Note: This content is accessible to all versions of every browser. However, this browser does not seem to support current Web standards, preventing the display of our site's design details.

  

Hurwitz Memorial Lecture Series

Hurwitz Lectures 2010 at ETH Zurich

Prof. Allen Tannenbaum



HURWITZ LECTURE (1): Mathematical Methods in Medical Imaging

Abstract:

In this talk, we will survey the use of various modern mathematical techniques for several key problems in medical imaging.
First we will describe the use of various geometric curvature based flows for fast reliable segmentation of medical imagery both in 2D and 3D. This will lead to modern approaches to the topic of geometric active contours. The term "active contours" does not refer to one specic technique, but rather to a broad family of related methods that can be tailored to a specic image processing task. Both local (edge-based) and global (statistics-based) information may be included in this framework. The underlying principle is based on the use of deformable contours that conform to various object shapes and motions. Active contours are used for edge detection, image segmentation, shape modeling, and for tracking. Further we will give some directional active contour models for extracting white matter fiber tracts in Diffusion Weighted Magnetic Resonance (MR) imagery. Related flows for smoothing and enhancement will also be considered. These flows have certain interesting invariance properties that can be tuned for several important tasks in general image processing and computer vision. We will provide some of the relevant results from the theory of curve and surface evolution in order to motivate our methods.
Next, we will outline some recent work using conformal mappings for the visualization of the cortical brain surface and automatic polyp detection in virtual colonoscopy. Regarding the former problem, we show how the method may be used in functional MR imagery to better visualize brain activity. Regarding the latter problem, we demonstrate how the conformal attening approach leads to a surface scan of the entire colon as a cine, and affords viewer the opportunity to examine each point on the surface without distortion. Both attening tasks (brain and colon) employ the theory angle-preserving mappings from differential geometry in order to derive an explicit method for surface attening. Indeed, we describe a general method based on a discretization of the Laplace-Beltrami operator for flattening a surface in a manner that preserves the local geometry. From a triangulated surface representation of the surface, we indicate how the procedure may be implemented using a finite element technique, which takes into account special boundary conditions.
The talk is designed to be accessible to a general audience with an interest in medical imaging. We will demonstrate our techniques with a wide variety of medical images including MR, CT, and ultrasound.

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:

2010-05-26  / 15:15 - 16:45 /
Location:

ETH, Rämistrasse 101, HG F 30 (Audimax)
Contact Person:

Prof. John Lygeros
File Download:

Request a copy of this publication.

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

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:

2010-05-27  / 15:15 - 16:45 /
Location:

ETH, Rämistrasse 101, HG F 7
Contact Person:

Prof. John Lygeros
File Download:

Request a copy of this publication.
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.



Other Hurwitz lectures: