Variational Image Segmentation Using Shape Priors (Fall 2009 - Spring 2010)
A butterfly shape and its corresponding signed distance function
When segmenting an object from an image, it is often very helpful to know what kind of shape one is trying to extract. Traditionally, segmentation algorithms have mostly relied on sharp boundaries in the image to extract shapes. In many real world applications, however, information from the image alone is insufficient to extract the correct object from the image. Building a statistical model of the shapes of desired objects helps to make image segmentation more robust by dealing effectively with noisy or partially occluded images, or images with low contrast.
I am implementing an algorithm for segmenting objects from images by biasing the segmented shape toward a desired shape, following the work of Tsai et al., I represent the training set of shapes as signed distance functions whose zero level sets are the bounaries of the shapes. I perform principal component analysis on the dataset of signed distance functions to extract the main directions of variability.
To find the optimal shape, I evolve an initial shape to minimize an energy functional (this measures the "badness" of the segmentation) while constraining the shape to be reasonably likely with respect to the distribution estimated from the dataset. This is accomplished by constraining the signed distance function of the shape to be a linear combination of only the first k principal components, or modes of variation, of the dataset.
 Andy Tsai, Anthony Yezzi Jr., William Wells, Claire Tempany, Dewey Tucker,Ayres Fan, W. Eric Grimson, Alan Willsky. A Shape-Based Approach to the Segmentation of Medical Imagery Using Level Sets
A butterfly shape embedded as the zero level set of a signed distance function
A butterfly shape and its corresponding signed distance function (left: 3D, right: 2D projection)
First principal component of a dataset of butterfly signed distance functions