Image decomposition using optimally sparse representations and a variational approach

In this paper, a new method which combines the basis pursuit denoising algorithm (BPDN) and the total variation (TV) regularization scheme is presented for separating images into texture and cartoon parts. It is a modification of the model [1]. In this process, two appropriate dictionaries are used, one for the representation of texture parts-the dual tree complex wavelet transform (DT CWT) and the other for the cartoon parts-the second generation of curvelet transform. To direct the separation process and reduce the pseudo-Gibbs phenomenon, the curvelet transform is followed by a projected regularization method for cartoon parts. Experimental results show that new method cannot only decompose better for a given image but also reduce the runtime, in comparison to the MCA approach.     

Higher degree total variation (HDTV) regularization for image recovery

We introduce novel image regularization penalties to overcome the practical problems associated with the classical total variation (TV) scheme. Motivated by novel reinterpretations of the classical TV regularizer, we derive two families of functionals involving higher degree partial image derivatives; we term these families as isotropic and anisotropic higher degree TV (HDTV) penalties, respectively. The isotropic penalty is the L(1) - L(2) mixed norm of the directional image derivatives, while the anisotropic penalty is the separable L(1) norm of directional derivatives. These functionals inherit the desirable properties of standard TV schemes such as invariance to rotations and translations, preservation of discontinuities, and convexity. The use of mixed norms in isotropic penalties encourages the joint sparsity of the directional derivatives at each pixel, thus encouraging isotropic smoothing. In contrast, the fully separable norm in the anisotropic penalty ensures the preservation of discontinuities, while continuing to smooth along the linelike features; this scheme thus enhances the linelike image characteristics analogous to standard TV. We also introduce efficient majorize-minimize algorithms to solve the resulting optimization problems. The numerical comparison of the proposed scheme with classical TV penalty, current second-degree methods, and wavelet algorithms clearly demonstrate the performance improvement. Specifically, the proposed algorithms minimize the staircase and ringing artifacts that are common with TV and wavelet schemes, while better preserving the singularities. We also observe that anisotropic HDTV penalty provides consistently improved reconstructions compared with the isotropic HDTV penalty.     

基于结构组全变分模型的图像压缩感知重建

针对基于传统全变分(TV)模型的图像压缩感知(CS)重建算法不能有效地恢复图像的细节和纹理,从而导致图像过平滑的问题,该文提出一种基于结构组全变分(SGTV)模型的图像压缩感知重建算法。该算法利用图像的非局部自相似性和结构稀疏特性,将图像的重建问题转化为由非局部自相似图像块构建的结构组全变分最小化问题。算法以结构组全变分模型为正则化约束项构建优化模型,利用分裂Bregman迭代将算法分离成多个子问题,并对每个子问题高效地求解。所提算法很好地利用了图像自身的信息和结构稀疏特性,保护了图像细节和纹理。实验结果表明,该文所提出的算法优于现有基于全变分模型的压缩感知重建算法,在PSNR和视觉效果方面取得了显著提升。     

基于L曲线调参的FOTV-ADMM超分辨率图像重建

基于图像高频细节的重构问题,建立了全变分(Total Variation,TV)约束重建模型,选取交替方向乘子法(Alternating Direction Method of Multipliers,ADMM)求解分析.TV-ADMM重建算法能够保持图像边缘信息,纹理细节的刻画却不够理想,图像平滑部分的重建出现阶梯效应和过平滑现象.为此,提出使用分数阶约束的模型算法FOTV-ADMM求解.该算法对图像纹理高频细节重建效果较好,能锐化图像边缘区域,同时为降低经验调节参数对图像重建的影响,减少调节参数的时间,引入L曲线调节参数,找出了正则化参数最优解.实验结果表明,基于L曲线调参的FOTV-ADMM算法能够更好地保留图像的纹理和平滑部分的细节特征,在峰值信噪比和结构相似度评价指标上,FOTV对高频细节的重建改善效果更佳.     

Variational Retinex model based on an extension of TV regularization with relative gradient and its application

In view of the shortcomings of the total variational Retinex model which use the total variation (TV) of the reflection as the regularization.An extension of TV regularization with the concept of relative gradient was introduced and finally a new variational Retinex model was proposed.Compared with variational Retinex and total variational Retinex model,the proposed model can preserve the estimated reflectance with more details as well as the more smoothed illumination.Further,a new integrated image enhancement model considering both the illumination and the reflectance was proposed.By adjusting the model parameters,the proposed model can be effectively applied to high dynamic range image tone mapping and non-uniform illumination enhancement.Compared with other algorithms,the proposed model can better handle the above image enhancement problems.     

