Adaptively weighted nonlocal means and TV minimization for speckle reduction in SAR images

Abstract

Speckle noise reduction is an important issue in synthetic aperture radar (SAR) imaging. Because SAR images are distinct in being complex valued and susceptible to corruption owing to multiplicative fluctuations, specialized methods for speckle reduction are needed. Techniques based on nonlocal means perform denoising by exploiting the natural redundancy of patterns within an image. They calculate a weighted average of pixels whose neighborhoods are close to one another, where this significantly reduces noise while preserving most image content. While this method performs well on flat areas and textures, its results are excessively smooth in low-contrast areas, and leave residual noise around edges and singular structures. Another variational denoising method uses total variation (TV) minimization to restore regular images but is prone to excessively smooth textures, the staircasing effect, and contrast losses. Our proposed model is intended for the logarithmic domain of SAR data, and combines the above two methods by minimizing an adaptive TV using a nonlocal data fidelity term. In the variational functionals developed here, weighted parameters of nonlocal regularization are adaptively tuned based on local heterogeneity information and noise in the images. A fast iterative shrinkage/thresholding algorithm (FISTA) is then used to solve the optimization problem. The results of experiments on real SAR images verify the effectiveness of the proposed method in terms of speckle reduction.

1 Introduction

Synthetic aperture radar (SAR) imaging is a widely used technique for observations of the Earth. It offers images of the same area in different temporal intervals and allows for the continuous monitoring of the Earth’s surface. However, a combination of several radar echoes within each resolution cell results in interference called the speckle phenomenon [10]. The direct estimation of the reflectivity, interferometric phase, and polarimetric properties is impractical given its high intensity of noise and prohibitively large variance. Therefore, speckle suppression through such techniques as feature extraction and target identification is important for applications of SAR images.

Some of the early techniques for speckle reduction [1, 25] use the homomorphic approach. They calculate the log of the data to obtain a tractable additive model and apply a method to eliminate additive white Gaussian noise (AWGN). Speckle statistics depart from the AGWN model widely used for processing optical images, because of which a more sophisticated denoising approach is needed to reduce speckle noise while preserving features of the image.

A number of despeckling methods based on the ideas of Wavelets have been proposed and have shown good performances. I. Erer et al. proposed the fast local SAR image despeckling by edge-avoiding wavelets [11], which provide the decomposition of the input image into several detail layers and a residual image. The performance of the method in terms of both speckle reduction and edge preservation ability. Tang et al. proposed skeletonization of ribbon-like shapes based on a new wavelet function [22], the scheme is capable of extracting the skeleton of the Ribbon-like shape with different width as well as different gray-levels. Liu et al. proposed an multi-scale correlated wavelet approach [14], which aims not only to significantly increase the perceptual visibility, but also to preserve more texture details and reduce the noise effect as well.

In recent years, variational methods have been widely studied to remove speckles in SAR images without smearing sharp edges. Rudin et al. proposed the first variational model [20] to reduce multiplicative Gaussian noise. Aubert and Aujol introduced a variational model [2] designed to suppress multiplicative gamma noise. This method can be adapted to denoising while preserving edges but has three major drawbacks: The textures tend to be excessively smooth, flat areas are approximated by a piecewise constant surface resulting in the staircasing effect, and the image suffers from losses of contrast. One solution to these problems is to spatially balance the regularization. Owing to the non-Gaussian distribution of the intensity of SAR images, the data-fitting term differs from the usual sum of squared differences that arises from a Gaussian assumption. Several regularization terms have been considered in the literature, e.g., the dictionary learning-based variational model [13], and the mth root transformed variational model [26].

Buade et al. proposed a nonlocal approach [4] about AWGN denoising. The basic idea is to take advantage of the self-similarity found in natural images. This similarity is measured by the Euclidean distance between patches surrounding the selected and the target pixels, which is extended for SAR image despeckling [6, 24, 27]. Several recent papers have focused on fast despeckling techniques for SAR images. A statistical approach has been introduced using a probabilistic patch-based (PPB) filter [8] that considers pixel selection as a detection problem and builds a statistical test to perform selection. Once similar pixels have been detected, denoising is performed by (weighted) maximum likelihood estimation. This idea has been recently extended, following the principles of the successful BM3D approach, with SAR-BM3D [18] that combines the nonlocal approach with several powerful tools, such as wavelet transforms, transform-domain shrinkage, and Wiener filtering, to deliver impressive performance at the cost of longer processing time. Cozzolino et al. proposed a fast adaptive nonlocal method for SAR despeckling (FANS) [7] that uses an activity-driven variable-size search area that is suited to time-critical applications. This method achieves excellent performance and has low complexity. Nonlocal methods achieve good performance on the whole but may select too many irrelevant candidates, resulting in structures in these areas that have been excessively smoothed. On the contrary, around singular structures or edges, it can be challenging to find a sufficient number of similar patches such that the pixels are not properly denoised, resulting in residual noise.

