Multiscale Filtering of SAR Images Using Scale and Space Consistency Samuel Foucher Research & Development Dept Computer Research Institute of Montreal Montreal, Canada Abstract—A new approach for speckle reduction in SAR images based on the stationary wavelet transform is proposed. Noisy wavelet coefficients are reduced via a shrinkage function that depends on a statistical modeling of the normalized wavelet coefficient probability density functions. Consistencies between coefficients across scales and space are also reinforced using consistency rules. The approach is particularly robust in cases of correlated speckle noise. Keywords-speckle filtering; SAR; wavelet; I. INTRODUCTION Most speckle reduction techniques assume the speckle noise as a white random process. Recently, numerous wavelet based methods have been proposed [1][2][3][4][5] where, like most of the mono-scale filters, a white speckle noise is assumed. However, the impact of the speckle second order statistics is expected to be significant as a non uniform speckle power spectral density function will strongly affect the distribution of the signal energy within the various wavelet sub-bands [5]. The proposed method is based on the work of Scharcanski et al. [6] on the filtering of optical images corrupted by Gaussian noise. In this approach, the amplitude of significant wavelet coefficients is modeled probabilistically and a shrinkage function is derived based on the model obtained. Scale consistency is applied and leads to a better preservation of strong coefficients present at successive scales. Identically, Geometric constraint (orientation preservation) aims at strengthening coefficients placed along edges. The present paper is an extension of the authors previous work on speckle filtering [1][5] where a multiscale MAP estimation was proposed (SWT-MMSE). The paper is organized as follows. Section II, considers the impact of speckle correlation on the normalized wavelet coefficient statistics. In Section III, a wavelet coefficient shrinkage function based on a two components mixture model is described. Finally, in section III the proposed approach is compared with other speckle reduction techniques on simulated and real images. II. INFLUENCE OF A CORRELATED SPECKLE ON THE WAVELET COEFFICIENTS A. General Considerations In the following, we assume a stationary wavelet transform (SWT) of a SAR image I composed of 3J high-frequency images { } 1,..., [] ,, j J j hvd W I ε ε = = and J low-frequency images { } 1,..., [] j J j A I = . An interesting property of the wavelet transform is that the autocorrelation function (acf) [] j W R of the wavelet coefficients [] j W I is the convolution of the acf I R of the SAR image I by the acf [] j R ψ of the wavelet filter [] j ψ : [] [] [j] [] W j j j I W I I R R R ψ ψ = ∗ ⇒ = ∗ (1) Note that the wavelet acf is also a wavelet. Consequently, the speckle noise power for a L looks SAR image within a homogeneous area of mean reflectivity I µ which is expressed by the following ( ) I 2 [j] [] W var (0, 0) j I S R L ψ µ ρ = ∗ (2) The linear filtering power gain is directly affected by the interaction of the speckle correlation function S ρ and the wavelet acf, hence ( ) 2 [] [] [] 2, 1 (0, 0) () () 2 j j j S S C S R d ψ ρ γ ω ω ω π ∆ = ∗ = Ψ ∫ (3) where S γ is the speckle power spectral density function and [] j Ψ is the wavelet Fourier transform. More particularly, for a white speckle ( () 1 S γ ω = ) the above relationship is simply the wavelet power gain [1]: ( ) 2 [] [] [] 2 2, j j j C k k S S ψ = = ∑ (4) For a correlated speckle noise, the impact will be an increased noise level on lower frequency bands (j>1) and a decrease on finest scale (j=1). This work has been supported in part by the NSERC of Canada (Discovery Grant) and the MDEIE of the “Gouvernement du Québec”.