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Abstract

Considering the objects by different granularity reflects the recognition common law of people, granular computing embodies the transformation between different granularity spaces. We present the image denoising algorithm by using the dictionary trained by granular computing with L∞-norm, which realizes three transformations, (1) the transformation from image space to patch granule space, (2) the transformation between granule spaces with different granularities, and (3) the transformation from patch granule space to image space. We demonstrate that the granular computing with L∞-norm achieved the comparable peak signal to noise ratio (PSNR) measure compared with BM3D and patch group prior based denoising for eight natural images.

Keywords

Granular computing patch granule image patch image denoising L∞-norm

Introduction

With the popularization of various digital instruments and digital products, image and video have become the main way to obtain the original information of the outside world and the most common information carriers in human activities. However, in the process of image acquisition, transmission, and storage, image quality is often affected by noise, and image preprocessing algorithm is directly related to the subsequent image processing, such as image segmentation, object recognition, edge extraction, and so on, it is necessary to obtain high-quality digital images.^1–3 Therefore, image denoising has been a hot spot in the research of image processing and computer vision.

As a classical problem in low level vision, image denoising is one of the well-studied problems in the field of image processing, yet it is still an active topic for that it provides an ideal test bed for image-modeling techniques.

The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics. In spite of the sophistication of the recently proposed methods, most algorithms have not yet attained a desirable level of applicability.⁴

Spatial filtering is a direct data operation on the original image, the gray value of the pixel is processed. The common spatial domain image denoising algorithm has the low pass filter, the neighborhood average method, the median filter, etc.^5,6

Transform domain image denoising method is a transform of the image. First, the image is transformed from spatial domain to transform domain. Second, the coefficients of transform are processed in the transform domain. Third, the inverse transform of the image from the transform domain into spatial domain to remove image noise. There are many transform domain method, such as, fast fourier transform/discrete fourier transform (FFT/DFT), wavelet transform, which are the common methods for image denoising.^7,8

A patch-based generalization of the bilateral filter⁹ was proposed in Buades et al.⁴ and Awate and Whitaker,¹⁰ where the concept of locality was extended to the entire image. Although the results were encouraging, the true potential for the nonlocal means (NLMs) method was only realized in Kervrann and Boulanger¹¹ and Kervrann and Boulanger.¹² Another patch redundancy-based framework, i.e., BM3D,¹³ adopts a hybrid approach of grouping similar patches and performing collaborative filtering in some transform domain. It ranks among the best performing methods that define the current state of the art. A significantly different approach to denoising was introduced in K-singular value decomposition (SVD).¹⁴ Building on the notion of image patches being sparse representation,¹⁵ Elad et al., proposed a greedy approach for dictionary learning tuned for denoising. In Chatterjee and Milanfar,¹⁶ P. Chatterjee and P. Milanfar proposed a hybrid approach K-locally learned dictionary (K-LLD) that bridged such dictionary-based approaches with the regression-based frameworks.^4,9,11 The motivation there was that similar patches shared similar subdictionaries, and such subdictionaries could be used for better image modeling. A similar observation was exploited in the form of a nonlocal sparse model (NLSM)¹⁷ to improve performance of the K-SVD¹⁵ framework. L. Zhang proposed the image denoising algorithm of patch group prior–based denoising (PGPD), in which patch groups are extracted from training images by putting nonlocal similar patches into groups, and a PG-based Gaussian Mixture Model (PG-GMM) learning algorithm is developed to learn the nonlocal self-similarity (NSS) prior.¹⁸

In this paper, we present the image denoising algorithm by using the dictionary trained by granular computing (GrC) with L∞-norm, which realizes three transformations, (1) the transformation from image space to granule space, (2) the transformation between granule spaces with different granularities, and (3) the transformation from granule space-to-image space. We demonstrate that the GrC with L∞-norm achieved the comparable PSNR measure compared with BM3D and PGPD for eight natural images.

