Multiscale MAP Filtering of SAR Images - CiteSeerX

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Multiscale MAP Filtering of SAR Images Samuel Foucher, Goze Bertin B´eni´e, Jean-Marc Boucher Abstract— SAR images are disturbed by a multiplicative noise depending on the signal (the ground reflectivity) due to the radar wave coherence. Images have a strong variability from one pixel to another reducing essentially the efficiency of the algorithms of detection and classification. In this study, we propose to filter this noise with a multiresolution analysis of the image. The wavelet coefficient of the reflectivity is estimated with a bayesian model, maximizing the a posteriori probability density function. The different probability density function are modeled with the Pearson system of distributions. The resulting filter combines the classical adaptive approach with wavelet decomposition where the local variance of high frequency images is used in order to segment and filter wavelet coefficients. Keywords— wavelet, adaptive filtering, SAR, speckle.

I. Introduction

W

ITH the advent of the Synthetic Aperture Radar (SAR), resolution performances keep improving reaching nominal resolutions of about 10 meters for RADARSAT. Unfortunately, in radar imagery, the high spatial resolution implies a poor radiometric resolution. The radar wave coherence produces a random aspect on the extended homogeneous targets of the image. Consequently, a value from a single pixel is not relevant and the radar backscattering coefficient cannot be derived from only one pixel information. In order to overcome this serious drawback, we can apply either the multilook technique during the radar signal processing, or image filtering methods. The latters are critical for monolook images when assigned to tasks of classification or segmentation. Up to now, speckle reduction remains a major issue in SAR imagery processing. Usual filtering techniques designed for an additive noise as the Wiener filter, fail due to the multiplicative nature of speckle. The first filtering techniques were heuristic as the median or Crimmins filters. Then, statistical adaptive approaches appeared using optimization criteria as the LMMSE (local minimum mean square error) for the Kuan filter [1] and the Lee filter [2]. The adaptive approach takes into account the natural non-stationarity of the image by adapting the filter to the local image information content. Later, Nezry introduced the Gamma-MAP filter ([3], [4]), both considering well established speckle and reflectivity Probability Density Functions (pdf). The ground reflectivity value is thereby estimated through a Maximum A Posteriori criteria (MAP). Most of these adaptive filters are based on a preliminary segmentation dividing the image into homoS. Foucher and G.B. B´ eni´ e are with the Centre d’applications et de Recherche en T´ el´ ed´ etection, Universit´ e de Sherbrooke, J1K2R1, Sherbrooke, Canada (e-mail: [email protected]). ´ J-M Boucher is with the Ecole Nationale Sup´ erieure des T´ el´ ecommunications de Bretagne (e-mail: [email protected]).

geneous areas and heterogeneous areas. For homogeneous surfaces, the reflectivity can be assessed with a simple local mean, whereas for heterogeneous surfaces, pixel values must be weighted from the local statistics or remain unchanged in order to preserve texture or edges. Usually, this segmentation is achieved with the help of the local estimation of the normalized standard deviation of the image which only needs the knowledge of the number of looks. Improvements have been proposed, using more efficient edge detectors such as the Touzi’s detector ([5], [4]). However, the detection performance is strongly related to the accurate determination of several thresholds. Independently, wavelet theory provides a new powerful tool for studies relying on the time-frequency signal analysis. The main advantage of wavelet transformation remains in its ability to locally describe signal frequency content. Unlike the short term spectral analysis, the wavelet transform provides localized frequential and spatial information about the signal. Small scale details corresponding to high frequency images are represented on a wavelet basis of small spatial support, whereas large scale variations are projected on wavelets with large spatial support. The wavelet transform has been applied by Mallat to image processing ([6], [7]), in the particular case of dyadic decomposition. The image is decomposed into several high frequency images containing wavelet coefficients representing details with increasing scale and different orientations. Wavelet filtering methods have been successful in the additive case using thresholding techniques developed by Donoho ([8], [9]). Attempts at speckle reduction using a wavelet decomposition exist, essentially by filtering the image logarithm transform. Recently, Gagnon et al. [10] proposed to apply a thresholding of the wavelet coefficients of the image logarithm employing a complex form of the Daubechies’ wavelet. Nevertheless, the logarithm transformation leads to a biased estimation of the reflectivity [11]. Moreover, most of these noise reduction methods are based on a pyramidal representation with undersampling, which was originally used for a image compression purpose. However, this representation presents a serious drawback by which the invariance by translation is not preserved. Artifacts like pseudo-Gibbs oscillations may appear near discontinuities in the reconstructed signal. Invariance by translation ensures that edges are similarly represented by wavelet coefficients independently of their position in the image. In order to allow this stationarity in the wavelet representation, we have to suppress the undersampling process. This derived transformation obtained, called “ trous” algorithm [12] or stationary wavelet transform [13], becomes redundant and no longer orthogonal. The stationary wavelet transform has been previously used with success in estima-

