^{1}

^{2}

^{1}

^{2}

We proposed an efficient image denoising scheme by fused lasso with dictionary learning. The scheme has two important contributions. The first one is that we learned the patch-based adaptive dictionary by principal component analysis (PCA) with clustering the image into many subsets, which can better preserve the local geometric structure. The second one is that we coded the patches in each subset by fused lasso with the clustering learned dictionary and proposed an iterative Split Bregman to solve it rapidly. We present the capabilities with several experiments. The results show that the proposed scheme is competitive to some excellent denoising algorithms.

As an essential low-level image processing procedure, image denoising is studied extensively, which is also a classical inverse problem. The general observation with additive noise is modeled as:

With the degradation model in (

More recently, a set of approaches with nonlocal techniques (NL) is used for removing noise. The idea of NL can be tracked to [

In this paper, we proposed a novel scheme for image denoising based on clustering dictionary learning. Firstly, we clustered the patches with similar geometric structure by taking a weight function as the feature. Secondly, we learned the patch-based dictionary by principle component analysis for each cluster. Lastly, we coded these patches by fused lasso and developed an iterative Split Bregman to solve it rapidly.

The rest of the paper is outlined as follows. In Section

Kernel regression is well studied in statistical signal processing. Recently, the KR is used to address many image restoration tasks, such as the denoising, interpolation, and deblurring [

By (

Introduced in [

Conveniently, the matrix

By (

With

We summarized the calculation of weights of each patch in Algorithm

Initialization: patch

(1) Calculate the local gradient matrix

(2) Implement the SVD to

(3) Compute the covariance matrix

(4) Compute the

(5) If

Then, we can cluster the ordering overlapping patches into subset

Once the clusters are formed, we can learn a dictionary with the principal component analysis suited to each cluster independently. To this end, according to the general dictionary learning algorithm, we need to solve the following minimization:

As the minimization in (

Then, assuming the

The patches in the same cluster have similar structure to each other, so we do not require the dictionary to be redundant enough. So, to simplify the problem (

The minimization problem in (

Our above learning method can train the dictionary with lower complexity. In additional, to make it more effective and compact, we also show a selection scenario in (

We summarized the dictionary learning with PCA in Algorithm

Input: Centered samples matrix

Output: Dictionary

(1) Compute the decomposition of

(2) Choose the

(3) Output the

With the preparation work in Sections

The reference [

To solve (

Note that the initial idea to get the solution of (

Furthermore, by the iterative Split Bregman, we can divide (

As to the quadratic minimization in (

While the

We summarized the coding algorithm in Algorithm

Initialization:

For

End

Output: the recovered patch

Now, with all the preparation works above, we can summarize the synthetic denoising algorithm in Algorithm

Input: A set of overlapping patches from noisy image

Output: The recovered image

(1) For

(2) Compute the weights vector

(3) Cluster the patches into

(4) For

(5) Learn the dictionary

(6) Recover all the patches in

(7) End

(8) Compute the

their indexes. Then, update

(9) End

We conducted various experiments on image denoising to demonstrate the performance of our proposed algorithm. We degrade the images by adding artificial zero mean Gaussian noise with different standard deviations. The test images are shown in Figure

Test images.

Lena

House

Couple

Man

We compared our proposed algorithm to several current excellent denoising approaches, including the FGTV method in [

Denoising results of Lena.

TV

BM3DW

KSVD

KR

TSPCA

Proposed

Denoising results of Couple.

TV

BM3DW

KSVD

KR

TSPCA

Proposed

PSNR results of test images.

PSNR results of Lena

PSNR results of House

PSNR results of Couple

PSNR results of Man

From the denoising results, we can note that the FGTV method has the worst visualization among the compared methods. Because it only constrains the total variation but does not consider the local structure adaptively, so it lost many details in original image and also showed the disadvantage in PSNR results. KR algorithm generates many mottled artifacts in denoised image, but it indeed preserves some texture by capturing the local structure with kernel function, such as the curtain in Couple. But its results declined rapidly with the increasing noise standard deviation both in visual quality and in PSNR. TSPCA generated specific smoothness in the recovered image and was weak to present the texture region. In addition, the other factor resulting in bad performance in texture is that the PCA dictionary is learned with neighboring patches, which will show weaker results than the nonlocal scheme. We can see the PSNR of BM3DW, DKSVD, and the proposed method are approximate to each other in Figure

In this paper, we propose a novel scheme for image denoising. To preserve the textures in image, we clustered the patches from the noisy image with the meaningful weights vector which can capture the underlying local structure. And then, we learned the dictionary to better present the patches for each cluster with PCA. Lastly, we coded the noisy patches with the learned dictionary by fused lasso and obtained recovery image. We compared our proposed scheme to some current excellent algorithms, and it can be seen that our method obtains the good performance both in visual quality and in PSNR among the compared methods. In addition, the dictionary learning and coding are performed independently in each cluster, so it can be easily parallelized by the processors with multiple cores when the image has been clustered. That means the proposed method can be used in the large-scale image denoising task and save more computational time effectively.

The authors declare that there is no conflict of interests regarding the publication of this paper.