- Research
- Open Access
A collaborative adaptive Wiener filter for image restoration using a spatial-domain multi-patch correlation model
- Khaled M Mohamed†^{1} and
- Russell C Hardie†^{1}Email author
https://doi.org/10.1186/s13634-014-0189-3
© Mohamed and Hardie; licensee Springer. 2015
- Received: 22 July 2014
- Accepted: 23 December 2014
- Published: 27 January 2015
Abstract
We present a new patch-based image restoration algorithm using an adaptive Wiener filter (AWF) with a novel spatial-domain multi-patch correlation model. The new filter structure is referred to as a collaborative adaptive Wiener filter (CAWF). The CAWF employs a finite size moving window. At each position, the current observation window represents the reference patch. We identify the most similar patches in the image within a given search window about the reference patch. A single-stage weighted sum of all of the pixels in the similar patches is used to estimate the center pixel in the reference patch. The weights are based on a new multi-patch correlation model that takes into account each pixel’s spatial distance to the center of its corresponding patch, as well as the intensity vector distances among the similar patches. One key advantage of the CAWF approach, compared with many other patch-based algorithms, is that it can jointly handle blur and noise. Furthermore, it can also readily treat spatially varying signal and noise statistics. To the best of our knowledge, this is the first multi-patch algorithm to use a single spatial-domain weighted sum of all pixels within multiple similar patches to form its estimate and the first to use a spatial-domain multi-patch correlation model to determine the weights. The experimental results presented show that the proposed method delivers high performance in image restoration in a variety of scenarios.
Keywords
- Image restoration
- Wiener filter
- Correlation model
- Patch-based processing
1 Introduction
1.1 Image restoration
During image acquisition, images are subject to a variety of degradations. These invariably include blurring from diffraction and noise from a variety of sources. Restoring such degraded images is a fundamental problem in image processing that has been researched since the earliest days of digital images [1,2]. A wide variety of linear and non-linear methods have been proposed. Many methods have focused exclusively on noise reduction, and others seek to address multiple degradations jointly, such as blur and noise.
A widely used method for image restoration, relevant to the current paper, is the classic Wiener filter [3]. The standard Wiener filter is a linear space-invariant filter designed to minimize mean squared error (MSE) between the desired signal and estimate, assuming stationary random signals and noise. It is important to note that there are many disparate variations of Wiener filters. These include finite impulse response, infinite impulse response, transform-domain, and spatially adaptive methods. Within each of these categories, a wide variety of statistical models may be employed. Some statistical models are very simple, such as the popular constant noise-to-signal power spectral density model, and others are far more complex. In the case of the empirical Wiener filter [4], no explicit statistical model is used at all. Rather, a pilot or prototype estimate is used in lieu of a parametric statistical model. While all of these methods may go by the name of ‘Wiener filter’, they can be quite different in their character.
Recently, a form of adaptive Wiener filter (AWF) has been developed and successfully applied to super-resolution (SR) and other restoration applications by one of the current authors [5]. This AWF approach employs a spatially varying weighted sum to form an estimate of each pixel. The Wiener weights are determined based on a spatially varying spatial-domain parametric correlation model. This particular brand of AWF SR emerged from earlier work, including that in [6-8]. This kind of AWF is capable of jointly addressing blur, noise, and undersampling and is well suited to dealing with a non-stationary signal and noise. The approach is also very well suited to dealing with non-uniformly sampled imagery and missing or bad pixels. This AWF SR method has been shown to provide best-in-class performance for nonuniform interpolation-based SR [5,9-11] and has also been used successfully for demosaicing [12,13] and Nyquist sampled video restoration [14]. Under certain conditions, the method can also be very computationally efficient [5]. The key to this method lies in the particular correlation model used and how it is employed for spatially adaptive filtering.
A different approach to image restoration, also relevant to the current paper, is based on fusing multiple similar patches within the observed image. This patch-based approach is used primarily for noise reduction applications and exploits spatial redundancy that may express itself within an image, locally and/or non-locally. The method of non-local means (NLM), introduced in [15], may be the first method to directly fuse non-local patches from within the observed image based on vector distances for the purpose of image denoising. A notable early precursor to the NLM is the vector detection method [16,17]. In the vector detection approach, a codebook of representative patches from training data is used, rather than patches from the observed image itself [16,17]. A number of NLM variations have been proposed, including [18-26]. The basic NLM method forms an estimate of a reference pixel as a weighted sum of non-local pixels. The weights are based on the vector distances of the patch intensities between various non-local patches and the reference patch. In particular, the center samples of a non-local patches are weighted in proportion to the negative exponential of the corresponding patch distance. The NLM algorithm can be viewed as an extension of the bilateral filter [27-31], which forms an estimate by weighting neighboring pixels based on both spatial proximity and intensity similarity of individual pixels (rather than patches).
Improved performance for noise reduction is obtained with the block matching and 3D filtering (BM3D) approach proposed in [32-34]. The BM3D method also uses vector distances between patches, but the filtering is performed using a transform-domain shrinkage operation. By utilizing all of the samples within selected patches and aggregating the results, excellent noise reduction performance can be achieved with BM3D. Another related patch-based image denoising algorithm is the total least squares method presented in [35]. In this method, each ideal patch is modeled as a linear combination of similar patches from the observed image. Another type of patch-based Wiener denoising filter is proposed in [36], and a globalized approach to patch-based denoising is proposed in [37]. While such patch-based methods perform well in noise reduction, most are not capable of addressing blur and noise jointly. However, there are a few recent methods that do treat both blur and noise and incorporate multi-patch fusion. These include BM3D deblurring (BM3DDEB) [38] and iterative decoupled deblurring-BM3D (IDD-BM3D) [39]. The deblurring in these algorithms is not achieved by patch fusion alone. Rather, the patch fusion component of these algorithms serves as a type of signal model used for regularization. Note that multi-patch methods have also been developed and applied to SR [40-45]. However, the focus of this paper is on image restoration without undersampling/aliasing.