Variational and nonlocal methods have been used together in the literature as “nonlocal” regularization terms [19, 21]. In [21], the authors combined both nonlocal means and TV minimization to reduce the defaults observed in each method, in particular the excessively smooth residual noise originating from nonlocal means and the staircasing effect, and the loss of contrast due to the TV minimization. Nie et al. proposed a nonlocal TV-based variational method [17] for PolSAR data speckle reduction. The method is obtained based on the Wishart fidelity term and the NLTV regularization, which can effectively reduce speckle noise and, better preserve the details and the repetitive structures.

In this paper, we focus on speckle noise deduction in SAR images by combining variational methods and nonlocal regularization. The main motivation behind the nonlocal regularization models proposed by Sutour et al. [21] and Ma et al [15] Our proposed model is derived for the logarithmic domain of SAR data, the weighted nonlocal regularization parameters are adaptively tuned based on local heterogeneity-related information and noise in the images. For the rapid implementation of the proposed model, we use the fast iterative shrinkage/thresholding algorithm (FISTA) [3].

This paper is organized as follows: In Section 2, we briefly review the statistical properties of SAR images, the related variational and nonlocal means models. In Section 3, the model for speckle reduction in SAR images is proposed. The results of experiments on real SAR images are presented in Section 4, and the conclusions of this paper are given in Section 5.

2 Related works

2.1 Statistics of SAR images

To measure similarity among pixels in the original SAR data domain, we use the method proposed by Deledalle et al [10] Speckle is signal dependent because the intensity of fluctuations varies with the underlying information. Given a positive integer L ∈ ℕ^∗, called the shape parameter, that acts according to the noise level, speckle usually follows a gamma distribution modeled by the following pdf:

$$ {p}_I\left(I|R\right)=\frac{L^L{I}^{L-1}}{\varGamma (L){R}^L}\exp \left(-L\frac{I}{R}\right) $$

(1)

where I ∈ ℝ₊is the observed intensity, R ∈ ℝ₊is the underlying reflectivity modeling the average variations in punctual back-scatters, L > 0 is the number of looks, and Γ is the gamma function. Note that we denote by ℝ₊ the set of positive real values. Intensity I can be decomposed as a product of reflectivity R and speckle component S distributed under a standard gamma distribution (S ∼ G(1; L)):I = R × S, $E\left[I\right]=R\ \mathrm{and}\ \mathrm{Var}\left[I\right]=\frac{R^2}{L}$. Figure 1a gives an illustration of gamma distributions. The relation Var[I] ∝ E[I]² indicates heteroscedastic noise that exhibits multiplicative behavior. When L = 1, the gamma distribution boils down to the exponential distribution known to have a heavy right tail and left head. Thus, the resulting image appears dark with very bright pixels.

Fig. 1

Two distributions modeling speckle noise: a The gamma distribution describes the speckle observed in intensity images. b The Fisher-Tippett distribution describes speckle observed in the logarithm images

Speckle in SAR images is modeled as multiplicative random noise, whereas most available filtering algorithms have been developed for AWGN in the context of image denoising and restoration. To take advantage of these algorithms, one way is to use a logarithmic transform y = log I ∈ ℝ. In this case, the resulting distribution is called a Fish-Tippett distribution [10] defined by:

$$ {p}_y\left(y|x\right)=\frac{L^L}{\varGamma (L)}{e}^{L\left(y-x\right)}\exp \left(-L{e}^{y-x}\right) $$

(2)

where x = log R ∈ ℝ. Figure 1b gives an illustration of the Fisher–Tippett distribution. The log-transform of signal-dependent multiplicative noise results in signal-independent additive noise. The Fisher-Tippet has a heavy left tail, because of which the resulting image has several dark pixels.

