Author

# Steven W. Zucker

Other affiliations: McGill University, University of Minnesota, St. Joseph Hospital ...read more

Bio: Steven W. Zucker is an academic researcher from Yale University. The author has contributed to research in topics: Curvature & Orientation (computer vision). The author has an hindex of 64, co-authored 310 publications receiving 20380 citations. Previous affiliations of Steven W. Zucker include McGill University & University of Minnesota.

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##### Papers

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Yale University

^{1}TL;DR: The process of iterating or diffusing the Markov matrix is seen as a generalization of some aspects of the Newtonian paradigm, in which local infinitesimal transitions of a system lead to global macroscopic descriptions by integration.

Abstract: We provide a framework for structural multiscale geometric organization of graphs and subsets of R(n). We use diffusion semigroups to generate multiscale geometries in order to organize and represent complex structures. We show that appropriately selected eigenfunctions or scaling functions of Markov matrices, which describe local transitions, lead to macroscopic descriptions at different scales. The process of iterating or diffusing the Markov matrix is seen as a generalization of some aspects of the Newtonian paradigm, in which local infinitesimal transitions of a system lead to global macroscopic descriptions by integration. We provide a unified view of ideas from data analysis, machine learning, and numerical analysis.

1,654 citations

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01 Jun 1976TL;DR: This paper formulates the ambiguity-reduction process in terms of iterated parallel operations (i.e., relaxation operations) performed on an array of object, identification data.

Abstract: Given a set of objects in a scene whose identifications are ambiguous, it is often possible to use relationships among the objects to reduce or eliminate the ambiguity. A striking example of this approach was given by Waltz [13]. This paper formulates the ambiguity-reduction process in terms of iterated parallel operations (i.e., relaxation operations) performed on an array of (object, identification) data. Several different models of the process are developed, convergence properties of these models are established, and simple examples are given.

1,513 citations

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TL;DR: It is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality, and a procedure nearly identical to the relaxation operator derived under restricted circum-stances serves in the more general setting.

Abstract: A large class of problems can be formulated in terms of the assignment of labels to objects. Frequently, processes are needed which reduce ambiguity and noise, and select the best label among several possible choices. Relaxation labeling processes are just such a class of algorithms. They are based on the parallel use of local constraints between labels. This paper develops a theory to characterize the goal of relaxation labeling. The theory is founded on a definition of con-sistency in labelings, extending the notion of constraint satisfaction. In certain restricted circumstances, an explicit functional exists that can be maximized to guide the search for consistent labelings. This functional is used to derive a new relaxation labeling operator. When the restrictions are not satisfied, the theory relies on variational cal-culus. It is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality. A procedure nearly identical to the relaxation operator derived under restricted circum-stances serves in the more general setting. Further, a local convergence result is established for this operator. The standard relaxation labeling formulas are shown to approximate our new operator, which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here. Observations about con-vergence and generalizations to higher order compatibility relations are described.

964 citations

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04 Jan 1998

TL;DR: A novel tree matching algorithm is introduced which finds the best set of corresponding nodes between two shock trees in polynomial time and is demonstrated under articulation, occlusion, and moderate changes in viewpoint.

Abstract: We have been developing a theory for the generic representation of 2-D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. We now apply the theory to the problem of shape matching. The shocks are organized into a directed, acyclic shock graph, and complexity is managed by attending to the most significant (central) shape components first. The space of all such graphs is highly structured and can be characterized by the rules of a shock graph grammar. The grammar permits a reduction of a shockgraph to a unique rooted shock tree. We introduce a novel tree matching algorithm which finds the best set of corresponding nodes between two shock trees in polynomial time. Using a diverse database of shapes, we demonstrate our system's performance under articulation, occlusion, and changes in viewpoint.

866 citations

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TL;DR: The decomposition of TPP is summarized by the equation TPP => PPP + VPP, and VPP is transformed to a two-dimensional PPP in path-time space with some additional constraints.

Abstract: We present a novel approach to solving the trajectory plan ning problem (TPP) in time-varying environments. The es sence of our approach lies in a heuristic but natural decom position of TPP into t...