The training process by GrC with L∞-norm

For a given clear training image I, we extract N patches with size p × p, and form the training set including N vectors, which are induced by transformation from patch into vector. There is redundancy in the training set S, the purpose of training process is to reduce redundancy by GrC, and then obtain the centers of granules, covariance matrices, mixed Gaussian components by likelihood estimation, such as GMM proposed in ZoranWeiss.¹⁹

In this section, we mainly discuss the GrC with L∞-norm, which includes the representation of granule, the join operation between two granules, and the GrC algorithms with L∞-norm.

The partition of image

The image I is partitioned into some patches with the patch size p × p and the overlapped parameter step by the feature values of pixel points, such as the gray value, RGB (stands for red, green, and blue), HSV (stands for hue, saturation, and value), HSL (stands for hue, saturation, and luminance), and gradient feature (first-order gradient, second gradient). Suppose p = 4 and step = 2, the gray image with 7 × 6 can be partitioned the four patches with 4 × 4 shown in Figure 1.

Figure 1.

The partition of image with patch size p = 4 and step = 2.

Granular computing with L∞-norm

As mentioned above, for the given image I, we can partition it into some image patches and form patch set. Each patch is represented as a vector, such as x_i = (x_i₁, x_i₂, …, x_in) and |x_i| = p × p for the image patch with size p × p, namely each patch corresponds to a vector.

Representation of granules. In the paper, for the patch set S, the patch granule is composed of the data x whose distance from the center C is less than the granularity. The distance between center vector C = (c₁, c₂ …, c_N) and vector x = (x₁, x₂ …, x_N) can be defined as follows

d_{\infty} (x, C) = max {| x_{1} - c_{1} |, | x_{2} - c_{2} |, \dots, | x_{N} - c_{N} |}

(1)

So we can see the patch granule is determined by two factors, such as the center and the granularity. The patch granule is defined as follows

G = (C, R)

(2)

In general, the patch granule includes some patches when R > 0, especially, the patch granule is a point and indivisible and regarded as atomic granule when R = 0.

For the patch set S = {x_i|i = 1,2 …, n} in N-dimensional space, GrC is to train the dictionary including some patch granules by the following steps. First, the granule representation is proposed for the patch. Second, join between two granules are designed, and the threshold of granularity controls the process of operations. Third, the dictionary including patch granules is obtained.

Figure 2 shows the process of image partition, granulation of patches from Figure 1. Each patch is represented as an atomic granule with the granularity 0. As mentioned in Figure 1, each patch is composed by 16(=4 × 4) feature values, which are sorted by indices, so each patch is transferred a vector and represented as a granule named as patch granule.

Figure 2.

The representation of patches by granules.

Operations between two granules. In reality, for the same image, the resolution changes with the distance between the image and the human eyes. If we recognize the image patch in detail, we must close to it. In GrC, the granularity is generally the size of granule, and reflects the degree that people recognize object. The less granularity means the more detail recognized by human, on the contrary, people understand objects only as a whole. The operations between two granules realize the transformation between granule space with different granularities. Operation ∨ unites the granules with small granularities to the granules with the large granularities. Inversely, operation ∧ divides the granules with large granularities into the granules with small granularities.

In the paper, we only discuss the operation ∨ which is used to train the dictionary composed by the image patches. A point is regarded as atomic granule with the granularity 0, which is indivisible, the join process is the key to obtain the larger granules compared with atomic granules.

For two patch granules G₁ = (C₁, R₁) and G₂ = (C₂, R₂), the join patch granule is

G = G₁∨G₂ = (C, R)

The center C and the granularity R of G are computed as follows.

First, the vector from C₁ to C₂ and vector from C₂ to C₁ are computed. If C₁ = C₂, then C₁₂ = 0 and C₂₁ = 0. If C₁ ≠ C₂, then C₁₂ = (C₂ − C₁)/d_∞(C₁,C₂) and C₂₁ = (C₁ − C₂)/d_∞(C₂,C₁), where d_∞(.,.) is induced by the formula (1), C₁₂ is the vector with the lenth 1 from C₁ to C₂, C₂₁ is the vector with the length 1 from C₂ to C₁.

Second, the crosspoints of G and G₁ are P₁ = C₁ − C₁₂R₁ and P₂ = C₁ + C₁₂R₁. The crosspoints of G and G₂ are Q₁ = C₂ − R₂C₂₁ and Q₂ = C₂ + R₂C₂₁.

