Project Description

In today's digital society, automatically recognizing objects is of great interest, especially in processing digital data in many branches of sciences, particularly in image analysis and computer vision. Analyzing shape of objects has increasingly been useful in detecting, tracking and recognizing objects in digital images. Many mathematical framework for features representing, differences measuring and modeling of those objects' shape have been developed. This approach has been especially useful in stochastic modeling of shapes in shape classes and other vision applications.

In object analysis, besides the shape property, there are other properties of objects helping us to identify their special attributes. These certain additional attributes, different from their geometries, are often available for analyzing objects besides the study of shape of them. This raise the question of how to combine these extra information with shape analysis to improve the analysis result on objects. For example, the texture information along the objects' contours in digital images show what the objects' material are; the expert marked landmark in medical images where the critical positions of organs are; the amino acid sequence marked on proteins backbone curves identifying what the proteins are; etc. From examples above, we can see there are valuable position dependent information associate with curves to identify the objects properties. This give us the idea of using them as auxiliary information to help us optimizing the result of shape analysis.

We define the curves with these auxiliary information along them as annotated curves. To achieve the analysis of annotated curves, we first need to formalize these information as auxiliary functions, and combine them with 3 dimensional curve functions to form higher dimensional composite curve functions. Then, we study the composite curve functions in a higher dimensional function space. Using defined Riemannian metric (usually this space is a nonlinear Hilbert manifold due to certain context-based constraints),form a joint analysis by removing shape-preserving transformations but preserving the auxiliary information, and derive numerical algorithms for computing geodesic paths between annotated curves in these function space. While this framework presents certain conceptual and computational complexities as the elastic shape framework, it also has the advantage of providing a comprehensive framework for statistical modeling and analysis of shapes.

References

  • [1] Wei Liu, Anuj Srivastava, and Jinfeng Zhang. A Mathematical Framework for Protein Structure Comparison. PLoS Computational Biology (Journal), Accepted. December, 2010.
  • [2] Wei Liu, Anuj Srivastava, and Jinfeng Zhang. Protein Structure Alignment Using Elastic Shape Analysis. ACM Conference on Bioinformatics and Computational Biology. August, 2010.
  • [3] Wei Liu, Anuj Srivastava. Shape Analysis of Annotated Curves Using An Elastic Framework. In progress.
  • [4] Wei Liu, Anuj Srivastava, and Eric Klassen. Joint Shape and Texture Analysis of Objects Boundaries in Images Using A Riemannian Approach. Asilomar Conference on Signals, Systems, and Computers. October, 2008.