Image classification uses a learning procedure to predict class labels based on quantitative characteristics of an image. Typical approaches involve first extracting features from a set of labeled images and using a subset of those images and their features as a "training set" to train a classifier. The classifier is then applied to the test images to predict their labels. In this thesis, we focus on shape classification and propose new features to be used for classifying binary images. Specifically, we propose new features based on the hitting time distribution of a Brownian motion process defined on the shape.
It has been shown that the expected time for a particle undergoing Brownian motion to hit the boundary of the shape given that the particle originated at a point x inside the shape can be determined from the solution to a Poisson equation with homogeneous Dirichlet boundary conditions on the shape boundary. We now consider Brownian motion that originates outside the shape. To prevent this Brownian motion from continuing infinitely, we introduce an exponential dying time for the particle, and we derive the distribution of the random variable representing the minimum of the dying time and the boundary hitting time, given that the Brownian motion originates at x. We show that the expected value, expressed as a function of x, satisfies the Helmholtz equation.
Finally, we show how moments of this new random variable can be used as quantitative features in two classification experiments, using natural silhouettes and handwritten numerals. We show that improved results are possible when the features computed based on Brownian motion originating outside a shape are appended to those inside the shape.
Library of Congress Subject Headings
Image processing--Digital techniques; Pattern recognition systems; Classification--Data processing; Machine learning
Applied and Computational Mathematics (MS)
Department, Program, or Center
School of Mathematical Sciences (COS)
Rolston, Laura A., "Shape Representation Using the Helmholtz Equation" (2016). Thesis. Rochester Institute of Technology. Accessed from
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