1.2 Proposed method and novel contribution
In this paper, we propose a novel multi-patch AWF algorithm for image restoration. In the spirit of [32], we refer to this new filter structure as a collaborative adaptive Wiener filter (CAWF). It can be viewed as an extension of the AWF in [5], with the powerful new feature of incorporating multiple patches for each pixel estimate. As with other patch-based algorithms, we employ a moving window. At each position, the current observation window represents the reference patch. Within a given search window about the reference, we identify the most similar patches to the reference patch. However, instead of simply weighting just the center pixels of these similar patches, as with NLM, or using transform-domain shrinkage like BM3D, we use a spatial-domain weighted sum of all of the pixels within all of the selected patches to estimate the one center pixel in the reference patch. The weights used are based on a novel spatially varying spatial-domain multi-patch correlation model. The correlation model takes into account each pixel’s spatial distance to the center of its corresponding patch, as well as the intensity vector distances among the similar patches. The ‘collaborative’ nature of the CAWF springs from the fusion of multiple, potentially non-local, patches. One key advantage of the CAWF approach is that it can jointly handle blur and noise. Furthermore, the CAWF method is able to accommodate spatially varying signal and noise statistics.
Our approach is novel in that we use a single-pass spatial-domain weighted sum of all pixels within all of the similar patches to form the estimate each desired pixel. In the case of NLM, only the center pixel of each similar patch is given a weight [15].This is simple and effective for denoising, but deconvolution is not possible within the basic NLM framework, and all of the available information in the patches may not be exploited. While BM3D does fuse all of the pixels in the similar patches, the fusion in BM3D is based on a wavelet shrinkage operation and not a spatial-domain correlation model [32]. Because of the nature of wavelet shrinkage, the standard BM3D is also unable to perform deblurring [32]. On the other hand, the CAWF structure can jointly address blur and noise and does not employ separate transform-domain inverse filtering as in [38] or iterative processing like that in [39]. To the best of our knowledge, this is the first multi-patch algorithm to use a single-pass weighted sum of all pixels within multiple similar patches to jointly address blur and noise. It is also the first to use a spatial-domain multi-patch correlation model.
The remainder of this paper is organized as follows. The observation model is described in Section 1. The CAWF algorithm is presented in Section 1. This includes the basic algorithm description as well as the new spatial-domain multi-patch correlation model. Computational complexity and implementation are also discussed in Section 1. Experimental results for simulated and real data are presented and discussed in Section 1. Finally, conclusions are offered in Section 1.
2 Observation model
where H is an N×N matrix containing values of PSF, and the vector f is the image f(n _{1},n _{2}) in lexicographical form. The PSF can be designed to model different types of blurring, such as diffraction from optics, spatial detection integration, atmospheric effects, and motion blurring. In the experimental results presented in this paper, we use a simple Gaussian PSF.
where g=[g _{1},g _{2},…,g _{ N }]^{ T } and η=[η _{1},η _{2},…,η _{ N }]^{ T } are the observed image and noise vectors, respectively. The random noise vector is Gaussian such that \(\boldsymbol {\eta } \sim \mathcal {N} \left ({\mathbf {0},{\sigma }^{2}_{\eta } \mathbf {I}} \right)\).
3 Collaborative adaptive Wiener filter
3.1 CAWF overview
The CAWF employs a moving window approach with a moving reference patch and corresponding moving search window, each centered about pixel i, where i=1,2,…,N. The reference patch spans K _{1}×K _{2}=K pixels symmetrically about pixel i. All of the pixels that lie within the span of this reference patch are placed into the reference patch vector defined as g _{ i }=[g _{ i,1},g _{ i,2},…,g _{ i,K }]^{ T }. The search window is of size L _{1}×L _{2}=L pixels. Let the set S _{ i }=[S _{ i }(1),S _{ i }(2),… S _{ i }(L)]^{ T } contain the indices of the pixels within the search window.
where \(\tilde {\mathbf {R}}_{i} = E \left \{ \tilde {\mathbf {g}}_{i} \tilde {\mathbf {g}}_{i}^{T} \right \}\) is a K M×K M autocorrelation matrix for the multi-patch observation vector \(\tilde {\mathbf {g}}_{i}\), and \(\tilde {\mathbf {p}}_{i} = E\left \{\tilde {\mathbf {g}}_{i} d_{i} \right \}\) is a K M×1 cross-correlation vector between the desired pixel d _{ i } and \(\tilde {\mathbf {g}}_{i}\). The statistics used to fill \(\tilde {\mathbf {R}}_{i}\) and \(\tilde {\mathbf {p}}_{i}\) are found using the new multi-patch correlation model described in Section 1.
3.2 Spatial-domain multi-patch correlation model
The multi-patch correlation model provides the values for \(\tilde {\mathbf {R}}_{i}\) and \(\tilde {\mathbf {p}}_{i}\), so that we may generate the weights in Equation 6. The model attempts to capture the spatial relationship among the pixels within a given patch, which is essential for deconvolution. Furthermore, it also seeks to incorporate knowledge of redundancy among the similar patches. Finally, the model captures the local desired signal variance, as well as the noise variance of each observed pixel.
Now, the problem reduces to modeling \(E\left \{\tilde {\mathbf {{f}}}_{i}\,d_{i}\right \} \) and \(E\left \{\tilde {\mathbf {{f}}}_{i}\,\tilde {\mathbf {{f}}}_{i}^{T}\right \}\).
where p is the K×1 cross-correlation vector for a single normalized patch, and [D _{ i }]_{1} is the first column of the distance matrix D _{ i }. The correlation models in Equations 11 and 12 capture the spatial correlations among pixels within each patch using R and p. The patch similarities, captured in the distance matrix, are used to ‘modulate’ these correlations with the Kronecker product to provide the full multi-patch correlation model. In this manner, pixels belonging to patches with smaller inter-patch distances will be modeled with higher correlations among them. Potential changes in the underlying desired image variance are captured in the model with the term \( \hat \sigma _{{d_{i}}}^{2} \). In addition, a spatially varying noise variance can easily be incorporated if appropriate.
where \({\bar f}_{\frac {K+1}{2}}\) corresponds to the spatial position of \({\bar d}\).
and \(\tilde h({n_{1}},{n_{2}}) = h({n_{1}},{n_{2}}) * h(- {n_{1}}, - {n_{2}})\). Thus, our desired signal variance estimate is \( {\hat \sigma }_{d_{i}}^{2} = {\hat \sigma }_{f_{i}}^{2} / C(\rho)\).
