

3D Shape Analysis and Visualization 
PhD Student: Aaron Ward 
Principal Investigator: Ghassan Hamarneh 
We are developing novel approaches and tools for the problem of the quantitative and qualitative analysis and visualization of 3D shapes. The aim is to apply these approaches to medical problems of anatomical shape analysis. 
Medial Patch Shape Representation
The Medial Patch shape representation is a means for representing 3D shapes in a medialbased fashion. At the core of the representation of each figure in an object is a medial sheet comprised of ordered nodes, where the 3D coordinates of each node are computable based on spherical coordinates stored at that node. The upper and lower object surfaces are implied by thickness vector magnitudes stored at each node. This results in a map from the 2D intrinsic coordinate space of the sheet to the 5D space containing the three spherical coordinates and the two thickness values at each node. The name for this shape representation was inspired by the similarity of this map to the definition of a patch in differential geometry.
Overview of the Medial Patch shape representation.
Details of Medial Patch values at each node (expanded from oval in previous figure).
Analysis in 2D and 3D of Medial Patch components illustrating a bend and a protrusion in a synthetic slab object.
Analysis in 3D of bending and protrusion in a caudate nucleus image. Note that the protrusion is difficult to see in conventional views (b), relying on a good viewing angle (c), but the Medial Patch representation enables intuitive colouring of the surface to highlight the protrusion.
3D Surface Parameterization Using Manifold Learning for Medial Shape Representation
In this project we set out to answer the question of how to compute a medial representation of an object represented as a binary volume. After performing skeletonization of the object, the two main questions are how to position medial nodes within the skeleton, and in which directions to emanate thickness vectors from each node. We evaluated the utility of nonlinear dimensionality reduction, or manifold learning, to discover the intrinsic 2D parameterization of the skeleton, and upper and lower surfaces of the object. Using this parameterization, we sample all three surfaces with equal geodesic spacing, and use correspondences established in the 2D domain in order to compute the destination of each thickness vector.
The surface parameterization process used, illustrating the flattening (b) of a surface (a) resulting from manifold learning, the parameterization of that surface (c), and the mapping back of the parameters to the 3D surface using interpolation (d).
Skeletons (green) of a thalamus and caudate nucleus, and the resulting parameterized medial sheets (yellow).
An artificial multisheet skeleton as a voxel surface (grey), clustered into multiple sheets, with each sheet parameterized in a different colour.
A caudate nucleus (left) and supraspinatus (right) with skeleton, upper, and lower surfaces parameterized, and thickness vectors computed and displayed

PublicationsConference
1.
3D Surface Parameterization Using Manifold Learning for Medial Shape Representation
Ward, A. D., Hamarneh, G.: "3D Surface Parameterization Using Manifold Learning for Medial Shape Representation", Conference on Image Processing, Proc. of SPIE Medical Imaging, 2007
2.
Quantification and Visualization of Localized and Intuitive Shape Variability using a Novel MedialBased Shape Representation
Hamarneh, G., Ward, A. D., Frank, R.: "Quantification and Visualization of Localized and Intuitive Shape Variability using a Novel MedialBased Shape Representation", Proceedings of IEEE International Symposium on Biomedical Imaging (ISBI), 2007







