Award Abstract #0133511
CAREER: Partial Differential Equation-based Image Processing with Applications to Radiation Oncology

NSF Org:	DMS Division of Mathematical Sciences
Initial Amendment Date:	February 27, 2002
Latest Amendment Date:	February 27, 2002
Award Number:	0133511
Award Instrument:	Standard Grant
Program Manager:	Junping Wang DMS Division of Mathematical Sciences MPS Directorate for Mathematical & Physical Sciences
Start Date:	July 1, 2002
Expires:	August 31, 2008 (Estimated)
Awarded Amount to Date:	$400000
Investigator(s):	Doron Levy dlevy@math.umd.edu (Principal Investigator)
Sponsor:	Stanford University 340 Panama Street STANFORD, CA 94305 650/723-2300
NSF Program(s):	SIGNAL PROCESSING SYS PROGRAM, COMPUTATIONAL MATHEMATICS, APPLIED MATHEMATICS
Field Application(s):	0000099 Other Applications NEC
Program Reference Code(s):	OTHR,9263,1187,1045,0000
Program Element Code(s):	4720,1271,1266

ABSTRACT

In the past decade, new nonlinear partial differential

equations (PDEs) have been developed for various image processing

applications, such as noise reduction, edge detection, image

segmentation and restoration. While the attention of the

scientific community in this area predominantly focused on

creating the new PDEs, very little attention was paid to

developing numerical algorithms that approximate their solutions.

The few numerical algorithms that are currently used suffer from

a variety of problems: they are not accurate enough, too slow,

and not fault-free. In this project, the investigator develops

accurate, efficient, and robust numerical algorithms for

nonlinear PDEs in image processing. The research activities are

based on the investigator's extensive work in the field of

hyperbolic conservation laws, and include numerical methods for

the Hamilton-Jacobi equations, fast algorithms for high-order

nonlinear PDEs, algorithms for computing steady-state solutions,

numerical homogenization of Hamilton-Jacobi equations and

multi-resolution analysis, analysis of nonlinear diffusion

equations, constrained morphing active contours and geodesic

flows, and "non-blind" algorithms for image processing. A

portion of the research activities focuses on improving existing

algorithms in order to solve a specific imaging problem in

radiation oncology treatment planning.

The investigator develops novel mathematical techniques for

image processing and uses these techniques for solving problems

in the field of radiation oncology imaging. Radiation oncology

treats cancer by delivering relatively small doses of radiation

to tumors in order to eliminate cancer without destroying or

chronically damaging healthy tissues in and around the growth.

CT and MRI scans are used as three-dimensional anatomical models

to ensure that the treatments conform geometrically to the tumor

target. This process depends critically upon identifying the

location of the tumor as well as the healthy organs (in order to

minimize the dose of radiation in these areas). Despite extended

research, the existing mathematical tools for image processing

are unsuitable for clinical medical applications. The

segmentation of the CT and MRI scans is still carried out by

manual tools, and consumes about one-half of the time required to

plan the treatments. The investigator designs accurate and

reliable automated algorithms that would significantly shorten

this time and have a big impact on radiation oncology. He

integrates into his work educational activities that demonstrate

the importance of applied mathematics in a broad spectrum of

sciences. Special emphasis is given to applications of

computational mathematics in biology and cutting-edge

technologies. The planned educational activities include

programs for junior-high, high-school, undergraduate, and

graduate students. The investigator works to increase the gender

and ethnic diversity in the mathematical sciences by encouraging

under-represented groups to study applied mathematics and choose

it as a future career.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

Next (Showing: 1 - 20 of 22).

A. Chertock and D. Levy.  "A Particle Method for the KdV Equation,"  Journal of Scientific Computing,  v.17,  2002,  p. 491.

A. Chertock and D. Levy.  "On Wavelet-Based Numerical Homogenization,"  Multiscale Modeling and Simulation,  v.3,  2004,  p. 65.

