Last update: July 29, 2002

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(Some) Publications by Steve Zucker et al. :

• Complexity, Confusion, and Perceptual Grouping. Part I: The Curve-like Representation, 2001.
• Complexity, Confusion, and Perceptual Grouping. Part II: Mapping Complexity, 2001.
• Categorical Features in Shape Perception, 1999.
• Logical/Linear Operators for Image Curves, 1995.
• Shapes, shocks, and deformations, 1995.
• The Effect of Contour Closure on the Rapid Discrimination of Two-Dimensional Shapes, 1993.
• The Integration of Figure Fragments into Representations of Planar Shape, 1993.
• Shape from Shading on a Cloudy Day, 1992 & 1994.
• Contour Closure and the Perception of Shape, 1992.
• Shading Flows and Scenel Bundles: A New Approach to Shape from Shading, 1992.
• Singularities of Principal Direction Fields from 3-D Images, 1992.
• Inferring Surface Trace and Differential Structure from 3-D Images, 1990.
• Cytochrome Oxidase and Functional Coding in Primate Striate Cortex: A Hypothesis, 1990.
• Evaluating the Fractal Dimension of Surfaces, 1989.
• Trace Inference, Curvature Consistency and Curve Detection, 1989.
• Radial Projection: An Efficient Update Rule for Relaxation Labeling, 1989.
• Two stages of Curve Detection Suggest Two Styles of Visual Computation, 1989.
• Endstopping and Curvature, 1989.
• Designing Endstopped Neurons for Curvature Selectivity, 1989.
• Size-Density Constraint and Dot Grouping, 1988.
• Corner Detection in Curvilinear Dot Grouping, 1988.
• The Organization of Curve Detection: Coarse Tangent Fields and Fine Spline Coverings, 1988.
• The Emerging Paradigm of Computational Vision, 1987.
• Endstopped Neurons in the Visual Cortex as a Substrate for Calculating Curvature, 1987.
• Tracing Surfaces for Surfacing Traces, 1987.
• Early Vision, 1986.
• Early Orientation Selection: Tangent Fields and the Dimensionality of Their Support, 1985.
• The Diversity of Perceptual Grouping, 1985.
• Proximiy, Brightness and Contrast Polarity in Dot Grouping, 1983.
• On the Foundations of Relaxation Labelling Processes, 1983.
• A Gradient Projection Algorithm for Relaxation Methods, 1983.
• A Three-Dimensional Edge Detector, 1981.
• Vertical and Horizontal Processes in Low Level Vision, 1978.
• An Application of Relaxation Labelling to Line and Curve Enhancement, 1977.

BibTeX references

Intermediate-level vision is central to form perception, and we outline an approach to intermediate-level segmentation based on complexity analysis. We focus on the problem of edge detection, and how edge elements might be grouped together. This is typical because, once the local structure is established, the transition to global structure must be effected and context is critical. To illustrate, consider an edge element inferred from an unknown image. Is this local edge part of a long curve, or part of a texture? If the former, which is the next element along the curve? If the latter, is the texture like a hair pattern, in which nearby elements are oriented similarly, or like a spaghetti pattern, in which they are not? Are there other natural possibilities? Such questions raise issues of dimensionality, since curves are 1-D and textures are 2-D, and also of complexity. Working toward a measure of representational complexity for vision, in this first of a pair of papers we develop a foundation based on geometric measure theory. The main result concerns the distribution of tangents in space and in orientation, which serves as a formal basis for the concrete measure of representational complexity developed in the companion paper.

Keywords: perceptual organization, segmentation, tangent maps, image curves texture, texture flow, complexity.

Intermediate-level vision is central to form perception, and we outline an approach to intermediate-level segmentation based on complexity analysis. In this second of a pair of papers, we continue the focus on edge-element grouping, and the motivating example of an edge element inferred from an unknown image. Is this local edge part of a long curve, or part of a texture? If the former, which is the next element along the curve? If the latter, is the texture like a well-combed hair pattern, in which nearby elements are oriented similarly, or more chaotic, as in a spaghetti pattern? In the previous paper we showed how these questions raise issues of complexity and dimensionality, and how context in both position and orientation are important. We now propose a measure based on tangential and normal complexities, and illustrate its computation. Tangential complexity is related to extension; normal complexity to density. Taken together they define four canonical classes of tangent distributions: those arising from curves, from texture flows, from turbulent textures, and from isolated "dust". Examples are included.

