Award Abstract #0820817
CAREER: Partial Differential Equation-based Image Processing with Applications to Radiation Oncology

NSF Org:	DMS Division of Mathematical Sciences
Initial Amendment Date:	July 9, 2008
Latest Amendment Date:	July 9, 2008
Award Number:	0820817
Award Instrument:	Standard Grant
Program Manager:	Junping Wang DMS Division of Mathematical Sciences MPS Directorate for Mathematical & Physical Sciences
Start Date:	October 1, 2007
Expires:	September 30, 2008 (Estimated)
Awarded Amount to Date:	$7349
Investigator(s):	Doron Levy dlevy@math.umd.edu (Principal Investigator)
Sponsor:	University of Maryland College Park 3112 LEE BLDG COLLEGE PARK, MD 20742 301/405-6269
NSF Program(s):	SIGNAL PROCESSING SYS PROGRAM
Field Application(s):	0000099 Other Applications NEC
Program Reference Code(s):	OTHR,1187,1045,0000
Program Element Code(s):	4720

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

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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.

Bhaya D., Levy D., and Requeijo T..  "Group Dynamics of Phototaxis: Interacting Stochastic Many-Particles Systems and their Continuum Limit,"  in S. Benzoni-Gavage and D. Serre (Eds.) "Hyperbolic Problems: theory, numerics, applications",  2008,  p. 145.

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, and L. Xing.  "Multiscale Deformable Registration for Noisy Medical Images,"  Mathematical Biosciences and Engineering,  v.5,  2008,  p. 125.

D. Paquin, D. Levy, and L. Xing.  "Hybrid Multiscale Landmark and Deformable Registration,"  Mathematical Biosciences and Engineering,  v.4,  2007,  p. 711.

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.

Kim P. , Lee P., Levy D..  "Modeling Imatinib-Treated Chronic Myelogenous Leukemia: Reducing the complexity of agent-based models,"  Bulletin of Mathematical Biology,  v.70,  2008,  p. 724.

Kim P., Lee P., and Levy D..  "Mini-transplants for chronic myelogenous leukemia: a modeling perspective,"  in "Biology and control theory: current challenges", Queinnec et al. (eds.),  v.LNCIS 3,  2007,  p. 3.

Kim P., Lee P., Levy D..  "Dynamics and Potential Impact of the Immune Response to Chronic Myelogenous Leukemia,"  PLOS Computational Biology,  v.4,  2008, 

Levy D. and Requeijo T..  "Modeling group dynamics of phototaxis: from particles to PDEs,"  Discrete and Continuous Dynamical Systems B,  v.9,  2008,  p. 108.

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.


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