Image denoising via 2D dictionary learning and adaptive hard thresholding
Introduction
Image denoising is an indispensable step in any imaging system, and it remains one of the most important problems in the field of image processing. It is now well realized that image priors (assumption about the principle that the image data complies with) play a central role in image denoising, and there is a long sequence of image priors accompanying the progress in the research of image restoration, ranging from global smoothness to sparsity with respect to a certain dictionary, to self-similarity of image content, etc.
Variational method assumed a smooth function space, such as bounded variation space and besov space, etc., as a prior and measured the global smoothness of nature images by the norm or semi-norm defined in this space (Rudin et al., 1992, Aubert and Kornprobst, 2006). This method has drawn considerable attention due to its mathematical fascination. However, natural images are too complex to be uniformly contained in a single smooth function space. As a consequence, this method tends to smooth out tiny structures of natural images, such as fine textures and weak edges.
The sparse representation method assumed that any informative image can be well represented by a few atoms drawn from a dictionary (the dictionary can be redundant or not). The key of this method is to construct or learn an appropriate dictionary which can accurately fit the local structures of images. Wavelet has ever been widely used due to its multiscale and localization properties compared to the Fourier transform (Sendur and Selesnick, 2002, Pizurica et al., 2002), but it cannot represent the anisotropic structures well due to its isotropic nature. To capture the anisotropic structures, a flood of directional extensions, such as curvelet, contourlet and bandelet (Mallat, 2008) have been constructed. These directional extensions show certain advantages over the classical wavelet. However, natural images always contain diverse and irregular patterns which cannot be well characterized by these predefined analytics dictionaries, and denoising algorithms built on these dictionaries can introduce visual artifacts in the denoising output. To alleviate this problem, several adaptive learning methods have been proposed. Specifically, Elad and Aharon proposed a dictionary learning paradigm, known as K-SVD, for sparse and redundant image representation (Aharon et al., 2006, Elad and Aharon, 2006). Priyam et al. and Zhang et al. proposed to learn local dictionaries from the group of similar patches by using principle component analysis (PCA) (Chatterjee and Milanfar, 2009, Zhang et al., 2010). All these adaptive learning methods show promising denoising performance.
Self-similarity image prior assumes that there are fruitful repeating patterns inherent in natural images. To the best of our knowledge, there are two methods available for exploring these self-similarity structures, namely, nonlocal method (Buades and Morel, 2005, Mahmoudi and Sapiro, 2005, Dabov et al., 2007, Rajwade et al., 2013) and graph-based method (Coifman and Maggioni, 2006, Hammond et al., 2011, Shuman et al., 2013). Nonlocal means (NLM) algorithm (Buades and Morel, 2005) is viewed as the seminal work on the nonlocal method. In this algorithm, each pixel is estimated by the weighted average of all pixels in the image, and the weights are evaluated according to similarity between two neighborhood patches. The BM3D algorithm (Dabov et al., 2007) is the most celebrated nonlocal method which jointly estimates the group of similar patches by using 3D wavelet approximation and leads to the state-of-the-art denoising performance. It is now well realized that the jointly estimation of a group similar patches (group-wise estimation) always leads to promising results at the expense of relatively high computational complexity. Indeed, most recent nonlocal denoising algorithms focus on how to jointly model or estimate a group of similar patches (Dabov et al., 2007, Mairal et al., 2009, Rajwade et al., 2013). The graph-based method constructs a graph from image data. The edge of the graph is usually assigned as the similarity measure between two neighborhood patches, and the images data is viewed as a function defined on the graph. The significance of this method is that the images is analyzed and processed in an inhomogeneous space or domain.
In this letter, we sail under the flag of nonlocal method and present a complementary view. When a group of similar patches are arranged into a matrix, there exist linear correlations among both columns and rows of this matrix. Under the spirit of exploiting as much as possible structures underlying images for image modeling, we simultaneously consider the correlations among both columns and rows, and take the merits of singular value decomposition (SVD) to learn the 2D dictionaries for image denoising. In this approach, each group of similar patches are estimated by sparse approximation with respect to the learned 2D dictionaries and the sparse approximation is implemented by using adaptive hard thresholding. Additionally, the proposed approach is developed in a similar framework of K-SVD, which facilitates the extension of it to other inverse problems in imaging. Experiments indicate that the proposed algorithm achieves comparable peak signal to noise ratio (PSNR) performance with the BM3D algorithm, and are very competitive in fine structures preservation.
The rest of the letter is arranged as the following. Section 2 presents the proposed 2DDL-AHT denoising algorithm in detail. Section 3 presents the experiment results and Section 4 concludes.
Section snippets
Notations
We consider the following image formation modelwhere u represents the ideal true image observed as f, v represents the independent identically distributed Gaussian noise of variance σ2.
We follow the notation style in (Aharon et al., 2006, Elad and Aharon, 2006). Denote the image u as , where i indexes the pixel location and N represents the total number of pixels. Let Ri be the operation that extracts the patch centered at position i from the image and vectorizes it
Experiment setup
Extensive experiments were conducted to evaluate the proposed 2DDL-AHT algorithm for image denoising. Due to the diversity and complexity of natural images, the same denoising algorithm may have different performance on different images. A way to remedy this case is to employ a large dataset which contains representative images of different content. We employ two groups of test images in the experiments. The first group includes eight benchmark test images (Fig. 4), in which image Baboon,
Conclusion
This letter presents an iterative denoising algorithm based on 2D dictionary learning and adaptive hard thresholding. The 2D dictionary is learned by exploring the two-direction correlation inherent in image data and the hard thresholding is designed such that the total energy filtered out by threholding is close to the noise energy. On the one hand, the algorithm enhances the traditional PCA-based methods in terms that the correlation among both columns and rows of the SDM is taken into
Acknowledgements
This work was supported by the National Science Foundation of China (Grants 61001156, 11261044, 61105011, 11101292, 60872138 and 61271294) and by the National Science Foundation of Ningxia Province (NZ13049).
References (19)
- et al.
Diffusion wavelets
Appl. Comput. Harmon. Anal.
(2006) - et al.
Wavelets on graphs via spectral graph theory
Appl. Comput. Harmon. Anal.
(2011) - et al.
Nonlinear total variation based noise removal algorithms
Phys. D
(1992) - et al.
Two-stage image denoising by principal component analysis with local pixel grouping
Pattern Recogn.
(2010) - et al.
Principal component analysis
Wiley Interdiscip. Rev. Comput. Stat.
(2010) - et al.
The K-SVD: an algorithm for designing of overcomplete dictionaries for sparse representation
IEEE Trans. Signal Process.
(2006) - et al.
Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations
(2006) - Buades, B.C., Morel, J.M., 2005. A non-local algorithm for image denoising. In: Proc. IEEE Conf. CVPR. pp....
- et al.
Clustering-based denoising with locally learned dictionaries
IEEE Trans. Image Process.
(2009)
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