Image Reconstruction

One-line definition:Image Reconstruction (or Restoration ) is generally an inverse problem, which intends to recover the original ideal image (or a general signal) from its given bad version, such as one that is snowed by noise, blurred by atmospheric turbulence (as in certain astronomic observations), or that has some regions damaged (like in a black-white photo in 1898, or an ancient painting).

* Image Inpainting (see the independent page)

* Image Decomposition & Restoration via TV & H^(-1) (Osher-Sole-Vese)
* Geometric Surface Denoising via Normal Diffusions (Tasdizen-Whitaker-Burchard-Osher)
* Image Statistics and Improved TV Denoising (Green)
* Weber's Law and Weberized TV Restoration (Shen)
* Denoising and Enhancement of Non-Flat Image Features (Chan-Shen)
* TV Denoising of Color Images By the CB and HSV Color Models (Chan-Kang-Shen)
* Vectorial and Multi-Spectral Images (Blomgren)
* Blind Deconvolution (Blomgren-Chan)

Image Decomposition & Restoration via TV & H^(-1)

From the abstract: In this paper, we propose a new model for image restoration and decomposition, abased on the total variation minimization of Rudin-Osher-Fatemi, and of the results of Y. Meyer on oscillatory functions. An initial image f is decomposed into a cartoon part u and a texture or noise part v. The u component is modeled by a function of bounded variation, while the v component by an oscillatory function, with bounded H^(-1) norm. After some transformation, the resulting PDE is of fourth order. Finally, image decomposition, denoising and deblurring numerical results are shown. CAM Report 02-57 (pdf file) by Stanley Osher, Andres Sole, and Luminita Vese. October 2002.

Geometric Surface Denoising via Normal Diffusions

From the abstract: This paper introduces a method for smoothing complex, noisy surfaces, while preserving (and enhancing) sharp, geometric features. It has two main advantages over previous approaches to feature preserving surface smoothing. First is the use of level set surface models, which allows us to process very complex shapes of arbitrary and changing topology. This generality makes it well suited for processing surfaces that are derived directly from measured data. The second advantage is that the proposed method derives from a well-founded formulation, which is a natural generalization of anisotropic diffusion, as used in image processing. This formulation is based on the proposition that the generalization of image filtering entails filtering the normals of the surface, rather than processing the positions of points on a mesh. CAM Report 02-56 (pdf file) by Tolga Tasdizen, Ross Whitaker, Paul Burchard, and Stanley Osher. October 2002.

Image Statistics and Improved TV Denoising

From the abstract: The goal of this paper is to present a new basic model for the joint density function for a broad class of spatial and time-series data. As evidence that this model is indeed useful in practical problems, an application to image denoising in the presence of textures will be explained. CAM Report 02-55 (pdf file) by Mark Green. October 2002.

Weber's Law and Weberized TV Restoration.

From the abstract: Most conventional image processors consider little the influence of human vision psychology. Weber's Law in psychology and psychophysics claims that humans' perception and response to the intensity fluctuation du of visual signals are weighted by the background stimulus u, instead of being plainly uniform. This paper attempts to integrate this well known perceptual law into the classical total variation (TV) image restoration model of Rudin, Osher, and Fatemi [ Physica D , 60:259-268, 1992]. We study the issues of existence and uniqueness for the proposed Weberized nonlinear TV restoration model, making use of the direct method in the space of functions with bounded variations. We also propose an iterative algorithm based on the linearization technique for the associated nonlinear Euler-Lagrange equations. [ Physica D, to appear. ATTN: the title has been changed to: On the foundations of vision modeling I. Weber's law and Weberized TV restoration.]

Denoising and Enhancement of Non-Flat Image Features.

Non-flat image features are those that live on Riemannian manifolds, instead of Euclidean Spaces. Familiar examples include the "orientation" feature that lives on the unit circle, the "alignment" feature that lives on the real projective line, and the "chromaticity" feature that lives on the unit sphere. We apply the variational method to the restoration of general non-flat image features. Mathematical models and their numerical implementations are carefully studied. This work extends the well-known TV (total variation) restoration model of Rudin, Osher, and Fetami (1992), and is closely inspired by the work of Perona at CalTech (IEEE Trans. Image Process. 7(3), 1998) and Sapiro's Minnesota group (Int. J. Comp. Vision, 36(2), 2000). The most recent related works (as of June 2000) are by Kimmel and Sochen (J. Visual Comm. Image Rep., to appear, 2000) on an enhancing model based on Polyakov actions, and by Chan, Kang, and Shen (preprint, 2000) on the TV model applied to the CB and HSV non-linear color models. (From CAM 99-20 Abstract, June 99). Report by Chan and Shen [SIAM J. Appl. Math., 61(4), 1338-1361, 2000].

Total Variation Denoising and Enhancement of Color Images Based on the CB and HSV Color Models.

Most denoising and enhancement methods for the color images have been formulated on linear color models, namely, the channel-by-channel model and vectorial model. In this paper, we study the total variation (TV) restoration based on the two nonlinear (or non-flat) color models: the Chromaticity - Brightness(CB) model and Hue-Saturation-Value (HSV) model. These models are known to be closer to human perception of colors. Recent works on the variational/PDE method for non-flat features by several authors (see the previous report) enable us to denoise the chromaticity and hue components directly. TV restoration results based on such nonlinear color models are generally better than those based on linear color models in terms of color control. (From CAM Report 00-25 Abstract, July 00). Report by Chan, Kang, and Shen [J. Visual Comm and Image Rep, 12(4), 422-435, 2001]

Figure 4, Comprison between Channel by Channel, Vectorial and CB (eps file)
Figure 5, Comprison between Channel by Channel, Vectorial and CB (eps file)
Figure 6, HSV channel by channel (eps file)
Figure 7, ZV model (eps.file)

Vectorial and Multi-Spectral Images

We extend total variation image restoration to vector valued, e.g. color and multi-spectral images. This has the desirable properties of not penalizing against edges in the image, it is rotationally invariant. (from CAM 98-30 Abstract, June 98). Reports on Vector Valued Images

Color TV : Total Variation Methods for restoration of Vector Valued Images

We propose a definition of the total variation norm for vector valued functions which can be applied to restore color and other vector valued images. The new TV norm has the desirable properties of 1) not penalizing discontinuities (edges) in the image, 2) rotationally invariant in the image space, and 3) reduces to the usual TV norm in the scalar case. (from CAM 96-5 Abstract, Feb 96) Report by Blomgren and Chan [IEEE Image Proc. March 1998].

Blind Deconvolution

Blind deconvolution refers to the image processing task of restoring the original image from a blurred version without knowledge of the blurring function. To find a local minimum of the objective function, we use an alternation minimization procedure in which we fix either the blur or the image and minimize respect to the other variable each step of which is a standard non-blind deconvolution problem. While the model is not convex and thus allows multiple solutions, we have found that the alternation minimization procedure always converges globally but with the converged solution depending on the initial guess. (from CAM 99-19 Abstract, June 99). Reports on Blind Deconvolution .

Reports on Image Restorations  

Tony Chan , Stanley Osher , Jianhong (Jackie) Shen , David Strong , Peter Blomgren , C. K. Wong, Sung Ha Kang