Third, the center C and granularity R of the join patch granule G is computed by algorithm 1.

Algorithm 1. Computing C and R of join patch granule G between G₁ and G₂
Input: G₁ = (C₁,R₁) and G₂ = (C₂,R₂)
Output: G = (C,R)
if R₁ ≥ R₂
if d_∞(C₁,C₂) ≥ R₁ − R₂ C = C₁ R = R₁
else C = (P₁ + Q₁)/2 R = d_∞(P₁,Q₁)/2
end
else
if d_∞(C₁,C₂) ≤ R₂ − R₁ C = C₂ R = R₂
else C = (P₁ + Q₁)/2 R = d_∞(P₁,Q₁)/2
end
end

The framework of GrC. For the patch set S, the GrC includes the following steps for image denoising. First, all the patches induced by the training image sequence are represented as the atomic granules. Second, the threshold of granularity is introduced to conditionally operate ∨ the atomic granules by the aforementioned join operation. Third, the patch granules are obtained and used to form image denoising algorithms. The GrC is described as follows.

Algorithm 2. GrCInput: Patch set S, threshold ρ of granularityOutput: Patch granule set
S1. Initialize the patch granule set PG = ∅
S2. i = 1
S3. For the ith patch x_i in S, form the corresponding atomic granule G_i
S4. j = 1
S5. Form the join patch granule G_i∨G_j of G_i and G_j ∈ PG, if the granularity of G_i∨G_j is ≤ρ, then G_j = G_i∨G_j, else
S6. j = j + 1
S7. If all the granularities of G_i∨G_j are greater than ρ, then PG = PG ∪ {G_i}
S8. Remove patch x_i from the patch set S until S is empty.

For algorithm 2, the patch set is composed of the image patch from an image or image sequence, each PG consists of some patches with the similarity. We explain the algorithm 2 by the selected image named t53 from the image sequence.²⁰ The image has the size 277 × 285, if the patch size is set to 20 × 20, namely p = 20, and the overlap size is set to 0, namely step = 0, the image is partitioned into 182 image patches which form the patch set. For the image patch marked the red lines shown in Figure 3, there are six image patches (Figure 3(a)) marked the green lines around it if we set ρ = 0.2, there are 24 image patches (Figure 3(b)) marked the green lines around it if we set ρ = 0.4. The situation coincides with the reality. In general, the PG with the larger granularity contains the more patches compared with the PG with the smaller granularity.

Figure 3.

The patch granules with the different granularities for the same image.

GrC–GMM learning

A GMM is a parametric probability density function represented as a weighted sum of Gaussian component densities. GMMs are commonly used as a parametric model of the probability distribution of continuous measurements or features. GMM parameters are estimated from training data using the iterative expectation–maximization (EM) algorithm or maximum A posteriori (MAP) estimation from a well-trained prior model. We estimate the parameters of GMM by GrC (GrC–GMM)

We take the learning problem in two-dimensional space as an example to illustrate the GrC–GMM learning process.

The mean and covariance of cluster 1 are [1 4] and $(\begin{matrix} 1 & - 0.1 \\ - 0.1 & 1 \end{matrix})$ , the mean and covariance of cluster 2 are [4 1] and $(\begin{matrix} 1 & 0.2 \\ 0.2 & 1 \end{matrix})$ , and the mean and covariance of cluster 3 are [5 5] and $(\begin{matrix} 1 & - 0.3 \\ - 0.3 & 1 \end{matrix})$ .

First, we mix all the training data into the data set X and perform GrC to obtain the three clusters by adjusting the parameter ρ, the three clusters are obtained by GrC when the parameter ρ is 4.9.

