We apply metric geometry tools to various applications like
computer aided diagnostics in medical imaging.
GIP Lab in a brief (2014 clips):
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
Fast computation of distances on surfaces.
See SIGGRAPH'08 trailer
Integral geometric measures, and variational techniques
for processing and analysis of imagees.
The Beltrami framework and the geodesic active contours models.
Modeling non-rigid surfaces as near isometries.
Treating images as geometric structures, and geometric structures
Shape reconstruction from various cues and priors.
Implicit formulations of propagating interfaces,
accurate segmentation, and optical flow computation.
Some of these models and tools were used
in our 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
Together with my students (at the time)
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
The embedding is performed using Multidimensional Scaling (MDS).
The result is an expression-invariant representation
of the face called canonical form.
Canonical forms, enabled accurate face recognition.
Prototype of our 3D face recognition system developed at the Technion.
(Photo: November 2004)
Next, we generalized the canonical forms approach by embedding into
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.
Later on, 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
Reuters article on CNN news:
Twins crack face recognition puzzle
R. Kimmel and G. Sapiro,
The mathematics of face recognition,
SIAM News, 36(3), 2003
Face recognition technology,
IEEE Magazine on Intelligent Systems, 18(3):4-7, 2003
Abel Prize lecture: Revolutionary work
in geometry and shape analysis,
SIAM News, 42(6), 2009
Metric geometry in action,
SIAM News, 44(8), 2011
Behind the scene:
3DFace recognition started as an undergraduate
project in the course
Numerical Geometry of Images
American Technion Society 2003 fund raising was based on the 3DFace project.
Asi Elad presented
the first mathematical engine of our
system during CVPR'2001 (Hawaii).
More than 400 companies showed (written) interest in the project.
About 100 expressed willingness to invest money.
About 30 companies were interested to integrate a prototype into their products.
About 15 expressed interest to jointly develop a product.
The startup Invision licensed the technology in 2009. It was subsequently acquired by Intel in 2011.
The first journalist to professionally cover the project
was Haim Rivlin from Israeli Channel 2.
have the same DNA, and almost identical fingerprints.
Both were invited to the International Achievement Summit
in Washington DC.
where Michael met
Our first 3D scanner was built out of LEGO parts and a
laser pointer by Gil Zigelman and Eyal Gordon.
The second generation scanner was built in two days before a
Science Fair in Jerusalem.
At the Science Fair,
Matan Vilnai, minister of science at the time,
Our 3D video scanner features
auto-calibration, rapid scan at 50 msec/frame and 0.5mm depth resolution.
The project received the Hershel Rich Innovation prize.
Sampling the world press:
3D Face Scan Distinguishes Twins
Panorama:Firma facciale contro il terrorismo
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
W-NBC (asf 3.6M)
Russian TV (asf 2.8M)
Israel Channel 2 (avi 1.6M)
Our research has been supported by:
Advanced ERC (EU), General Motors, BSF, ISF, Intel, and US-ONR