Regularized pointwise map recovery from functional correspondence

The concept of using functional maps for representing dense correspondences between deformable shapes has proven to be extremely effective in many applications. However, despite the impact of this framework, the problem of recovering the point-to-point correspondence from a given functional map has received surprisingly little interest. In this paper, we analyse the aforementioned problem and propose a novel method for reconstructing pointwise correspondences from a given functional map. The proposed algorithm phrases the matching problem as a regularized alignment problem of the spectral embeddings of the two shapes. Opposed to established methods, our approach does not require the input shapes to be nearly-isometric, and easily extends to recovering the point-to-point correspondence in part-to-whole shape matching problems. Our numerical experiments demonstrate that the proposed approach leads to a significant improvement in accuracy in several challenging cases

Shape Analysis Using Spectral Geometry

Shape analysis is a fundamental research topic in computer graphics and computer vision. To date, more and more 3D data is produced by those advanced acquisition capture devices, e.g., laser scanners, depth cameras, and CT/MRI scanners. The increasing data demands advanced analysis tools including shape matching, retrieval, deformation, etc. Nevertheless, 3D Shapes are represented with Euclidean transformations such as translation, scaling, and rotation and digital mesh representations are irregularly sampled. The shape can also deform non-linearly and the sampling may vary. In order to address these challenging problems, we investigate Laplace-Beltrami shape spectra from the differential geometry perspective, focusing more on the intrinsic properties. In this dissertation, the shapes are represented with 2 manifolds, which are differentiable. First, we discuss in detail about the salient geometric feature points in the Laplace-Beltrami spectral domain instead of traditional spatial domains. Simultaneously, the local shape descriptor of a feature point is the Laplace-Beltrami spectrum of the spatial region associated to the point, which are stable and distinctive. The salient spectral geometric features are invariant to spatial Euclidean transforms, isometric deformations and mesh triangulations. Both global and partial matching can be achieved with these salient feature points. Next, we introduce a novel method to analyze a set of poses, i.e., near-isometric deformations, of 3D models that are unregistered. Different shapes of poses are transformed from the 3D spatial domain to a geometry spectral one where all near isometric deformations, mesh triangulations and Euclidean transformations are filtered away. Semantic parts of that model are then determined based on the computed geometric properties of all the mapped vertices in the geometry spectral domain while semantic skeleton can be automatically built with joints detected. Finally we prove the shape spectrum is a continuous function to a scale function on the conformal factor of the manifold. The derivatives of the eigenvalues are analytically expressed with those of the scale function. The property applies to both continuous domain and discrete triangle meshes. On the triangle meshes, a spectrum alignment algorithm is developed. Given two closed triangle meshes, the eigenvalues can be aligned from one to the other and the eigenfunction distributions are aligned as well. This extends the shape spectra across non-isometric deformations, supporting a registration-free analysis of general motion data

Calculating Sparse and Dense Correspondences for Near-Isometric Shapes

Comparing and analysing digital models are basic techniques of geometric shape processing. These techniques have a variety of applications, such as extracting the domain knowledge contained in the growing number of digital models to simplify shape modelling. Another example application is the analysis of real-world objects, which itself has a variety of applications, such as medical examinations, medical and agricultural research, and infrastructure maintenance. As methods to digitalize physical objects mature, any advances in the analysis of digital shapes lead to progress in the analysis of real-world objects. Global shape properties, like volume and surface area, are simple to compare but contain only very limited information. Much more information is contained in local shape differences, such as where and how a plant grew. Sadly the computation of local shape differences is hard as it requires knowledge of corresponding point pairs, i.e. points on both shapes that correspond to each other. The following article thesis (cumulative dissertation) discusses several recent publications for the computation of corresponding points: - Geodesic distances between points, i.e. distances along the surface, are fundamental for several shape processing tasks as well as several shape matching techniques. Chapter 3 introduces and analyses fast and accurate bounds on geodesic distances. - When building a shape space on a set of shapes, misaligned correspondences lead to points moving along the surfaces and finally to a larger shape space. Chapter 4 shows that this also works the other way around, that is good correspondences are obtain by optimizing them to generate a compact shape space. - Representing correspondences with a “functional map” has a variety of advantages. Chapter 5 shows that representing the correspondence map as an alignment of Green’s functions of the Laplace operator has similar advantages, but is much less dependent on the number of eigenvectors used for the computations. - Quadratic assignment problems were recently shown to reliably yield sparse correspondences. Chapter 6 compares state-of-the-art convex relaxations of graphics and vision with methods from discrete optimization on typical quadratic assignment problems emerging in shape matching