Second, for each cluster, we estimate the mean and covariance. For the training data, the learned mean and covariance of cluster 1, by GrC with L∞-norm, are m₁ = [0.8917 4.1745] and $c_{1} = (\begin{matrix} 0.8729 & - 0.1741 \\ - 0.1741 & 1.0292 \end{matrix})$ , the learned mean and covariance of cluster 2 are m₂ = [5.0274 5.0960] and $c_{2} = (\begin{matrix} 0.9437 & - 0.2462 \\ - 0.2462 & 1.3486 \end{matrix})$ , the learned mean and covariance of cluster 3 are m₃ = [4.0449 1.0998] and $c_{3} = (\begin{matrix} 1.3295 & 0.1426 \\ 0.1426 & 1.1790 \end{matrix})$ , the weights of principal components are w₁ = 0.3212, w₂ = 0.3560, and w₃ = 0.3229.

Third, by SVD, covariance matrix Σ can be factorized as $Σ = D Λ D^{T}$ , namely

\begin{array}{l} D_{1} = (\begin{matrix} - 0.3482 & 0.9374 \\ 0.9374 & 0.3482 \end{matrix}), \\ Λ_{1} = (\begin{matrix} 1.3058 & 0 \\ 0 & 0.7233 \end{matrix}), \\ D_{2} = (\begin{matrix} 0.4505 & 0.8928 \\ 0.8928 & - 0.4505 \end{matrix}) \end{array}

\begin{array}{l} Λ_{2} = (\begin{matrix} 1.3775 & 0 \\ 0 & 0.7376 \end{matrix}), \\ D_{3} = (\begin{matrix} - 0.7328 & 0.6805 \\ 0.6805 & 0.7328 \end{matrix}), \\ Λ_{3} = (\begin{matrix} 1.3541 & 0 \\ 0 & 0.6801 \end{matrix}) \end{array}

For the input noisy datum x, the clean x is estimated by the following formula

\hat{x} = x (w_{1} D_{1} Λ_{1} D_{1}^{- 1} + w_{2} D_{2} Λ_{2} D_{2}^{- 1} + w_{3} D_{3} Λ_{3} D_{3}^{- 1})

When $w = (w_{1}, w_{2}, w_{3})$ is sparse, such as $w_{2} \approx 0$ , $w_{3} \approx 0$ , the clean x is estimated by

\hat{x} \approx x w_{1} D_{1} Λ_{1} D_{1}^{- 1}

Image denoising via GrC with L∞-norm

For a noisy image y, we partition y into some patches with the same patch size as the patch in learned patch granule dictionary obtained by the aforementioned GrC–GMM, for each patch we search its similar patch granule in the W × W window, denoted by Y = {y₁, y₂ …, y_M}. Then the group mean of Y, denoted by u_y, is calculated and subtracted from each patch, leading to the mean subtracted PG $\bar{Y}$ ={y₁ − u_y, y₂ − u_y …, y_M − u_y}. We can write $\bar{Y}$ as $\bar{Y} = \bar{X} + V$ , where $\bar{X}$ is the corresponding clean patch granule and V contains the corrupted noise. The problem then turns to how to recover $\bar{X}$ from $\bar{Y}$ by using the learned GrC–GMM priors. Note that the mean u_y of Y is very close to the mean of X since the mean vector of noise V is nearly zero. u_y will be added back to the denoised patch granule to obtain the denoised image.

In practice, the clean patch $x_{m}$ is estimated by the incremental process, namely we perform the denoising process by several iterations, and the noise is gradually reduced during the iteration process described by algorithm 3.

Algorithm 3. Patch granule prior–based denoising
Input: Noisy image y, GMM model by GrC
Output: The denoised image x
S1. Initialization ${\hat{x}}^{(0)} = y$ , $y^{(0)} = y$
S2. For t = 1: MaxIter $y^{(t)} = {\hat{x}}^{(t - 1)} + δ (y - y^{(t - 1)})$
S3. Estimate the standard deviation of noise for $y^{(t)}$
S4. Find Y for each patch of $y^{(t)}$
S5. For i = 1: M
S6. Compute mean u_y and $\bar{Y}$ for Y
S7. Select Gaussian component GrC–GMM
S8. Compute the weights of centers of patch granules by GrC–GMM
S9. Recover the estimated clean patch ${\hat{x}}_{m}$
S10. Aggregate the estimated clean patches to form the estimated clean image ${\hat{x}}_{t}$
S11. End for i
S12. End for t

EM is often combined with the GMM for data clustering in machine learning and computer vision. For algorithm 3, the GrC clustering algorithm is used to perform GMM for image denoising in XuZhang et al.,¹⁸ instead of EM. Since the scanning data set is completed one time, the computational complexity of GrC is O(n), where n is the number of data set, for the image denoising, n is the number of patches from the training image sequence.

