SAR Image Denoising Using MMSE Techniques

Abstract

Synthetic aperture radar (SAR) provides many advantages over optical remote sensing, principally the all-weather and all-day acquisition capability. For this reason, SAR images have been exploited for many applications such as forestry, agriculture, disaster monitoring, sea/ice monitoring. However, the main limitation in SAR images is the contamination with the multiplicative speckle noise. The speckle damages the radiometric quality of SAR images and contracts the performance of information extraction techniques. Many methods have been proposed in the literature to reduce speckle noise. These methods, however, must avoid degrading the useful information in the SAR images, such as textures, local mean of backscatter, and point targets. The minimum mean square error (MMSE) techniques have been largely applied in SAR image speckle denoising. The objective of this chapter is to review and give new insights into the MMSE denoising of SAR images. In particular, the performances of three MMSE-based techniques which are the commonly applied Lee sigma filter and the newly introduced iterative MMSE (IMMSE) filter, and the infinite number of looks prediction (INLP) filter are studied.

1. Introduction

Remote sensing imagery constitutes nowadays an important source of information for the characterization of the Earth’s surface. The potentiality of synthetic aperture radar (SAR) systems is recognized for geoscience and remote sensing applications due to their operation in all-time and all-weather conditions. However, due to the coherent nature of the scattering mechanisms, SAR data are affected by the multiplicative speckle noise. The presence of speckle noise disturbs human interpretation of the images and reduces the accuracy of postprocessing such as image classification [1].

The multi-looking process (i.e., boxcar filter) reduces speckles by averaging the intensities of neighboring pixels [2]. Nevertheless, the spatial resolution is degraded. Many other denoising techniques have been introduced in the literature to alleviate this limitation by using other estimation domains including spatial [3] and wavelet [4]. In the intensity-driven adaptive-neighborhood (IDAN) filter a region-growing technique is applied to produce an adaptive neighborhood [5]. The total variation (TV) techniques [6] have been widely applied for SAR image denoising due to their efficiency to preserve spatial details and speckle reduction. The nonlocal NL filtering represents one of the powerful speckle reduction techniques. Zhong et al. [7] applied the NL Means (NLM) to filter SAR images by adapting the use of Euclidean distance to multiplicative noise. The probabilistic patch based (PPB) filter introduces a patch-based weight to generalize the Euclidean distance-based weight used in the NL means algorithm [8]. A hybrid NL-wavelet domain denoising technique has been proposed [9]. Penna et al. replaced the Euclidean distance in the NLM filter with stochastic distances in the Haar wavelet domain [10]. The NL-based filters improved significantly the denoising performance of SAR images. However, their main disadvantage resides in the high computing cost. Deep Learning techniques constitute a recent trend of PolSAR speckle filtering [11, 12, 13].

The minimum mean square error (MMSE) based filters that account for the local statistics of the image constitute an important branch of speckle filtering techniques. Since the introduction of the Lee sigma filter in early 1980 [14, 15], many versions have been elaborated such as Frost [16], Kuan [17], the improved Lee [18, 19], etc. Due to their effectiveness in speckle reduction, simplicity and low computational demand, many MMSE-based filters have been implemented in remote sensing software. To mitigate the drawbacks of the Lee sigma filter, various versions of the iterative MMSE (IMMSE) filter have been introduced recently [20, 21, 22, 23, 24, 25, 26]. Based on the MMSE principle, it has been demonstrated that the filtered pixels and their variances are linearly related. Then, a linear regression of means and variances for different window sizes is applied to estimate the infinite number of looks prediction (INLP) filtered pixels [27, 28, 29, 30]. In this chapter, the improved MMSE-based Lee sigma, the IMMSE, and the INLP denoising techniques are studied.

This paper is organized as follows: Section 2 reviews the classical MMSE-based denoising technique and presents the updated versions, i.e., INLP and IMMSE techniques. The results are shown in Section 3. Finally, Section 4 presents the conclusions of this paper.