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tion/detection problems ([14], [24]. The purpose of this paper, is an extension of the classical adaptive methods and particularly an extension of the Gamma-MAP filter to the image wavelet representation. The application of a bayesian analysis requires models for the different probability density functions. Consequently we propose an application of the Pearson system of distributions in order to approximate wavelet coefficient pdf, assuming gamma distributions for both radar reflectivity and speckle. After a brief review of the usual statistical hypothesis for SAR images (section I) and a presentation of the discrete wavelet decomposition (section III), we describe the wavelet coefficients behavior using the second order moments for a speckled image (section IV). The use of the multiplicative model allow a segmentation of the high frequency images. Before applying a MAP criteria, we also demonstrate in section IV that the wavelet probability density function (pdf) of a gamma distributed image is well approximated by a Pearson type IV distribution. Therefore, the local bayesian estimate of the wavelet coefficient of the ground reflectivity is the solution of a third degree equation. Results of this method are then compared to the Gamma-MAP filter with edge detection (Touzi’s detector).

B. Statistical models of SAR images In this paper, we only considered an intensity image. Results for a square-root of intensity image can be easily deduced by a change of variable. We note X and Y as random processes of the observed intensity and the ground reflectivity, respectively. B.1 Speckle probability density function For a L looks intensity image, the conditional observed intensity to the underlying radar reflectivity is Γ(L, L/y) distributed [15]: PX|Y (x|y) =

LL y L Γ(L)

xL−1 e−Lx/y

B.2 Multiplicative model Usually the speckle random process is normalized, which gives a random process Z of mean E[Z] = µZ = 1 and the density is Γ(L, L): PZ (z) =

LL L−1 −Lz z e Γ(L)

This normalization leads to the multiplicative model largely employed in the literature: X =YZ

II. Statistical image model and hypothesis A. Nature and origin of the speckle Speckle noise is a consequence of radar wave coherence illuminating the scene [15]. Each ground resolution cell is composed of a large number of elementary reflectors backscattering the radar wave in the sensor direction. For a rough surface in comparaison with the radar wavelength, these elementary reflectors are present in a large enough number to ensure the statistical independence in phase and amplitude of these backscattered elementary contributions. For this type of target, the speckle is fully developed. Elementary phases are then uniformly distributed random variables. The total component backscattered by the resolution cell is the vectorial sum of these elementary backscattered electrical fields. The energy registrated by the sensor from a resolution cell can be either nil or significant according to the constructive or destructive interferences between the elementary contributions. Consequently, the backscattered energy can randomly fluctuate from one resolution cell to another independently of the radar backscattering coefficient. The homogeneous areas within the SAR image present a particular texture called speckle. The ground radar reflectivity proportional to the surface backscattering coefficient σ 0 cannot then be derived from a single pixel digital number. The latter is only estimated by an average of a pixel set. One way to reduce the radiometric variability due to the speckle is the multilook processing which consists of taking an average of L looks of the same scene produced by a signal bandwith sub-band extraction.

(1)

Random variables Y and Z are considered to be independent, when the speckle is assumed to be fully developed. The multiplicative model is considered valid within homogeneous and weakly textured areas. The relation (1) leads to the following relation between the different normalized standard deviations of the ground reflectivity, the speckle and the intensity: µX

=

CY2

=

µY µZ = µY 2 CX − CZ2 1 + CZ2

(2) (3)

σY σX σZ 1 , CX = , CZ = =√ . µY µX µZ L In most adaptive filtering techniques ([16], [2], [4]), the local estimate of CX allows to distinguish homogeneous regions (CˆX ≤ CZ ) where CY = 0 from heterogeneous areas (CˆX > CZ ) where CY > 0. In the section IV-B, these relations will be extended to the wavelet coefficients, in order to permit the segmentation of high frequency images.

with CY =

B.3 Reflectivity and intensity probability density functions The first MAP speckle filter [16] was based on a gaussian hypothesis for the reflectivity: 1 PY (y) = √ 2πσY

(y − µY )2 2σY2 e −

The Nezry’s gamma-MAP filter [4] assumes a more realistic gamma distribution for the reflectivity which supposed

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All functions f of L2 (R) can be decomposed in the following way:

a Poisson distribution for the elementary reflectors: PY (y) =

νν y ν−1 e−νy/µY µνY Γ(ν)

Where µY is the mean reflectivity in the considered zone, µ2 the degree of heterogeneity is measured by ν = σY2 = 1/CY2 Y . Presuming a gamma pdf for Y , we obtained a K pdf for the observed intensity: s !  (L+ν)/2 2 Lνx Lνx PX (x) = KL−ν 2 xΓ(ν)Γ(L) µY µY Where Kα is the modified Bessel function of the third kind of order α. This distribution proves to be well adapted at describing reality [17].

