Implementation of high‐order variational models made easy for image processing

High‐order variational models are powerful methods for image processing and analysis, but they can lead to complicated high‐order nonlinear partial differential equations that are difficult to discretise to solve computationally. In this paper, we present some representative high‐order variational models and provide detailed descretisation of these models and numerical implementation of the split Bregman algorithm for solving these models using the fast Fourier transform. We demonstrate the advantages and disadvantages of these high‐order models in the context of image denoising through extensive experiments. The methods and techniques can also be used for other applications, such as image decomposition, inpainting and segmentation. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlocal patches based Gaussian mixture model for image inpainting

We consider the inpainting problem for noisy images. It is very challenge to suppress noise when image inpainting is processed. An image patches based nonlocal variational method is proposed to simultaneously inpainting and denoising in this paper. Our approach is developed on an assumption that the small image patches should be obeyed a distribution which can be described by a high dimension Gaussian Mixture Model. By a maximum a posteriori (MAP) estimation, we formulate a new regularization term according to the log-likelihood function of the mixture model. To optimize this regularization term efficiently, we adopt the idea of the Expectation Maximization (EM) algorithm. In which, the expectation step can give an adaptive weighting function which can be regarded as a nonlocal connections among pixels. Using this fact, we built a framework for non-local image inpainting under noise. Moreover, we mathematically prove the existence of minimizer for the proposed inpainting model. By using a splitting algorithm, the proposed model are able to realize image inpainting and denoising simultaneously. Numerical results show that the proposed method can produce impressive reconstructed results when the inpainting region is rather large.     

A review of Bandlet methods for geometrical image representation

This article reviews bandlet approaches to geometric image representations. Orthogonal bandlets using an adaptive segmentation and a local geometric flow well suited to capture the anisotropic regularity of edge structures. They are constructed with a “bandletization” which is a local orthogonal transformation applied to wavelet coefficients. The approximation in these bandlet bases exhibits an asymptotically optimal decay for images that are regular outside a set of regular edges. These bandlets can be used to perform image compression and noise removal. More flexible orthogonal bandlets with less vanishing moments are constructed with orthogonal grouplets that group wavelet coefficients alon a multiscale association field. Applying a translation invariant grouplet transform over a translation invariant wavelet frame leads to state of the art results for image denoising and super-resolution.     

基于四点插值细分小波的SAR图像去噪

针对SAR图像去噪过程中存在降低相干斑与保持有效细节这一矛盾,提出了一种基于四点插值细分小波的SAR图像去噪算法,该方法将小波和细分方法相融合,将四点插值细分规则应用到细分小波中,提出了图像去噪的新方法.该算法先用四点插值细分小波对原始图像进行分解,然后用Bayes自适应阈值及阈值函数对图像进行去噪,最后对去噪的小波系数进行重构,并通过等效视数、边缘保持指数等评价指标对去噪结果进行了评价.实验结果表明,算法的等效视数、边缘保持指数都有所提高,去噪效果得到了优化.     

An edge driven wavelet frame model for image restoration

Wavelet frame systems are known to be effective in capturing singularities from noisy and degraded images. In this paper, we introduce a new edge driven wavelet frame model for image restoration by approximating images as piecewise smooth functions. With an implicit representation of image singularities sets, the proposed model inflicts different strength of regularization on smooth and singular image regions and edges. The proposed edge driven model is robust to both image approximation and singularity estimation. The implicit formulation also enables an asymptotic analysis of the proposed models and a rigorous connection between the discrete model and a general continuous variational model. Finally, numerical results on image inpainting and deblurring show that the proposed model is compared favorably against several popular image restoration models.     

过抽样M通道小波星图去噪方法研究

小波图像去噪已经成为目前图像去噪的主要方法之一,在分析了小波变换的基本理论和小波变换的多尺度分析基础上,根据多尺度小波变换的多分辨特性,提出了过抽样M通道小波变换去噪方法,并将此方法用于星图降噪处理中,收到良好的效果.     

A Parameter Selection Method for Wavelet Shrinkage Denoising

Thresholding estimators in an orthonormal wavelet basis are well established tools for Gaussian noise removal. However, the universal threshold choice, suggested by Donoho and Johnstone, sometimes leads to over-smoothed approximations.For the denoising problem this paper uses the deterministic approach proposed by Chambolle et al., which handles it as a variational problem, whose solution can be formulated in terms of wavelet shrinkage. This allows us to use wavelet shrinkage successfully for more general denoising problems and to propose a new criterion for the choice of the shrinkage parameter, which we call H-curve criterion. It is based on the plot, for different parameter values, of the B ₁ ¹(L ₁)-norm of the computed solution versus the L ₂-norm of the residual, considered in logarithmic scale. Extensive numerical experimentation shows that this new choice of shrinkage parameter yields good results both for Gaussian and other kinds of noise.     