2. MMSE-based filters: classical version

The intensity pixel y(i) of a SAR image is affected by a multiplicative noise [2]

$$y\left(i\right)=x\left(i\right)\nu \left(i\right),$$E1

x(i) is the noise-free pixel and ν(i) is the speckle noise with unit mean and standard deviation ${\sigma }_{\nu }$. It is assumed that x(i) and ν(i) are statistically independent. In the rest of the chapter, the index (i) will be omitted. Let $\widehat{x}$ and $\overline{x}$ be the estimated and the a priori mean of x, respectively.

From (1) we have

$$E\left(y\right)=E\left(x\nu \right)=E\left(x\right)\mathrm{E}\left(\nu \right)=E\left(x\right)$$E2

where E() is a mathematical expectation (i.e., statistical mean). By exploiting the ergodicity of the SAR data, the statistical mean is substituted by the spatial mean, i.e.,

$$\overline{y}=\mathrm{E}\left(\mathrm{y}\right)=\mathrm{E}\left(\mathrm{x}\right)=\overline{x}$$E3

Nevertheless, it has been demonstrated recently that the statistical and spatial averaging statistics are quite different [27] since in the spatial averaging process the processing windows are overlapping, and a spatial correlation is introduced. Hence, replacing the statistical mean with the spatial one should be taken with caution.

The MMSE filter is assumed to be a linear combination of x and $\overline{x}$ [1]

$$\widehat{x}=a\overline{x}+\mathit{by}$$E4

The parameters a and b are selected optimally to minimize the MSE

$$I=E\left[{\left(x‐\stackrel{⌢}{x}\right)}^2\right]$$E5

then

$$\frac{\partial I}{\partial a}=0$$E6

and

$$\frac{\partial I}{\partial b}=0$$E7

This minimization leads to [1]

$$a=1-b$$E8

and

$$b=\frac{var\left(x\right)}{var\left(y\right)}$$E9

For a given X and Y two independent random variables, we have

$$var.\left(\mathrm{XY}\right)=\mathrm{E}{\left(\mathrm{Y}\right)}^{2}var\left(\mathrm{X}\right)+var.\left(\mathrm{Y}\right)var.\left(\mathrm{X}\right)+var.\left(\mathrm{Y}\right)\mathrm{E}{\left(\mathrm{X}\right)}^2$$E10

Then

$$var\left(\mathrm{y}\right)=var\left(\mathrm{x}\nu \right)={\overline{x}}^2{\sigma }_v^2+var\left(\mathrm{x}\right){\sigma }_v^2+{\overline{v}}^{2}var\left(x\right)$$E11

Eq. (2) gives

$$\overline{y}=\overline{x}$$E12

then

$$var\left(x\right)=\frac{var\left(y\right)-{\overline{y}}^2{\sigma }_{\nu }^2}{1+{\sigma }_{\nu }^2}$$E13

finally

$$\widehat{x}=\overline{y}+b\left(y-\overline{y}\right)$$E14

$$b=\frac{var\left(y\right)-{\overline{y}}^2{\sigma }_{\nu }^2}{var\left(y\right)\left(1+{\sigma }_{\nu }^2\right)}$$E15

var(y) and $\overline{y}$ are estimated using a moving window W. ${\sigma }_{\nu }^2$ is assumed to be a constant (i.e., ${\sigma }_{\nu }^2=1$ for single look SAR data).

2.1 Lee sigma filter

The Lee sigma filter has been implemented in several geographic information system (GIS) software due to its effectiveness in speckle reduction, its simplicity, and its computational efficiency. However, in amplitude and intensity SAR data, the probability density functions (pdf) are not symmetrical, because they follow the Rayleigh and the negative exponential distributions, respectively. This asymmetry produces biased estimates since the original sigma range was derived based on Gaussian distribution. Hence, to remove the bias and to preserve the mean value, the sigma ranges were recomputed based on the corresponding pdf. The sigma ranges of amplitude and intensity SAR data are given in [18]. In [28], the performance of the improved Lee sigma filter is revised.