795 citations

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TL;DR: The analogy between images and statistical mechanics systems is made and the analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations, creating a highly parallel ``relaxation'' algorithm for MAP estimation.

Abstract: We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.

18,761 citations

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TL;DR: This work uses snakes for interactive interpretation, in which user-imposed constraint forces guide the snake near features of interest, and uses scale-space continuation to enlarge the capture region surrounding a feature.

Abstract: A snake is an energy-minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges. Snakes are active contour models: they lock onto nearby edges, localizing them accurately. Scale-space continuation can be used to enlarge the capture region surrounding a feature. Snakes provide a unified account of a number of visual problems, including detection of edges, lines, and subjective contours; motion tracking; and stereo matching. We have used snakes successfully for interactive interpretation, in which user-imposed constraint forces guide the snake near features of interest.

18,095 citations

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01 Jan 1988TL;DR: Probabilistic Reasoning in Intelligent Systems as mentioned in this paper is a complete and accessible account of the theoretical foundations and computational methods that underlie plausible reasoning under uncertainty, and provides a coherent explication of probability as a language for reasoning with partial belief.

Abstract: From the Publisher:
Probabilistic Reasoning in Intelligent Systems is a complete andaccessible account of the theoretical foundations and computational methods that underlie plausible reasoning under uncertainty. The author provides a coherent explication of probability as a language for reasoning with partial belief and offers a unifying perspective on other AI approaches to uncertainty, such as the Dempster-Shafer formalism, truth maintenance systems, and nonmonotonic logic. The author distinguishes syntactic and semantic approaches to uncertaintyand offers techniques, based on belief networks, that provide a mechanism for making semantics-based systems operational. Specifically, network-propagation techniques serve as a mechanism for combining the theoretical coherence of probability theory with modern demands of reasoning-systems technology: modular declarative inputs, conceptually meaningful inferences, and parallel distributed computation. Application areas include diagnosis, forecasting, image interpretation, multi-sensor fusion, decision support systems, plan recognition, planning, speech recognitionin short, almost every task requiring that conclusions be drawn from uncertain clues and incomplete information.
Probabilistic Reasoning in Intelligent Systems will be of special interest to scholars and researchers in AI, decision theory, statistics, logic, philosophy, cognitive psychology, and the management sciences. Professionals in the areas of knowledge-based systems, operations research, engineering, and statistics will find theoretical and computational tools of immediate practical use. The book can also be used as an excellent text for graduate-level courses in AI, operations research, or applied probability.

15,671 citations

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General Electric

^{1}TL;DR: In this paper, a divide-and-conquer approach is used to generate inter-slice connectivity, and then a case table is created to define triangle topology using linear interpolation.

Abstract: We present a new algorithm, called marching cubes, that creates triangle models of constant density surfaces from 3D medical data. Using a divide-and-conquer approach to generate inter-slice connectivity, we create a case table that defines triangle topology. The algorithm processes the 3D medical data in scan-line order and calculates triangle vertices using linear interpolation. We find the gradient of the original data, normalize it, and use it as a basis for shading the models. The detail in images produced from the generated surface models is the result of maintaining the inter-slice connectivity, surface data, and gradient information present in the original 3D data. Results from computed tomography (CT), magnetic resonance (MR), and single-photon emission computed tomography (SPECT) illustrate the quality and functionality of marching cubes. We also discuss improvements that decrease processing time and add solid modeling capabilities.

13,231 citations

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TL;DR: A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced, chosen to vary spatially in such a way as to encourage intra Region smoothing rather than interregion smoothing.

Abstract: A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced. The diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing. It is shown that the 'no new maxima should be generated at coarse scales' property of conventional scale space is preserved. As the region boundaries in the approach remain sharp, a high-quality edge detector which successfully exploits global information is obtained. Experimental results are shown on a number of images. Parallel hardware implementations are made feasible because the algorithm involves elementary, local operations replicated over the image. >

12,560 citations