Non-convex hybrid total variation for image denoising

Image restoration problems, such as image denoising, are important steps in various image processing method, such as image segmentation and object recognition. Due to the edge preserving property of the convex total variation (TV), variational model with TV is commonly used in image restoration. However, staircase artifacts are frequently observed in restored smoothed region. To remove the staircase artifacts in smoothed region, convex higher-order TV (HOTV) regularization methods are introduced. But the valuable edge information of the image is also attenuated. In this paper, we propose non-convex hybrid TV regularization method to significantly reduce staircase artifacts while well preserving the valuable edge information of the image. To efficiently find a solution of the variation model with the proposed regularizer, we use the iterative reweighted method with the augmented Lagrangian based algorithm. The proposed model shows the best performance in terms of the signal-to-noise ratio (SNR) and the structure similarity index measure (SSIM) with comparable computational complexity.     

Alternating minimization algorithm for speckle reduction with a shifting technique

Speckles (multiplicative noise) in synthetic aperture radar (SAR) make it difficult to interpret the observed image. Due to the edge-preserving feature of total variation (TV), variational models with TV regularization have attracted much interest in reducing speckles. Algorithms based on the augmented Lagrangian function have been proposed to efficiently solve speckle-reduction variational models with TV regularization. However, these algorithms require inner iterations or inverses involving the Laplacian operator at each iteration. In this paper, we adapt Tseng's alternating minimization algorithm with a shifting technique to efficiently remove the speckle without any inner iterations or inverses involving the Laplacian operator. The proposed method is very simple and highly parallelizable; therefore, it is very efficient to despeckle huge-size SAR images. Numerical results show that our proposed method outperforms the state-of-the-art algorithms for speckle-reduction variational models with a TV regularizer in terms of central-processing-unit time.     

Example-based single document image super-resolution: a global MAP approach with outlier rejection

Regularization plays a vital role in inverse problems, and especially in ill-posed ones. Along with classical regularization techniques based on smoothness, entropy, and sparsity, an emerging powerful regularization is one that leans on image examples. In this paper, we propose an efficient scheme for using image examples as driving a powerful regularization, applied to the image scale-up (super-resolution) problem. In this work, we target specifically scanned documents containing written text, graphics, and equations. Our algorithm starts by assigning per each location in the degraded image several candidate high-quality patches. Those are found as the nearest-neighbors (NN) in an image-database that contains pairs of corresponding low- and high-quality image patches. The found examples are used for the definition of an image prior expression, merged into a global MAP penalty function. We use this penalty function both for rejecting some of the irrelevant outlier examples, and then for reconstructing the desired image. We demonstrate our algorithm on several scanned documents with promising results.     

多形态稀疏性正则化的图像超分辨率算法

 如何设计更加高效并能保持图像几何和纹理结构的多幅图像超分辨模型和算法是目前该领域有待解决的难点问题.针对图像的几何、纹理结构形态,分别建立符合类内强稀疏而类间强不相干的几何结构和纹理分量稀疏表示子成份字典,形成图像的多形态稀疏表示模型,进而提出一种新的基于多形态稀疏性正则化的多帧图像超分辨凸变分模型,模型中的正则项刻画了理想图像在多成份字典下的稀疏性先验约束,保真项度量其在退化模型下与观测信号的一致性,采用交替迭代法对该多变量优化问题进行数值求解,每一子问题采用前向后向的算法分裂法进行快速求解.针对可见光与红外图像序列进行了数值仿真,实验结果验证了本文模型与数值算法的有效性.     

An alternating direction algorithm for total variation reconstruction of distributed parameters

Augmented Lagrangian variational formulations and alternating optimization have been adopted to solve distributed parameter estimation problems. The alternating direction method of multipliers (ADMM) is one of such formulations/optimization methods. Very recently, the number of applications of the ADMM, or variants of it, to solve inverse problems in image and signal processing has increased at an exponential rate. The reason for this interest is that ADMM decomposes a difficult optimization problem into a sequence of much simpler problems. In this paper, we use the ADMM to reconstruct piecewise-smooth distributed parameters of elliptical partial differential equations from noisy and linear (blurred) observations of the underlying field. The distributed parameters are estimated by solving an inverse problem with total variation (TV) regularization. The proposed instance of the ADMM solves, in each iteration, an l(2) and a decoupled l(2) - l(1) optimization problems. An operator splitting is used to simplify the treatment of the TV regularizer, avoiding its smooth approximation and yielding a simple yet effective ADMM reconstruction method compared with previously proposed approaches. The competitiveness of the proposed method, with respect to the state-of-the-art, is illustrated in simulated 1-D and 2-D elliptical equation problems, which are representative of many real applications.     