2.2 Variational denoising models

Both the variational and the Bayesian maximum a posteriori (MAP) formulations of image denoising lead to optimization problems with two terms: a data fidelity term (log-likelihood) and a regularization term. Variational methods consist of searching for an image that minimizes a given energy to fit the data while respecting some smoothness constraints. The restored image is obtained by minimizing the following energy function:

$$ {u}^{\mathrm{TV}}=\underset{u\in {\mathbb{R}}^N}{\mathrm{argmin}}-\sum \limits_{i\in \varOmega}\log p\left({g}_i|{u}_i\right)+\lambda \mathrm{TV}(u) $$

(3)

where p(g_i| u_i) is the conditional likelihood of the true pixel value u_i given the observation of the noisy value g_i. The term $-\sum \limits_{i\in \varOmega}\log p\left({g}_i|{u}_i\right)$ is a data fidelity term, $\mathrm{TV}(u)=\sum \limits_{i\in \varOmega}\left\Vert {\left(\nabla u\right)}_i\right\Vert$ is a regularization term, and λ > 0 is the parameter that sets the compromise between data fidelity and smoothness.

Minimizing the total variation forces the solution to be piecewise regular, which renders it well adapted to denoising while preserving edges. However, a compromise needs to be found between regularity in flat areas and the preservation of textures, based on the choice of parameter λ.This trade-off makes the choice of λ challenging and strongly image dependent [21].

2.3 Nonlocal means models

The nonlocal means algorithm compares patches using small windows extracted around each pixel to average those whose surroundings are similar. Weights are computed to reflect the extent to which two noisy pixels are likely to represent the same true gray level. For each pixel i ∈ Ω, the solution of the nonlocal means is given by the following weighted maximum likelihood estimation minimization problem:

$$ {u}^{\mathrm{NL}}=\underset{u\in {\mathbb{R}}^N}{\mathrm{argmin}}-\sum \limits_{i\in \varOmega}\sum \limits_{j\in \varOmega }{w}_{i,j}\log p\left({g}_j|{u}_i\right) $$

(4)

and the distance between noisy patches g₁ and g₂ can be adapted using the following likelihood ratio [21]:

$$ d\left({g}_1,{g}_2\right)=-\log \frac{\sup_up\left({g}_1|{u}_1=u\right)p\left({g}_2|{u}_2=u\right)}{\sup_up\left({g}_1|{u}_1=u\right){\sup}_up\left({g}_2|{u}_2=u\right)} $$

(5)

where u₁ and u₂ refer to the underlying noise-free patches. In (4) the weights w_{i, j} ≥ 0 are computed to select pixels j whose surrounding patches are similar to that extracted around the pixel of interest i as:

$$ {w}_{i,j}=\varphi \left[-\frac{d\left(g\left({P}_i\right),g\left({P}_j\right)\right)}{2\mid P\mid {h}^2}\right] $$

(6)

where φ is a kernel decay function and d a distance function that measures the similarity between patches P_i and P_j of size ∣P∣, extracted around pixels i and j, respectively.

Nonlocal means methods achieve good performance overall, but suffer from two drawbacks [21]. First, the algorithm can select too many irrelevant candidates, resulting in structures in these areas that are overly smoothed due to a combination of candidates with different underlying values. Second, around singular structures or edges, it can be difficult to find a sufficient number of similar patches such that the pixels are not properly denoised, resulting in residual noise. These two problems are conflicting, and are controlled by the filtering parameter h that acts as a trade-off between bias and variance.

2.4 The proposed model

Under the hypothesis of a fully developed speckle (Goodman model [12]), SAR images are typically analyzed by the multiplicative model, which can be expressed as

$$ g=u\cdot n $$

(7)

where g is the observed noisy image, u is the noise-free reflection, and n is speckle noise. It is well known that the speckle of a multi-look SAR intensity image can be modeled by a gamma distribution. In the case of L-look, fully developed speckle noise, $p(n)=\frac{L^L}{\varGamma (L)}{n}^{L-1}{e}^{- nL}$ is the gamma density. The scattering vector g is distributed according to the likelihood law given by:

$$ p\left(g|u\right)=\frac{L^L{g}^{L-1}}{\varGamma (L){u}^L}\exp \left(-\frac{Lg}{u}\right) $$

(8)

When dealing with this kind of multiplicative noise, we consider both variational and nonlocal means models. Our proposed model combines nonlocal means and TV minimization to reduce defaults observed in each method, in particular the excessively smoothed and residual noise originating from nonlocal means, and the staircasing effect and loss of contrast due to TV minimization.

