Frame Based Image Restoration Research Articles

Approximation theory of wavelet frame based image restoration

In this paper, we analyze the error estimate of a wavelet frame based image restoration method from degraded and incomplete measurements. We present the error between the underlying original discrete image and the approximate solution which has the minimal ℓ1-norm of the canonical wavelet frame coefficients among all possible solutions. Then we further connect the error estimate for the discrete model to the approximation to the underlying function from which the underlying image comes.

Approximation Theory of Wavelet Frame Based Image Restoration

Approximation Theory of Wavelet Frame Based Image Restoration

Wavelet frame-based image restoration using sparsity, nonlocal, and support prior of frame coefficients

The wavelet frame systems have been widely investigated and applied for image restoration and many other image processing problems over the past decades, attributing to their good capability of sparsely approximating piece-wise smooth functions such as images. Most wavelet frame based models exploit the $l_1$ norm of frame coefficients for a sparsity constraint in the past. The authors in \cite{ZhangY2013, Dong2013} proposed an $l_0$ minimization model, where the $l_0$ norm of wavelet frame coefficients is penalized instead, and have demonstrated that significant improvements can be achieved compared to the commonly used $l_1$ minimization model. Very recently, the authors in \cite{Chen2015} proposed $l_0$-$l_2$ minimization model, where the nonlocal prior of frame coefficients is incorporated. This model proved to outperform the single $l_0$ minimization based model in terms of better recovered image quality. In this paper, we propose a truncated $l_0$-$l_2$ minimization model which combines sparsity, nonlocal and support prior of the frame coefficients. The extensive experiments have shown that the recovery results from the proposed regularization method performs better than existing state-of-the-art wavelet frame based methods, in terms of edge enhancement and texture preserving performance.

Image Restoration: A General Wavelet Frame Based Model and Its Asymptotic Analysis

Image restoration is one of the most important areas in imaging science. Mathematical tools have been widely used in image restoration, where the wavelet frame based approach is one of the successful examples. In this paper, we introduce a generic wavelet frame based image restoration model, called the “general model,” which includes most of the existing wavelet frame based models as special cases. Moreover, the general model also includes examples that are new to the literature. Motivated by our earlier studies [R. Adams, Sobolev Spaces, Academic Press, New York, 1975; D. Bertsekas and J. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, New York, 1989; K. Bredies, K. Kunisch, and T. Pock, SIAM J. Imaging Sci., 3 (2010), p. 492], we provide an asymptotic analysis of the general model as image resolution goes to infinity, which establishes a connection between the general model in discrete setting and a new variatonal model in continuum setting. The variational model also...

Image restoration by minimizing zero norm of wavelet frame coefficients

In this paper, we propose two algorithms, namely the extrapolated proximal iterative hard thresholding (EPIHT) algorithm and the EPIHT algorithm with line-search, for solving the -norm regularized wavelet frame balanced approach for image restoration. Under the theoretical framework of Kurdyka–Łojasiewicz property, we show that the sequences generated by the two algorithms converge to a local minimizer with linear convergence rate. Moreover, extensive numerical experiments on sparse signal reconstruction and wavelet frame based image restoration problems including CT reconstruction, image deblur, demonstrate the improvement of -norm based regularization models over some prevailing ones, as well as the computational efficiency of the proposed algorithms.

Composite analysis‐based model for image restoration solved by alternating direction method of multipliers fast iterative shrinkage‐thresholding algorithm

A composite analysis-based model for image restoration by utilising the image edge's well preserving property of the total variation (TV) and the sparsely representation capability of the wavelet frame-based image restoration model is proposed. This model is an analysis-based approach that the wavelet frame coefficient is quite often able to be linked with the smoothness of the image and is more in accord with the model's essence of TV model. This model is solved using the alternating direction method of multipliers fast iterative shrinkage-thresholding algorithm and a faster alternating minimisation algorithm, and numerical experiments show that the proposed method is more suitable for image restoration.

