A general non-parametric technique is proposed for the analysis of a complex multimodal feature space and to delineate arbitrarily shaped clusters in it. The basic computational module of the technique is an old pattern recognition procedure: the mean shift. For discrete data, we prove the convergence of a recursive mean shift procedure to the nearest stationary point of the underlying density function and, thus, its utility in detecting the modes of the density. The relation of the mean shift procedure to the Nadaraya-Watson estimator from kernel regression and the robust M-estimators; of location is also established. Algorithms for two low-level vision tasks discontinuity-preserving smoothing and image segmentation - are described as applications. In these algorithms, the only user-set parameter is the resolution of the analysis, and either gray-level or color images are accepted as input. Extensive experimental results illustrate their excellent performance.

Mean shift: a robust approach toward feature space analysis

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained l1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms l1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted l1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the l1 norm of the coefficient sequence as is common, but by reweighting the l1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as Compressive Sensing.

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Enhancing Sparsity by Reweighted ℓ 1 Minimization

Humans perceive the three-dimensional structure of the world with apparent ease. However, despite all of the recent advances in computer vision research, the dream of having a computer interpret an image at the same level as a two-year old remains elusive. Why is computer vision such a challenging problem and what is the current state of the art? Computer Vision: Algorithms and Applications explores the variety of techniques commonly used to analyze and interpret images. It also describes challenging real-world applications where vision is being successfully used, both for specialized applications such as medical imaging, and for fun, consumer-level tasks such as image editing and stitching, which students can apply to their own personal photos and videos. More than just a source of recipes, this exceptionally authoritative and comprehensive textbook/reference also takes a scientific approach to basic vision problems, formulating physical models of the imaging process before inverting them to produce descriptions of a scene. These problems are also analyzed using statistical models and solved using rigorous engineering techniques Topics and features: structured to support active curricula and project-oriented courses, with tips in the Introduction for using the book in a variety of customized courses; presents exercises at the end of each chapter with a heavy emphasis on testing algorithms and containing numerous suggestions for small mid-term projects; provides additional material and more detailed mathematical topics in the Appendices, which cover linear algebra, numerical techniques, and Bayesian estimation theory; suggests additional reading at the end of each chapter, including the latest research in each sub-field, in addition to a full Bibliography at the end of the book; supplies supplementary course material for students at the associated website, http://szeliski.org/Book/. Suitable for an upper-level undergraduate or graduate-level course in computer science or engineering, this textbook focuses on basic techniques that work under real-world conditions and encourages students to push their creative boundaries. Its design and exposition also make it eminently suitable as a unique reference to the fundamental techniques and current research literature in computer vision.

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Computer Vision: Algorithms and Applications

A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with user-defined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels. By assigning each pixel to the label for which the greatest probability is calculated, a high-quality image segmentation may be obtained. Theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits. This algorithm is formulated in discrete space (i.e., on a graph) using combinatorial analogues of standard operators and principles from continuous potential theory, allowing it to be applied in arbitrary dimension on arbitrary graphs

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Random Walks for Image Segmentation

We present a new technique for the display of high-dynamic-range images, which reduces the contrast while preserving detail. It is based on a two-scale decomposition of the image into a base layer,...

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Fast bilateral filtering for the display of high-dynamic-range images

Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edge-stopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edge-stopping" function based on Tukey's biweight robust estimator that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in an image that has been smoothed with anisotropic diffusion. Additionally, we derive a relationship between anisotropic diffusion and regularization with line processes. Adding constraints on the spatial organization of the line processes allows us to develop new anisotropic diffusion equations that result in a qualitative improvement in the continuity of edges.

Robust anisotropic diffusion

We describe procedures for creating efficient spectral representations for color. The representations generalize conventional tristimulus representations, which are based on the peripheral encoding by the human eye. We use low-dimensional linear models to approximate the spectral properties of surfaces and illuminants with respect to a collection of sensing devices. We choose the linear-model basis functions by minimizing the error in approximating sensor responses for collections of surfaces and illuminants. These linear models offer some conceptual simplifications for applications such as printer calibration; they also perform substantially better than principal-components approximations for computer-graphics applications.

Linear models of surface and illuminant spectra

We show how to compute and to use the wavelength-dependent optical transfer function (OTF) to create color matches between spatially patterned images. We model the human OTF as a defocused optical system with a circular aperture. In our model the defocus arises from axial chromatic aberration and wavelength-independent aberrations. From the computed OTF it is apparent that high-spatial-frequency components of the image can play little role in contrast and color appearance and that in the spatial-frequency range from 5 to 20 cycles/deg the visual system is dichromatic, because there is no contrast in the short-wavelength receptor signal. We show how to use the wavelength-dependent OTF to match color images across displays by setting matches in corresponding spatial-frequency bands. Because chromatic aberration so affects the OTF, this new procedure is a significant improvement over the conventional procedure of setting matches point by point.

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Matching color images: the effects of axial chromatic aberration

A method of rendering a color image on a designated output medium is disclosed which maps colors to the gamut of the designated output medium while preserving the semantic consistency of the object color and illumination information in the image The method performs gamut mapping earlier in the image synthesis process than current gamut method methods, at the point where information about object primitives and their spectral attributes in a scene description is available, but after the fixed scene geometry has been determined by the rendering system The method makes use of the output of a symbolic rendering system which produces symbolic pixel expressions, having basis spectra variables which represent the interplay of light and object primitives in the scene description, and spectral data having color information about the light and object primitives in the scene, and which is indexed to the basis spectra variables The method performs spectral change calculations using the symbolic pixel expressions, the spectral data, and spectral information about the gamut of the specific output medium to determine the modifications to the original spectral data that need to be made to the individual object primitives in the scene in order to produce image pixel colors which are in the gamut of the output medium The resulting image colors are locally and globally consistent with the semantics of the image, make effective use of as much of the available gamut of the display device as is possible, and require no further post-rendering gamut mapping prior to display or reproduction on the designated medium

Method of rendering a color image for an output medium from symbolic image data

We describe a robust, dynamic algorithm to compute the arrangement of a set of line segments in the plane, and its implementation. The algorithm is robust because, following Greene7 and Hobby,11 it rounds the endpoints and intersections of all line segments to representable points, but in a way that is globally topologically consistent. The algorithm is dynamic because, following Mulmuley,16 it uses a randomized hierarchy of vertical cell decompositions to make locating points, and inserting and deleting line segments, efficient. Our algorithm is novel because it marries the robustness of the Greene and Hobby algorithms with Mulmuley's dynamic algorithm in a way that preserves the desirable properties of each.

David H. Marimont

Papers

Robust anisotropic diffusion

Linear models of surface and illuminant spectra

Matching color images: the effects of axial chromatic aberration

Method of rendering a color image for an output medium from symbolic image data

Rounding arrangements dynamically