Hessian-based norm regularization for image restoration with biomedical applications

We present nonquadratic Hessian-based regularization methods that can be effectively used for image restoration problems in a variational framework. Motivated by the great success of the total-variation (TV) functional, we extend it to also include second-order differential operators. Specifically, we derive second-order regularizers that involve matrix norms of the Hessian operator. The definition of these functionals is based on an alternative interpretation of TV that relies on mixed norms of directional derivatives. We show that the resulting regularizers retain some of the most favorable properties of TV, i.e., convexity, homogeneity, rotation, and translation invariance, while dealing effectively with the staircase effect. We further develop an efficient minimization scheme for the corresponding objective functions. The proposed algorithm is of the iteratively reweighted least-square type and results from a majorization-minimization approach. It relies on a problem-specific preconditioned conjugate gradient method, which makes the overall minimization scheme very attractive since it can be applied effectively to large images in a reasonable computational time. We validate the overall proposed regularization framework through deblurring experiments under additive Gaussian noise on standard and biomedical images.     

Sparsity-driven reconstruction for FDOT with anatomical priors

In this paper we propose a method based on (2, 1)-mixed-norm penalization for incorporating a structural prior in FDOT image reconstruction. The effect of (2, 1)-mixed-norm penalization is twofold: first, a sparsifying effect which isolates few anatomical regions where the fluorescent probe has accumulated, and second, a regularization effect inside the selected anatomical regions. After formulating the reconstruction in a variational framework, we analyze the resulting optimization problem and derive a practical numerical method tailored to (2, 1)-mixed-norm regularization. The proposed method includes as particular cases other sparsity promoting regularization methods such as l(1)-norm penalization and total variation penalization. Results on synthetic and experimental data are presented.     

Variational models for image colorization via chromaticity and brightness decomposition.

Colorization refers to an image processing task which recovers color in grayscale images when only small regions with color are given. We propose a couple of variational models using chromaticity color components to colorize black and white images. We first consider total variation minimizing (TV) colorization which is an extension from TV inpainting to color using chromaticity model. Second, we further modify our model to weighted harmonic maps for colorization. This model adds edge information from the brightness data, while it reconstructs smooth color values for each homogeneous region. We introduce penalized versions of the variational models, we analyze their convergence properties, and we present numerical results including extension to texture colorization.     

Compressive near-field millimeter wave imaging algorithm based on Gini index and total variation mixed regularization

A compressive near-field millimeter wave (MMW) imaging algorithm is proposed. From the compressed sensing (CS) theory, the compressive near-field MMW imaging process can be considered to reconstruct an image from the under-sampled sparse data. The Gini index (GI) has been founded that it is the only sparsity measure that has all sparsity attributes that are called Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates, and Babies. By combining the total variation (TV) operator, the GI-TV mixed regularization introduced compressive near-field MMW imaging model is proposed. In addition, the corresponding algorithm based on a primal-dual framework is also proposed. Experimental results demonstrate that the proposed GI-TV mixed regularization algorithm has superior convergence and stability performance compared with the widely used l₁-TV mixed regularization algorithm.     

Expectation–maximization algorithm with total variation regularization for vector-valued image segmentation

We integrate the total variation (TV) minimization into the expectation–maximization (EM) algorithm to perform the task of image segmentation for general vector-valued images. We first propose a unified variational method to bring together the EM and the TV regularization and to take advantages from both approaches. The idea is based on operator interchange and constraint optimization. In the second part of the paper we propose a simple two-phase approach by splitting the above functional into two steps. In the first phase, a typical EM method can classify pixels into different classes based on the similarity in their measurements. However, since no local geometric information of the image has yet been incorporated into the process, such classification in practice gives unsatisfactory segmentation results. In the second phase, the TV-step obtains the segmentation of the image by applying a TV regularization directly to the clustering result from EM.     

Wavelet-domain approximation and compression of piecewise smooth images.