X

f (x) =

aJ,k φ˜J,k (x) +

j≤J

k

The connection between filter banks and wavelets stems from dilation equations allowing us to pass from a finer scale to a coarser one [18]: =

ψ(x) =

A. Principles

= √ In the same way, basis functions ψj,k (x) −j −j 2 ψ(2 x − k) are the result of dilations and translations ofRthe same function ψ(x) called the wavelet, which verifies ψ(x)dx = 0. The family {ψj,k }k span a sub-space Wj of L2 (R). The projection of f on Wj gives the wavelet coefficients {wj,k =< f, ψj,k >}k∈Z of f representing the details between two successive approximations. Consequently, sub-space Wj+1 is the complement of Vj+1 in Vj : Vj = Vj+1 ⊕ Wj+1

(4)

Sub-spaces Vj realizes a multiresolution analysis. They present the following properties [18]: 1. Vj+1 ⊂ Vj , ∀j ∈ Z. 2. f (x) ∈ Vj+1 ⇔ f (2x) ∈ Vj . 3. f (x) ∈ Vj ⇔ f (2j x − k) ∈ V0 , ∀k ∈ Z. +∞ +∞ \ [ Vj = 0. Vj is dense in L2 (R) and 4. −∞ −∞ R 5. A scale function φ ∈ V0 ( φ(x)dx = 1) exists as: {φ(x− k)}k∈Z is a basis of V0 . Consequently, a multiresolution analysis with J levels gives a decomposition of L2 (R) as depicted:   M L2 (R) =  W j  ⊕ VJ (5) j≤J

(6)

k

B. Filter bank

III. Multiscale analysis

A multiresolution analysis with J levels of a signal f of finite energy is a projection of f on a basis {{φ √ J,k }k , {ψj,k }j≤J }k∈Z [18]. Basis functions φj,k (x) = 2−j φ(2−j x − k) results from translations and dilations of R a same function φ(x) called scale function, verifying φ(x)dx = 1. The family {φj,k }k∈Z span a sub-space Vj in L2 (R). The projection of f on Vj gives an approximation {aj,k =< f, φj,k >}k∈Z of f at the scale 2j .

wj,k ψ˜j,k (x)

Duals functions φ˜j,k (x) and ψ˜j,k (x) have to be defined in order to ensure a perfect reconstruction.

φ(x)

In the following sections, we note L2 (R) the Hilbert space of real square R summable functions, with a scalar product < f, g >= f (x)g(x)dx.

XX

√ X 2 hi φ(2x − i) i X √ 2 gi φ(2x − i)

(7)

i

with hi =< φ, φ−1,i > and gi =< ψ, φ−1,i >.

X The normalisation of the scale function implies hi = i X √ R 2. In the same way, ψ(x)dx = 0 implies gi = 0. i

A signal multiresolution analysis can be performed with a filter bank composed of a low-pass analysis filter {hi } and a high-pass analysis filter {gi }: aj+1,k wj+1,k

= =

< f, φj+1,k >= < f, ψj+1,k >=

X

i X

hi−2k aj,i gi−2k aj,i

(8)

i

As a result, successive coarser approximations of f at scale 2j are provided by successive low-pass filtering (an undersampling operation is applied on each filter output). Wavelet coefficients at scale 2j are obtained by a highpass filtering of an approximation of f at the scale 2j−1 , followed by an undersampling. The signal reconstruction is directly derived from relation (4): aj,k

= =

< f, φj,k > X X ˜ k−2i aj+1,i + h g˜k−2i wj+1,i i

(9)

i

˜ i } and {˜ Where the coefficients {h gi } define the synthesis filters. B.1 Orthogonal wavelet We can construct ψ(x) and φ(x) in order to realize an orthogonal decomposition of the signal, then Wj+1 is the orthogonal complement of Vj+1 in Vj . Quadrature mirror filters (QMF filters) satisfy all these constraints ([19], [18]) with gn = (−1)n h−n+1 . Despite the mathematical elegance of the decomposition, constraints imposed on filters do not allow a symmetric design and establish the bandwith value.