Practical implementation

1. Define a square window W

2. Define ${\sigma }_{\nu }^2$ (${\sigma }_{\nu }^2=1/\mathrm{N}$ where is the Initial number of looks (N = 1 in our study))

3. Compute the statistics of the pixel ($\overline{y}$ and var(y))

4. Compute the parameter b using (15)

5. Compute the filtered pixel $\widehat{x}$ using (14).

3. MMSE-based filters: New insights

3.1 The IMMSE filter

The objective of SAR speckle filtering is:

• Reduce speckle noise in the homogeneous areas (i.e., averaging all pixels $\widehat{x}=\overline{y}$).

• Maintain the spatial details (i.e., $\widehat{x}=y$).

Hence in general cases, we have.

$$\widehat{x}\in \left[y,\overline{y}\right]\text{or}\widehat{x}\in \left[\overline{y},y\right]$$E16

The principle of the IMMSE filter is to scan the range of $\widehat{x}$ in $\left[y,\overline{y}\right]$ by the following iterative procedure.

$${\widehat{x}}_0=\overline{y},$$E17

$${\widehat{x}}_{k+1}={\widehat{x}}_k+b_k^{\text{'}}\left(y‐{\widehat{x}}_k\right).$$E18

If $0<b_k^{\text{'}}<1$, then ${\widehat{x}}_{\infty }=y$ and ${\widehat{x}}_k\in \left[\overline{y},y\right]$.

The performance of the IMMSE denoising technique is the function of:

• The choice of the initial filtered image ${\widehat{x}}_0$: The initial filtered image ${\widehat{x}}_0$ must ensure a high speckle reduction level. In [26], the boxcar filter was selected as an initial filter. However, it has been demonstrated that the use of a more sophisticated filter ensured better filtering performance [20, 21, 22, 23, 24, 25].

• The choice of the parameter $b_k^{\text{'}}$. In fact, this parameter is a tuning factor that controls the performance of the filtering process as the parameter b in (15). To ensure robust speckle denoising, this parameter must satisfy three important properties:

  1. 0 ≤ b′_k ≤ 1.

  2. $b_k^{\text{'}}\approx 0$ in homogeneous areas,

  3. $b_k^{\text{'}}\approx 1$ in heterogeneous areas.

Hence, by implementing N iterations (N is sufficiently low), the denoising procedure maintained the filtered homogeneous areas (i. e.${\widehat{x}}_N\approx {\widehat{x}}_0$ since $b_k^{\text{'}}\approx 0$) and preserved spatial details (i. e.${\widehat{x}}_N=y$ since $b_k^{\text{'}}\approx 1$).

By the analogy of the MMSE expression of the parameter b (15), the parameter b’ has been expressed as

$$b_k^{\text{'}}=\frac{var\left({\widehat{x}}_k\right)}{\left(\left(1+{\sigma }_{\nu }^2\right)var\left({\widehat{x}}_k\right)+{{\widehat{x}}_k}^2{\sigma }_{\nu }^2\right)}$$E19

In [21], the authors proposed a more sophisticated version expressed as

$$b_k^{\text{'}}=tanh\left(\frac{{\mathrm{CV}}_{{\widehat{x}}_k}^2{\mathrm{CV}}_y^2}{\mathrm{C}}\right)$$E20

$$\mathrm{C}=\frac{1}{{\left({\mathit{ENL}}_0\right)}^2},$$E21

where CV is the coefficient variation and ${\mathit{ENL}}_0$ is the equivalent number of looks of the original image y estimated in a homogenous area,

$$\mathit{CV}\left(y\right)=\frac{\mathit{std}\left(y\right)}{\overline{y}}$$E22

$${\mathit{ENL}}_0=\frac{{\left(\overline{y}\right)}^2}{var\left(y\right)}$$E23

where std is the standard deviation.

3.2 Practical implementation

For a given SAR image y

1. Compute the parameter C in (21).

2. Compute ${\widehat{x}}_0$ image by applying a filter ensuring high speckle reduction level.

3. For a given pixel,

4. From the selected pixels of the filtered image ${\widehat{x}}_0$, compute ${\mathrm{CV}}_{\widehat{x}}$ [21].

5. From the selected pixel of the original image y, compute ${\mathrm{CV}}_y$ using the same process in iii.

6. Compute $b_k^{\text{'}}$ using (20).

7. Update the filtered pixel using (18).

8. Apply the process for all pixels of the image.

Repeat iii to vii K iterations. K is a tuning parameter to control the speckle reduction and spatial detail preservation.

