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Kalman's shrinkage for wavelet-based despeckling of SAR images

In this paper, a new probability density function (pdf) is proposed to model the statistics of wavelet coefficients, and a simple Kalman's filter is derived from the new pdf using Bayesian estimation theory. Specifically, we decompose the speckled image into wavelet subbands, we apply the Kalman's filter to the high subbands, and reconstruct a despeckled image from the modified detail coefficients. Experimental results demonstrate that our method compares favorably to several other despeckling methods on test synthetic aperture radar (SAR) images.


I. INTRODUCTION
HE despeckling of a SAR image corrupted by speckle noise is an important problem in image processing. If the wavelet transform and maximum a-posteriori (MAP) estimator are used for this problem, the solution requires a priori knowledge about wavelet coefficients. Therefore, two problems arise: 1) What kind of distribution represents the wavelet coefficients ? 2) What is the corresponding estimator (shrinkage function) ?
Statistical models pretend wavelet coefficients to be random realizations from a distribution function. For the first problem, mostly these models assume the coefficients are time-invariant, and try to characterize them by using Gaussian, Laplacian, or Gaussian scale mixtures. For example, classical soft thresholding operator suggested by Donoho [1] can be obtained by a Laplacian assumption. Bayesian methods for image denosing using other distributions have also been proposed [2]- [6].
The wavelet transform has become an important tool for this problem due to its energy compaction property. Basically, it states that the wavelet transform yields a large number of small coefficients and a small number of large coefficients.
Simple denoising algorithms that use the wavelet transform consist of three steps: 1) Calculate the wavelet transform of the noisy image. 2) Modify the noisy wavelet coefficients according to some rule. 3) Compute the inverse transform using the modified coefficients.
Manuscript received April 12, 2005 Besides, two properties are very important for the following deduction: 1) If a wavelet coefficient is large/small, the adjacent coefficients are likely to be large/small, and 2) large/small coefficients tend to propagate across the scales [7]- [9]. In this paper, we use a new time-variant pdf and derive a Kalmanian shrinkage (KalmanShrink) function using Bayesian estimation theory.

II. BAYESIAN DENOISING
In this section, the denoising of an image corrupted by white Gaussian noise will be considered, i.e., n x g + = (1) where n is independent Gaussian noise. We observe g (a noisy signal) and wish to estimate the desired signal x as accurately as possible according to some criteria. In the wavelet domain, if we use an orthogonal wavelet transform, the problem can be formulated as n w y + = (2) where y noisy wavelet coefficient, w true coefficient, and n noise, which is independent Gaussian. This is a classical problem in estimation theory. Our aim is to estimate from the noisy observation. The MAP estimator will be used for this purpose. Time-variant and time-invariant models will be discussed for this problem in Sections II-A and B, and new MAP estimators are derived.

A. Time-Invariant Models
The classical MAP estimator for (2) is  (5) It has been observed that wavelet coefficients of natural images have highly non-Gaussian statistics [7]- [9]. The pdf for wavelet coefficients is often modeled as a generalized (heavytailed) Gaussian [7]- [9].
p (6) where s, p are the parameters for this model, and K(s, p) is the parameter-dependent normalization constant. Other pdf models have also been proposed [7]- [9]. In practice, generally, two problems arise with the Bayesian approach when an accurate but complicated pdf pw(w) is used: 1) It can be difficult to estimate the parameters of pw for a specific image, especially from noisy data, and 2) the estimators for these models may not have simple closed form solution and can be difficult to obtain. The solution for these problems usually requires numerical techniques.
Let us continue developing the MAP estimator and show it for Gaussian and Laplacian cases. Equation (4) is also equivalent to As in [7]- [9], let us define f(w) = log(pw(w)). By using (5), (7) becomes This is equivalent to solving the following equation for ŵ if pw(w) is assumed to be strictly convex and differentiable. .
If pw(w) is assumed to be a zero mean gaussian density with variance σ 2 , then f(w) = -log( σ π 2 ) -w 2 /2σ 2 , and the estimator can be written as )w 2 /σ , and the estimator will be (12) Here, (g)+ is defined as g g (13) Equation (12) is the classical soft shrinkage function. Let us define the soft operator as The soft shrinkage function (12) can be written as