An analysis of wavelet frame based scattered data reconstruction

In real world applications many signals contain singularities, like edges in images. Recent wavelet frame based approaches were successfully applied to reconstruct scattered data from such functions while preserving these features. In this paper we present a recent approach which determines the approximant from shift invariant subspaces by minimizing an ℓ 1 -regularized least squares problem which makes additional use of the wavelet frame transform in order to preserve sharp edges. We give a detailed analysis of this approach, i.e., how the approximation error behaves dependent on data density and noise level. Moreover, a link to wavelet frame based image restoration models is established and the convergence of these models is analyzed. In the end, we present some numerical examples, for instance how to apply this approach to handle coarse-grained models in molecular dynamics.

Image restoration: A wavelet frame based model for piecewise smooth functions and beyond

In this paper, we propose a new wavelet frame based image restoration model that explicitly treats images as piecewise smooth functions. It estimates both the image to be restored and its singularity set. It can well protect singularities, which are important image features, and provide enough regularization in smooth regions at the same time. This model penalizes the ℓ2-norm of the wavelet frame coefficients away from the singularity set, while penalizes the ℓ1-norm of the coefficients on the singularity set. This model explicitly models images as piecewise smooth functions with a general smoothness regularization and characterizes rather general singularity set, which includes both jump discontinuities and jumps after certain orders of differentiations. As we know, all types of singularities are important image features and need to be recovered. Furthermore, the singularity set can be robustly estimated by wavelet frame transform during the image recovery procedure, which makes our model easy to solve numerically; hence, the model is insensitive to the estimation of the singularity set.The proposed model is in discrete setting and is a wavelet frame based approach. To further understand the piecewise smooth nature of the obtained solutions, we connect it to a variational model on the space of piecewise smooth functions and prove rigorously that the discrete model converges to the variational model as image resolution goes to infinity. Also, we show that the approximate solutions of the discrete model can be regarded as an approximation of those of the variational model. Through these theoretical analysis, we manage to connect the proposed discrete wavelet frame based model with the variational model. Such connection not only enables us to acquire deeper understandings of the discrete model, but also leads us to the discovery of a variational model new to the literature, which is more general and works better than the Mumford–Shah model [1] for image restoration problems. Although the focus of the paper is to propose the new model and provide theoretical studies, we still conduct numerical simulations to support our claims and theoretical findings. Our numerical studies show that the proposed model is the right one for image restorations, when the underlying solutions are piecewise smooth. Generally speaking, this model combines the merits of the PDE based approach [1–7] and the wavelet frame based approach [8–10].

Wavelet Frame Based Image Restoration via Combined Sparsity and Nonlocal Prior of Coefficients

Owing to the good ability of sparsely approximating piece-wise smooth functions like images, the (tight) wavelet frame has been widely investigated and applied for image restoration and other image processing problems over the past few years. Most of the variational models based on wavelet frame proposed in the past utilize the $$l_{1}$$l1 norm of frame coefficients as a sparsity prior. Very recently, the variational model which penalizes the $$l_{0}$$l0 "norm" of frame coefficients was proposed for image restoration, and proved to outperform the commonly used $$l_{1}$$l1 minimization methods in the quality of restored images. Though the $$l_{0}$$l0 "norm" has the ability of preserving sharp edges and smooth regions, textures and small details may be mistakenly removed at the same time. Therefore, we introduce a $$l_0-l_2$$l0-l2 regularization model which contains a nonlocal prior of frame coefficients to avoid this issue in this paper. Meanwhile, a narrow-band technique is introduced to further improve the computational efficiency of the proposed algorithm. Numerical experiments demonstrate that the propose algorithm is superior to the recently proposed algorithm for $$l_{0}$$l0 "norm" minimization in both iterative time and recovery quality.

Iterative support detection-based split Bregman method for wavelet frame-based image inpainting.

The wavelet frame systems have been extensively studied due to their capability of sparsely approximating piece-wise smooth functions, such as images, and the corresponding wavelet frame-based image restoration models are mostly based on the penalization of the l1 norm of wavelet frame coefficients for sparsity enforcement. In this paper, we focus on the image inpainting problem based on the wavelet frame, propose a weighted sparse restoration model, and develop a corresponding efficient algorithm. The new algorithm combines the idea of iterative support detection method, first proposed by Wang and Yin for sparse signal reconstruction, and the split Bregman method for wavelet frame l1 model of image inpainting, and more important, naturally makes use of the specific multilevel structure of the wavelet frame coefficients to enhance the recovery quality. This new algorithm can be considered as the incorporation of prior structural information of the wavelet frame coefficients into the traditional l1 model. Our numerical experiments show that the proposed method is superior to the original split Bregman method for wavelet frame-based l1 norm image inpainting model as well as some typical l(p) (0 ≤ p < 1) norm-based nonconvex algorithms such as mean doubly augmented Lagrangian method, in terms of better preservation of sharp edges, due to their failing to make use of the structure of the wavelet frame coefficients.

