Surface registration, which transforms different sets of surface data into one common reference space, is an important process which allows us to compare or integrate the surface data effectively. If a nonrigid transformation is required, surface registration is commonly done by parameterizing the surfaces onto a simple parameter domain, such as the unit square or sphere. In this work, we are interested in looking for meaningful registrations between surfaces through parameterizations, using prior features in the form of landmark curves on the surfaces. In particular, we generate optimized conformal parameterizations which match landmark curves exactly with shape-based correspondences between them. We propose a variational method to minimize a compound energy functional that measures the harmonic energy of the parameterization maps and the shape dissimilarity between mapped points on the landmark curves. The novelty is that the computed maps are guaranteed to align the landmark features consistently and give a shape-based diffeomorphism between the landmark curves. We achieve this by intrinsically modeling our search space of maps as flows of smooth vector fields that do not flow across the landmark curves. By using the local surface geometry on the curves to define a shape measure, we compute registrations that ensure consistent correspondences between anatomical features. We test our algorithm on synthetic surface data. An application of our model to medical imaging research is shown, using experiments on brain cortical surfaces, with anatomical (sulcal) landmarks delineated, which show that our computed maps give a shape-based alignment of the sulcal curves without significantly impairing conformality. This ensures correct averaging and comparison of data across subjects.

Adolescent Idiopathic Scoliosis (AIS) characterized by the 3D spine deformity affects about 4% schoolchildren worldwide. One of the prominent theories of the etiopathogenesis of AIS was proposed to be the poor postural balance control due to the impaired vestibular function. Thus, the morphometry of the vestibular system (VS) is of great importance for studying AIS. The VS is a genus-3 structure situated in the inner ear and consists of three semicircular canals lying perpendicular to each other. The high-genus topology of the surface poses great challenge for shape analysis. In this work, we propose an effective method to analyze shapes of high-genus surfaces by considering their geodesic spectra. The key is to compute the canonical hyperbolic geodesic loops of the surface, using the Ricci flow method. The Fuchsian group generators are then computed which can be used to determine the geodesic spectra. The geodesic spectra effectively measure shape differences between high-genus surfaces up to the hyperbolic isometry. We applied the proposed algorithm to the VS of 12 normal and 15 AIS subjects. Experimental results show the effectiveness of our algorithm and reveal statistical shape difference in the VS between right-thoracic AIS and normal subjects.

We address the registration problem of genus-one surfaces (such as vertebrae bones) with prescribed landmark constraints. The high-genus topology of the surfaces makes it challenging to obtain a unique and bijective surface mapping that matches landmarks consistently. This work proposes to tackle this registration problem using a special class of quasi-conformal maps called Teichmüller maps (T-Maps). A landmark constrained T-Map is the unique mapping between genus-1 surfaces that minimizes the maximal conformality distortion while matching the prescribed feature landmarks. Existence and uniqueness of the landmark constrained T-Map are theoretically guaranteed. This work presents an iterative algorithm to compute the T-Map. The main idea is to represent the set of diffeomorphism using the Beltrami coefficients (BC). The BC is iteratively adjusted to an optimal one, which corresponds to our desired T-Map that matches the prescribed landmarks and satisfies the periodic boundary condition on the universal covering space. Numerical experiments demonstrate the effectiveness of our proposed algorithm. The method has also been applied to register vertebrae bones with prescribed landmark points and curves, which gives accurate surface registrations.

Surface parameterizations have been widely applied in the computer-aided design for the geometric processing tasks of surface registration, remeshing, texture mapping, and so on. In this paper, we present an efficient balanced energy minimization (BEM) algorithm for the computation of simply connected open surface parameterizations with balanced angle and area distortions. The existence of a nontrivial accumulation function of the BEM algorithm is guaranteed under some mild conditions and the limiting function is shown to be one-to-one. Comparisons of the BEM algorithm with the angle- and the area-preserving parameterizations show that the angular distortion is close to that of the angle-preserving parameterization while the area distortion is significantly improved. An application of the BEM on the Chinese virtual broadcasting technique is demonstrated thereafter, which is consisted of surface remeshing, registration, and morphing.

The medial axis is an important shape representation that finds a wide range of applications in shape analysis. For large-scale shapes of high resolution, a progressive medial axis representation that starts with the lowest resolution and gradually adds more details is desired. In this paper, we propose a fast and robust geometric algorithm that computes progressive medial axes of a large-scale planar shape. The key ingredient of our method is a novel structural analysis of merging medial axes of two planar shapes along a shared boundary. Our method is robust by separating the analysis of topological structure from numerical computation. Our method is also fast and we show that the time complexity of merging two medial axes is O(n log nv), where n is the number of total boundary generators, nv is strictly smaller than n and behaves as a small constant in all our experiments. Experiments on large-scale polygonal data and comparison with state-of-the-art methods show the efficiency and effectiveness of the proposed method.