3.3 Multi-pixel estimation and aggregation
In the CAWF algorithm described in Section 1, one pixel is estimated for each reference patch. However, in a manner similar to that in [5], it is possible to estimate multiple desired pixels from each multi-patch observation vector \(\tilde {\mathbf {g}}_{i}\). In fact, all of the desired pixels corresponding to \(\tilde {\mathbf {g}}_{i}\) can be estimated. Let this full K M×1 vector of desired pixels be denoted \(\tilde {\mathbf {d}}_{i}\). If all multi-patch observation vectors are used in this fashion, many estimates of each desired pixel are obtained. These can be aggregated by a simple average. In the case of noise only, we have observed that aggregation yields improved results. For joint deblurring and denoising with any significant amount of blur, the aggregation does not appear to provide any advantage. However, this multi-pixel estimation approach can be used to reduce the computational complexity, since not every multi-patch observation vector must be processed in order to form a complete image estimate.
\( \mathbf {P} = E\left \{\bar {\mathbf {f}} \bar {\mathbf {d}}^{T} \right \}\) is a K×K normalized cross-correlation matrix, and \(\bar {\mathbf {d}}\) is the K×1 desired vector corresponding to \(\bar {\mathbf {f}}\).
3.4 Computational complexity and implementation
Here, we briefly address the computational complexity of the CAWF filter by tracking the number of floating point operations (flops), where a flop is defined as one multiply plus add operation. The first action of the CAWF filter is finding similar patches. This requires computing L distances of K dimensional vectors (note that L is the search window size, and K is the patch size in pixels). The next step is computing the distance matrix based on Equation 13 for the M selected patches. This requires computing M ^{2}/2 scaled and shifted distances for K dimensional vectors. The Kronecker products for \(\tilde {\mathbf {R}}_{i}\) and \(\tilde {\mathbf {p}}_{i}\) require (K M)^{2} and KM multiplies, respectively. However, the main computational burden of the CAWF filter comes next with the computation of the weights in Equation 6. This can be done using Cholesky factorization, which requires (K M)^{3}/3 flops to perform LU decomposition for the K M×K M autocorrelation matrix \(\tilde {\mathbf {R}}_{i}\). Computing the weights from the LU decomposition requires 2(K M)^{2} flops using forward and backward substitution. The final weighted sum operation is accomplished with KM flops. Since the dominant term in the computational complexity is the Cholesky factorization, we might conclude that the complexity of the CAWF filter is O((K M)^{3}). Thus, the complexity of the CAWF algorithm goes up significantly with larger windows sizes, K, and more similar patches, M. However, an important thing to note is that the CAWF algorithm is completely parallel at the output pixel level. Unlike most variational image restoration methods, each output pixel can be computed independently and in parallel. Also, the CAWF approach requires only one pass over the data.
To put the CAWF computational complexity into context, consider that the AWF method employed here, with a spatially varying signal-to-noise ratio (SNR) estimate, may be viewed as a special case of the CAWF with M=1. Thus, increasing M for CAWF causes a corresponding increase in complexity according to O((K M)^{3}). The NLM method shares the same distance computations and comparisons and CAWF. However, in contrast to CAWF, NLM only requires L flops per output in the weighted sum, since it only weights the center sample of each patch in the search window. Although significantly simpler computationally, NLM does not fully exploit all of the information in the patches and it cannot perform deconvolution. Also, AWF is not able to exploit multi-patch information.
For pure denoising application, we have found that good results can be obtained with CAWF for M=10, and K=3×3=9 for light noise and K=5×5=25 for moderate to heavy noise. In the case of joint deblurring and denoising, a larger window size is needed for adequate deconvolution. We have found that K=9×9=81 is a reasonable choice for light to moderate blurring. Our implementation uses MATLAB with no parallel acceleration or mex files, and processing is done on a PC with Intel®Xeon®Processor 3.7 GHz. CAWF processing time for a pure denoising application with a 512×512 image using K=9 and M=10 is 155 s. For context, the AWF processing takes 33 s, and NLM takes 3.2 s.
4 Experimental results
CAWF parameters used in experimental results
Case 1: noise only | Case 2: blur and noise | |||
---|---|---|---|---|
Parameter name | Variable | Selected value | Selected value | Selected value |
σ _{ η } <20 | σ _{ η } ≥20 | |||
Patch size | K | 3×3=9 | 5×5=25 | 9×9=81 |
Search window size | L | 17×17=289 | 11×11=121 | 9×9=81 |
Number of patches | M | 10 | 10 | 8 |
Autocorrelation decay | ρ | 0.65 | 0.70 | 0.65 |
Patch similarity decay | α | 2.00 | 1.40 | 1.20 |
Distance offset | D _{0} | 0.25 | 0.50 | 0.00 |
Aggregation | N/A | Averaging | Averaging | None |
4.1 Simulated data
In this section, we present quantitative results using simulated data. We consider two cases: noise only and blur with noise. For each case, we consider four specific scenarios and use six test images. Also, for each case, we compare against state-of-the art methods for which MATLAB implementations are publicly available.
4.1.1 Additive Gaussian noise
In our first case, we consider additive Gaussian noise with no PSF blur (i.e., h(n _{1},n _{2})=δ(n _{1},n _{2})). We consider four different noise standard deviations. The denoising benchmark methods are NLM [15], Globalized NLM (GLIDE-NLM) [37], PLOW [36], BM3D [32], and the single patch AWF [5]. Note that the NLM implementation is from [37], and AWF used is the same as CAWF with no aggregation and M=1.