Image denoising using the Gaussian curvature of the image surface

A number of high‐order variational models for image denoising have been proposed within the last few years. The main motivation behind these models is to fix problems such as the staircase effect and the loss of image contrast that the classical Rudin–Osher–Fatemi model [Leonid I. Rudin, Stanley Osher and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), pp. 259–268] and others also based on the gradient of the image do have. In this work, we propose a new variational model for image denoising based on the Gaussian curvature of the image surface of a given image. We analytically study the proposed model to show why it preserves image contrast, recovers sharp edges, does not transform piecewise smooth functions into piecewise constant functions and is also able to preserve corners. In addition, we also provide two fast solvers for its numerical realization. Numerical experiments are shown to illustrate the good performance of the algorithms and test results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1066–1089, 2016     

On some new variational problems for image denoising

This paper is devoted to image denoising problems using multiresolution schemes related to variational problems. We start with the linear approach of Donoho and Johnstone, that is related to a well known diffusion‐type variational problem. In order to improve the behavior of this approach, we propose some new nonlinear variational problems more adapted to the problem of denoising. Moreover, the discretization is performed using nonlinear multiresolution schemes. In particular, we obtain some fast and well adapted schemes for the considered problem of denoising.     

Compression and denoising using <Emphasis Type="Italic">l</Emphasis><Subscript>0</Subscript>-norm

In this paper, we deal with l ₀-norm data fitting and total variation regularization for image compression and denoising. The l ₀-norm data fitting is used for measuring the number of non-zero wavelet coefficients to be employed to represent an image. The regularization term given by the total variation is to recover image edges. Due to intensive numerical computation of using l ₀-norm, it is usually approximated by other functions such as the l ₁-norm in many image processing applications. The main goal of this paper is to develop a fast and effective algorithm to solve the l ₀-norm data fitting and total variation minimization problem. Our idea is to apply an alternating minimization technique to solve this problem, and employ a graph-cuts algorithm to solve the subproblem related to the total variation minimization. Numerical examples in image compression and denoising are given to demonstrate the effectiveness of the proposed algorithm.     

Construction of parameterizations of masks for tight wavelet frames with two symmetric/antisymmetric generators and applications in image compression and denoising

In this paper, we present a general construction framework of parameterizations of masks for tight wavelet frames with two symmetric/antisymmetric generators which are of arbitrary lengths and centers. Based on this idea, we establish the explicit formulas of masks of tight wavelet frames. Additionally, we explore the transform applicability of tight wavelet frames in image compression and denoising. We bring forward an optimal model of masks of tight wavelet frames aiming at image compression with more efficiency, which can be obtained through SQP (Sequential Quadratic Programming) and a GA (Genetic Algorithm). Meanwhile, we present a new model called Cross-Local Contextual Hidden Markov Model (CLCHMM), which can effectively characterize the intrascale and cross-orientation correlations of the coefficients in the wavelet frame domain, and do research into the corresponding algorithm. Using the presented CLCHMM, we propose a new image denoising algorithm which has better performance as proved by the experiments.     

A multiscale fourth‐order model for the image inpainting and low‐dimensional sets recovery

We consider a fourth‐order variational model, to solve the image inpainting problem, with the emphasis on the recovery of low‐dimensional sets (edges and corners) and the curvature of the edges. The model permits also to perform simultaneously the restoration (filtering) of the initial image where this one is available. The multiscale character of the model follows from an adaptive selection of the diffusion parameters that allows us to optimize the regularization effects in the neighborhoods of the small features that we aim to preserve. In addition, because the model is based on the high‐order derivatives, it favors naturally the accurate capture of the curvature of the edges, hence to balance the tasks of obtaining long curved edges and the obtention of short edges, tip points, and corners. We analyze the method in the framework of the calculus of variations and the Γ‐convergence to show that it leads to a convergent algorithm. In particular, we obtain a simple discrete numerical method based on a standard mixed‐finite elements with well‐established approximation properties. We compare the method to the Cahn–Hilliard model for the inpainting, and we present several numerical examples to show its performances. Copyright © 2016 John Wiley & Sons, Ltd.     

Using a level set approach for image segmentation under interpolation conditions

In this paper, we propose a new 2D segmentation model including geometric constraints, namely interpolation conditions, to detect objects in a given image. We propose to apply the deformable models to an explicit function using the level set approach (Osher and Sethian [24]); so, we avoid the classical problem of parameterization of both segmentation representation and interpolation conditions. Furthermore, we allow this representation to have topological changes. A problem of energy minimization on a closed subspace of a Hilbert space is defined and introducing Lagrange multipliers enables us to formulate the corresponding variational problem with interpolation conditions. Thus the explicit function evolves, while minimizing the energy and it stops evolving when the desired outlines of the object to detect are reached. The stopping term, as in the classical deformable models, is related to the gradient of the image. Numerical results are given. AMS subject classification 74G65, 46-xx, 92C55     

Image inpainting using complex 2-D dual-tree wavelet transform

The dual-tree complex wavelet transform is a useful tool in signal and image processing. In this paper, we propose a dual-tree complex wavelet transform (CWT) based algorithm for image inpainting problem. Our approach is based on Cai, Chan, Shen and Shen’s framelet-based algorithm. The complex wavelet transform outperforms the standard real wavelet transform in the sense of shift-invariance, directionality and anti-aliasing. Numerical results illustrate the good performance of our algorithm.     