2.5 Data fidelity term in adaptively weighted nonlocal means (AWNL)

We focus on gamma multiplicative noise, which models speckle fluctuations encountered in SAR images. The statistical model defined by Eq. (2) that leads to the negative log-likelihood of u > 0 for an observed intensity g > 0 is given by:

$$ -\log \left(g|u\right)=\frac{Lg}{u}+L\log u+\log \varGamma (L)-L\log L-\left(L-1\right)\log g $$

(9)

where L is the “number of looks” that sets the level of noise, and Γ is the gamma function.

A peculiarity of SAR images is their very high dynamic range. Bright targets have intensities that are several orders of magnitude larger than their background. Although a patch in the nonlocal means method containing such a bright target at pixel i is dissimilar to one with the background only, and the corresponding weights w(i, j) are very low, the weighted mean given in (6) excessively smooths the bright target. To avoid these problems, we choose to replace the log-likelihood term of the nonlocal means. This term considers not only the scattering vector g_i, but also all g_j for j spanning all pixel indices of an extended neighborhood N_i centered on pixel i, as

$$ {D}_i=-\sum \limits_{i\in \varOmega}\sum \limits_{j\in \varOmega }{w}_{i,j}\log p\left({g}_j|{u}_i\right) $$

(10)

with a weight w_{i, j} given to g_j for estimation at pixel i. Such weights are typically chosen in a data-driven way to select only samples relevant to the subsequent estimation. To avoid excessive smoothing, residual noise, and the staircasing effect originating from nonlocal means, we derive the data fidelity term for adaptively weighted nonlocal means.

We assume that in the nonlocal neighborhood of pixel i, observations g_i are all realizations of the random variable g_i = f_i + ε_i, where f_i and ε_i are independent random variables. The quantity f_i models signal fluctuations and ε_i noise fluctuations. We assume that the signal f_i has a mean ${u}_i^{\mathrm{NL}}$ and a standard deviation ${\sigma}_i^{\mathrm{signal}}$, and that ε_i has a zero mean with a known standard deviation ${\sigma}_i^{\mathrm{noise}}$. A significant value ${\sigma}_i^{\mathrm{signal}}$ is an indicator of small rapid variation; it assesses whether observations in the nonlocal neighborhood of i belong to different populations.

Under the inspiration of the local linear minimum mean square estimator (LLMMSE) strategy [9], we locally perform a convex combination of nonlocal estimation u^NL and noisy data g, according to the following formula:

$$ {u}_i^{\mathrm{AWNL}}=\left(1-{\alpha}_i\right){u}_i^{\mathrm{NL}}+{\alpha}_i{g}_i $$

(11)

where α_i is a confidence index defined by:

$$ {\alpha}_i=\frac{{\left({\sigma}_i^{\mathrm{signal}}\right)}^2}{{\left({\sigma}_i^{\mathrm{signal}}\right)}^2+{\left({\sigma}_i^{\mathrm{noise}}\right)}^2}\approx \frac{\mid {\left({\hat{\sigma}}_i^{\mathrm{NL}}\right)}^2-{\left({\sigma}_i^{\mathrm{noise}}\right)}^2\mid }{\mid {\left({\hat{\sigma}}_i^{\mathrm{NL}}\right)}^2-{\left({\sigma}_i^{\mathrm{noise}}\right)}^2\mid +{\left({\sigma}_i^{\mathrm{noise}}\right)}^2} $$

(12)

Because gamma corruption data are signal dependent, but with a standard deviation proportional to the expectation, the expected nonlocal variance is chosen as ${\left({\sigma}_i^{\mathrm{noise}}\right)}^2={\left({u}_i^{\mathrm{NL}}\right)}^2/L$. Thus, $\mathrm{Var}\left[{g}_i\right]={\left({\sigma}_i^{\mathrm{signal}}\right)}^2+{\left({\sigma}_i^{\mathrm{noise}}\right)}^2$. The signal variance Var[g_i] can be estimated directly from data g_j located in the nonlocal neighborhood using the following formula:

$$ {\left({\hat{\sigma}}_i^{\mathrm{NL}}\right)}^2=\sum \limits_{j\in \varOmega }{w}_{i,j}^{\mathrm{NL}}{g}_j^2-{\left(\sum \limits_{j\in \varOmega }{w}_{i,j}^{\mathrm{NL}}{g}_j\right)}^2 $$

(13)

Pixels in the same nonlocal neighborhood should belong to the same population. Therefore, the estimated standard deviation ${\hat{\sigma}}_i^{\mathrm{NL}}$ should be close to that of noise ${\sigma}_i^{\mathrm{noise}}$. Thus, if the estimated standard deviation ${\hat{\sigma}}_i^{\mathrm{NL}}$ is close to the expected ${\sigma}_i^{\mathrm{noise}}$, the index α_i is close to zero. According to (11), nonlocal estimation ${u}_i^{\mathrm{NL}}$ is unchanged. On the contrary, if the nonlocal variance is found to be far from the expected value, α is closer to one. This results in the re-injection of some noise to balance the bias–variance trade-off.