PSNR comparison for additive Gaussian noise
Image | Method | PSNR (ISNR) | |||
---|---|---|---|---|---|
σ _{ η } =10 | σ _{ η } =20 | σ _{ η } =30 | σ _{ η } =40 | ||
Aerial | Corrupted | 28.13 | 22.11 | 18.59 | 16.09 |
491×434 | NLM | 30.14 (2.01) | 25.56 (3.45) | 23.62 (5.03) | 22.35 (6.26) |
GLIDE-NLM | 30.25 (2.12) | 25.82 (3.71) | 23.79 (5.19) | 22.58 (6.49) | |
PLOW | 28.76 (0.63) | 25.30 (3.19) | 23.68 (5.09) | 22.64 (6.55) | |
BM3D | 30.61 (2.48) | 26.48 (4.37) | 24.38 (5.79) | 22.92 (6.82) | |
AWF | 30.19 (2.06) | 26.09 (3.98) | 24.05 (5.46) | 22.73 (6.63) | |
CAWF | 30.67 (2.54) | 26.55 (4.44) | 24.50 (5.90) | 23.13 (7.04) | |
Bridge | Corrupted | 28.14 | 22.11 | 18.59 | 16.09 |
512^{2} | NLM | 30.56 (2.42) | 26.36 (4.24) | 24.69 (6.09) | 23.62 (7.52) |
GLIDE-NLM | 30.64 (2.51) | 26.55 (4.43) | 24.82 (6.22) | 23.73 (7.64) | |
PLOW | 30.00 (1.87) | 26.70 (4.58) | 25.23 (6.63) | 24.25 (8.16) | |
BM3D | 31.17 (3.04) | 27.27 (5.16) | 25.46 (6.87) | 24.31 (8.21) | |
AWF | 30.58 (2.45) | 26.87 (4.76) | 25.09 (6.50) | 23.90 (7.80) | |
CAWF | 31.11 (2.97) | 27.31 (5.19) | 25.50 (6.91) | 24.33 (8.23) | |
River | Corrupted | 28.13 | 22.11 | 18.59 | 16.09 |
(Kodak 11) | NLM | 29.95 (1.82) | 25.46 (3.35) | 23.63 (5.04) | 22.47 (6.37) |
768×512 | GLIDE-NLM | 29.84 (1.71) | 25.71 (3.60) | 23.78 (5.19) | 22.62 (6.53) |
PLOW | 28.53 (0.40) | 24.70 (2.58) | 22.94 (4.35) | 22.10 (6.01) | |
BM3D | 30.37 (2.24) | 26.16 (4.05) | 24.16 (5.57) | 22.86 (6.77) | |
AWF | 29.96 (1.82) | 25.90 (3.78) | 23.90 (5.31) | 22.61 (6.52) | |
CAWF | 30.48 (2.35) | 26.32 (4.20) | 24.35 (5.76) | 23.11 (7.02) | |
Bones | Corrupted | 28.13 | 22.11 | 18.59 | 16.09 |
512×768 | NLM | 30.36 (2.23) | 26.65 (4.54) | 25.50 (6.91) | 24.72 (8.63) |
GLIDE-NLM | 30.48 (2.35) | 26.86 (4.75) | 25.50 (6.91) | 24.78 (8.68) | |
PLOW | 29.33 (1.20) | 26.54 (4.43) | 25.60 (7.01) | 24.94 (8.85) | |
BM3D | 30.85 (2.72) | 27.18 (5.07) | 25.81 (7.22) | 25.05 (8.96) | |
AWF | 30.46 (2.33) | 27.13 (5.02) | 25.60 (7.01) | 24.51 (8.41) | |
CAWF | 30.93 (2.79) | 27.46 (5.35) | 26.03 (7.44) | 25.11 (9.02) | |
Building | Corrupted | 28.13 | 22.11 | 18.59 | 16.09 |
768×512 | NLM | 30.61 (2.48) | 26.35 (4.24) | 24.50 (5.91) | 23.30 (7.21) |
GLIDE-NLM | 30.32 (2.19) | 26.43 (4.32) | 24.64 (6.05) | 23.45 (7.36) | |
PLOW | 29.25 (1.12) | 25.04 (2.93) | 23.32 (4.73) | 22.51 (6.42) | |
BM3D | 30.97 (2.84) | 26.91 (4.79) | 24.88 (6.29) | 23.62 (7.53) | |
AWF | 30.46 (2.33) | 26.51 (4.40) | 24.51 (5.92) | 23.17 (7.08) | |
CAWF | 31.12 (2.98) | 27.05 (4.94) | 25.09 (6.50) | 23.84 (7.74) | |
Gazebo | Corrupted | 28.13 | 22.11 | 18.59 | 16.09 |
768×512 | NLM | 31.11 (2.98) | 26.71 (4.59) | 24.93 (6.34) | 23.77 (7.68) |
GLIDE-NLM | 31.11 (2.98) | 26.97 (4.86) | 25.05 (6.46) | 23.95 (7.86) | |
PLOW | 29.82 (1.68) | 25.69 (3.58) | 24.20 (5.61) | 23.46 (7.36) | |
BM3D | 31.57 (3.44) | 27.40 (5.29) | 25.48 (6.89) | 24.29 (8.19) | |
AWF | 30.78 (2.64) | 26.88 (4.77) | 24.91 (6.32) | 23.59 (7.50) | |
CAWF | 31.60 (3.47) | 27.49 (5.37) | 25.55 (6.96) | 24.30 (8.21) |
SSIM for additive Gaussian noise
Image | Method | SSIM | |||
---|---|---|---|---|---|
σ _{ η } =10 | σ _{ η } =20 | σ _{ η } =30 | σ _{ η } =40 | ||
Aerial | Corrupted | 0.9719 | 0.9046 | 0.8224 | 0.7388 |
491×434 | NLM | 0.9753 | 0.9058 | 0.8412 | 0.7872 |
GLIDE-NLM | 0.9747 | 0.9058 | 0.8427 | 0.7916 | |
PLOW | 0.9734 | 0.9252 | 0.8760 | 0.8282 | |
BM3D | 0.9783 | 0.9340 | 0.8835 | 0.8287 | |
AWF | 0.9762 | 0.9304 | 0.8804 | 0.8322 | |
CAWF | 0.9787 | 0.9351 | 0.8859 | 0.8365 | |
Bridge | Corrupted | 0.9542 | 0.8512 | 0.7365 | 0.6305 |
512^{2} | NLM | 0.9611 | 0.8651 | 0.7925 | 0.7379 |
GLIDE-NLM | 0.9621 | 0.8757 | 0.8010 | 0.7363 | |
PLOW | 0.9587 | 0.9051 | 0.8485 | 0.7943 | |
BM3D | 0.9692 | 0.9126 | 0.8539 | 0.7963 | |
AWF | 0.9652 | 0.9061 | 0.8477 | 0.7927 | |
CAWF | 0.9676 | 0.9109 | 0.8523 | 0.7985 | |
River | Corrupted | 0.9512 | 0.8429 | 0.7252 | 0.6196 |
(Kodak 11) | NLM | 0.9525 | 0.8343 | 0.7588 | 0.7042 |
768×512 | GLIDE-NLM | 0.9548 | 0.8461 | 0.7617 | 0.7007 |
PLOW | 0.9505 | 0.8732 | 0.7921 | 0.7299 | |