Experiments

Implementation details

The five training images selected from Yang et al.²⁰ are used to perform the training process (see Figure 4). If the size of patch is set to 7, there are 576,732 atomic granules induced by the patches. The GrC is performed to reduce the redundancy by the suitable parameter, and the centers of patch granules are used to perform the denoising process.

Figure 4.

The training image.

Parameter setting

The patch group–based GrC with L∞-norm for image denoising contains two stages, the prior learning stage by GrC and the denoising stage. In learning stage, there are p and ρ. We set the patch size (p) as p = 7 for σ = 10, σ = 20, σ = 30, σ = 40 in order to compared the proposed image denoising algorithm with state-of-the-art algorithms.¹⁸ In the denoising stage, there are three parameters: c, δ, and η. We set the same c, δ, and η as PGPD in our implementation, namely (c, δ, η) are set to (0:32; 0:08; 0:77), (0:27; 0:10; 0:71), (0:17; 0:07; 0:86), (0:13; 0:05; 0:96) when σ = 10; 20; 30; 40, respectively. In addition, on all noise levels, we stop Algorithm 1 in four iterations.

Comparison methods

We compare the proposed patch group–based GrC for image denoising algorithm with BM3D,¹³ EPLL,¹⁹ and PGPD,¹⁸ which represent the state-of-the-arts of modern image denoising techniques and all of them exploit image nonlocal self-similarity (NSS). The source codes of all competing algorithms are downloaded from the authors’ websites and we use the default parameter settings.

Result analysis

We evaluate the competing methods from PSNR shown in Table 1, σ is the noise level, PSNR is the peak signal-to-noise ratio between the denoising image and the original image, the denoising image with the higher PSNR is closer to the original image than the denoising image with lower PSNR. We can analyze the data from the table vertically and horizontally, from a vertical point of view, the PSNR value decreases as the σ value increases, that is why we choose lower noise levels for the four selected denoising algorithms, such as σ = 10, 20, 30, 40, the performances of denoising algorithms are compared from a horizontal point of view.

Table 1.

PSNR (dB) results of different denoising algorithms on eight natural images.

Image	σ	BM3D¹³	EPLL¹⁹	ZL¹⁸	GrC
Barbara	σ = 10	32.6472	31.3465	32.2942	32.3989
	σ = 20	28.8689	26.9237	28.4281	28.3947
	σ = 30	26.7208	24.6637	26.1219	26.2235
	σ = 40	24.8852	23.4426	24.6074	24.8058
Bloodcell	σ = 10	34.7578	34.4092	34.9418	34.8733
	σ = 20	31.8389	31.3107	31.9166	31.9991
	σ = 30	29.9021	29.3521	30.0177	30.0199
	σ = 40	28.0651	27.8942	28.5102	28.4343
Cameraman	σ = 10	34.3082	34.0705	34.1878	34.2293
	σ = 20	30.6944	30.2818	30.4687	30.4058
	σ = 30	28.5812	28.1870	28.3278	28.4729
	σ = 40	26.8519	26.7524	26.8527	27.0290
House	σ = 10	36.0934	35.1149	35.9433	35.8605
	σ = 20	33.1796	32.3566	33.2322	33.2398
	σ = 30	31.4950	30.6077	31.6088	31.5748
	σ = 40	29.9775	29.2688	30.2535	30.2225
Lena	σ = 10	33.7598	33.5324	33.7378	33.8451
	σ = 20	30.1178	29.9547	30.2123	30.1425
	σ = 30	28.1090	27.9784	28.1657	27.9980
	σ = 40	26.5259	26.5928	26.7877	26.5495
Monarch	σ = 10	33.7069	33.8532	34.2099	34.2560
	σ = 20	29.8712	29.9870	30.2112	30.1302
	σ = 30	27.7928	27.7979	27.8859	27.8792
	σ = 40	25.9720	26.3096	26.3271	26.3824
Paint	σ = 10	33.5949	33.5962	33.9275	33.8840
	σ = 20	29.8702	29.9507	30.1703	30.0482
	σ = 30	27.7826	27.8848	27.9073	27.8017
	σ = 40	26.1517	26.4371	26.3963	26.1575
Peppers	σ = 10	33.5202	33.4252	33.4338	33.5132
	σ = 20	30.4827	30.4099	30.4766	30.3801
	σ = 30	28.6742	28.6368	28.6817	28.5526
	σ = 40	27.2696	27.3443	27.4568	27.2493