3.3 The INLP filter

The INLP is based on the statistics of the SAR intensity (i.e., multiplicative noise model (1) and the MMSE expression (14). In [26, 29, 30, 31, 32], it has been demonstrated that.

$$\widehat{x}=avar\left(\widehat{x}\right)+d$$E24

where

$$a=\left(y-x\right)/var\left(x\right),$$E25

and

$$d=x.$$E26

Eq. (24) shows that the filtered pixel $\widehat{x}$ is linearly related to its variance $var\left(\widehat{x}\right)$. This rule is applied to estimate the INLP-filtered pixel (i.e., the parameter d or the noise-free pixel x). In the extended homogeneous area, the MMSE filtered pixel is $\widehat{x}=\overline{x}\approx x$ while in the INLP filter $\widehat{x}=d\approx x$ where d is estimated using a linear regression between means and their variances and not using a simple mean (i.e., $\widehat{x}=\overline{x}\approx x$) as in the original MMSE denoising technique.

3.4 Practical implementation

For each pixel of the image:

1. Define a window W having N samples.

2. Uniformly select N_i samples from W, where N_i = N_min,…, N. N_min is the smallest number of samples.

3. Apply the original filter to all sets of N_i samples. We obtain $\widehat{X}=\left({\widehat{x}}_N,{\widehat{x}}_{N-1},\dots ,{\widehat{x}}_{{\mathrm{N}}_{\min }}\right)$.

4. Repeat steps ii and iii L times to obtain sufficient samples.

5. For each pixel of the filtered image, compute the vector $var\widehat{X}=\left(var\left({\widehat{x}}_M\right),var\left({\widehat{x}}_{M-1}\right),.\dots ,var\left({\widehat{x}}_1\right)\right)$ using the window W.

6. For each pixel of the image, perform a linear regression between $var\widehat{X}=\left(var\left({\widehat{x}}_M\right),var\left({\widehat{x}}_{M-1}\right),.\dots ,var\left({\widehat{x}}_1\right)\right)$, $\widehat{X}=\left({\widehat{x}}_N,{\widehat{x}}_{N-1},\dots ,{\widehat{x}}_{{\mathrm{N}}_{\min }}\right)$ and compute the filtered value (i.e., the constant d in (24)).

4. Results

EPD-ROATo assesses the performance of the studied denoising techniques, airborne and spaceborne SAR images were used (see Figure 1a). For the spaceborne SAR data, the Sentinel 1 C-band vv SAR image of Dubai UAE is considered. The airborne SAR is the hh image of Les-Landes site, France acquired by NASA JPL AIRSAR sensor (see Figure 2a)).

Figure 1.

(a) Original spaceborne SAR image, (b) Boxcar filter, (c) MMSE filter (i. e. improved sigma filter [18], (d) IMMSE filter [21].

Figure 2.

(a) Original airborne SAR image, (b) Boxcar filter, (c) MMSE filter (i. e. improved sigma filter [18], (d) INLP filter [31].

4.1 Evaluation criteria

In addition to visual inspections, quantitative parameters have been employed to assess the performance of the studied denoising techniques. The ENL was employed to evaluate speckle reduction level

$$\mathit{ENL}\left(i\right)=\frac{{\left(\overline{\widehat{x}}\left(i\right)\right)}^2}{var\left(\widehat{x}\left(i\right)\right)}.$$E27

The edge preservation degree based on the ratio of averages (EPD-ROA) [33] is used to assess the preservation of spatial details. The EPD-ROA in horizontal direction is:

$$\mathit{EPD}-{\mathit{ROA}}_H\left(i\right)=\frac{{\sum }_{m,n}\left|\widehat{x}\left(m,n\right)/\widehat{x}\left(m,n+1\right)\right|}{{\sum }_{m,n}\left|y\left(m,n\right)/y\left(m,n+1\right)\right|},$$E28

where m and n are the xy coordinates of the pixel in the selected zone, respectively. EPD-ROA_V is calculated by replacing in (28) the indexes (m,n + 1) by (m + 1,n). In general cases, EPD-ROA < 1. High EPD-ROA means a high ability for spatial detail preservation.

