Image Restoration Using One-Dimensional Sobolev Norm Profiles of Noise and Texture

This work is devoted to image restoration (denoising and deblurring) by variational models. As in our prior work [Inverse Probl. Imaging, 3 (2009), pp. 43-68], the image (f) over tilde to be restored is assumed to be the sum of a cartoon component u (a function of bounded variation) and a texture component v (an oscillatory function in a Sobolev space with negative degree of differentiability). In order to separate noise from texture in a blurred noisy textured image, we need to collect some information that helps distinguish noise, especially Gaussian noise, from texture. We know that homogeneous Sobolev spaces of negative differentiability help capture oscillations in images very well; however, these spaces do not directly provide clear distinction between texture and noise, which is also highly oscillatory, especially when the blurring effect is noticeable. Here, we propose a new method for distinguishing noise from texture by considering a family of Sobolev norms corresponding to noise and texture. It turns out that the two Sobolev norm profiles for texture and noise are different, and this enables us to better separate noise from texture during the deblurring process.open0

A total variation-undecimated wavelet approach to chest radiograph image enhancement

Most often medical images such as X-Rays have a low dynamic range and many of their targeted features are difficult to identify. Intensity transformations that improve image quality usually rely onwavelet denoising and enhancement typically use the technique of thresholding to obtain better quality medical images. A disadvantage of wavelet thresholding is that even though it adequately removes noise in an image, it introduces unwanted artifacts into the image near discontinuities. We utilize a total variation method and an undecimated wavelet image enhancing algorithm for improving the image quality of chest radiographs. Our approach achieves a high level chest radiograph image deniosing in lung nodules detection while preserving the important features. Moreover, our method results in a high image sensitivity that reduces the average number of false positives on a test set of medical data

A Coordinate Descent Method for Total Variation Minimization

Total variation (TV) is a well-known image model with extensive applications in various images and vision tasks, for example, denoising, deblurring, superresolution, inpainting, and compressed sensing. In this paper, we systematically study the coordinate descent (CoD) method for solving general total variation (TV) minimization problems. Based on multidirectional gradients representation, the proposed CoD method provides a unified solution for both anisotropic and isotropic TV-based denoising (CoDenoise). With sequential sweeping and small random perturbations, CoDenoise is efficient in denoising and empirically converges to optimal solution. Moreover, CoDenoise also delivers new perspective on understanding recursive weighted median filtering. By incorporating with the Augmented Lagrangian Method (ALM), CoD was further extended to TV-based image deblurring (ALMCD). The results on denoising and deblurring validate the efficiency and effectiveness of the CoD-based methods

ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation

Total variation denoising in l1l^1 anisotropy

We aim at constructing solutions to the minimizing problem for the variant of Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying anisotropic total variation functional. We consider a naturally defined class of functions piecewise constant on rectangles (PCR). This class forms a strictly dense subset of the space of functions of bounded variation with an anisotropic norm. The main result shows that if the given noisy image is a PCR function, then solutions to both considered problems also have this property. For PCR data the problem of finding the solution is reduced to a finite algorithm. We discuss some implications of this result, for instance we use it to prove that continuity is preserved by both considered problems.Comment: 34 pages, 9 figure

Image Denoising via Asymptotic Nonlocal Filtering

The nonlocal means algorithm is widely used in image denoising, but this algorithm does not work well for high-intensity noise. To overcome this shortcoming, we establish a coupled iterative nonlocal means model in this paper. Considering the computation complexity of the new model, we realize it by using multiscale wavelet transform and propose an asymptotic nonlocal filtering algorithm which can reduce the influence of noise on similarity estimation and computation complexity. Moreover, we build a new nonlocal weight function based on the structure similarity index. Simulation results indicate that the proposed approach cannot only remove the noise but also preserve the structure of image and has good visual effects, especially for highly degenerated images

An Iterative Shrinkage Approach to Total-Variation Image Restoration

The problem of restoration of digital images from their degraded measurements plays a central role in a multitude of practically important applications. A particularly challenging instance of this problem occurs in the case when the degradation phenomenon is modeled by an ill-conditioned operator. In such a case, the presence of noise makes it impossible to recover a valuable approximation of the image of interest without using some a priori information about its properties. Such a priori information is essential for image restoration, rendering it stable and robust to noise. Particularly, if the original image is known to be a piecewise smooth function, one of the standard priors used in this case is defined by the Rudin-Osher-Fatemi model, which results in total variation (TV) based image restoration. The current arsenal of algorithms for TV-based image restoration is vast. In the present paper, a different approach to the solution of the problem is proposed based on the method of iterative shrinkage (aka iterated thresholding). In the proposed method, the TV-based image restoration is performed through a recursive application of two simple procedures, viz. linear filtering and soft thresholding. Therefore, the method can be identified as belonging to the group of first-order algorithms which are efficient in dealing with images of relatively large sizes. Another valuable feature of the proposed method consists in its working directly with the TV functional, rather then with its smoothed versions. Moreover, the method provides a single solution for both isotropic and anisotropic definitions of the TV functional, thereby establishing a useful connection between the two formulae.Comment: The paper was submitted to the IEEE Transactions on Image Processing on October 22nd, 200