Title :
Characterizing the shape of anatomical structures with Poisson´s equation
Author :
Haidar, Haissam ; Bouix, Sylvain ; Levitt, James J. ; McCarley, Robert W. ; Shenton, Martha E. ; Soul, Janet S.
Author_Institution :
Dept. of Neurology, Harvard Med. Sch., Boston, MA
Abstract :
Poisson\´s equation, a fundamental partial differential equation in classical physics, has a number of properties that are interesting for shape analysis. In particular, the equipotential sets of the solution graph become smoother as the potential increases. We use the displacement map, the length of the streamlines formed by the gradient field of the solution, to measure the "complexity" (or smoothness) of the equipotential sets, and study its behavior as the potential increases. We believe that this function complexity=f(potential), which we call the shape characteristic, is a very natural way to express shape. Robust algorithms are presented to compute the solution to Poisson\´s equation, the displacement map, and the shape characteristic. We first illustrate our technique on two-dimensional synthetic examples and natural silhouettes. We then perform two shape analysis studies on three-dimensional neuroanatomical data extracted from magnetic resonance (MR) images of the brain. In the first study, we investigate changes in the caudate nucleus in Schizotypal Personality Disorder (SPD) and confirm previously published results on this structure . In the second study, we present a data set of caudate nuclei of premature infants with asymmetric white matter injury. Our method shows structural shape differences that volumetric measurements were unable to detect
Keywords :
Poisson equation; biomedical MRI; brain; diseases; medical image processing; paediatrics; Poisson equation; anatomical structures; asymmetric white matter injury; brain; caudate nucleus; displacement map; equipotential set smoothness; function complexity; magnetic resonance images; partial differential equation; premature infants; robust algorithms; schizotypal personality disorder; shape analysis; solution graph; structural shape differences; three-dimensional neuroanatomical data; Anatomical structure; Displacement measurement; Image analysis; Length measurement; Magnetic field measurement; Partial differential equations; Physics; Poisson equations; Robustness; Shape; Brain morphometry; Poisson´s equation; caudate nucleus; shape analysis;
Journal_Title :
Medical Imaging, IEEE Transactions on
DOI :
10.1109/TMI.2006.881378