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B.2 Biorthogonal wavelets Leaving out the orthogonality constraint, we can have symmetric filters which are suitable for image processing. Synthesis filters are derived from analysis filters with the help of the following relations [20]: ˜ n = (−1)n+1 g−n h g˜n = (−1)n+1 h−n C. Stationary wavelet transform The pyramidal algorithm described above, does not preserve the translation invariance. In other words, a translation of the original signal does not necessarily imply a translation of the corresponding wavelet coefficients. This property is essential in image processing. On the contrary, wavelet coefficients generated by an image discontinuity could disapeared arbitrarely. This non-stationarity in the representation is a direct consequence of the undersampling operations following each filtration. In order to preserve the translation invariance property, some authors have introduced the concept of stationary wavelet transforms [13] originally called “ trous “ algorithm [12]. The undersampling operation is then suppressed but filters are dilated by inserting 2j−1 zeros between low-pass and high-pass filter coefficients of a given level j. This is performed, in order to reduce the bandwidth by a factor of 2 from one level to another.  hk/2j k = 2j m, if m ∈ Z [j] hk = 0 else [j] gk

=



gk/2j 0

k = 2j m, if m ∈ Z else

Image multiresolution analysis was introduced by Mallat [7]. The unidimensional filter bank used for the stationary wavelet decomposition can be applied in the two dimensional case. Image rows and columns are then filtered separately. Filtering equations to obtain the level j + 1 from the level j are the following (where (a, b) is for the pixel position): X [j] [j] AX,j+1 (a, b) = hk hl AX,j (a + k, b + l) k,l

[j] = [H X,j ]a,b X A [j] [j] h WX,j+1 (a, b) = gk hl AX,j (a + k, b + l) k,l [j]

= d WX,j+1 (a, b) =

k,l [j] [G X,j ]a,b Xv A [j] [j] gk gl AX,j (a

W[j]

=

G[j]

j−1 Y

H[i]

i=1

  WX (a, b) = W[j] X a,b X WX (a, b) = Hm1 ,m2 ,...,mj Gn1 ,n2 ,...,nj

(11)

m1 ,n1 ,...,mj ,nj

×X(a + u, b + v)

with u = m1 + m2 + . . . + mj , v = n1 + n2 + . . . + nj and: Hm1 ,m2 ,...,mj

=

[j−2] h[j−1] mj hmj−1 . . . hm1

(12)

Gn1 ,n2 ,...,nj

=

gn[j−1] h[j−2] nj−1 j

(13)

. . . hn1

IV. Speckle influence on wavelet coefficients A. Wavelet coefficient behavior in presence of speckle The moment generating function of a PWX distributed random variable WX is by definition the integral [21]: Z +∞ Φ(s) = PWX (x)esx dx (14) −∞

D. Stationary wavelet decomposition of an image

= [G X,j ]a,b Xh A [j] [j] v hk gl AX,j (a + k, b + l) WX,j+1 (a, b) =

[0, π/2j ]. Image details are contained in three high freh v d quency images WX,j ,WX,j and WX,j corresponding to horizontal, vertical and diagonal detail orientations respectively. Wavelet coefficients of the j level give high frequency information in the sub-band [π/2j , π/2j−1 ]. For each decomposition level, images preserve the original size in so far as undersampling operations after each filter have been suppressed. We can define a filtering operator Wj permitting to obtain the j level high frequency image for any orientation. This operator is the result of j successive convolutions:

If s is changed to jω we obtain the characteristic function. The derivatives of Φ(s) at the origin equal the moments of WX . In our case, it is more covenient to use the second (generating) moment function defined by: Ψ(s) = ln (Φ(s))

Considering the wavelet coefficients at position (a, b) on a high frequency image of level j and with any orientation; according to the relation (11), the random variable WX (a, b) is a linear combination of random variables {X(u + a, v + b)}u,v assumed independent and equally distributed. Therefore, using the second moment function of the X pdf, we obtain the second moment function of WX :

(10)

+ k, b + l)

k,l [j]

= [Gd AX,j ]a,b

AX,j is the approximation of the original image at the scale 2j , giving the low-frequency content in the subband

(15)

ΨWX (s) =

X

m1 ,n1 ,...mj ,nj

ΨX (Hm1 ,m2 ,...,mj Gn1 ,n2 ,...,nj s)

The value of the nth order derivative of ΨWX (s) at the origin, gives the WX cumulants of order n [21]: X
n (n) ΨWX (0) = Hm1 ,m2 ,...,mj Gn1 ,n2 ,...,nj (16) m1 ,n1 ,...mj ,nj (n) ×ΨX (0)

5

We note: Sn[j]

B. High frequency image segmentation X

=

mj ,nj ,...m1 ,n1

X

=

(gk )

n

k

Hm1 ,m2 ,...,mj Gn1 ,n2 ,...,nj

!

