July 29, 2002

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Publications by Song Chun Zhu :

• Stochastic Jump-Diffusion process for computing Medial Axes in Markov Random Fields, 1999.
• Embedding Gestalt Laws in Markov Random Fields, 1999.
• Stochastic Computation of Medial Axis in Markov Random Fields, CVPR'98.
• Embedding Gestalt Laws in the Markov Random Fields -- A theory for shape modeling and perceptual organization, POCV'98.
• FORMS: A Flexible Object Recognition and Modeling System, 1996.
• Statistical and Computational Theories for Image Segmentation, Texture Modeling, and Object Recognition, 1996.

BibTeX references.

Web links:

• at Harvard (when a post-doc) : http://hrl.harvard.edu/people/postdocs/zhu/publication.html
• at Ohio State University : http://www.cis.ohio-state.edu/~szhu/
• at UCLA (most recently): http://www.cs.ucla.edu/~sczhu/

This paper proposes a statistical framework for computing medial axis of 2D shapes. In this paper the computation of medial axis is posed as a statistical inference problem not as a mathematical transform. Our method provides answers to two essential problems in the medial axis representation. I) Prior models are adopted for axes and junctions so that the axes around junctions become well defined. II) A stochastic algorithm is proposed for estimating medial axis in a Markov random field, therefore our algorithm for computing medial axis is compatible with existing algorithms for image segmentation, such as region growing \cite{Zucker76}, snake \cite{Kass87} and region competition \cite{Zhu_region}. Thus our algorithm provides a new direction for computing medial axis from real textured images. Experiments are demonstrated on both synthetic and real shapes, and relationship to medial axis detection in early human vision is also discussed.

An important goal of the research in the middle level vision and perceptual organization is to bridge the gap between low level representation, such as raw images and edge maps and the high level descriptions such as deformable templates for object recognition. This task requires a generic probability model of shapes, which characterizes the most common features of real objects, and which is compatible with the existing statistical theories in both the low level and the high level vision. In this paper, a novel theoretic framework is proposed for shape modeling based on the maximum entropy principle, and it learns generic probabilistic shape models from observed realistic object shapes. The learned models are of the forms of Gibbs distributions defined on the Markov random fields. The neighborhood structures of the random fields corresponds to the Gestalt laws --co-linearity, co-circularity, proximity, parallelism, and symmetry, and thus both contour-based and region-based features are accounted for. The potential functions of the Gibbs distributions are learned so that the shape models duplicate the observed statistics of natural shapes, and multiple shape features are weighted by the potential functions to form a single probability measure. Stochastic algorithms are proposed for learning the shape distributions and for sampling random shapes from the models. The paper also provides a quantitative measure for the non-accidental arrangement by comparing the observed statistics and the statistics of the random shapes. Our experiments demonstrate that global shape properties can arise from the local interactions of local features.

This paper proposes a statistical framework for computing medial axis of 2D shapes. In this paper the computation of medial axis is posed as a statistical inference problem not as a mathematical transform. Our method provides answers to two essential problems in the medial axis representation.

1. Prior models are adopted for axes and junctions so that the axes around junctions become well defined.
   
2. A stochastic algorithm is proposed for estimating medial axis in a Markov random field.

Therefore our algorithm for computing medial axis is compatible with existing algorithms for image segmentation, such as region growing [Zucker76], snake [Kass87] and region competition [Zhu96PAMI]. Thus our algorithm provides a new direction for computing medial axis from real textured images. Experiments are demonstrated on both synthetic and real shapes, and relationship to medial axis detection in early human vision is also discussed.

Motivations

• Perceptual relevance: M.Leyton
• Physiological relevance: neurons in V1 detect symmetry axis
• Although the sensitivity to boundary noise has been addressed, 2 important problems have not:
  1. Medial axes at corners or junctions are often ill-defined and semantically meaningless (!).
  2. Computation should be simultaneous with segmentation, but the present deterministic approach is too sensitive to boundary deformations in textured areas: e.g. when part of a contour deforms, the medial axis changes drastically.

Proposition:

• Define medial axis locally, on a Markov random field (MRF).

Advantages:

• Prior knowledge can be introduced, e.g. prior models or energy terms, to favor axis smoothness, symmetry and alignement at junctions, ...
  
• Inference through stochastic algorithms can be integrated with image segmentation.
  
• Multiple solutions, with assigned probability measure can be computed.

An important goal of the research in the middle level vision and perceptual organization is to bridge the gap between low level representation, such as raw images and edge maps and the high level descriptions such as deformable templates for object recognition. This task requires a generic probability model of shapes, which characterizes the most common features of real objects, and which is compatible with the existing statistical theories in both the low level and the high level vision. In this paper, a novel theoretic framework is proposed for shape modeling based on the maximum entropy principle, and it learns generic probabilistic shape models from observed realistic object shapes. The learned models are of the forms of Gibbs distributions defined on the Markov random fields. The neighborhood structures of the random fields corresponds to the Gestalt laws -- co-linearity, co-circularity, proximity, parallelism, and symmetry, and thus both contour-based and region-based features are accounted for. The potential functions of the Gibbs distributions are learned so that the shape models duplicate the observed statistics of natural shapes, and multiple shape features are weighted by the potential functions to form a single probability measure. Stochastic algorithms are proposed for learning the shape distributions and for sampling random shapes from the models. The paper also provides a quantitative measure for the non-accidental arrangement by comparing the observed statistics and the statistics of the random shapes. Our experiments demonstrate that global shape properties can arise from the local interactions of local features.

Motivation

This project aims at answering the following questions:

• Does there exist a generic probability model for object shapes?
• Can we prove the ecologic reason for the Gestalt laws of organization?
• Can will learn such shape model from observation?
• Why is medial axis an important representation?

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