Purpose: The shape correspondence is critical for boundary conditions and external forces in biomechanical studies. One way to ensure shape correspondence is using the intrinsic properties of graphs based on the Laplace-Beltrami theory and functional maps, i.e., a map between functions defined on shapes. We try to attract attentions of AAPM community to its applications with a few experimental examples.
Methods: Spectral graph theory is profound in analyzing the intrinsic properties of graph structures. However, it’s difficult to gain an intuition about their performance or about their applicability in real-life. With functional maps and available tools, problems can be solved with a unifying treatment and assuming minimal background knowledge. A function defined on a shape assigns a value to its nodes with a heatmap of the shape that can be obtained by pyFM. Handling mappings using their action on functions is both easy and flexible than the classical theory. The functional map is a m×n (bijection) matrix, where m basis functions are defined on the source shape and n basis functions are defined on the target shape. Using eigen-functions of the Laplace-Beltrami operator, the matrix represents how individual m basis functions of the source shape are mapped onto the set of n basis functions of the target shape. With a reduced set of basis functions, the corresponding functional map can be found extremely fast (usually in seconds) by solving the linear system of equations.
Results: Using an adjoint map representation and ICP we demonstrated the shape correspondence can be correctly established between two complicated body shapes. We constructed a simple porous model with linear elastic property to show the shape correspondence between compressed and expanded shapes can help to build the motion with FEM.
Conclusion: Using functional maps is efficient in establishing shape correspondence.
Not Applicable / None Entered.