Note that the solution ${u}_i^{\mathrm{AWNL}}$ can be rewritten as the following weighted sum:

$$ {u}_i^{\mathrm{AWNL}}=\sum \limits_{j\in \varOmega }{w}_{i,j}^{\mathrm{AWNL}}{g}_j $$

(14)

where ${w}_{i,j}^{\mathrm{AWNL}}=\left(1-{\alpha}_i\right){w}_{i,j}^{\mathrm{NL}}+{\alpha}_i{\delta}_{i,j}$, δ_{i, j} = 1 if i = j, and is zero otherwise. The residual variance of the despeckling solution u^AWNL can be approached at pixel i by:

$$ {\left({\hat{\sigma}}_i^{\mathrm{residual}}\right)}^2=\left[\sum \limits_{j\in \varOmega }{\left({w}_{i,j}^{\mathrm{AWNL}}\right)}^2\right]{\left({\sigma}_i^{\mathrm{noise}}\right)}^2 $$

(15)

The quantity ${\hat{\sigma}}_i^{\mathrm{residual}}$ is an indicator of the level of residual variance at pixel i obtained following despeckling, i.e., once a bias–variance trade-off has been achieved. This indicator can assess the quality of the denoising: it accounts for both bias and variance because bias in u^NL leads to residual variance in u^AWNL.

We define the adaptive weighted nonlocal means estimator u^AWNL as the variance matrix that minimizes D_i:

$$ {\displaystyle \begin{array}{c}{u}^{\mathrm{AWNL}}=\underset{u\in {\mathbb{R}}^N}{\mathrm{argmin}}-\sum \limits_{i\in \varOmega}\sum \limits_{j\in \varOmega }{w}_{i,j}\log p\left({g}_j|{u}_i\right)\\ {}=\underset{u\in {\mathbb{R}}^N}{\mathrm{argmin}}-\sum \limits_{i\in \varOmega}\left(\sum \limits_{j\in \varOmega }{w}_{i,j}\right)\log p\left({g}_j|{u}_i\right)\\ {}=\underset{u\in {\mathbb{R}}^N}{\mathrm{argmin}}\sum \limits_{i\in \varOmega}\left(\sum \limits_{j\in \varOmega }{w}_{i,j}\right)\left[\begin{array}{l}\frac{L{g}_j}{u_i}+L\log {u}_i+\log \varGamma (L)\\ {}-L\log L-\left(L-1\right)\log {g}_j\end{array}\right]\\ {}=\underset{u\in {\mathbb{R}}^N}{\mathrm{argmin}}\sum \limits_{i\in \varOmega }{\tau}_i\left[L\log \left({u}_i\right)+L\frac{u_i^{\mathrm{NL}}}{u_i}\right]\end{array}} $$

(16)

where τ_i > 0 are spatially varying regularization parameters, with nonlocal weights ${w}_{i,j}^{\mathrm{AWNL}}$. Parameter τ_i can be chosen as a (non-negative) decreasing function of ${\hat{\sigma}}_i^{\mathrm{residual}}$. The choice of this regularization function is critical to ensure restoration without excessively smoothing the edges and textures. A simple dimensional analysis indicates that τ_i should be chosen as inversely proportional to ${\hat{\sigma}}_i^{\mathrm{residual}}$. More precisely, we find that τ_i should be chosen proportional to our approximation of the amount of noise reduction, i.e.,

$$ {\tau}_i=\gamma {\left(\frac{{\hat{\sigma}}_i^{\mathrm{residual}}}{\sigma_i^{\mathrm{noise}}}\right)}^{-1}=\gamma {\left(\sum \limits_{j\in \varOmega }{w}_{i,j}^2\right)}^{-1/2} $$

(17)

where γ > 0 is a fixed parameter that sets the strength of adaptive regularization, which influential only locally where the nonlocal means cannot significantly reduce noise.