A. Kurganov and D. Levy.  "Central-Upwind Schemes for the Saint-Venant System With a Source Term,"  Mathematical Modelling and Numerical Analysis,  v.36,  2002,  p. 397.

A.L. Boyer, C. Cardenas, F. Gibou, P. Liu, T. Koumrian, D. Levy.  "Evaluation of a semi-automated segmentation technique using partial differential equations,"  Int J Radiat Oncol Biol Phys,  v.57,  2003,  p. 206.

D. Levy, C.-W. Shu, and J. Yan.  "Local Discontinuous Galerkin Methods for Nonlinear Dispersive Equations,"  Journal of Computational Physics,  v.196,  2004,  p. 751.

D. Paquin, D. Levy, E. Schreibmann, and L. Xing.  "Multiscale Image Registration,"  Mathematical Biosciences and Engineering,  v.3,  2006, 

Doron Levy.  "A Stable Semi-Discrete Central Scheme for the Two-Dimensional Incompressible Euler Equations,"  IMA Journal of Numerical Analysis,  v.25,  2005,  p. 507.

Doron Levy and Yuan-Nan Young.  "Registration-based Morphing of Active Contours for Segmentation of CT Scans,"  Mathematical Biosciences and Engineering,  v.2,  2005,  p. 79.

F. Gibou, D. Levy, C. Cardenas, P. Liu, and A. Boyer.  "PDE-based Segmentation for Radiation Therapy Treatment Planning,"  Mathematical Bioscinces and Engineering,  v.25,  2005,  p. 209.

P. Kim, P. Lee, and D. Levy.  "Modeling Regulation Mechanisms in the Immune System,"  Journal of Theoretical Biology,  v.246,  2007, 

R. De Conde, P. Kim, P. Lee, and D. Levy.  "Post Transplantation Dynamics of the Immune Response to Chronic Myelogenous Leukemia,"  Journal of Theoretical Biology,  v.236,  2005, 

Razvan Fetecau and Doron Levy.  "Approximate Model Equations for Water Waves,"  Communications in Mathematical Sciences,  v.3,  2005,  p. 159.

S. Bryson and D. Levy.  "High-Order Central WENO Schemes for Multi-dimensional Hamilton-Jacobi Equations,"  SIAM J. Numer. Anal,  v.41,  2003,  p. 1339.

S. Bryson and D. Levy.  "High-Order Semi-Discrete Central-Upwind Schemes for Multi-dimensional Hamilton-Jacobi Equations,"  Journal of Computational Physics,  v.189,  2003,  p. 63.

S. Bryson and D. Levy.  "Central Schemes for Multi-dimensional Hamilton-Jacobi Equations,"  SIAM J. on Sci. Comp.,  v.25,  2003,  p. 767.

S. Bryson, A. Kurganov, D. Levy, and G. Petrova.  "Semi-Discrete Central-Upwind Schemes with Reduced Dissipation for Hamilton-Jacobi Equations,"  IMA J. Numerical Analysis,  v.25,  2005,  p. 113.

S.-I. Niculescu, P. Kim, K. Gu, D. Levy.  "On the Stability Crossing Boundaries of Some Delay Systems Modeling Immune Dynamics in Leukemia,"  Proc. of MTNS, 2006,  2006, 

Steve Bryson and Doron Levy.  "Mapped WENO and Weighted Power ENO Reconstructions in Semi-Discrete Central Schemes for Hamilton-Jacobi Equations,"  Applied Numerical Mathematics,  v.56,  2006,  p. 1211.

Steve Bryson and Doron Levy.  "On the Total Variaion of High-Order Semi-Discrete Central Schemes for Conservation Laws,"  Journal of Scientific Computing,  v.27,  2006,  p. 163.

Steve Bryson and Doron Levy.  "Balanced Central Schemes for the Shallow Water Equations on Unstructured Grids,"  SIAM Journal on Scientific Computing,  v.27,  2005,  p. 532.


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