BM3D | 0.9597 | 0.8792 | 0.8066 | 0.7437 | |
AWF | 0.9574 | 0.8823 | 0.8142 | 0.7512 | |
CAWF | 0.9609 | 0.8873 | 0.8207 | 0.7634 | |
Bones | Corrupted | 0.9259 | 0.7735 | 0.6224 | 0.4979 |
512×768 | NLM | 0.9329 | 0.7946 | 0.7269 | 0.6821 |
GLIDE-NLM | 0.9379 | 0.8163 | 0.7351 | 0.6725 | |
PLOW | 0.9240 | 0.8295 | 0.7587 | 0.7047 | |
BM3D | 0.9417 | 0.8404 | 0.7641 | 0.7091 | |
AWF | 0.9389 | 0.8459 | 0.7669 | 0.6970 | |
CAWF | 0.9439 | 0.8551 | 0.7839 | 0.7277 | |
Building | Corrupted | 0.9153 | 0.7660 | 0.6349 | 0.5318 |
768×512 | NLM | 0.9395 | 0.8370 | 0.7808 | 0.7386 |
GLIDE-NLM | 0.9426 | 0.8416 | 0.7835 | 0.7385 | |
PLOW | 0.9379 | 0.8577 | 0.7823 | 0.7262 | |
BM3D | 0.9482 | 0.8748 | 0.8141 | 0.7637 | |
AWF | 0.9398 | 0.8583 | 0.7797 | 0.7072 | |
CAWF | 0.9502 | 0.8788 | 0.8190 | 0.7665 | |
Gazebo | Corrupted | 0.9130 | 0.7678 | 0.6428 | 0.5435 |
768×512 | NLM | 0.9561 | 0.8653 | 0.8090 | 0.7649 |
GLIDE-NLM | 0.9578 | 0.8712 | 0.8117 | 0.7686 | |
PLOW | 0.9525 | 0.8698 | 0.8046 | 0.7559 | |
BM3D | 0.9620 | 0.8963 | 0.8397 | 0.7940 | |
AWF | 0.9452 | 0.8740 | 0.7950 | 0.7230 | |
CAWF | 0.9611 | 0.8988 | 0.8387 | 0.7837 |
4.1.2 Gaussian blur plus Gaussian noise
Scenarios of Gaussian blur and Gaussian noise
Scenario | PSF | σ _{ η } |
---|---|---|
I | Gaussian std. = 1.0 | 10 |
II | Gaussian std. = 1.5 | 10 |
III | Gaussian std. = 1.0 | 20 |
IV | Gaussian std. = 1.5 | 20 |
PSNR comparison for Gaussian blur plus Gaussian noise
Image | Method | PSNR (ISNR) | |||
---|---|---|---|---|---|
Scenario | Scenario | Scenario | Scenario | ||
I | II | III | IV | ||
Aerial | Corrupted | 22.64 | 20.79 | 19.98 | 18.88 |
491×434 | L0-Abs | 24.02 (1.38) | 21.88 (1.10) | 21.80 (1.82) | 20.36 (1.48) |
TVMM | 24.18 (1.54) | 22.18 (1.40) | 21.94 (1.95) | 20.49 (1.60) | |
BM3DDEB | 24.41 (1.76) | 22.56 (1.78) | 22.54 (2.56) | 21.32 (2.44) | |
IDD-BM3D | 24.71 (2.06) | 22.69 (1.90) | 22.71 (2.73) | 21.32 (2.44) | |
AWF | 24.25 (1.61) | 22.33 (1.55) | 22.64 (2.66) | 21.29 (2.41) | |
CAWF | 24.47 (1.83) | 22.56 (1.78) | 22.83 (2.85) | 21.43 (2.55) | |
Bridge | Corrupted | 24.19 | 22.80 | 20.76 | 20.07 |
512^{2} | L0-Abs | 25.63 (1.44) | 24.08 (1.28) | 23.52 (2.77) | 22.53 (2.47) |
TVMM | 25.73 (1.53) | 24.27 (1.47) | 23.57 (2.82) | 22.84 (2.77) | |
BM3DDEB | 26.02 (1.82) | 24.71 (1.91) | 24.47 (3.72) | 23.62 (3.55) | |
IDD-BM3D | 26.23 (2.03) | 24.79 (2.00) | 24.52 (3.76) | 23.63 (3.56) | |
AWF | 25.78 (1.59) | 24.43 (1.63) | 24.39 (3.63) | 23.45 (3.38) | |
CAWF | 25.94 (1.74) | 24.56 (1.77) | 24.48 (3.72) | 23.48 (3.41) | |
River | Corrupted | 22.14 | 20.66 | 19.69 | 18.79 |
(Kodak 11) | L0-Abs | 23.22 (1.08) | 21.43 (0.77) | 21.45 (1.76) | 20.33 (1.54) |
768×512 | TVMM | 23.13 (1.00) | 21.52 (0.86) | 21.14 (1.45) | 20.33 (1.53) |
BM3DDEB | 23.24 (1.11) | 21.78 (1.12) | 21.72 (2.03) | 20.90 (2.11) | |
IDD-BM3D | 23.63 (1.49) | 21.90 (1.24) | 21.95 (2.26) | 20.96 (2.17) | |
AWF | 23.29 (1.15) | 21.78 (1.12) | 22.06 (2.36) | 21.06 (2.27) | |
CAWF | 23.48 (1.34) | 21.93 (1.28) | 22.21 (2.52) | 21.15 (2.36) | |
Bones | Corrupted | 24.80 | 23.78 | 21.01 | 20.56 |
512×768 | L0-Abs | 26.07 (1.27) | 25.14 (1.36) | 24.65 (3.64) | 24.17 (3.62) |
TVMM | 25.94 (1.15) | 25.11 (1.33) | 24.52 (3.51) | 24.24 (3.68) | |
BM3DDEB | 26.33 (1.54) | 25.48 (1.71) | 25.23 (4.22) | 24.79 (4.23) | |
IDD-BM3D | 26.46 (1.66) | 25.51 (1.74) | 25.26 (4.24) | 24.80 (4.24) | |
AWF | 26.31 (1.51) | 25.37 (1.59) | 25.25 (4.24) | 24.69 (4.14) | |
CAWF | 26.44 (1.64) | 25.43 (1.66) | 25.28 (4.27) | 24.67 (4.12) | |
Building | Corrupted | 22.60 | 21.27 | 19.96 | 19.18 |
768×512 | L0-Abs | 23.88 (1.28) | 22.24 (0.97) | 22.25 (2.29) | 21.22 (2.04) |
TVMM | 23.82 (1.22) | 22.42 (1.16) | 22.11 (2.15) | 21.57 (2.39) | |
BM3DDEB | 23.86 (1.26) | 22.53 (1.27) | 22.49 (2.53) | 21.73 (2.55) | |
IDD-BM3D | 24.25 (1.65) | 22.69 (1.43) | 22.75 (2.79) | 21.85 (2.66) | |
AWF | 23.89 (1.28) | 22.52 (1.26) | 22.76 (2.80) | 21.85 (2.66) | |
CAWF | 24.05 (1.45) | 22.65 (1.38) | 22.89 (2.93) | 21.92 (2.74) | |
Gazebo | Corrupted | 23.29 | 21.90 | 20.31 | 19.55 |
768×512 | L0-Abs | 24.82 (1.53) | 23.11 (1.21) | 23.02 (2.71) | 21.93 (2.38) |
TVMM | 24.80 (1.50) | 23.41 (1.50) | 23.34 (3.03) | 22.17 (2.61) | |
BM3DDEB | 24.91 (1.61) | 23.45 (1.55) | 23.39 (3.08) | 22.52 (2.97) | |
IDD-BM3D | 25.29 (2.00) | 23.65 (1.75) | 23.68 (3.37) | 22.66 (3.11) | |
AWF | 24.83 (1.53) | 23.37 (1.47) | 23.58 (3.26) | 22.58 (3.03) | |
CAWF | 24.99 (1.69) | 23.51 (1.61) | 23.70 (3.39) | 22.66 (3.10) |
SSIM comparison for Gaussian blur plus Gaussian noise
Image | Method | SSIM | |||
---|---|---|---|---|---|
Scenario | Scenario | Scenario | Scenario | ||
I | II | III | IV | ||
Aerial | Corrupted | 0.8891 | 0.7794 | 0.8176 | 0.7117 |
491×434 | L0-Abs | 0.9087 | 0.8050 | 0.7821 | 0.6448 |
TVMM | 0.9215 | 0.8325 | 0.7885 | 0.6604 | |
BM3DDEB | 0.9198 | 0.8571 | 0.8400 | 0.7710 | |
IDD-BM3D | 0.9292 | 0.8591 | 0.8406 | 0.7577 | |
AWF | 0.9199 | 0.8519 | 0.8517 | 0.7733 | |
CAWF | 0.9273 | 0.8662 | 0.8649 | 0.7957 | |
Bridge | Corrupted | 0.8836 | 0.7975 | 0.7745 | 0.6929 |
512^{2} | L0-Abs | 0.8902 | 0.8044 | 0.7425 | 0.6524 |
TVMM | 0.8995 | 0.8156 | 0.7323 | 0.6689 | |
BM3DDEB | 0.9099 | 0.8579 | 0.8294 | 0.7796 | |
IDD-BM3D | 0.9173 | 0.8573 | 0.8252 | 0.7657 | |
AWF | 0.9080 | 0.8503 | 0.8344 | 0.7712 | |
CAWF | 0.9155 | 0.8637 | 0.8462 | 0.7892 | |
River | Corrupted | 0.8373 | 0.7198 | 0.7246 | 0.6161 |
(Kodak 11) | L0-Abs | 0.8328 | 0.7159 | 0.6670 | 0.5560 |
768×512 | TVMM | 0.8452 | 0.7219 | 0.6344 | 0.5421 |
BM3DDEB | 0.8487 | 0.7713 | 0.7377 | 0.6745 | |
IDD-BM3D | 0.8672 | 0.7767 | 0.7463 | 0.6666 | |
AWF | 0.8573 | 0.7736 | 0.7676 | 0.6878 | |
CAWF | 0.8707 | 0.7949 | 0.7837 | 0.7105 | |
Bones | Corrupted | 0.8231 | 0.7323 | 0.6681 | 0.5884 |
512×768 | L0-Abs | 0.8051 | 0.7287 | 0.6652 | 0.6181 |
TVMM | 0.7954 | 0.7192 | 0.6427 | 0.6172 | |
BM3DDEB | 0.8320 | 0.7768 | 0.7420 | 0.7064 | |
IDD-BM3D | 0.8421 | 0.7753 | 0.7389 | 0.6969 | |
AWF | 0.8358 | 0.7699 | 0.7526 | 0.7000 | |
CAWF | 0.8476 | 0.7866 | 0.7615 | 0.7108 | |
Building | Corrupted | 0.8119 | 0.7162 | 0.6609 | 0.5753 |
768×512 | L0-Abs | 0.8356 | 0.7491 | 0.7142 | 0.6397 |
TVMM | 0.8454 | 0.7592 | 0.6936 | 0.6615 | |
BM3DDEB | 0.8520 | 0.7913 | 0.7633 | 0.7138 | |
IDD-BM3D | 0.8662 | 0.7953 | 0.7715 | 0.7130 | |
AWF | 0.8509 | 0.7848 | 0.7721 | 0.7147 | |
CAWF | 0.8607 | 0.7974 | 0.7816 | 0.7268 | |
Gazebo | Corrupted | 0.8279 | 0.7413 | 0.6819 | 0.6030 |
768×512 | L0-Abs | 0.8651 | 0.7867 | 0.7527 | 0.6788 |
TVMM | 0.8741 | 0.8057 | 0.7785 | 0.6826 | |
BM3DDEB | 0.8805 | 0.8249 | 0.8001 | 0.7500 | |
IDD-BM3D | 0.8898 | 0.8282 | 0.8064 | 0.7504 | |
AWF | 0.8796 | 0.8190 | 0.8035 | 0.7481 | |
CAWF | 0.8801 | 0.8238 | 0.8072 | 0.7561 |
4.2 Real data
Real video frames have been acquired of an outdoor natural scene on the campus of the University of Dayton using an Imaging Source 8 bit grayscale camera (DMK 23U618) with Sony ICX618ALA sensor. A short exposure time is used, proving a low SNR. A sequence of 500 frames is acquired for the static scene. This allows us to form a temporal average as a type of reference with which to compare the noise reduction estimates. Since this real noise will have both a signal-dependent and signal-independent component, we apply an Anscombe transform to stabilize the local noise variance prior to applying all denoising methods [49]. After the transform and scaling, an effective constant noise standard deviation of σ _{ η }=9.11 is estimated and used for all methods.
4.3 Parameter and distance metric sensitivity
Next, we examine the autocorrelation decay constant, ρ, and the patch similarity decay, α, for the image aerial with additive Gaussian noise. We have evaluated CAWF PSNR for ρ ranging from 0.6 to 0.75, with all other parameter values as listed in Table 1. The maximum change in PSNR as a function of ρ, for noise levels ranging from σ _{ η }=10 to σ _{ η }=40, is only 0.13%. Similarly, we have evaluated PSNR values for α ranging from 1.0 to 2.0. The maximum change in PSNR as a function of α is observed to be 0.23%. Thus, we conclude that the CAWF method is not highly sensitive to these tuning parameters within these operating ranges.
