Characterizing the shape of anatomical structures with Poisson's equation.
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.
Haidar
Haissam
H
Department of Neurology, Children's Hospital and Harvard Medical School, Boston, MA 02115, USA.
Bouix
Sylvain
S
Levitt
James J
JJ
McCarley
Robert W
RW
Shenton
Martha E
ME
Soul
Janet S
JS
D000328:Adult; D000465:Algorithms; D019412:Anatomy, Cross-Sectional / Q000379:methods*; D001185:Artificial Intelligence; D001921:Brain / Q000473:pathology*; D003627:Data Interpretation, Statistical; D003711:Demyelinating Diseases / Q000175:diagnosis*; D005260:Female; D006801:Humans; D007089:Image Enhancement / Q000379:methods; D007090:Image Interpretation, Computer-Assisted / Q000379:methods*; D021621:Imaging, Three-Dimensional / Q000379:methods*; D007231:Infant, Newborn; D016247:Information Storage and Retrieval / Q000379:methods; D008279:Magnetic Resonance Imaging / Q000379:methods; D008297:Male; D010363:Pattern Recognition, Automated / Q000379:methods*; D016012:Poisson Distribution; D015203:Reproducibility of Results; D012559:Schizophrenia / Q000175:diagnosis*; D012680:Sensitivity and Specificity