2.6 Prior term

In urban areas at meter resolutions, height is typically constant from one pixel to a neighboring pixel, or varies strongly when two pixels belong to different structures, e.g., the ground and the roof. We therefore select a prior term that favors piecewise constant images: this is TV regularization defined by

$$ \sum \limits_{i\in \varOmega}\left\Vert {\left({\nabla}_wu\right)}_i\right\Vert $$

(18)

with

$$ \sum \limits_{i\in \varOmega}\left\Vert {\left({\nabla}_wu\right)}_i\right\Vert =\sum \limits_{i\in \varOmega}\sqrt{\sum \limits_{j\in \varOmega }{\left({u}_i-{u}_j\right)}^2{w}_{i,j}} $$

2.7 The proposed energy model

We propose a SAR speckle reduction model that combines nonlocal means with TV minimization to reduce defaults observed in each method, in particular excessively smooth residual noise and the staircasing effect originating in nonlocal means, and the loss of contrast due to TV minimization. We perform TV regularization with a nonlocal data fidelity term as follows:

$$ {\displaystyle \begin{array}{c}E(u)=\sum \limits_{i\in \varOmega }{D}_i\left({u}_i\right)+\lambda \sum \limits_{i\in \varOmega}\left\Vert {\left({\nabla}_wu\right)}_i\right\Vert \\ {}=\sum \limits_{i\in \varOmega }{\tau}_i\left[L\log \left({u}_i\right)+L\frac{u_i^{\mathrm{NL}}}{u_i}\right]+\lambda \sum \limits_{i\in \varOmega}\left\Vert {\left({\nabla}_wu\right)}_i\right\Vert \end{array}} $$

(19)

where λ is a hyperparameter that balances the relative importance of the fidelity to the observation as well as the smoothness of the a priori term. L is the “number of looks” that sets the level of the noise. Speckle noise is signal dependent, but with a standard deviation proportional to expectation. The variance of noise is chosen as ${\left({\sigma}_i^{\mathrm{noise}}\right)}^2={\left({u}_i^{\mathrm{NL}}\right)}^2/L$. Then, adaptive regularization parameters ${\tau}_i=\gamma {\left({\hat{\sigma}}_i^{\mathrm{residual}}/{\sigma}_i^{\mathrm{noise}}\right)}^{-1}$ can be computed accordingly.

With the proposed model, nonlocal means and TV regularization complete each other: On areas where redundancy is important (homogeneous areas, for example), nonlocal means select many candidates such that residual variance is low. In terms of energy to minimize, the data fidelity term is then prominent over the regularization term, because of which the solution is close to the nonlocal mean. This provides adequate smoothing, and prevents the staircasing effect observed on smooth areas when treated with TV minimization. Around singular structures and edges where redundancy is low, nonlocal means select fewer candidates so that the residual variance is high. This is also the case if incorrect candidates are selected because the despeckling procedure reintroduces noise, thus leading to higher residual variance. The regularization term comes to dominate the data fidelity term, because of which it costs less to minimize the total variation in the image. The solution tends to a TV solution, preserving edges while reducing the rare-patch effect.

2.8 Algorithm and implementation

The functional in (19) is not convex, because of which there is no guarantee of the existence of an unique minimizer. The algorithm is shown as Algorithm 1, where the data fidelity term is differentiable and we cannot directly compute the proximal operator of total variation. Therefore, we need to proceed to an inner loop that iteratively calculates this proximal operator through the FISTA [3].

We describe here the general scheme of our method for the nonlocal adaptation regularization of speckle reduction in SAR images. Figure 2 shows the main steps of the method. Starting from a SAR image, we begin by computing the log-likelihood of the SAR signal. To achieve a good bias/variance trade-off, adaptively weighted nonlocal estimation and TV regularization are combined. A key ingredient to the performance and robustness of the method is unsupervised selection at each pixel of the best estimate among several estimates obtained by varying parameters of the nonlocal estimation.