$\ell _0$ Minimization for wavelet frame based image restoration

The theory of (tight) wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems. The success of wavelet frame based models, including balanced approach and analysis based approach, is due to their capability of sparsely approximating piecewise smooth functions like images. Motivated by the balanced approach and analysis based approach, we shall propose a wavelet frame based $\ell_0$ minimization model, where the $\ell_0$ "norm" of the frame coefficients is penalized. We adapt the penalty decomposition (PD) method to solve the proposed optimization problem. Numerical results showed that the proposed model solved by the PD method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images. Some convergence analysis of the PD method will also be provided.

Alternating Direction Method for Balanced Image Restoration

This paper presents an efficient algorithm for solving a balanced regularization problem in the frame-based image restoration. The balanced regularization is usually formulated as a minimization problem, involving an l(2) data-fidelity term, an l(1) regularizer on sparsity of frame coefficients, and a penalty on distance of sparse frame coefficients to the range of the frame operator. In image restoration, the balanced regularization approach bridges the synthesis-based and analysis-based approaches, and balances the fidelity, sparsity, and smoothness of the solution. Our proposed algorithm for solving the balanced optimal problem is based on a variable splitting strategy and the classical alternating direction method. This paper shows that the proposed algorithm is fast and efficient in solving the standard image restoration with balanced regularization. More precisely, a regularized version of the Hessian matrix of the l(2) data-fidelity term is involved, and by exploiting the related fast tight Parseval frame and the special structures of the observation matrices, the regularized Hessian matrix can perform quite efficiently for the frame-based standard image restoration applications, such as circular deconvolution in image deblurring and missing samples in image inpainting. Numerical simulations illustrate the efficiency of our proposed algorithm in the frame-based image restoration with balanced regularization.

An Efficient Algorithm for ℓ 0 Minimization in Wavelet Frame Based Image Restoration

Wavelet frame based models for image restoration have been extensively studied for the past decade (Chan et al. in SIAM J. Sci. Comput. 24(4):1408---1432, 2003; Cai et al. in Multiscale Model. Simul. 8(2):337---369, 2009; Elad et al. in Appl. Comput. Harmon. Anal. 19(3):340---358, 2005; Starck et al. in IEEE Trans. Image Process. 14(10):1570---1582, 2005; Shen in Proceedings of the international congress of mathematicians, vol. 4, pp. 2834---2863, 2010; Dong and Shen in IAS lecture notes series, Summer program on "The mathematics of image processing", Park City Mathematics Institute, 2010). The success of wavelet frames in image restoration is mainly due to their capability of sparsely approximating piecewise smooth functions like images. Most of the wavelet frame based models designed in the past are based on the penalization of the ? 1 norm of wavelet frame coefficients, which, under certain conditions, is the right choice, as supported by theories of compressed sensing (Candes et al. in Appl. Comput. Harmon. Anal., 2010; Candes et al. in IEEE Trans. Inf. Theory 52(2):489---509, 2006; Donoho in IEEE Trans. Inf. Theory 52:1289---1306, 2006). However, the assumptions of compressed sensing may not be satisfied in practice (e.g. for image deblurring and CT image reconstruction). Recently in Zhang et al. (UCLA CAM Report, vol. 11-32, 2011), the authors propose to penalize the l 0 "norm" of the wavelet frame coefficients instead, and they have demonstrated significant improvements of their method over some commonly used l 1 minimization models in terms of quality of the recovered images. In this paper, we propose a new algorithm, called the mean doubly augmented Lagrangian (MDAL) method, for l 0 minimizations based on the classical doubly augmented Lagrangian (DAL) method (Rockafellar in Math. Oper. Res. 97---116, 1976). Our numerical experiments show that the proposed MDAL method is not only more efficient than the method proposed by Zhang et al. (UCLA CAM Report, vol. 11-32, 2011), but can also generate recovered images with even higher quality. This study reassures the feasibility of using the l 0 "norm" for image restoration problems.

