07-05-2014, 11:41 AM
Nonlocally Centralized Sparse Representation for Image Restoration
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Abstract:
The sparse representation models code an image patch as a linear combination of a few atoms
chosen out from an over-complete dictionary, and they have shown promising results in various image
restoration applications. However, due to the degradation of the observed image (e.g., noisy, blurred and/or
downsampled), the sparse representations by conventional models may not be accurate enough for a faithful
reconstruction of the original image. To improve the performance of sparse representation based image
restoration, in this paper the concept of sparse coding noise is introduced, and the goal of image restoration
turns to how to suppress the sparse coding noise. To this end, we exploit the image nonlocal self-similarity to
obtain good estimates of the sparse coding coefficients of the original image, and then centralize the sparse
coding coefficients of the observed image to those estimates. The so-called nonlocally centralized sparse
representation (NCSR) model is as simple as the standard sparse representation model, while our extensive
experiments on various types of image restoration problems, including denoising, deblurring and
super-resolution, validate the generality and state-of-the-art performance of the proposed NCSR algorithm.
Introduction
Reconstructing a high quality image from one or several of its degraded (e.g., noisy, blurred and/or
down-sampled) versions has many important applications, such as medical imaging, remote sensing,
surveillance and entertainment, etc. For an observed image y, the problem of image restoration (IR) can be
generally formulated by
y = Hx + υ ,
(1)
where H is a degradation matrix, x is the original image vector and υ is the additive noise vector. With
different settings of matrix H, Eq. (1) can represent different IR problems; for example, image denoising
when H is an identity matrix, image deblurring when H is a blurring operator, image superresolution when
H is a composite operator of blurring and down-sampling, and compressive sensing when H is a random
projection matrix [1-3]. In the past decades, extensive studies have been conducted on developing various IR
approaches [4-23]. Due to the ill-posed nature of IR, the regularization-based techniques have been widely
used by regularizing the solution spaces [5-9, 12-22]. In order for an effective regularizer, it is of great
importance to find and model the appropriate prior knowledge of natural images, and various image prior
models have been developed [5-8, 14, 17-18, 22].
Algorithm of NCSR
A. Parameters determination
In Eq. (8) or Eq. (11) the parameter λ that balances the fidelity term and the centralized sparsity term should
be adaptively determined for better IR performance. In this subsection we provide a Bayesian interpretation
of the NCSR model, which also provides us an explicit way to set the regularization parameter λ. In the
literature of wavelet denoising, the connection between Maximum a Posterior (MAP) estimator and sparse
representation has been established, and here we extend the connection from the local sparsity to nonlocally
centralized sparsity.
Image super-resolution
In image super-resolution, the low-resolution (LR) image is obtained by first blurring the high-resolution
(HR) image with a blur kernel and then downsampling by a scaling factor. Hence, recovering the HR image
from a single LR image is more severely underdetermined than image deblurring. In this subsection, we test
the proposed NCSR based IR for image super-resolution. The simulated LR image is generated by first
blurring an HR image with a 7×7 Gaussian kernel with standard deviation 1.6, and then downsampling the
blurred image by a scaling factor 3 in both horizontal and vertical directions. The additive Gaussian noises
of standard deviation 5 are also added to the LR images, making the IR problem more challenging. Since
human visual system is more sensitive to luminance changes, we only apply the IR methods to the
luminance component and use the simple bicubic interpolator for the chromatic components.
Conclusion
In this paper we presented a novel nonlocally centralized sparse representation (NCSR) model for image
restoration. The sparse coding noise (SCN), which is defined as the difference between the sparse code of
the degraded image and the sparse code of the unknown original image, should be minimized to improve the
performance of sparsity-based image restoration. To this end, we proposed a centralized sparse constraint,
which exploits the image nonlocal redundancy, to reduce the SCN. The Bayesian interpretation of the NCSR
model was provided and this endows the NCSR model an iteratively reweighted implementation. An
efficient iterative shrinkage function was presented for solving the l1-regularized NCSR minimization
problem. Experimental results on image denoising, deblurring and super-resolution demonstrated that the
NCSR approach can achieve highly competitive performance to other leading denoising methods, and
outperform much other leading image deblurring and super-resolution methods.