Conformal Geometry and Its Applications on 3D Shape Matching, Recognition, and Stitching
IEEE Transactions on Pattern Analysis and Machine Intelligence
Elastic Shape Models for Face Analysis Using Curvilinear Coordinates
Journal of Mathematical Imaging and Vision
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
3D point correspondence by minimum description length in feature space
ECCV'10 Proceedings of the 11th European conference on computer vision conference on Computer vision: Part III
Human brain mapping with conformal geometry and multivariate tensor-based morphometry
MBIA'11 Proceedings of the First international conference on Multimodal brain image analysis
Distinguishing Facial Features for Ethnicity-Based 3D Face Recognition
ACM Transactions on Intelligent Systems and Technology (TIST)
Characterizing shape using conformal factors
EG 3DOR'08 Proceedings of the 1st Eurographics conference on 3D Object Retrieval
Face recognition by SVMs classification and manifold learning of 2D and 3D radial geodesic distances
EG 3DOR'08 Proceedings of the 1st Eurographics conference on 3D Object Retrieval
Hyperbolic ricci flow and its application in studying lateral ventricle morphometry
MBIA'12 Proceedings of the Second international conference on Multimodal Brain Image Analysis
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Face recognition and many medical imaging applications require the computation of dense correspondence vector fields that match one surface with another. In brain imaging, surface-based registration is useful for tracking brain change, and for creating statistical shape models of anatomy. Based on surface correspondences, metrics can also be designed to measure differences in facial geometry and expressions. To avoid the need for a large set of manually-defined landmarks to constrain these surface correspondences, we developed an algorithm to automate the matching of surface features. It extends the mutual information method to automatically match general 3D surfaces (including surfaces with a branching topology). We use diffeomorphic flows to optimally align the Riemann surface structures of two surfaces. First, we use holomorphic1-forms to induce consistent conformal grids on both surfaces. High genus surfaces are mapped to a set of rectangles in the Euclidean plane, and closed genus-zero surfaces are mapped to the sphere. Next, we compute stable geometric features (mean curvature and conformal factor) and pull them back as scalar fields onto the 2D parameter domains. Mutual information is used as a cost functional to drive a fluid flow in the parameter domain that optimally aligns these surface features. A diffeomorphic surface-to-surface mapping is then recovered that matches surfaces in 3D. Lastly, we present a spectral method that ensures that the grids induced on the target surface remain conformal when pulled through the correspondence field. Using the chain rule, we express the gradient of the mutual information between surfaces in the conformal basis of the source surface. This finite-dimensional linear space generates all conformal reparameterizations of the surface. Illustrative experiments apply the method to face recognition and to the registration of brain structures, such as the hippocampus in 3D MRI scans, a key step in understanding brain shape alterations in Alzheimer驴s disease and schizophrenia.