Non-rigid Motion and Correspondence

Abstract

I am developing a theory of non-rigid motion that obtains motion by matching "similarly shaped" parts of the curves. Experimental results are shown below.

The main reference is [1]. Reference [2] is shorter and may be more accessible. Reference [3] has applications to heart motion analysis

Mathematical Theory

The key part of the work is a geometrically consistent way of comparing shapes which are non-rigidly related.

The key steps in theory are

[1] Motion estimation is equivalent to finding a non-rigid correspondence between succesive pairs of curves,

[2] Investigation of the topology of correspondences shows that "smooth" correspondences have the following topological structure -- they are regular curves in the product space of the two curves.

[3] The shape of the two curves can be compared non-rigidly from the vantage point of the correspondence. The resulting differential geometry has all the right properties -- it does not use rigid shape properties such as curvature but when rigidity is imposed it reduces to the a comparison of curvatures.

[4] All of the above can be expressed as an objective function such that the minima of the objective function occurs at the desired correspondence.

[5] The correspondence gives the motion.

Experimental Results

Heart motion. The following images show an MRI of a left-ventricle at systole, at diastole, and the non-rigid motion computed by the algorithm.

NOTE: The algorithm gives motion as a dense map from one curve to the other. For simplicity, only some elements of the map are shown.

Neural Growth Here the algorithm is processing images of a neuron grown on a glass slide.

Visual Curve Perception

This theory appears to explain some aspects of human curve perception.

References

[1] The Topology and Geometry of Shape-based Non-rigid Correspondence, Hemant D. Tagare, Don O'Shea Tech. Rep. 95-3, Division of Imaging Science, Yale University, 1995.

[2] A geometric criterion for shape based non-rigid correspondence, H. D. Tagare, D. O'Shea and A. Rangarajan, Fifth Intl. Conf. on Computer Vision (ICCV) , pp. 434--439, 1995.

[3] Non-rigid Curve Correspondence for Estimating Heart Motion, H. D. Tagare, Submitted to the XVth IPMI,1997.

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