Algorithm 1: Adaptively Weighted Nonlocal Means and TV SAR Speckle Reduction

Inpute g: noisy input image h:filtering parameter ∣P∣:patch size, N: size and shape of the search neighborhood λ: regularization parameter u0 = uNL fori ∈ Ωdo Compute wi, j ← φ[d(g(Pi), g(Pj))], ∀ j ∈ Ni Normalize wi, j ← wi, j/∑jwi, j, ∀ j ∈ Ni Compute ${u}_i^{\mathrm{NL}}\leftarrow {\sum}_j{w}_{i,j}T\left({g}_j\right)$ Compute ${\left({\hat{\sigma}}_i^{\mathrm{NL}}\right)}^2\leftarrow {\sum}_j{w}_{i,j}T{\left({g}_j\right)}^2-{\left({u}_i^{\mathrm{NL}}\right)}^2$ Compute ${\left({\sigma}_i^{\mathrm{noise}}\right)}^2$ according to the type of noise Compute ${\alpha}_i\leftarrow \frac{\mid {\left({\hat{\sigma}}_i^{\mathrm{NL}}\right)}^2-{\left({\sigma}_i^{\mathrm{noise}}\right)}^2\mid }{\mid {\left({\hat{\sigma}}_i^{\mathrm{NL}}\right)}^2-{\left({\sigma}_i^{\mathrm{noise}}\right)}^2\mid +{\left({\sigma}_i^{\mathrm{noise}}\right)}^2}$ Update ${u}_i^{\mathrm{NL}}\leftarrow \left(1-{\alpha}_i\right){u}_i^{\mathrm{NL}}+{\alpha}_i{g}_i$ Update wi, j ← (1 − αi)wi, j + αiδi, j Compute ${\lambda}_i\leftarrow \gamma {\left({\sum}_j{w}_{i,j}^2\right)}^{-1/2}$ end for uk + 1 = (I + λ∂F)−1(uk − γ ∇ Lw(uk)) return uk + 1

Fig. 2

The proposed scheme based on adaptively weighted nonlocal means combined with TV regularization

In addition to the filtering performance, the computational complexity is also important when assessing the applicability of a despeckling algorithm. Therefore, computationally efficient algorithms will be required for the processing of SAR data. We let the search window size be w × w and the target patch size be p × p, then the complexity of the proposed filter is about O(mnwp² + mntr), where t is the iteration times, r is the number of selected nearest neighbors in the search area.

2.9 Experiments and results

In this section, to illustrate the performance of the proposed AWNL model, results using eight real SAR images are reported. In this paper, the size of the searching window for the proposed method was fixed to 21 × 21 for all tested images. The constants were kept in log-likelihoods used for TV minimization to obtain a regularization scaling parameter γ of the scale for each type of noise. It was set to 55 for low noise level, and to 99 for medium and high noise, for images quantified in eight bits. We compared the proposed AWNL model with state-of-the-art techniques: the Pretest NLM method [5], the probabilistic patch-based (PPB) filter [8], the SAR block-matching 3D (SAR-BM3D) algorithm [18], the intensity-driven adaptive neighborhood (IDAN) filter [23], and the PolSAR NLTV method [17].

The performance of the filters with the following indexes is analyzed in this study:

• 1) To evaluate the availability of the filtering techniques in speckle reduction equivalent number of looks (ENL) is used. ENL of the see surface is calculated. The higher value shows better denoising. For SAR intensity images, the ENL is generally computed as

$$ \mathrm{ENL}=1/{\sigma}_v^2 $$

(20)

where σ_v denotes the coefficient of variation in homogeneous areas.

• 2) The target-to-clutter ratio (TCR) measures the difference in intensity ratios between point targets and the surrounding area before and after despeckling by

$$ \mathrm{TCR}=\mid 20{\log}_{10}\frac{\max_p\left({I}_d\right)}{{\mathrm{mean}}_p\left({I}_d\right)}-20{\log}_{10}\frac{\max_p\left({I}_s\right)}{{\mathrm{mean}}_p\left({I}_s\right)}\mid $$

(21)

where I_s and I_d are, respectively, the speckled intensity image and the despeckled intensity image. Subscript p denotes the patch containing a point target, and max_p and mean_p are computed over the patch.

• 3) EPD-ROA: The EPD-ROA indicator is given by

  $$ \mathrm{EPD}-\mathrm{ROA}=\frac{\sum_{i=1}^N\left|{I}_{\mathrm{d}1}(i)/{I}_{\mathrm{d}2}(i)\right|}{\sum_{i=1}^N\left|{I}_{s1}(i)/{I}_{\mathrm{s}2}(i)\right|}. $$

  (22)

  where I_d1 and I_d2 denote the adjacent pixel values of the despeckled image along the horizontal or vertical direction, respectively. Similarly, I_s1 and I_s2denote the corresponding adjacent pixel values of the speckled image. An EPD-ROA valued closer to one indicates a better edge-preservation ability.