X

n

(hl )

l

!2j−1


n (17)

h v defined for images WX,j et WX,j . For the high frequency d images WX,j with a diagonal orientation we obtain:

Sn[j]

X

=

(gk )

n

k

!2

X l

n

(hl )

!2(j−1)

The relation (23), in a similar way to the SAR image segmentation using the grey level normalized standard deviation, allows to segment high frequency images with the help of the normalized standard deviation CWX . To do this, we locally estimate within a neighbourhood V(a, b) centered on each pixel (a, b) the standard deviation of wavelet coefficients σ ˆWX (a, b) and the mean µ ˆX (a, b) on the original SAR image. The neighbourhood is defined by a window of size Nj = Dj × Dj enlarging in relation to the level j in order to take into account the scale increase:

(18) 2 (a, b) = σ ˆW X

R The condition on the wavelet function ( ψ(x)dx = 0) [j] implies S1 = 0. The relation (16) will be used in section IV to establish moments of the wavelet coefficients useful for the computation of different parameters of the Pearson distribution system. Assuming a Γ(µY , µY /ν) distribution for the reflectivity pdf and a Γ(L, L) distribution for the speckle pdf, we obtain: (n)

µn = ΦX (0) =

µnY ν n Ln

Γ(L + n)Γ(ν + n) Γ(L)Γ(ν)

(19)

Applying the relation (16), WX second order cumulant can be derived from the X second order cumulant:
(2) [j] (2) [j] (20) ΨWX (0) = S2 ΨX (0) = S2 µ2 − µ21  2  µY Γ(L + 2)Γ(ν + 2) (2) [j] ΨWX (0) = S2 − µ2Y (21) ν 2 L2 Γ(L)Γ(ν)

µ ˆX (a, b) =

(2)

[j]

= ΨWX (0) = S2 µ2Y ((1 + CZ2 )CY2 + CZ2 ) (22)

Consequently, we have a relation equivalent to (3), in defining a normalized standard deviation on wavelet coefficients σ CWX = µWXX : [j]

CY2 =

2 CW − S2 CZ2 X [j]

S2 (1 + CZ2 )

(23)

Analogous to the relation (3), q we can define a normalized [j] standard deviation CWZ = S2 CZ of the speckle wavelet coefficient. Within image homogeneous areas without texture (CY2 = 0), we have: CWX = C WZ Therefore, the multiplicative model on the original image involves that the standard deviation of the wavelet coefficients σWX is proportional to the mean intensity E[X] = µY of the homogeneous region under consideration.

X

2 (u, v) wX

(u,v)∈V(a,b)

X

1 Nj

x(u, v)

(u,v)∈V(a,b)

The window size on the level j can follow a similar progression to wavelet support, i.e. Dj = 2j−1 (D0 − 1) + 1 (where D0 is the original size on the first level). B.1 Highly heterogeneous region detection When the resolution cell has only a few strong dominant reflectors, the imaging system response is deterministic, consequently the pixel value must be preserved. We choose to use an upper threshold CX,max beyond which the region will be considered highly heterogeneous. Previous work provides a threshold derived from the maximum likelihood detector of Frost ([3], [5]): CX,max =

p 1 + 2/L

A similar threshold can be defined for CWX :

In fact, the second order cumulant of WX equals its variance, which can be expressed as a function of CZ2 = 1/L and CY2 = 1/ν: 2 σW X

1 Nj

CWX ,max =

q

[j]

S2

p

1 + 2/L

(24)

Images depicted on Figure 1 give an example of segmentation between homogeneous, heterogeneous and highly heterogeneous regions. V. MAP wavelet coefficient filtering A. Weighting of the wavelet coefficient using a MAP criteria The wavelet decomposition operation can be written as: WX

= = =

W[j] X = W[j] Y Z W[j] Y + W[j] Y (Z − 1) = W[j] Y + W[j] Y Zc WY + WB

with WY = W[j] Y and WB = W[j] (Y Zc ) = WX − WY which are centered and uncorrelated random processes: E[WY ] = 0, E[WB ] = 0 E[WY WB ] = W[j] W[j] E[Y ]E[Zc ] = 0

6

The A Posteriori probability density function conditional to the observation can be expressed from the Bayes relation: PWX |WY (wX |wY )PWY (wY ) PWX (wX ) PWB |WY (wB |wY )PWY (wY ) PWX (wX )

PWY |WX (wY |wX ) = =

The estimate w ˆY maximizing the A Posteriori pdf, is:

w ˆY = arg maxwY PWY |WX (wY |wX )
d (25) ln PWB |WY (wB |wY ) dwY +ln (PWY (wY ))) |wY =wˆY = 0 In order to apply this Bayesian estimation, we have to establish a model for the different pdf. B. Probability density function of the wavelet coefficient The grey level distribution of the original images, acquired with a small number of looks, are highly asymmetric. The extreme case for monolook images, results in an exponential distribution. Consequently, we can still expect asymmetric pdf for the high frequency images of the first levels. When the considered level increases, the pdf becomes symmetric and can be assumed gaussian (consequence of the central limit theorem). In first approximation, we assume a gaussian model for PWB |WY (wB |wY ) and PWY (wY ). The MAP criteria is then equivalent to applying a MMSE criteria. Then, a more realistic model, introducing a certain amount of asymmetry in the wavelet coefficient pdf, will be established in the framework of the Pearson distribution system (c.f. section B.2).

