Generalized multidimensional scaling (GMDS) is an extension of traditional metric MDS, in which the target space is nonEuclidean. Particularly important setting of the problem is the isometric embedding problem, when one wishes to represent the intrinsic metric structure of one surface using the instrinsic geometry of another surface.
We have developed an efficient theoretical and numerical framework for the solution of the GMDS problem. Using GMDS, many fundamental problems pattern analysis can be solved. First, GMDS allows to establish intrinsic geometric correspondence between two similar objects, e.g. two nearisometric deformations of the same object. Secondly, the average (or the maximum) metric distortion, referred to as "stress", serves as a measure of shape dissimilarity. Particularly, using GMDS it is possible to compute the GromovHausdorff distance between two surfaces or the partial embedding distance, which allows for partial matching of surfaces. Finally, "local stress" obtained as a byproduct of the GMDS procedure allows to find local differences between two shapes.
GMDS allows computing the GromovHausdorff distance between nonrigid shapes, which accurately captures their intrinsic similarity.

Currently, we explore the use of the GMDS framework in applications such as 3D face recognition, expressioninvariant texture for animated 3D facial surfaces, expression exaggeration and face morphing.


Publications
A. M. Bronstein, M. M. Bronstein, R. Kimmel,
"Efficient computation of isometryinvariant distances between surfaces", SIAM J. Scientific Computing, Vol. 28/5, pp. 18121836, 2006.
A. M. Bronstein, M. M. Bronstein, R. Kimmel,
"Generalized multidimensional scaling: a framework for isometryinvariant partial surface matching", Proc. National Academy of Sciences (PNAS), Vol. 103/5, pp. 11681172, January 2006.
A. M. Bronstein, M. M. Bronstein, R. Kimmel,
"Robust expressioninvariant face recognition from partially missing data",
Proc. European Conf. on Computer Vision (ECCV), pp. 396408, 2006.
A. M. Bronstein, M. M. Bronstein, A. M. Bruckstein, R. Kimmel,
"Matching twodimensional articulated shapes using generalized multidimensional scaling",
Proc. Conf. on Articulated Motion and Deformable Objects (AMDO), pp. 4857, 2006.
A. M. Bronstein, M. M. Bronstein, R. Kimmel,
"Face2Face: an isometric model for facial animation",
Proc. Conf. on Articulated Motion and Deformable Objects (AMDO), pp. 3847, 2006.
A. M. Bronstein, M. M. Bronstein, R. Kimmel,
"Calculus of nonrigid surfaces for geometry and texture manipulation", IEEE Trans. Visualization and Computer Graphics, Vol 13/5, pp. 902913, SeptemberOctober 2007.
A. M. Bronstein, M. M. Bronstein, A. M. Bruckstein, R. Kimmel,
"Analysis of twodimensional nonrigid shapes", Intl. Journal of Computer Vision, to appear.
See also
Multigrid MDS
Face recognition