Finally, we explore CAWF performance using different distance metrics in the correlation model. Our standard metric uses the l ^{2}-norm as defined in Equation 13. To test distance metric sensitivity, we compare distances with the l ^{1}-, l ^{2}-, and l ^{10}-norms. For aerial with σ _{ η }=10, the PSNRs are 30.67, 30.67, and 30.60, respectively (with tuned scaling parameters). For aerial with σ _{ η }=40, the PSNRs are 23.01, 23.13, and 23.05, respectively (also with tuned scaling parameters). As with the other parameters, we do not see a strong sensitivity to the choice of distance metric. However, the l ^{2}-norm generally provides the best results.
5 Conclusions
We have proposed a novel CAWF method for image restoration, which can be thought of an extension of the AWF [5] using multiple patches. For each reference window, M similar patches are identified. The output is formed as a single-pass weighted sum of all of the pixels from the multiple selected patches. Wiener weights are used to provide a minimum MSE estimate for this filter structure. A key aspect of the method is the new spatial-domain multi-patch correlation model, presented in Section 1. This model attempts to capture the spatial correlation among the samples within a given patch and also the correlations among the patches.
The CAWF is able to jointly perform denoising and deblurring. We believe this type of joint restoration is advantageous, compared with decoupling these operations. The CAWF algorithm is also capable of adapting to local signal and noise variance. Bad or missing pixels can easily be accommodated by leaving them out of the multi-patch observation vector and corresponding correlation statistics. The weights will adapt in a non-trivial way to the missing pixels [5,9,10].
In simulated and real data for Gaussian noise, the CAWF outperforms the benchmark methods in our experiments in Sections 1 and 1, both in PSNR and in SSIM. With blur and noise, CAWF produces the highest SSIM in more cases than the benchmark methods. However, IDD-BM3D does provide a higher PSNR in more instances. Our results show that the CAWF method consistently outperforms the AWF. This clearly demonstrates that incorporating multiple patches within this filter structure is advantageous. From the results in Section 1, we also conclude that CAWF performance is not highly sensitive to the tuning parameter values within a given operating range.
We believe the single-pass weighted-sum structure of the CAWF method is conceptually simple and versatile. It is also highly parallel. In principle, each output pixel can be computed in parallel. We have demonstrated that the method provides excellent performance in image restoration with noise and blur and noise. This method may be beneficial in numerous other applications as well, including those where its predecessor, the AWF, is successful [5,9-14]. We think there may also be an opportunity for further improvements in the parametric correlation model that could boost filter performance. Thus, we hope this approach will be of interest to the signal and image processing community.
Declarations
Authors’ Affiliations
References
- G Demoment, Image reconstruction and restoration: overview of common estimation structures and problems. IEEE Trans Acoustics, Speech Signal Process. 37(12), 2024–36 (1989).View ArticleGoogle Scholar
- MI Sezan, AM Tekalp, Survey of recent developments in digital image restoration. Opt Eng. 29(5), 393–404 (1990).View ArticleGoogle Scholar
- AK Jain, Fundamentals of Digital Image Processing (Prentice Hall, New Jersey, 1989).Google Scholar
- SP Ghael, AM Sayeed, RG Baraniuk, in Proc of SPIE, 3169. Improved wavelet denoising via empirical Wiener filtering (SPIESan Diego, 1997), pp. 389–99.Google Scholar
- RC Hardie, A fast image super-resolution algorithm using an adaptive Wiener filter. IEEE Transactions on Image Process. 16, 2953–64 (Dec. 2007).View ArticleMathSciNetGoogle Scholar
- KE Barner, AM Sarhan, RC Hardie, Partition-based weighted sum filters for image restoration. IEEE Trans Image Process. 8, 740–745 (May 1999).View ArticleGoogle Scholar
- M Shao, KE Barner, RC Hardie, Partition-based interpolation for image demosaicing and super-resolution reconstruction. Opt Eng. 44, 107003–1–107003–14 (Oct 2005).View ArticleGoogle Scholar
- B Narayanan, RC Hardie, KE Barner, M Shao, A computationally efficient super-resolution algorithm for video processing using partition filters. IEEE Trans Circuits Syst Video Technol. 17, 621–34 (May 2007).View ArticleGoogle Scholar
- RC Hardie, KJ Barnard, R Ordonez, Fast super-resolution with affine motion using an adaptive Wiener filter and its application to airborne imaging. Opt Express, 1926208–31 (Dec 2011).Google Scholar
- RC Hardie, KJ Barnard, Fast super-resolution using an adaptive Wiener filter with robustness to local motion. Opt Express. 20, 21053–73 (Sep 2012).View ArticleGoogle Scholar
- B Narayanan, RC Hardie, E Balster, Multiframe adaptive Wiener filter super-resolution with JPEG2000-compressed images. EURASIP J Adv Signal Process. 2014(1), 55 (2014).View ArticleGoogle Scholar
- RC Hardie, DA LeMaster, BM Ratliff, Super-resolution for imagery from integrated microgrid polarimeters. Opt Express. 19, 12937–60 (Jul 2011).View ArticleGoogle Scholar
- BK Karch, RC Hardie, Adaptive Wiener filter super-resolution of color filter array images. Opt Express, 2118820–41 (Aug 2013).Google Scholar
- M Rucci, RC Hardie, KJ Barnard, 53. Appl Opt, C1–13 (May 2014).Google Scholar
- A Buades, B Coll, JM Morel, A review of image denoising algorithms, with a new one. Multiscale Model Simul. 4, 490–530 (2005).View ArticleMATHMathSciNetGoogle Scholar
- KE Barner, GR Arce, J-H Lin, On the performance of stack filters and vector detection in image restoration. Circuits Syst Signal Process. 11, No. 1 (Jan 1992).View ArticleGoogle Scholar