2 N (0, σW Y

We assume gaussian distributions ) and 2 N (0, σW ) for both W and W respectively. From reY B B lation (22), we obtain: [j]

S2 µ2Y CY2

=

(26)

Similarly, for the variance of B = Zc Y , we have: 2 σW B

=

(2)

[j]

ΨWB (0) = S2 µ2Y CZ2 (1 + CY2 )

(27)

In the gaussian case, the uncorrelation (E[WY WB ] = 0) is sufficient to ensure the independence between WY and WB . Consequently, the likelihood term PWB |WY (wB |wY ) in the Bayesian relation is equal to the pdf PWB (wB ) of WB . The MAP equation in the gaussian assumption can be written as follows: −

1 wX − wˆY wˆY + =0 2 2 σW σW Y B

(28)

This leads to: wˆY

=

wˆY

=

2 2 CW − CW X Z wX 2 CWX (1 + CZ2 )

(30)

This expression is similar to the Kuan’s filter [1] determined according to a local MMSE criteria. B.2 Pearson system of distributions The distribution family of Pearson verifies the following differential equation: df (x − a)f = dx b0 + b1 x + b2 x2

(31)

Distributions f verifying this relation are unimodal and have a tangential contact with the x axis on extremities (df /dx = 0) [22]. By integrating this differential equation, Pearson parameters can be derived directly from moments of the observation: b1 = a

=

b0

=

b2

=

A

=

µ3 (µ4 + 3µ22 ) A µ2 (4µ2 µ4 − 3µ23 ) − A (2µ2 µ4 − 3µ23 − 6µ32 ) − A 10µ2 µ4 − 18µ32 − 12µ23 −

(32)

The distribution f is unimodal with a maximum at x = a. With an origin translation X = x − a, the differential equation (31) can take a simple form: d ln f X = dX B0 + B1 X + B2 X 2

(33)

b0 + a2 (1 + b2 ) a(1 + 2b2 ) b2

(34) (35) (36)

with:

B.1 Gaussian assumption

2 σW Y

using the relation (3), we obtain:

2 σW Y wX 2 + σW Y

2 σW B

(29)

B0 B1 B2

= = =

The behavior of zeros of B0 +B1 X +B2 X 2 can be examined through the values of the indice K = B12 /(4B0 B2 ). From this, we can identify three main types for f : • roots are reals with opposite signs, K ≤ 0; • roots are reals with same signs, K ≥ 1; • roots are complexes, 0 ≤ K < 1. The indice κ = b21 /(4b0 b2 ) is equivalent to K from which we can identify, in the same way, the type of distribution obtained. The advantage of using κ, is that we can derive its value directly from the Pearson parameters (32), and then from the moments. However, the previous translation origin x = X + a can affect the root signs: 1 β˜2 > β˜1 > 0 (see table I). Consequently, η verifies: 0 < η(α) < 1

[j] [j] 1 > β˜2 > β˜1 > 0 α > αmin

(44) (45)

When α takes large values, the distribution tends to the gaussian case (where κ(α) = 0). According to the κ value, the wavelet coefficient pdf for a gamma distributed image is a Pearson type IV [22]:   x−a+γ − λ arctan  m 2 δ f (x) = k (x − a + γ) + δ 2 e (46) with:

m=

γ B1 B0 1 ,λ= ,γ= , δ2 = τ − γ 2, τ = 2B2 B2 δ 2B2 B2

B.4 Probability density function of the speckle wavelet coefficients When the ground radar reflectivity is constant (pdf of Y is a dirac), the image pdf conditionally to Y is the speckle pdf. Applying relations (38) with α = L and β = µY we obtain:

Pearson parameter expressions (32) lead to the following formulation for κ: 1 κ = (38) β(α) (2 − η(α)) (1 − η(α))

η(α) =

[j] β˜1 >0 (43) 8 The minimal value αmin for α is the positive solution of κ(α) = 1, which leads to a second degree equation. Table I gives some values for αmin which are all largely inferior to 1. Therefore, we have 0 < κ(α) < 1 when the following conditions apply:

κ(α) >

(42)

µWX ,1

=

0

µWX ,2

=

S2

µWX ,3

=

2 [j] µY

L

3 [j] 2µY S3 L24 [j] 6µY S4 L3

(47)

µ4Y L2 The lower bound (45) on α is largely verified since in practice α = L ≥ 1. So, the type IV model is always valid for the pdf of the random process WX|Y . Figure 2 shows an example of an image of pure speckle filtered with a biorthogonal wavelet (B5). Table II gives Pearson and distribution parameters for different L values and for the decomposition level j. As expected, the type IV distribution leads to a gaussian when j or L increases. µWX ,4

=

[j]

+ 3(S2 )2

B.5 Probability density function for the wavelet coefficients of the reflectivity In a gamma assumption for the reflectivity and from relations (38) with α = ν and β = µY , we obtain the first fourth moments: µWY ,1

=

0

µWY ,2

=

S2

µWY ,3

=

µWY ,4

=

2 [j] µY

ν

3 [j] 2µY S3 ν 24 [j] 6µY S4 ν3

(48) [j]

+ 3(S2 )2

µ4Y ν2

8

with:

B.6 Probability density function of WB We have the following moments for the noisy term B = Y (Z − 1): µWB ,1

=

0

µWB ,2

=

2 [j] µ S2 Y

µWB ,3

=

µWB ,4

=

  1 1+ L  ν  3 3 2 [j] 2µY S3 1+ + 2 L2  ν ν 4 6 11 [j] 6µY S4 1+ + 2 3 L ν ν  6 + 3 (1 + L/2) + 3µ2WB ,2 ν

β˜1 (ν+2) (ν+1) + Lν β˜2 (ν+2)(ν+3) (1 + L/2) + Lν (ν+1)

Assuming a type IV model for the probability function of WY and WB , the MAP equation (25) can be expressed as follows:   1 2 fM AP (wY ) = ln (wY − aY + γY ) + δY2 2BY,2   γY w Y − a Y + γY − arctan BY,2 δY δY   1 2 2 + ln (wX − wY − aB + γB ) + δB 2BB,2   γB wX − wY − aB + γB − arctan BB,2 δB δB Derivating according to wY :

+

γY /δY

2

BY,2 δY (wY − aY + γY )

/δY2

wX − wY − aB + γB

+1

2

2 BB,2 (wX − wY − aB + γB ) + δB



= = = =

wX − aB + γB wX − aB a Y − γY aY

=

X

γB /δB 2

2 +1 BB,2 δB (wX − wY − aB + γB ) /δB

(52)

=

X

(gk )n

k

CWX ,max

=

CWZ

=

! !2

X

(hk )

k

X

q p [j] S2 1 + 2/L q [j] S 2 CZ

k

n

!2j−1

(hk )n

!2(j−1)

We estimate σWX and µY within a neighbourhood Dj × Dj around each pixel of the high frequency and original image respectively. The value of CˆWX = σ ˆWX /ˆ µY measures the degree of local homogeneity, and therefore, determines the type of estimator to apply: • If CWZ < CˆWX < CWX ,max : The considered neighbourhood is textured, the normalized standard deviation of Y can be estimated with: 2 2 CˆW − CW X Z [j]

2 S 2 + CW Z

νˆ = 1/CˆY2







The single point MAP estimate w ˆY is the solution of a third degree equation: A3 w ˆY3 + A2 w ˆY2 + A1 w ˆ Y + A0 = 0

(gk )

n

k

CˆY2 =

d fM AP (wY ) w − a Y + γY   Y = 2 dwY BY,2 (wY − aY + γY ) + δY2



Sn[j] or Sn[j]

C. Maximum a posteriori equation

−

UB VB UY VY

for n = 2, 3, 4 ∀j = 1, . . . , J:

4Lν(ν + 1) β˜1 (ν + 2)2



=

(51)

First, the image is decomposed in J levels, each level having three high frequency images with a particular orientation. For each high frequency image, the algorithm is as following:

Conditions for a type IV distribution (0 < κ < 1) are easily verified for all the L and ν values.