4.2 MMSE vs IMMSE

Figure 1 displays the filtered denoised spaceborne SAR images using the boxcar filter (i.e., mean filter), the MMSE (improved Lee filter), and the IMMSE filters. It can be observed that the boxcar filter reduced the speckle noise but blurred spatial details. The MMSE filter improved the filtering performance. The IMMSE maintained the high speckle reduction of the initial filter and enhanced considerably the spatial details such as lines (see rectangles) and points (see arrows). It can be seen that the IMMSE outperforms the MMSE-based filter in terms of speckle reduction and spatial detail preservation. Quantitative results in Table 1 confirmed visual interpretations where the IMMSE filter maintained the high speckle reduction level of the initially applied filter and enhanced spatial details. Quantitative results show also that the MMSE filter outperformed the boxcar filter in terms of speckle reduction and spatial detail preservation. The IMMSE gave better filtering results than the MMSE filter in terms of speckle reduction (ENL_IMMSE(1124) > ENL_MMSE(410)) and spatial detail preservation (EPD_IMMSE(0.98) > EPD_MMSE(0.94)).

	ENL	EPDH	EPDV
Boxcar 9$\times$9	296	0.94	0.94
MMSE 11$\times$11	410	0.94	0.94
IMMSE	1124	0.98	0.98

Table 1.

Performances of the filters using spaceborne data.

4.3 MMSE vs INLP

Figure 2 displays the filtered airborne images using the studied filters. It can be observed visually that the MMSE filter outperformed the boxcar filter. The INLP reduced the blurring effects introduced by the MMSE filter. This can be easily seen in lines (see rectangles). Concerning the speckle reduction, it is observed that the INLP filter ensured better filtering performance. In fact, the homogeneous areas appear smoother than the ones ensured by the MMSE filter (see circles). These results are recorded quantitatively in Table 2 from which it is observed that the INLP filter outperformed the MMSE filter in terms of speckle reduction (ENL_INLP(10) > ENL_MMSE(9)) and spatial detail preservation (EPD_INLP(0.85) > ENL_MMSE(0.84)).

	ENL	EPDH	EPDV
Boxcar 5$\times$5	10	0.83	0.83
MMSE 5$\times$5	9	0.84	0.84
INLP	10	0.85	0.85

Table 2.

Performances of the filters using airborne data.

5. Conclusion

In this chapter, the authors reviewed the use of the MMSE-based speckle denoising techniques in SAR images. It has been shown that the MMSE-based filters (i.e., the improved Lee sigma filter) ensured high speckle denoising performance. Based on the MMSE principle, two improved MMSE versions have been introduced recently in the literature, i.e., the IMMSE and the INLP filters. The results showed that when the IMMSE is initialized with an image ensuring high speckle reduction, it ensures better denoising performance than the classical MMSE-based filters in terms of speckle filtering and spatial detail preservation. In the INLP filter, unlike the MMSE-based filters which estimated the noise-free pixels using spatial means, linear regressions between the filtered pixels and their variances for different window sizes are applied. Results show that this new strategy increased the filtering performance. Future researches will focus on the extension of the IMMSE and the INLP on additive image denoising.

Nomenclature

b	tuning factor of the MMSE filter
b’	tuning factor of the IMMSE filter
C	a normalizing factor
CV	the coefficient variation
CVx̂	the coefficient variation of x̂k
CVy	the coefficient variation of y
d	the INLP filtered pixel
E()	mathematical expectation (i. e. statistical mean).
ENL0	the equivalent number of looks of the original image y
EPD-ROA	the edge preservation degree based on the ratio of averages
I	mean square error
K	number of iterations.
N	the Initial number of looks
Ni	number samples selected from W
std	the standard deviation
ν	the speckle noise
W	square window
y	the intensity pixel
y¯	spatial mean of y
x	the noise-free pixel
x¯	spatial mean of x
x̂0	the initial filtered image
x̂	filtered image
σν	standard deviation of the speckle.