A fractional partial differential equation based multiscale denoising model for texture image

Traditional integer‐order partial differential equation based image denoising approach can easily lead edge and complex texture detail blur, thus its denoising effect for texture image is always not well. To solve the problem, we propose to implement a fractional partial differential equation (FPDE) based denoising model for texture image by applying a novel mathematical method—fractional calculus to image processing from the view of system evolution. Previous studies show that fractional calculus has some unique properties that it can nonlinearly enhance complex texture detail in digital image processing, which is obvious different with integer‐order differential calculus. The goal of the modeling is to overcome the problems of the existed denoising approaches by utilizing the aforementioned properties of fractional differential calculus. Using classic definition and property of fractional differential calculus, we extend integer‐order steepest descent approach to fractional field to implement fractional steepest descent approach. Then, based on the earlier fractional formulas, a FPDE based multiscale denoising model for texture image is proposed and further analyze optimal parameters value for FPDE based denoising model. The experimental results prove that the ability for preserving high‐frequency edge and complex texture information of the proposed fractional denoising model are obviously superior to traditional integral based algorithms, as for texture detail rich images. Copyright © 2013 John Wiley & Sons, Ltd.     

Real-Time Semiparametric Regression

We develop algorithms for performing semiparametric regression analysis in real time, with data processed as it is collected and made immediately available via modern telecommunications technologies. Our definition of semiparametric regression is quite broad and includes, as special cases, generalized linear mixed models, generalized additive models, geostatistical models, wavelet nonparametric regression models and their various combinations. Fast updating of regression fits is achieved by couching semiparametric regression into a Bayesian hierarchical model or, equivalently, graphical model framework and employing online mean field variational ideas. An Internet site attached to this article, realtime-semiparametric-regression.net, illustrates the methodology for continually arriving stock market, real estate, and airline data. Flexible real-time analyses based on increasingly ubiquitous streaming data sources stand to benefit. This article has online supplementary material.     

Orthogonal bandelet bases for geometric images approximation

This paper introduces orthogonal bandelet bases to approximate images having some geometrical regularity. These bandelet bases are computed by applying parametrized Alpert transform operators over an orthogonal wavelet basis. These bandeletization operators depend upon a multiscale geometric flow that is adapted to the image at each wavelet scale. This bandelet construction has a hierarchical structure over wavelet coefficients taking advantage of existing regularity among these coefficients. It is proved that C^α‐images having singularities along C^α‐curves are approximated in a best orthogonal bandelet basis with an optimal asymptotic error decay. Fast algorithms and compression applications are described. © 2008 Wiley Periodicals, Inc.     

Ridgelet bi-frame

This article introduces a new bi-frame called ridgelet bi-frame. The ridgelet bi-frame consists of two ridgelet frames that are dual to each other. The construction of the ridgelet bi-frame starts with a bi-frame built on a biorthogonal wavelet system in the Radon domain. The image of the resulting bi-frame under an isometric map from the Radon domain to L²(R²) is also a bi-frame, which we refer to as the ridgelet bi-frame. The ridgelet bi-frame can be thought of as an extension of the notion of the orthonormal ridgelet, which provides a more flexible and effective tool for image analysis and processing applications. An algorithm for image denoising based on the new bi-frame is developed. Experimental examples have demonstrated that the excellent performance can be achieved when using the ridgelet bi-frame for image denoising.     

Algorithms and software for total variation image reconstruction via first-order methods

This paper describes new algorithms and related software for total variation (TV) image reconstruction, more specifically: denoising, inpainting, and deblurring. The algorithms are based on one of Nesterov’s first-order methods, tailored to the image processing applications in such a way that, except for the mandatory regularization parameter, the user needs not specify any parameters in the algorithms. The software is written in C with interface to Matlab (version 7.5 or later), and we demonstrate its performance and use with examples.     

Wavelet-based image processing

The 1990s witnessed an explosion of wavelet-based methods in the field of image processing. This article will focus primarily on wavelet-based image compression. We shall describe the connection between wavelets and vision, and how wavelet techniques provide image compression algorithms that are clearly superior to the present JPEG standard. In particular, the wavelet-based algorithms known as SPIHT, ASWDR, and the new standard JPEG2000, will be described and compared. Our comparison will show that, in many respects, ASWDR is the best algorithm. Applications to denoising will also be briefly referenced and pointers supplied to other references on wavelet-based image processing.