−

A0

BB,2 + BY,2 −BB,2 (2UB + VY ) − BY,2 (2UY + VB ) 2 BB,2 (UB (UB + 2VY ) + δB ) +BY,2 (UY (UY + 2VB ) + δY2 ) 2 −BB,2 VY (UB2 + δB ) − BY,2 VB (UY2 + δY2 )

VI. Filter Implementation

2

β(L, ν) =

= = =

(49)

η and β giving κ expression are:

η(L, ν) =

A3 A2 A1

(50)

Then, the point estimate of WY requires the following steps: 1. Calculate the first fourth moments of WY and WB (relations (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: A3 w ˆY3 + A2 w ˆY2 + A1 w ˆY + A0 = 0. wX = w ˆY • If CˆWX ≥ CWX ,max : The local neighbourhood is strongly heterogeneous, and may contain a point target or a strong edge:

9

VIII. Conclusion wX = wX • If CˆWX ≤ CWZ : The local neighbourhood is considered homogeneous, consequently: wX = 0 The resulting filtered image is obtained by applying the reconstruction algorithm from the previously filtered high frequency images. VII. Results A. Weighting behavior Figure 3 represents the weighing of WX obtained by solving the MAP equation in relation to the normalized standard deviation of the wavelet coefficients for a monolook image (L = 1) with WX = 1.5σWX . The gaussian case established in section V-B.1 is represented with a dotted line. We observe as expected a difference between the weighting and the gaussian case on the first levels, due to the asymmetry introduced by the type IV models. This asymmetry diminishes when the level increases, the weighting becomes similar to the gaussian case beyond the fourth level. B. Results on SAR images The proposed filter is tested on a monolook RADARSAT image of the town of Sorel (Qu´ebec, Canada). The resolution cell is about 9m × 9m. The image contains a lot of large homogeneous areas (see Figure 4). Wavelets used are the orthogonal Daubechies wavelet [18] (noted D4) and the biorthonal wavelet of Cohen et al. [20] (noted B5). Best results are obtained with the shortest filters. Beyond 3 levels of decomposition (J = 3), speckle reduction performances remain unchanged. The size of the windows used for the local statistics estimation is 7 × 7 (D0 = 7) on the first level, hence the size increases 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 estimation window size decreases from 11 × 11 to 5 × 5 while there is an edge detection inside. In terms of speckle reduction in homogeneous areas, performances are measured with an estimation of the equivalent number of looks (ENL) within three homogeneous windows selected in the image Figure 4. Smoothing increases with the low-pass filter size, however for short filters, performances are equivalent to the Gamma-MAP filter (Table III). In terms of edges and meaningful detail preservation, the proposed multiscale filter retains more information in homogeneous areas and preserves well strong edges (roads, agricultural boundaries,...). Strong reflectors are slightly smoothed compared to the one scale Gamma-MAP filter but without target blurring.

The proposed filter combines image multiscale analysis and classical techniques of adaptive filtering. The multiplicative model introduced in high frequency images, permits to retain coefficients produced by significant structures present in the image and suppress those produced by the speckle noise. From gamma assumptions for the pdf of the reflectivity and the speckle, and with the help of second generating moment functions, we have expressed wavelet coefficients moments up to the fourth moment. On these results, we applied the Pearson distribution system, leading to a type IV model for the wavelet coefficient pdf. This asymmetric distribution becomes naturally gaussian when the level increases. The Pearson type IV model combined with the explicit calculation of the first fourth moments of the wavelet coefficients, gives us an entirely parametric model for the wavelet coefficient pdf. Parameters depend only on the local estimation of the reflectivity normalized standard deviation and the number of looks on the original image. From this statistical modeling, the application of a MAP criteria leads to an estimate of the wavelet coefficient of the reflectivity as the solution of a simple third degree equation. The resulting multiscale filter shows interesting results compared to the classical Gamma-MAP filter. The main advantage of the multiscale approach is the multiscale detection of discontinuities along different orientations provided by the wavelet decomposition. High frequency images of the first level are sensitive to small scale variations of the image, so grey level variations due to the speckle are mainly concentrated on the first level. Whereas, meaningful image discontinuities can be represented on many scales, and therefore influence high frequency images on all levels. Consequently, the probability of a significant discontinuity detection increases with the level (the signal to noise ratio increase). A multiscale decision on the presence or not of a discontinuity is less brutal than in a one scale detection, since a wavelet coefficient can be suppressed at one scale but preserved on a higher scale. Acknowledgments The authors acknowledge NSERC (Natural Sciences and Engineering Research Council) of Canada, FDT (Fonds de d´eveloppement technologique du Qu´ebec) and Viasat Gotechnologie Inc. (Montr´eal, Qu´ebec) who provided financial support. The authors also thank E. Rosenberg, F. Zagolsky, B. Burdsall and P. Gagnon for their linguistic support.

10

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11

List of table captions ˜[j] , β˜[j] and αmin for the first level of • Table I Values of β 1 2 the multiscale decomposition (j = 1) and for two wavelets used in this study: Daubechies’ wavelets with 4 and 8 coefficients (D4,D8), biorthogonals with 5 and 9 coefficients [1] [1] (B5,B9). We observe that 0 < β˜1 < β˜2 < 1 and αmin