Means of 2D and 3D Shapes and Their Application in Anatomical Atlas Building

This works deals with the concept of mean when applied to 2D or 3D shapes and with its applicability to the construction of digital atlases to be used in digital anatomy. Unlike numerical data, there are several possible definitions of the mean of a shape distribution and procedures for its estimation from a sample of shapes. Most popular definitions are based in the distance function or in the coverage function, each with its strengths and limitations. Closely related to the concept of mean shape is the concept of atlas, here understood as a probability or membership map that tells how likely is that a point belongs to a shape drawn from the shape distribution at hand. We devise a procedure to build probabilistic atlases from a sample of similar segmented shapes using information simultaneously from both functions: the distance and the coverage. Applications of the method in digital anatomy are provided as well as experiments to show the advantages of the proposed method regarding state of the art techniques based on the coverage function.