The wavelet transform provides a sparse representation for smooth images, enabling efficient approximation and compression using techniques such as zerotrees. Unfortunately, this sparsity does not extend to piecewise smooth images, where edge discontinuities separating smooth regions persist along smooth contours. This lack of sparsity hampers the efficiency of wavelet-based approximation and compression. On the class of images containing smooth C2 regions separated by edges along smooth C2 contours, for example, the asymptotic rate-distortion (R-D) performance of zerotree-based wavelet coding is limited to D(R) (< or = 1/R, well below the optimal rate of 1/R2. In this paper, we develop a geometric modeling framework for wavelets that addresses this shortcoming. The framework can be interpreted either as 1) an extension to the "zerotree model" for wavelet coefficients that explicitly accounts for edge structure at fine scales, or as 2) a new atomic representation that synthesizes images using a sparse combination of wavelets and wedgeprints--anisotropic atoms that are adapted to edge singularities. Our approach enables a new type of quadtree pruning for piecewise smooth images, using zerotrees in uniformly smooth regions and wedgeprints in regions containing geometry. Using this framework, we develop a prototype image coder that has near-optimal asymptotic R-D performance D(R) < or = (log R)2 /R2 for piecewise smooth C2/C2 images. In addition, we extend the algorithm to compress natural images, exploring the practical problems that arise and attaining promising results in terms of mean-square error and visual quality.     

Image reconstruction with locally adaptive sparsity and nonlocal robust regularization

Sparse representation based modeling has been successfully used in many image-related inverse problems such as deblurring, super-resolution and compressive sensing. The heart of sparse representations lies on how to find a space (spanned by a dictionary of atoms) where the local image patch exhibits high sparsity and how to determine the image local sparsity. To identify the locally varying sparsity, it is necessary to locally adapt the dictionary learning process and the sparsity-regularization parameters. However, spatial adaptation alone runs into the risk of over-fitting the data because variation and invariance are two sides of the same coin. In this work, we propose two sets of complementary ideas for regularizing image reconstruction process: (1) the sparsity regularization parameters are locally estimated for each coefficient and updated along with adaptive learning of PCA-based dictionaries; (2) a nonlocal self-similarity constraint is introduced into the overall cost functional to improve the robustness of the model. An efficient alternative minimization algorithm is present to solve the proposed objective function and then an effective image reconstruction algorithm is presented. The experimental results on image deblurring, super-resolution and compressive sensing demonstrate that the proposed image reconstruct method outperforms many existing image reconstruction methods in both PSNR and visual quality assessment.     

基于约束空间光谱联合的亚像素定位方法北大核心CSCD

针对高光谱遥感图像,提出了一种约束空间光谱的亚像素定位方法。传统的亚像素定位方法以解混的结果作为输入,可能无法充分利用高光谱图像丰富的光谱信息。本文所提出的基于约束空间光谱联合的亚像素定位方法(constraint spatial-spectral subpixel mapping,CSSSM),利用下采样将像素丰度与亚像素丰度显式联系起来,代入线性解混模型得到亚像素丰度求解的新模型。在求解过程中,通过添加稀疏性约束与平滑性约束,以限制亚像素丰度的解空间,亚像素丰度求解更精确。其中,针对亚像素丰度稀疏性先验采用重加权1范数作为新的约束,并自适应地更新权重;针对亚像素丰度空间先验信息则采用全变分(total variational,TV)正则化作为约束,然后使用乘法迭代算法求解亚像素丰度,最后利用赢者通吃的策略进行类别确定。在两个合成数据集上进行了实验,结果表明,本方法能够进一步提高亚像素定位的精度。     

基于压缩感知理论的高能闪光照相密度反演方法

高能闪光照相中需要研究少数投影数据条件下的非轴对称客体的的密度反演问题。现有利用压缩感知思想的全变差TV类算法虽然考虑了图像的局部相似性，但没有考虑图像的非局部相似性。针对上述问题，文中提出了一种基于组稀疏正则化的全变分重建技术TV-GSR。该技术将组稀疏模型集成于TV框架之下，同时考虑了客体图像的局部相似性和非局部自相似性，充分利用了图像的先验稀疏信息，并利用客体的上、下、左、右4点对称性来降低图像重建的规模，重构精度有所增加，重建速度也更快。仿真实验表明，文中提出的TV-GSR算法提升了图像在无噪声和有噪声情况下的重建精度，对于高能闪光图像和纹理细节丰富的CT图像都有较好的效果，具有普适性。