2.10 Results on real SAR images

This section presents an overview of the results obtained on five real SAR images with the same state-of-the-art speckle filters, and our proposed method. Two single-look SAR acquisitions were identified as Bayard and Cheminot from Saint-Pol-sur-Mer (France), sensed in 1996 by RAMSES of ONERA. There were two single-look SAR acquisitions identified as Toulouse and Lelystadt of the CNES in Toulouse (France), sensed also by RAMSES and provided by the CNES. There was also a multilook SAR image, identified as GSN, of an agriculture region in Lelystadt (Netherlands), sensed by ERS-1 with identical numbers of looks (PRI data), provided by the European Space Agency (ESA). All these images were assumed to follow Goodman’s multiplicative speckle noise model. These five images provided a testing set that presented diversity: different sensors (RAMSES/ERS), different scenes (urban/agricultural), and different types of noise (single-look/multi-looks).

Figure 3 shows the denoised images obtained for different real SAR images and different denoising methods. The outputs of the pretest NLM method are shown in Fig. 3b, where the filter effectively reduces the speckle and enhances edges. However, the image is slightly over-smoothed, and loses some point signatures. The PPB filter suppresses the speckle to the highest degree, and its outputs are shown in Fig. 3c. The SAR-BM3D performed well in terms of preserving edges, and its outputs are shown in Fig. 3d. The PolSAR NLTV filter obtains a better result in retaining point targets, the results are shown in Fig. 3e. The results obtained using our method appear appropriately smoothed, with better edges and shape preservation than the other methods. The speckle effect was significantly reduced, and the spatial resolution appears to have been well preserved. The outputs are shown in Fig. 3f.

Fig. 3

From top to bottom: SAR images of Bayard ©DGA ©ONERA, Cheminot ©DGA ©ONERA, Toulouse ©DGA ©ONERA, Lelystadt ©ESA, and CNES ©DLR, taken by a multiplicative GSN with L = 2. Denoised images using a noisy image, b pretest, c PPB, d SAR-BM3D, e PolSAR NLTV, and f our results

2.11 Results on high-resolution PolSAR images

In this section, we consider a very high-resolution airborne image, the L-band San Francisco dataset acquired by the AIRSAR project of the National Aeronautics and Space Administration/Jet Propulsion Laboratory. This dataset was processed by the European Space Agency as a four-look dataset. Figure 4 illustrates the capability of four methods in estimating airborne polarimetric images. The denoising methods considered are as follows: Fig. 4b shows the iterative version of PPB denoising, Fig. 4c shows the intensity-driven region-growing method IDAN, Fig. 4d shows the pretest nonlocal method, Fig. 4e shows the PolSAR NLTV method, and Fig. 4f shows the proposed method. As shown in Fig. 4, all four filters obtain positive results on the San Francisco images. However, compared with the other three methods, the proposed method can effectively reduce speckle, especially for the homogeneous ocean area. Compared with pretest, the proposed method provided slightly over-smoothed results, but was free of systematic oscillating artifacts inherent of the PPB and IDAN. In this section, we use the ENL and TCR index to measure the capability of filters in terms of speckle reduction in homogeneous areas and scene feature preservation [16]. From the quantitative evaluation in Table 1, it can be concluded that the proposed method delivered the best performance on this dataset because it obtained robust performance in all aspects.

Fig. 4

From top to bottom, SAR images of Sanfrancisco: @JPL-NASA-Caltech, AIRSAR image of San Francisco, polarimatric, 2-looks, L-band, resolution 10 m × 10 m; kaufbeuren2: @DLR, F-SAR image of Kaufbeuren, polarimatric, 1-look, band FIXME, resolution FIXME (correlated noise); kaufbeuren3: @DLR, F-SAR image of Kaufbeuren, polarimatric, 1-look, S or X-band, resolution FIXME. From left to right, denoised images using a noisy; b PPB; c IDAN; d Pretest; e PolSAR NLTV, and f our results

Table 1 Results of quantitative assessment of real SAR datasets filtered by different methods

3 Conclusion

In this paper, we proposed an approach that combines nonlocal means with TV regularization. We used a weighted, nonlocal data-fidelity term whose magnitude is driven by the estimation of the denoising performed by nonlocal means. The reduction in variance offers a measure of confidence in the denoising performed by the nonlocal means, and TV regularization is automatically used according to the bias value. This leads to a flexible algorithm that locally uses both redundant and smooth properties of images while offering both a reduction in excessive smoothing and noise associated with nonlocal means, as well as a mitigation of the staircasing effect that occurs in TV minimization. Experimental results on synthetic and real SAR images show that the proposed method can reduce speckle and effectively avoid the staircase effect in homogeneous regions while preserving fine details, textures, and repetitive structures.