Doron Shaked

A Variational Framework for Retinex

Retinex theory addresses the problem of separating the illumination from the reflectance in a given image and thereby compensating for non-uniform lighting. This is in general an ill-posed problem. In this paper we propose a variational model for the Retinex problem that unifies previous methods. Similar to previous algorithms, it assumes spatial smoothness of the illumination field. In addition, knowledge of the limited dynamic range of the reflectance is used as a constraint in the recovery process. A penalty term is also included, exploiting a-priori knowledge of the nature of the reflectance image. The proposed formulation adopts a Bayesian view point of the estimation problem, which leads to an algebraic regularization term, that contributes to better conditioning of the reconstruction problem.Based on the proposed variational model, we show that the illumination estimation problem can be formulated as a Quadratic Programming optimization problem. An efficient multi-resolution algorithm is proposed. It exploits the spatial correlation in the reflectance and illumination images. Applications of the algorithm to various color images yield promising results.

Skeletonization via distance maps and level sets

The medial axis transform (MAT) of a shape, better known as its skeleton, is frequently used in shape analysis and related areas. In this paper a new approach for determining the skeleton of an object is presented. The boundary is segmented at points of maximal positive curvature and a distance map from each of the segments is calculated. The skeleton is then located by applying simple rules to the zero sets of distance map differences. A framework is proposed for numerical approximation of distance maps that is consistent with the continuous case and hence does not suffer from digitization bias due to metrication errors of the implementation on the grid. Subpixel accuracy in distance map calculation is obtained by using gray-level information along the boundary of the shape in the numerical scheme. The accuracy of the resulting efficient skeletonization algorithm is demonstrated by several examples.

IMPLEMENTING CONTINUOUS-SCALE MORPHOLOGY VIA CURVE EVOLUTION

Abstract A new approach to digital implementation of continuous-scale mathematical morphology is presented. The approach is based on discretization of evolution equations associated with continuous multiscale morphological operations. Those equations, and their corresponding numerical implementation, can be derived either directly from mathematical morphology definitions or from curve evolution theory. The advantages of the proposed approach over the classical discrete morphology are demonstrated.

Sharpness measure: towards automatic image enhancement

We propose a measure for image sharpness, which facilitates automatic image sharpness enhancement. This way blurry images will be sharpened more whereas sufficiently sharp images will not be sharpened at all. The measure employs localized frequency content analysis in a feature-based context. The proposed sharpness measure correlates well with perceived sharpness, and is to a large degree invariant to image content. Furthermore, we show that the proposed measure can be used to drive an enhancement algorithm, which will sharpen an input image to a nominal measure. Last but not least, the proposed sharpness measure is computationally efficient, and requires fewer computations than a 3/spl times/3 convolution.

Deriving Stopping Rules for the Probabilistic Hough Transform by Sequential Analysis

It is known that Hough transform computation can be significantly accelerated by polling instead of voting. A small part of the data set is selected at random and used as input to the algorithm. The performance of these probabilistic Hough transforms depends on the poll size. Most probabilistic Hough algorithms use a fixed poll size, which is far from optimal since conservative design requires the fixed poll size to be much larger than necessary under average conditions. It has recently been experimentally demonstrated that adaptive termination of voting can lead to improved performance in terms of the error rate versus average poll size tradeoff. However, the lack of a solid theoretical foundation made general performance evaluation and optimal design of adaptive stopping rules nearly impossible. In this paper it is shown that the statistical theory of sequential hypotheses testing can provide a useful theoretical framework for the analysis and development of adaptive stopping rules for the probabilistic Hough transform. The algorithm is restated in statistical terms and two novel rules for adaptive termination of the polling are developed. The performance of the suggested stopping rules is verified using synthetic data as well as real images. It is shown that the extension suggested in this paper to A. Walds one-sided alternative sequential test (Sequential Analysis,Wiley, New York, 1947) performs better than previously available adaptive (or fixed) stopping rules.