*Bold values indicate the best results.

In Table 1, we present the PSNR results on four noise levels σ = 10, 20, 30, 40, the best PSNRs are highlighted by boldface. From Table 1, we have several observations. First, denoised image by GrC is much better than by EPLL because there are 31 denoised images by GrC, which are better than the denoised images by EPLL for 32 images with different noise level. Second, PGPD achieves much better PSNR results than EPLL, BM3D, and GrC, this demonstrates that the dictionary learned by GrC needs to further optimization. Third, GrC achieved the best results for 7 images of 32 images compared with the other three state-or-the-art image denoising techniques, this shows that the denoising performance can be improved by the more training images. Considering that human subjects are the ultimate judge of the image quality, the visual quality of denoised images is also critical to evaluate a denoising algorithm. The denoised images with the best PSNR compared with the other three methods are shown from Figures 5 to 9.

Figure 5.

Denoised images of bloodcell by different methods (σ = 20). (a) Original image. (b) Noisy image. (c) BM3D. (d) EPLL. (e) PGPD. (f) GrC.

Figure 6.

Denoised images of cameraman by different methods (σ = 40). (a) Original image. (b) Noisy image. (c) BM3D. (d) EPLL. (e) PGPD. (f) GrC.

Figure 7.

Denoised images of house by different methods (σ = 20). (a) Original image. (b) Noisy image. (c) BM3D. (d) EPLL. (e) PGPD. (f) GrC.

Figure 8.

Denoised images of lena by different methods (σ = 10). (a) Original image. (b) Noisy image. (c) BM3D. (d) EPLL. (e) PGPD. (f) GrC.

Figure 9.

Denoised images of monarch by different methods (σ = 40). (a) Original image. (b) Noisy image. (c) BM3D. (d) EPLL. (e) PGPD. (f) GrC.

Conclusion

NSS prior for image restoration is a class of popular image denoising method and an open problem, and we made a good attempt using the dictionary trained by GrC with L∞-norm on the basis of patch group. After group mean subtraction, a granule induced by image patch can naturally represent the group of image patches. A GrC-based GMM learning algorithm was developed to learn the NSS prior from natural images, and an associated weighted sparse coding algorithm was developed for high-performance image denoising. The feasibility of image denoising is demonstrated by eight natural images and measured by PSNR. On the one hand, in the experiment section of this paper, we only discuss the denoising of gray image, the denoising of color image is our future work because the color image includes multiple features, such as RGB image includes red, green, and blue channels, the denoising algorithm can be performed on three channels respectively, the clean color image is combination of the clean images from three channels. On the other hand, the patch granule space is induced by GrC with L∞-norm in the paper, the GrC can be extended to the other shape of granule, such as granule induced by L1 norm and L2 norm, for image denoising.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the Natural Science Foundation of China (Grant No. 61170202, 61202287), Henan Natural Science Foundation Project (182300410145), and the Teacher Education Curriculum Reform Project of Henan Province in China (2016-JSJYZD-027).

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Yang

Wright

Huang

et al . Image super-resolution via sparse representation. IEEE Trans Image Process 2010; 19: 2861–2873.

Training dictionary by granular computing with L∞-norm for patch granule–based image denoising

Abstract

Keywords

Introduction

The training process by GrC with L∞-norm

The partition of image

Granular computing with L∞-norm

GrC–GMM learning

Image denoising via GrC with L∞-norm

Experiments

Implementation details

Parameter setting

Comparison methods

Result analysis

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References