B. Time-Variant Model (KalmanShrink)
Time-invariant models cannot model the temporal variation of wavelet coefficients. However, the coefficients depend strongly on the time. This paper suggests such dependency and derives the corresponding Kalmanian MAP estimator based on noisy wavelet coefficients in detail.
Here, we modify the Bayesian estimation problem as to take into account the time dependency of coefficients. If pdf is Replacing (18) into (9), the estimator can be written as where Kt is the Kalman's gain of the Kalman's filter [10]- [15], being is the norm of error covariance, that is to say , E{•} means the expectation of {•} and ||(•)||2 means the L2-norm of (•). Finally, the original system is formed by (2) and . w w These equations can be described as a Kalman filter loop [10] as follows: 1) Calculate 2 n σ as the variance of (2) 2) Initialize ŵ and P The true radiometric values of the image are represented by I, and the values measured by the radar instrument are represented by Is . The speckle noise is represented by S. The parameters r and c means row and column of the respective pixel of the image. If and we begin with a multiplicative speckle S and finish with an additive speckle N [16], which avoid the log-transform, because the mean of log-transformed speckle noise does not equal to zero [17] and thus requires correction to avoid extra distortion in the restored image. For single-look SAR images, S is Rayleigh distributed (for amplitude images) or negative exponentially distributed (for intensity images) with a mean of 1. For multi-look SAR images with independent looks, S has a gamma distribution with a mean of 1. Further details on this noise model are given in [18].

IV. ASSESSMENT PARAMETERS
In this work, the assessment parameters that are used to evaluate the performance of speckle reduction are Noise Variance, Mean Square Difference, Noise Mean Value, Noise Standard Deviation, Equivalent Number of Looks, Deflection Ratio [19], [20], and Pratt's figure of Merit [21].

A. Noise Mean Value (NMV), Noise Variance (NV), and Noise Standard Deviation (NSD)
NV determines the contents of the speckle in the image. A lower variance gives a "cleaner" image as more speckle is reduced, although, it not necessarily depends on the intensity. The formulas for the NMV, NV and NSD calculation are where R-by-C pixels is the size of the despeckled image Id . On the other hand, the estimated noise variance is used to determine the amount of smoothing needed for each case for all filters.

B. Mean Square Difference (MSD)
MSD indicates average square difference of the pixels throughout the image between the original image (with speckle) Is and Id , see Fig. 1. A lower MSD indicates a smaller difference between the original (with speckle) and despeckled image. This means that there is a significant filter performance. Nevertheless, it is necessary to be very careful with the edges. The formula for the MSD calculation is

C. Equivalent Numbers of Looks (ENL)
Another good approach of estimating the speckle noise level in a SAR image is to measure the ENL over a uniform image region [20]. A larger value of ENL usually corresponds to a better quantitative performance. The value of ENL also depends on the size of the tested region, theoretically a larger region will produces a higher ENL value than over a smaller region but it also tradeoff the accuracy of the readings. Due to the difficulty in identifying uniform areas in the image, we proposed to divide the image into smaller areas of 25x25 pixels, obtain the ENL for each of these smaller areas and finally take the average of these ENL values. The formula for the ENL calculation is The significance of obtaining both MSD and ENL measurements in this work is to analyze the performance of the filter on the overall region as well as in smaller uniform regions.

D. Deflection Ratio (DR)
A fourth performance estimator that we used in this work is the DR proposed by H. Guo et al (1994), [22]. The formula for the deflection calculation is The ratio DR should be higher at pixels with stronger reflector points and lower elsewhere. In H. Guo et al's paper, this ratio is used to measure the performance between different wavelet shrinkage techniques. In this paper, we apply the ratio approach to all techniques after despeckling in the same way [19].

E. Pratt's figure of merit (FOM)
To compare edge preservation performances of different speckle reduction schemes, we adopt the Pratt's figure of merit edge pixels, respectively, di is the Euclidean distance between the ith detected edge pixel and the nearest ideal edge pixel, and α is a constant typically set to 1/9. FOM ranges between 0 and 1, with unity for ideal edge detection.
We compute and compare the NMV and NSD over six different homogeneous regions in our SAR image, before and after filtering, for all filters. The KalmanAShrink has obtained the best mean preservation and variance reduction, as shown in Table I. Since a successful speckle reducing filter will not significantly affect the mean intensity within a homogeneous region, KalmanShrink demonstrated to be the best in this sense too. The quantitative results of Table I show that the KalmanShrink technique can eliminate speckle without distorting useful image information and without destroying the important image edges. In fact, the KalmanShrink outperformed the conventional and no conventional speckle reducing filters in terms of edge preservation measured by  Pratt's figure of merit [21], as shown in Table I. On the other hand, all filters were implemented in MATLAB® (Mathworks, Natick, MA) on a PC with an Athlon (2.4 GHz) processor.

VI. CONCLUSION
We have presented a new speckle filter for SAR images based on wavelet denoising. In order to convert the multiplicative speckle model into an additive noise model, Argenti et al's approach is applied. The simulations show that the KalmanShrink have better performance than the most commonly used filters for SAR imagery (for the studied benchmark parameters) which include statistical filters and several wavelets techniques in terms of smoothing uniform regions and preserving edges and features. The effectiveness of the technique encourages the possibility of using the approach in a number of ultrasound and radar applications. In fact, cleaner images suggest potential improvements for classification and recognition. Besides, considerably increased deflection ratio strongly indicates improvement in detection performance.
Finally, the method is computationally efficient and can significantly reduce the speckle while preserving the resolution of the original image, and avoiding several levels of decomposition and block effect.