Wavelet frame based surface reconstruction from unorganized points

Applications of wavelet frames to image restoration problems (e.g. image deblurring and inpainting) have been successful due to their redundancy and capability of sparsely approximating piecewise smooth functions like images (see e.g. [13,14,8]). However, wavelet frames have not yet been used for surface reconstruction problems. Recently in [6], connections between one of the wavelet frame based image restoration model [8,28,48] and variational models (e.g. the ROF model [45]) were rigorously established. Such connections not only grant new insights to wavelet frame based image restorations, it also case a geometric explanation to wavelet frame based approaches. This leads us to a wavelet frame based model, as well as a fast algorithm, to reconstruct implicit surfaces from unorganized point sets in R^3. We will demonstrate the effectiveness of the proposed model using several commonly used examples.

An Accelerated Proximal Gradient Algorithm for Frame-Based Image Restoration via the Balanced Approach

Frame-based image restoration by using the balanced approach has been developed over the last decade. Many recently developed algorithms for image restoration can be viewed as an acceleration of the proximal forward-backward splitting algorithm. Accelerated proximal gradient (APG) algorithms studied by Nesterov, Nemirovski, and others have been demonstrated to be efficient in solving various regularized convex optimization problems arising in compressed sensing, machine learning, and control. In this paper, we adapt the APG algorithm to solve the $\ell_1$-regularized linear least squares problem in the balanced approach in frame-based image restoration. This algorithm terminates in $O(1/\sqrt{\epsilon})$ iterations with an $\epsilon$-optimal solution, and we demonstrate that this single algorithmic framework can universally handle several image restoration problems, such as image deblurring, denoising, inpainting, and cartoon-texture decomposition. Our numerical results suggest that the APG algorithms are efficient and robust in solving large-scale image restoration problems. The algorithms we implemented are able to restore $512\times512$ images in various image restoration problems in less than 50 seconds on a modest PC. We also compare the numerical performance of our proposed algorithms applied to image restoration problems by using one frame-based system with that by using cartoon and texture systems for image deblurring, denoising, and inpainting.

Frame-based segmentation for medical images

Medical image segmentation is an important but difficult problem that attracts tremendous attentions of researchers from various fields. In this paper, we propose a frame based model, as well as a fast implementation, for general medical image segmentation problems. Our model combines ideas of the frame based image restoration model of [1] with ideas of the total variation based segmentation model of [2, 3, 4, 5]. Numerical experiments show that the proposed frame based model outperforms total variation based model in terms of capturing key features of biological structures. Successful segmentations of blood vessels and aneurysms in 3D CT angiography images are also presented.

Split Bregman Methods and Frame Based Image Restoration

Split Bregman methods introduced in [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323–343] have been demonstrated to be efficient tools for solving total variation norm minimization problems, which arise from partial differential equation based image restoration such as image denoising and magnetic resonance imaging reconstruction from sparse samples. In this paper, we prove the convergence of the split Bregman iterations, where the number of inner iterations is fixed to be one. Furthermore, we show that these split Bregman iterations can be used to solve minimization problems arising from the analysis based approach for image restoration in the literature. We apply these split Bregman iterations to the analysis based image restoration approach whose analysis operator is derived from tight framelets constructed in [A. Ron and Z. Shen, J. Funct. Anal., 148 (1997), pp. 408–447]. This gives a set of new frame based image restoration algorithms that cover several topics in image restorations...

A variational formulation for frame-based inverse problems

A convex variational framework is proposed for solving inverse problems in Hilbert spaces with a priori information on the representation of the target solution in a frame. The objective function to be minimized consists of a separable term penalizing each frame coefficient individually, and a smooth term modelling the data formation model as well as other constraints. Sparsity-constrained and Bayesian formulations are examined as special cases. A splitting algorithm is presented to solve this problem and its convergence is established in infinite-dimensional spaces under mild conditions on the penalization functions, which need not be differentiable. Numerical simulations demonstrate applications to frame-based image restoration.