A geometric approach for 3D object segmentation and representation is presented. The segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3D image. The model is based on curvature motion and the computation of surfaces with minimal areas, better known as minimal surfaces. The space where the surfaces are computed is induced from the 3D image (volumetric data) in which the objects are to be detected. The model links between classical deformable surfaces obtained via energy minimization, and intrinsic ones derived from curvature based flows. The new approach is stable, robust, and automatically handles changes in the surface topology during the deformation.

W e discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. W e propose a set of basic assu,mptions to be satisfied by the interpolation algorith*ms which lead to U set of models in terms of possibly degenerate elliptic partial differential equations. The Absolute Minimal Lipschitz Extension model ( A M L E ) is singled out and studied in more detail. W e show experiments suggesting a possible application, the restoration of images with poor dynamic range.

We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The absolute minimal Lipschitz extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range.

While the retinex theory aimed at explaining human color perception, its derivations have led to efficient algorithms enhancing local image contrast, thus permitting among other features, to "see in the shadows". Among these derived algorithms, Multiscale Retinex is probably the most successful center-surround image filter. In this paper, we offer an analysis and implementation of Multiscale Retinex. We point out and resolve some ambiguities of the method. In particular, we show that the important color correction final step of the method can be seriously improved. This analysis permits to formulate an automatic implementation of Multiscale Retinex which is as faithful as possible to the one described in the original paper. Overall, this implementation delivers excellent results and confirms the validity of Multiscale Retinex for image color restoration and contrast enhancement. Nevertheless, while the method parameters can be fixed, we show that a crucial choice must be left to the user, depending on the lightning condition of the image: the method must either be applied to each color independently if a color balance is required, or to the luminance only if the goal is to achieve local contrast enhancement. Thus, we propose two slightly different algorithms to deal with both cases. Source CodeThe source code, the code documentation, and the online demo are accessible at the IPOL web page of this article 1 . In this link an implementation is available for download. Compilation and usage instruction are included in the README.txt file of the compressed archive. This software includes the implementations of algorithms potentially linkable to patents.

Abstract-Demosaicking is the process by which from a matrix of colored pixels measuring only one color component per pixel, red, green or blue, one can infer a whole color information at each pixel. This inference requires a deep understanding of the interaction between colors, and the involvement of image local geometry. Although quite successful in making such inferences with very small relative error, state of the art demosaicking methods fail when the local geometry cannot be inferred from the neighboring pixels. In such a case, which occurs when thin structures or fine periodic patterns were present in the original, state of the art methods can create disturbing artifacts, known as zipper effect, blur, and color spots. The aim of this paper is to show that these artifacts can be avoided by involving the image self-similarity to infer missing colors. Detailed experiments show that a satisfactory solution can be found, even for the most critical cases. Extensive comparisons with state of the art algorithms will be performed on two different classic image databases.

In 1964 Edwin H. Land formulated the Retinex theory, the first attempt to simulate and explain how the human visual system perceives color. His theory and an extension, the "reset Retinex" were further formalized by Land and McCann. Several Retinex algorithms have been developed ever since. These color constancy algorithms modify the RGB values at each pixel to give an estimate of the color sensation without a priori information on the illumination. Unfortunately, the Retinex Land-McCann original algorithm is both complex and not fully specified. Indeed, this algorithm computes at each pixel an average of a very large set of paths on the image. For this reason, Retinex has received several interpretations and implementations which, among other aims, attempt to tune down its excessive complexity. In this paper, it is proved that if the paths are assumed to be symmetric random walks, the Retinex solutions satisfy a discrete screened Poisson equation. This formalization yields an exact and fast implementation using only two FFTs. Several experiments on color images illustrate the effectiveness of the Retinex original theory.

Summary.A novel geometric approach for three dimensional object segmentation is presented. The scheme is based on geometric deformable surfaces moving towards the objects to be detected. We show that this model is related to the computation of surfaces of minimal area (local minimal surfaces). The space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected. The general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework. The scheme is stable, robust, and automatically handles changes in the surface topology during the deformation. Results related to existence, uniqueness, stability, and correctness of the solution to this geometric deformable model are presented as well. Based on an efficient numerical algorithm for surface evolution, we present a number of examples of object detection in real and synthetic images.

Even after over two decades, the total variation (TV) remains one of the most popular regularizations for image processing problems and has sparked a tremendous amount of research, particularly to move from scalar to vector-valued functions. In this paper, we consider the gradient of a color image as a three dimensional matrix or tensor with dimensions corresponding to the spatial extend, the differences to other pixels, and the spectral channels. The smoothness of this tensor is then measured by taking different norms along the different dimensions. Depending on the type of these norms one obtains very different properties of the regularization, leading to novel models for color images. We call this class of regularizations collaborative total variation (CTV). On the theoretical side, we characterize the dual norm, the subdifferential and the proximal mapping of the proposed regularizers. We further prove, with the help of the generalized concept of singular vectors, that an ∞ channel coupling makes the most prior assumptions and has the greatest potential to reduce color artifacts. Our practical contributions consist of an extensive experimental section where we compare the performance of a large number of collaborative TV methods for inverse problems like denoising, deblurring and inpainting.