IIIEQUIVALENT NUMBER OF LOOKS (ENL) ESTIMATED WITHIN THREE

HOMOGENEOUSREGIONS ONDIFFERENTFILTERING RESULTS OFFIG. 4(a). THE

GREATER ENL IS, THE SMOOTHER IS THECONSIDEREDREGION

level (the signal to noise ratio increase). A multiscale decisionon the presence or not of a discontinuity is less brutal than

in a one-scale detection, since a wavelet coefficient can besuppressed at one-scale but preserved on a higher scale.

ACKNOWLEDGMENT

The authors would like to thank E. Rosenberg, F. Zagolsky,B. Burdsall and P. Gagnon for their linguistic support.

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Samuel Foucher (S’00) was born in Nantes,France, in 1969. He received the B.S. degree inphysics from the University of Nantes in 1989,the telecommunication engineering degree fromthe École Nationale des Télécommunications deBretagne, France, and the M.S. degree in imageprocessing from the University of Rennes, Rennes,France, in December 1996. He is currently pursuingthe Ph.D. degree at the University of Sherbrooke,Sherbrooke, QC, Canada.

His research includes radar filtering and segmen-tation.

Goze Bertin Béniéwas born in Daloa, Cöte d’Ivoire.From 1977 to 1987 he received the B.A.Sc. degreein surveying and the M.Sc. and the Ph.D. degrees inphotogrammetry and remote sensing from Universit´eLaval, Sainte-Foy, QC, Canada.

He was a Postdoctoral Fellow at the CanadaCentre for Remote Sensing, Digim, Inc., Lavalin,Montreal, QC, and at Intera Information Technolo-gies, Inc., Calgary, Alta., Canada, from 1987 to 1990.In 1990, he joined the Department of Geographyand Remote Sensing and the Centre d’applications

et de recherches en télédétection (CARTEL) of the Université de Sherbrooke,Sherbrooke, QC, as an Assistant Professor. He was the head of CARTELfrom 1995 to 2000. He is currently Full Professor in image processing andgeomatics. His research interests include image filtering, segmentation andclassification methodology and spatial modeling in GIS.

Jean-Marc Boucherwas born in 1952. He receivedthe engineering degree in telecommunications fromthe Ecole Nationale Supérieure des Telecommunica-tions, Paris, France, in 1975 and the Habilitation áDiriger des Recherches degree in 1995 from the Uni-versity of Rennes 1, Rennes, France.

He is currently Professor with the Departmentof Signal and Communications, Ecole NationaleSuperieure des Telecommunications de Bretagne,France, where he is also Education Deputy Director.His current research interests include estimation

theory, Markov models and Gibbs fields, blind deconvolution, wavelets andmultiscale image analysis with applications to radar and sonar image filteringand classification, multisensor seismic signal deconvolution, electrocardio-graphic signal processing, and speech coding. He has published 100 technicalarticles in these areas in international journals and conferences.

multiscale map filtering of sar images - image processing ...ieee transactions on image processing,...

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