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Deformable and non-rigid objects, both natural and artificial, surround us at all scales from nano to macro,
and play an important role in many applications ranging from medical image analysis to robotics and gaming.
Such applications require the ability to acquire, reconstruct, analyze, and synthesize non-rigid three-dimensional shapes.
These procedures pose challenging problems both theoretically and practically due to the vast number of degrees
of freedom involved in non-rigid deformations.
While modelling and analysis of non-rigid shapes has greatly advanced in the past decade, existing solutions
are largely based on parametric models restricting the objects of interest to a narrow class of similar shapes.
Broadly speaking, reconstruction, analysis, and synthesis of arbitrary deformable shapes remain unsolved problems,
a practical solution of which would be a major milestone in computer vision and related fields.
My research aims at answering these fundamental questions by adopting tools from modern metric geometry,
a field of theoretical mathematics which in the past few decades has undergone a series of revolutions that
remained largely unnoticed and unused in applied sciences.
We believe that metric geometry tools could systematically answer these questions, and, coupled with modern numerical
optimization techniques and novel hardware architectures, pave the computational way to the next generation in
deformable shape analysis.
Our goal is to develop such numerical tools while demonstrating their efficiency on several challenging real-life applications
such as surgery prediction and planning, biometry, and computer-aided diagnosis.
So far, while exploring metric and differential geometry we developed computational tools like
Some of these models and tools were used in the 3DFACE project that deals with face recognition. Three-dimensional (3D) face recognition is the process of using the geometric structure of the face for accurate identification of the subject. While traditional two-dimensional (2D) face recognition methods are sensitive to variations in illumination, pose, makeup and cosmetics, 3D methods are more robust to these factors. Yet, facial expressions introduce a major challenge to 3D face recognition, as the geometry of the face changes significantly. Together with my students Alex and Michael Bronstein, we developed an expression-invariant 3D face recognition approach based on the isometric model of facial expressions. According to this model, a person's identity is associated with the intrinsic geometry of his or her facial surface, while the facial expressions are associated with the extrinsic geometry. Our first attempt was to represent the intrinsic geometry of the surface by isometrically embedding it into a low-dimensional Euclidean space. The embedding is performed using Multidimensional Scaling (MDS). The result is an expression-invariant representation of the face called canonical form. Using canonical forms, we perform very accurate face recognition. |
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Prototype of our 3D face recognition system developed at the Technion.
(Photo: November 2004) |
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Next, we generalized the canonical forms approach by embedding into non-Euclidean spaces. Particularly, two- and three-dimensional spaces with spherical geometry were found to be appealing for the representation of faces, as the resulting metric distortion is usually smaller compared to a Euclidean space. | |||
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More recently, we introduced the concept of Generalized Multidimensional Scaling (GMDS), which allows embedding into manifolds with an arbitrary geometric structure. Instead of embedding the facial surfaces into a common embedding space, we embed one surface into the other and use the metric distortion as a measure of their dissimilarity. The GMDS approach is more accurate compared to canonical forms and allows face recognition even when parts of the surfaces are missing. |
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COVER STORIES Reuters article on CNN news: Twins crack face recognition puzzle R. Kimmel and G. Sapiro, The mathematics of face recognition, SIAM News, 36(3) April 2003 D. Voth, Face recognition technology, IEEE Magazine on Intelligent Systems, 18(3):4-7, 2003 G. Sapiro, Abel Prize science lecture: Revolutionary work in geometry and shape analysis, SIAM News, 42(6) July/August, 2009 |
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Behind the scene: Sampling the world press: 3D Face Scan Distinguishes Twins Panorama:Firma facciale contro il terrorismo Face recognition Gemelos israelíes revolucionan identificación de rostros Gêmeos criam novo sistema de identificação facial BIOMETRÍA Dos gemelos israelíes revolucionan la técnica de identificación de rostros Revolucionan la biometría por unas buenas notas Desarrollan tecnolog?a en identificaci?n de rostros Oblicejový podpis je vytesán z nul a jednicek Gemelli identici creano tecnologia di video-riconoscimento volti On TV: Progress: We have fingerprints-like recognition rates, robustness to partial occlusions and expressions, and are searching for partners that would license our technology. Our research has been supported by: Advanced ERC, General Motors, BSF, ISF, Intel, and the US-ONR | |||
