A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image. Piecewise planar images look more natural than step images which anisotropic diffusion (second order PDEs) uses to approximate an observed image. So the proposed PDEs are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion, while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion. Although both approaches seem to be comparable in removing speckles in the observed images, speckles are more visible in images processed by the proposed PDEs, because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges. Speckles can be easily removed by simple algorithms such as the one presented in this paper.

This paper presents anisotropic regularization techniques to exploit the piecewise smoothness of the image and the point spread function (PSF) in order to mitigate the severe lack of information encountered in blind restoration of shift-invariantly and shift-variantly blurred images. The new techniques, which are derived from anisotropic diffusion, adapt both the degree and direction of regularization to the spatial activities and orientations of the image and the PSF. This matches the piecewise smoothness of the image and the PSF which may be characterized by sharp transitions in magnitude and by the anisotropic nature of these transitions. For shift-variantly blurred images whose underlying PSFs may differ from one pixel to another, we parameterize the PSF and then apply the anisotropic regularization techniques. This is demonstrated for linear motion blur and out-of-focus blur. Alternating minimization is used to reduce the computational load and algorithmic complexity.

This paper analyzes the behavior of the one-dimensional discretized anisotropic diffusion in order to explain the manner in which false step edges are created. The creation of these edges are related to the characteristics of the local convexity of the image intensity function. An approach is proposed to prevent the appearance of false edges.

In pyramidal coding, the key to significant bit rate reduction is the reduction of interpolation error. This is usually accomplished through well-designed nonlinear interpolation filters, such as median and morphological filters. These filters are usually isotropic, hence cannot account for the anisotropic nature of the images. In addition, these filters base their interpolation solely on the image at the immediate higher level (lower resolution), hence a smaller interpolation error can be expected if the already-transmitted error signals are used for the interpolation of the remaining pixels. In view of these two problems, this paper proposes to decrease the interpolation error through the introduction of anisotropic techniques and the use of the transmitted error signals to improve the interpolation of the remaining pixels. This error is further reduced through the usage of anisotropic diffusion as the low-pass filter. Finally, an anisotropic DPCM coder is presented to code the image at the top of the pyramid.

Anisotropic diffusion is proposed as a technique to regularize joint blur identification and image restoration. In comparison with previously proposed space-adaptive regularization methods, it shares the feature of space-adaptive degree of regularization, but it has the unique feature of adapting its direction of regularization to the orientation of edges. Consequently, good restoration quality was observed with both numerically and photographically degraded images.

The success of ringing artifact reduction in image restoration depends mainly on how the restoration methods accommodate the piecewise smoothness of the image. The article presents a regularization method that adapts both its degree and orientation of regularization operation to the spatial activities of the image intensity and edge orientations. This is achieved by anisotropic diffusion. Its restoration quality compares favorably with that of a previously proposed regularization method that is space adaptive only in the degree of regularization.

The primary difficulty with blind image restoration, or joint blur identification and image restoration, is insufficient information. This calls for proper incorporation of a priori knowledge about the image and the point-spread function (PSF). A well-known space-adaptive regularization method for image restoration is extended to address this problem. This new method effectively utilizes, among others, the piecewise smoothness of both the image and the PSF. It attempts to minimize a cost function consisting of a restoration error measure and two regularization terms (one for the image and the other for the blur) subject to other hard constraints. A scale problem inherent to the cost function is identified, which, if not properly treated, may hinder the minimization/blind restoration process. Alternating minimization is proposed to solve this problem so that algorithmic efficiency as well as simplicity is significantly increased. Two implementations of alternating minimization based on steepest descent and conjugate gradient methods are presented. Good performance is observed with numerically and photographically blurred images, even though no stringent assumptions about the structure of the underlying blur operator is made.

In this paper, we analyze the behavior of the anisotropic diffusion model of Perona and Malik (1990). The main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem, so its behavior can be analyzed based on the shape of the corresponding energy surface. We show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem. It is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space. We give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima. The mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts: one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator, and the other that flows normally to the edges and thus acts as an enhancement operator.

This paper presents shift-adaptive blind image restoration algorithms which can deal with realistic shift-variant blurs and which integrate the usually separate tasks of blur identification and image restoration. The key to success is the effective utilization of the piecewise smoothness of both the image and the PSF to compensate for the severe lack of information in this type of problems. This is achieved through regularization of the image and the point spread function (PSF) by anisotropic diffusion which has the property that smoothing is allowed only in the direction of edges.

This paper describes a class of ill-posed anisotropic diffusion models of the type presented by Perona and Malik (1990). The analysis is based on a previous result that anisotropic diffusion is a steepest descent motion on an energy surface and its behavior is thus determined by the shape of this energy surface. We show that the class of diffusion models are ill-posed because the energy surface is discontinuous at all continuous images, and all step images, which are dense in the space of piecewise images, are global minima of the energy surface.

In this paper we experiments with geometric algorithms for image smoothing. Examples are given for MRI and ATR data. We emphasize experiments with the affine invariant geometric smoother or affine heat equation, originally developed for binary shape smoothing, and found to be efficient for gray-level images as well. Efficient numerical implementations of these flows give anisotropic diffusion processes which preserve edges.>

Earlier work on blur identification suffers from using space-invariant image models to match space-variant (piecewise smooth) images. It is expected that an effective space-adaptive image model or estimate would improve the joint blur and image estimation. A well-known space-adaptive regularization method for image restoration is extended to solve this problem. The resulting scheme has two regularization terms, one for the image and the other for the blur. Very good performance is observed even though no stringent assumption about the structure of the underlying blur operator is made. The computational overhead for blur identification is less than the computational load required for regularized image restoration when the blur operator is exactly known.>

Anisotropic diffusion is posed as a process of minimizing an energy function. Its global convergence behavior is determined by the shape of the energy surface, and its local behavior is described by an orthogonal decomposition with the decomposition coefficients being the eigenvalues of the local energy function. A sufficient condition for its convergence to a global minimum is given and is identified to be the same as the condition previously proposed for the well-posedness of 1-D diffusions. Some behavior conjectures are made for anisotropic diffusions not satisfying the sufficient condition. Finally, some well-behaved anisotropic diffusions are proposed and simulation results are shown.>