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HURWITZ LECTURE (2): Optimal Mass Transport Mappings for Surface Warping and Image Registration

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Abstract:
In this talk, we will outline some recent work using the theory of optimal mass transport for surface warping and image registration.
The mass transport problem was first formulated by Gaspar Monge in 1781, and concerned finding the optimal way, in the sense of minimal transportation cost, of moving a pile of soil from one site to another. This problem was given a modern formulation in the work of Kantorovich, and is now known as the "Monge-Kantorovich problem." The registration problem is one of the great challenges that must be addressed in order to make image-guided surgery a practical reality. Registration is the process of establishing a common geometric reference frame between two or more data sets obtained by possibly different imaging modalities. In the context of medical imaging, this is an essential technique for improving preoperative and intraoperative information for diagnosis and image-guided therapy. Registration has a substantial recent literature devoted to it, with numerous approaches effective in varying situations, and ranging from optical flow to computational fluid dynamics to various types of warping methodologies.
The method we discuss in this talk is designed for elastic registration, and is based on an optimization problem built around the L2 Monge-Kantorovich distance taken as a similarity measure. The constraint that we put on the transformations considered is that they obey a mass preservation property. Thus, we are matching "mass densities" in this method, which may be thought of as weighted areas in 2D or weighted volumes in 3D. We will assume that a rigid (non-elastic) registration process has already been applied before applying our scheme.
Our method has a number of distinguishing characteristics. It is parameter free. It utilizes all of the gray-scale data in both images, and places the two images on equal footing. It is thus symmetrical, the optimal mapping from image A to image B being the inverse of the optimal mapping from B to A. It does not require that landmarks be speciffied. The minimizer of the distance functional involved is unique; there are no other local minimizers. Finally, it is specifically designed to take into account changes in density that result from changes in area or volume.
We believe that this type of elastic warping methodology is quite natural in the medical context where density can be a key measure of similarity, e.g.,when registering the proton density based imagery provided by MR. It also occurs in functional imaging, where one may want to compare the degree of activity in various features deforming over time, and obtain a corresponding elastic registration map. A special case of this problem occurs in any application where volume or area preserving mappings are considered.

On-demand video.

Type of Seminar:
Public Seminar
Speaker:
Prof. Allen Tannenbaum
Georgia Institute of Technology, Schools of ECE and BME; Technion-IIT, Department of EE; Emory University, Department of Radiology
Date/Time:
May 27, 2010   15:15 - 16:45 /
Location:

ETH, Rämistrasse 101, HG F 7
Contact Person:

Prof. John Lygeros
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Biographical Sketch:
Dr. Tannenbaum was born in New York City in 1953. He attended Columbia University where he received his B.A. in 1973, and then moved to Massachusetts to attend Harvard University where he earned a Ph.D. in mathematics 1976. He has held faculty positions at the Weizmann Institute of Science, Ben-Gurion University of the Negev, the Technion (Israel Institute of Technology), and the University of Minnesota. In August 1999, he joined the ECE Department of the Georgia Institute of Technology where he set up the Laboratory for Computational Computer Vision. Dr. Tannenbaum has over 230 publications and has authored or co-authored three research texts on systems and control. He has played a leading role in developing new mathematical techniques for various engineering problems in systems and control, vision, signal processing, and cryptography. Dr. Tannenbaum has received a number of awards for his research, and has given plenary talks at a number of conferences in engineering and mathematics. Dr.Tannenbaum is also a professor with the GT/Emory Department of Biomedical Engineering.