Shapes provide a rich set of clues on the identity and topological properties of an object. In many imaging environments, however, the same object appears to have different shapes due to such distortions as translation, rotation, reflection, anisotropic scaling, skewing, or shearing. These distortions are generally captured by affine transformations. Further, the order by which the object’s feature points are scanned changes. we refer to these as permutation distortions. We show below on the left two images of the same airplane that are affine distorted. When scanned, say lexicographically (top to bottom, left to right), the pixels are not in correspondence.

To relate shapes like these, i.e., of the same object and that are distorted by different affine and permutation transformations is a challenge. Overcoming the permutation distortions, i.e., the unknown scanning order, is combinatorial–the correspondence problem. Our work is concerned with developing algorithms that are invariant to these affine-permutation distortions. We have introduced the concept of intrinsic shape of an object. It is a uniquely defined representative of the equivalence class of all affine-permuted distortions of the same object. The shape of the object is essentially the shape that results after we factor out the distortions. The figure of the airplane above on the right is the intrinsic shape of the two distorted images on the left as obtained by the BLAISER, a blind algorithm described in the references below by Ha and Moura. The distortions are interpreted as actions of the group of distortions (affine-permutation group as a subgroup of the general linear group) on the space of configurations (distorted shapes). We developed a blind algorithm that recovers the intrinsic shape from any arbitrarily unknown affine-permutation distorted image of the object. We are pursuing the definition of shape space and studying the geometry of this shape space, for example, the notions of distance and geodesics in shape space.

Selected Journal Papers (additional papers in Journal Publications)

• Victor Ha and José M. F. Moura, “Affine-permutation Invariance of 2D Shapes,” IEEE Transactions on Image Processing, 14(11), pp. 1687-1700, 2005.

Selected Conference Papers (additional papers in Conference Publications)

• Victor Ha and José M. F. Moura, “Robust Reorientation of 2D Shapes Using the Orientation Indicator Index,” ICASSP’05, IEEE International Conference on Signal Processing, Philadelphia, PA, March 18-23, 2005.
• Victor Ha and José M. F. Moura, “Three-dimensional Intrinsic Shapes,” ICIP’04, IEEE International Symposium on Image Processing, Singapore, October 24-27, 2004.
• David Sepaishvili, José M. F. Moura, and Vitor Ha, “Affine Permutation Symmetry: Invariance and Shape Space,” IEEE Workshop on Statistical Signal Processing, St. Louis, MI, Spetmber 2003.
• Victor Ha and José M. F. Moura, “Efficient 2D Shape Orientation,” ICIP’03, IEEE International Conference on Image Processing, Barcelona, Spain, September 2003.
• Viktor Ha and José M. F. Moura, “Intrinsic Shape,” 36th Asilomar Conference on Signals, Systems, and Communications, vol.2: 993-997, Monterey, CA, November 2002. Invited paper, Special Session on Statistical Image Processing.
• Victor H. Ha and José M. F. Moura, “Affine Invariant Wavelet Transform,” ICASSP’01, IEEE International Conference on Signal Processing, vol. 3, 1937-1940, Salt Lake City, Utah, May 2001.

Lab Members

1. Bernado Pires
2. David Sepiashvili
3. Hyeong-Seok Viktor Ha