Space-dependent color gamut mapping: a variational approach

Gamut mapping deals with the need to adjust a color image to fit into the constrained color gamut of a given rendering medium. A typical use for this tool is the reproduction of a color image prior to its printing, such that it exploits best the given printer/medium color gamut, namely the colors the printer can produce on the given medium. Most of the classical gamut mapping methods involve a pixel-by-pixel mapping and ignore the spatial color configuration. Recently proposed spatial-dependent approaches for gamut mapping are either based on heuristic assumptions or involve a high computational cost. In this paper, we present a new variational approach for space-dependent gamut mapping. Our treatment starts with the presentation of a new measure for the problem, closely related to a recent measure proposed for Retinex. We also link our method to recent measures that attempt to couple spectral and spatial perceptual measures. It is shown that the gamut mapping problem leads to a quadratic programming formulation, guaranteed to have a unique solution if the gamut of the target device is convex. An efficient numerical solution is proposed with promising results.

Watermarking of dither halftoned images

Image watermarking concerns embedding information in images, in a manner that does not affect the visual quality of the image. This paper focusses on watermarking of dither halftone images. The basic idea is to use a sequence of two dither matrices (instead of one) to encode the watermark information. Analyzing a specific statistical model of input images, leads to an optimal decoding algorithm in term of the rate- distortion trade-off. Furthermore, we characterize optimal dither matrix pairs (i.e.: dither matrix pairs whose use results in the most favorable rate-distortion). Finally, the results are demonstrated in a synthetic example. The example is synthetic in the sense that it does not resort to printing and re-scanning of the image.

EVOLUTIONS OF PLANAR POLYGONS

Evolutions of closed planar polygons are studied in this work. In the first part of the paper, the general theory of linear polygon evolutions is presented, and two specific problems are analyzed. The first one is a polygonal analog of a novel affine-invariant differential curve evolution, for which the convergence of planar curves to ellipses was proved. In the polygon case, convergence to polygonal approximation of ellipses, polygo nal ellipses, is proven. The second one is related to cyclic pursuit problems, and convergence, either to polygonal ellipses or to polygonal circles, is proven. In the second part, two possible polygonal analogues of the well-known Euclidean curve shortening flow are presented. The models follow from geometric considerations. Experimental results show that an arbitrary initial polygon converges to either regular or irregular polygonal approximations of circles when evolving according to the proposed Euclidean flows.

A Discriminative Approach for Wavelet Denoising

This paper suggests a discriminative approach for wavelet denoising where a set of mapping functions (MFs) are applied to the transform coefficients in an attempt to produce a noise free image. As opposed to the descriptive approaches, modeling image or noise priors is not required here and the MFs are learned directly from an ensemble of example images using least-squares fitting. The suggested scheme generates a novel set of MFs that are essentially different from the traditional soft/hard thresholding in the over-complete case. These MFs are demonstrated to obtain comparable performance to the state-of-the-art denoising approaches. Additionally, this framework enables a seamless customization of the shrinkage operation to a new set of restoration problems that were not addressed previously with shrinkage techniques, such as deblurring, JPEG artifact removal, and various types of additive noise that are not necessarily Gaussian white noise.

Variational famework for Retinex

Retinex theory addresses the problem of separating the illumination from the reflectance in a given image and thereby compensating for non-uniform lighting. This is in general an ill-posed problem. In this paper we propose a variational model for the Retinex problem that unifies previous methods. Similar to previous algorithms, it assumes spatial smoothness of the illumination field. In addition, knowledge of the limited dynamic range of the reflectance is used as a constraint in the recovery process. A penalty term is also included, exploiting a-priori knowledge of the nature of the reflectance image. The proposed formulation adopts a Bayesian view point of the estimation problem, which leads to an algebraic regularization term, that contributes to better conditioning of the reconstruction problem. Based on the proposed variational model, we show that the illumination estimation problem can be formulated as a Quadratic Programming optimization problem. An efficient multi-resolution algorithm is proposed. It exploits the spatial correlation in the reflectance and illumination images. Applications of the algorithm to various color images yield promising results.