- KE Barner, RC Hardie, GR Arce, in Proceedings of the 1994 CISS. On the permutation and quantization partitioning of R N and the filtering problem (New Jersey, Princeton, Mar 1994).Google Scholar
- A Buades, B Coll, JM Morel, in Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Comput Soc Conference on, vol. 2. A non-local algorithm for image denoising (IEEE, June 2005), pp. 60–65.Google Scholar
- C Kervrann, J Boulanger, Optimal spatial adaptation for patch-based image denoising. IEEE Trans Image Process. 15(10), 2866–78 (2006).View ArticleGoogle Scholar
- A Buades, B Coll, J-M Morel, Nonlocal image and movie denoising. Int J Comput Vision. 76, 123–39 (Feb 2008).View ArticleGoogle Scholar
- Y Han, R Chen, Efficient video denoising based on dynamic nonlocal means. Image Vision Comput. 30, 78–85 (Feb 2012).View ArticleGoogle Scholar
- Tasdizen, Principal neighborhood dictionaries for nonlocal means image denoising. IEEE Trans Image Process. 18, 2649–60 (July 2009).View ArticleMathSciNetGoogle Scholar
- H Bhujle, S Chaudhuri, Novel speed-up strategies for non-local means denoising with patch and edge patch based dictionaries. IEEE Trans Image Process. 23, 356–365 (Jan 2014).View ArticleMathSciNetGoogle Scholar
- Y Wu, B Tracey, P Natarajan, JP Noonan, SUSAN controlled decay parameter adaption for non-local means image denoising. Electron Lett. 49, 807–8 (June 2013).View ArticleGoogle Scholar
- WL Zeng, XB Lu, Region-based non-local means algorithm for noise removal. Electron Lett. 47, 1125–7 (September 2011).View ArticleGoogle Scholar
- WF Sun, YH Peng, WL Hwang, Modified similarity metric for non-local means algorithm. Electron Lett. 45, 1307–9 (Dec 2009).View ArticleGoogle Scholar
- C Tomasi, R Manduchi, in Proceedings of the 1998 IEEE International Conference on Computer Vision, Bombay. Bilateral filtering for gray and color images (IEEEIndia, 1998).Google Scholar
- H Kishan, CS Seelamantula, Sure-fast bilateral filters. Acoustics, Speech and Signal Processing (ICASSP) 2012 IEEE International Conference on (IEEE, Kyoto, 2012).Google Scholar
- W Kesjindatanawaj, S Srisuk, in Communications and Information Technologies (ISCIT), 2013 13th International Symposium on. Deciles-based bilateral filtering (IEEESurat Thani, 2013), pp. 429–33.Google Scholar
- X Changzhen, C Licong, P Yigui, An adaptive bilateral filtering algorithm and its application in edge detection. Measuring Technology and Mechatronics Automation (ICMTMA), 2010 International Conference on. vol. 1 (IEEE, Changsha City, 2010).Google Scholar
- H Peng, R Rao, SA Dianat, Multispectral image denoising with optimized vector bilateral filter. IEEE Trans Image Process. 23, 264–73 (Jan 2014).View ArticleMathSciNetGoogle Scholar
- K Dabov, A Foi, V Katkovnik, K Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering. IEEE Trans Image Process. 16, 2080–95 (Aug 2007).View ArticleMathSciNetGoogle Scholar
- K Dabov, A Foi, K Egiazarian, Video denoising by sparse 3d transform-domain collaborative filtering. Proc 15th Eur Signal Process Conference. 1, 7 (2007).Google Scholar
- M Maggioni, G Boracchi, A Foi, K Egiazarian, Video denoising, deblocking, and enhancement through separable 4-d nonlocal spatiotemporal transforms. IEEE Trans Image Process. 21(9), 3952–66 (2012).View ArticleMathSciNetGoogle Scholar
- K Hirakawa, T Parks, Image denoising using total least squares. IEEE Trans Image Process. 15(9), 2730–42 (2006).View ArticleGoogle Scholar
- P Chatterjee, P Milanfar, Patch-based near-optimal image denoising. IEEE Trans Image Process. 21, 1635–49 (April 2012).View ArticleMathSciNetGoogle Scholar
- H Talebi, P Milanfar, Global image denoising. IEEE Trans Image Process. 23, 755–768 (Feb 2014).View ArticleMathSciNetGoogle Scholar
- K Dabov, A Foi, V Katkovnik, K Egiazarian, in SPIE Electronic Imaging, 6812. Image restoration by sparse 3d transform-domain collaborative filtering (San Jose, Jan 2008).Google Scholar
- A Danielyan, V Katkovnik, K Egiazarian, BM3D frames and variational image deblurring. IEEE Trans Image Process. 21, 1715–28 (April 2012).View ArticleMathSciNetGoogle Scholar
- K Nasrollahi, TB Moeslund, Super-resolution: a comprehensive survey. Mach Vision Appl. 25(6), 1423–68 (Aug 2014).View ArticleGoogle Scholar
- M Protter, M Elad, Super-resolution with probabilistic motion estimation. IEEE Trans Image Process. 18(8), 1899–904 (2009).View ArticleMathSciNetGoogle Scholar
- M Protter, M Elad, H Takeda, P Milanfar, Generalizing the nonlocal-means to super-resolution reconstruction. IEEE Trans Image Process. 18(1), 36–51 (2009).View ArticleMathSciNetGoogle Scholar
- MH Cheng, HY Chen, JJ Leou, Video super-resolution reconstruction using a mobile search strategy and adaptive patch size. Signal Process. 91, 1284–97 (2011).View ArticleGoogle Scholar
- B Huhle, T Schairer, P Jenke, W Straber, Fusion of range and color images for denoising and resolution enhancement with a non-local filter. Comput Vision Image Understanding. 114, 1336–45 (2012).View ArticleGoogle Scholar
- K-W Hung, W-C Siu, Single image super-resolution using iterative Wiener filter. Proc IEEE Int Conference Acoustics, Speech and Signal Process, 1269–72 (2012).Google Scholar
- Z Wang, A Bovik, H Sheikh, E Simoncelli, Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process. 13, 600–12 (April 2004).View ArticleGoogle Scholar
- J Portilla, Image restoration through l0 analysis-based sparse optimization in tight frames. Image Process (ICIP), 2009 16th IEEE Int Conference on, 3909–912 (Nov 2009).Google Scholar
- J Oliveira, JM Bioucas-Dias, MA Figueire, Adaptive total variation image deblurring: A majorization-minimization approach. Signal Process. 89, 1683–93 (Sep 2009).View ArticleMATHGoogle Scholar
- M Makitalo, F Foi, Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise. IEEE Trans Image Process. 22(1), 91–103 (2013).View ArticleMathSciNetGoogle Scholar
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