Keywords: perceptual organization, segmentation, complexity, curve detection, texture.

Web link.

We propose a language for designing image measurement operators suitable for early vision. We refer to them as Logical/Linear (L/L) operators, since they unify aspects of linear operator theory and boolean logic. A family of these operators appropriate for measuring the low-order differential structure of image curves is developed. These L/L operators are derived by decomposing a linear model into logical components to ensure that certain structural preconditions for the existence of an image curve are upheld. Tangential conditions guarantee continuity, while normal conditions select and categorize contrast profiles. The resulting operators allow for coarse measurement of curvilinear differential structure (orientation and curvature) while successfully segregating edge- and line-like features. By thus reducing the incidence of false-positive responses, these operators are a substantial improvement over (thresholded) linear operators which attempt to resolve the same class of features.

Index Terms: Edge detection, feature extraction, image processing, computer vision, nonlinear operators.

Generic singularities can provide position-independent information about the qualitative shape of surfaces. The authors determine the singularities of the principal direction fields of a surface (its umbilic points) from a computation of the index of the fields. The authors present examples both for 3-D synthetic images to which noise has been added and for clinical magnetic resonance images.

Index Terms: 3D images; generic singularity; picture processing; pattern recognition; principal direction fields; qualitative shape; clinical magnetic resonance images; computational geometry; pattern recognition; picture processing

Early image understanding seeks to derive analytic representations from image intensities. The authors present steps towards this goal by considering the inference of surfaces from three-dimensional images. Only smooth surfaces are considered and the focus is on the coupled problems of inferring the trace points (the points through which the surface passes) and estimating the associated differential structure given by the principal curvature and direction fields over the estimated smooth surfaces. Computation of these fields is based on determining an atlas of local charts or parameterizations at estimated surface points. Algorithm robustness and the stability of results are essential for analyzing real images; to this end, the authors present a functional minimization algorithm utilizing overlapping local charts to refine surface points and curvature estimates, and develop an implementation as an iterative constraint satisfaction procedure based on local surface smoothness properties. Examples of the recovery of local structure are presented for synthetic images degraded by noise and for clinical magnetic resonance images.

Index Terms: 3D image inference; surface trace; differential structure; image understanding; trace points; principal curvature; robustness; functional minimization algorithm; overlapping local charts; surface smoothness; clinical magnetic resonance images; inference mechanisms; pattern recognition; picture processing.

An approach is described for curve inference that is based on curvature information. The inference procedure is divided into two stages: a trace inference stage, which is the subject of the present work, and a curve synthesis stage. It is shown that recovery of the trace of a curve requires estimating local models for the curve at the same time, and that tangent and curvature information are sufficient. These make it possible to specify powerful constraints between estimated tangents to a curve, in terms of a neighborhood relationship called cocircularity, and between curvature estimates, in terms of a curvature consistency relation. Because all curve information is quantized, special care must be taken to obtain accurate estimates of trace points, tangents, and curvatures. This issue is addressed specifically to the introduction of a smoothness constraint and a maximum curvature constraint. The procedure is applied to two types of images: artificial images designed to evaluate curvature and noise sensitivity, and natural images.

Index Terms: picture processing; pattern recognition; curvature consistency; curve detection; curve inference; curvature information; tangent; cocircularity; trace points; artificial images; natural images; inference mechanisms; pattern recognition; picture processing

Relaxation labeling uses contextual information for finding consistent labelings of graphs. Although relaxation labeling is parallel and iterative, the complexity of updating with standard rules is too costly for practical implementation. A description is given of a computationally more efficient updating rule that utilizes radial projection instead of normal projection to avoid the complexities incurred by previous update rules when boundaries to the labeling space are encountered. This reduction in complexity is achieved by first restricting support vectors to the positive quadrant, and then using radial projection onto the constraint instead of normal projection. Crucial order information is conserved through smooth convergence towards the optimum and a rate of convergence proportional to the magnitudes of the support functions.

Index Terms: picture processing; pattern recognition; update rule; relaxation labeling; contextual information; graphs; radial projection; convergence; convergence of numerical methods; graph theory; iterative methods; pattern recognition; picture processing

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