Learning shape correspondence with anisotropic convolutional neural networks

Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks

3D shape matching and registration : a probabilistic perspective

Dense correspondence is a key area in computer vision and medical image analysis. It has applications in registration and shape analysis. In this thesis, we develop a technique to recover dense correspondences between the surfaces of neuroanatomical objects over heterogeneous populations of individuals. We recover dense correspondences based on 3D shape matching. In this thesis, the 3D shape matching problem is formulated under the framework of Markov Random Fields (MRFs). We represent the surfaces of neuroanatomical objects as genus zero voxel-based meshes. The surface meshes are projected into a Markov random field space. The projection carries both geometric and topological information in terms of Gaussian curvature and mesh neighbourhood from the original space to the random field space. Gaussian curvature is projected to the nodes of the MRF, and the mesh neighbourhood structure is projected to the edges. 3D shape matching between two surface meshes is then performed by solving an energy function minimisation problem formulated with MRFs. The outcome of the 3D shape matching is dense point-to-point correspondences. However, the minimisation of the energy function is NP hard. In this thesis, we use belief propagation to perform the probabilistic inference for 3D shape matching. A sparse update loopy belief propagation algorithm adapted to the 3D shape matching is proposed to obtain an approximate global solution for the 3D shape matching problem. The sparse update loopy belief propagation algorithm demonstrates significant efficiency gain compared to standard belief propagation. The computational complexity and convergence property analysis for the sparse update loopy belief propagation algorithm are also conducted in the thesis. We also investigate randomised algorithms to minimise the energy function. In order to enhance the shape matching rate and increase the inlier support set, we propose a novel clamping technique. The clamping technique is realized by combining the loopy belief propagation message updating rule with the feedback from 3D rigid body registration. By using this clamping technique, the correct shape matching rate is increased significantly. Finally, we investigate 3D shape registration techniques based on the 3D shape matching result. Based on the point-to-point dense correspondences obtained from the 3D shape matching, a three-point based transformation estimation technique is combined with the RANdom SAmple Consensus (RANSAC) algorithm to obtain the inlier support set. The global registration approach is purely dependent on point-wise correspondences between two meshed surfaces. It has the advantage that the need for orientation initialisation is eliminated and that all shapes of spherical topology. The comparison of our MRF based 3D registration approach with a state-of-the-art registration algorithm, the first order ellipsoid template, is conducted in the experiments. These show dense correspondence for pairs of hippocampi from two different data sets, each of around 20 60+ year old healthy individuals

Multi-View Face Recognition From Single RGBD Models of the Faces

This work takes important steps towards solving the following problem of current interest: Assuming that each individual in a population can be modeled by a single frontal RGBD face image, is it possible to carry out face recognition for such a population using multiple 2D images captured from arbitrary viewpoints? Although the general problem as stated above is extremely challenging, it encompasses subproblems that can be addressed today. The subproblems addressed in this work relate to: (1) Generating a large set of viewpoint dependent face images from a single RGBD frontal image for each individual; (2) using hierarchical approaches based on view-partitioned subspaces to represent the training data; and (3) based on these hierarchical approaches, using a weighted voting algorithm to integrate the evidence collected from multiple images of the same face as recorded from different viewpoints. We evaluate our methods on three datasets: a dataset of 10 people that we created and two publicly available datasets which include a total of 48 people. In addition to providing important insights into the nature of this problem, our results show that we are able to successfully recognize faces with accuracies of 95% or higher, outperforming existing state-of-the-art face recognition approaches based on deep convolutional neural networks

Machine Intelligence for Advanced Medical Data Analysis: Manifold Learning Approach

In the current work, linear and non-linear manifold learning techniques, specifically Principle Component Analysis (PCA) and Laplacian Eigenmaps, are studied in detail. Their applications in medical image and shape analysis are investigated. In the first contribution, a manifold learning-based multi-modal image registration technique is developed, which results in a unified intensity system through intensity transformation between the reference and sensed images. The transformation eliminates intensity variations in multi-modal medical scans and hence facilitates employing well-studied mono-modal registration techniques. The method can be used for registering multi-modal images with full and partial data. Next, a manifold learning-based scale invariant global shape descriptor is introduced. The proposed descriptor benefits from the capability of Laplacian Eigenmap in dealing with high dimensional data by introducing an exponential weighting scheme. It eliminates the limitations tied to the well-known cotangent weighting scheme, namely dependency on triangular mesh representation and high intra-class quality of 3D models. In the end, a novel descriptive model for diagnostic classification of pulmonary nodules is presented. The descriptive model benefits from structural differences between benign and malignant nodules for automatic and accurate prediction of a candidate nodule. It extracts concise and discriminative features automatically from the 3D surface structure of a nodule using spectral features studied in the previous work combined with a point cloud-based deep learning network. Extensive experiments have been conducted and have shown that the proposed algorithms based on manifold learning outperform several state-of-the-art methods. Advanced computational techniques with a combination of manifold learning and deep networks can play a vital role in effective healthcare delivery by providing a framework for several fundamental tasks in image and shape processing, namely, registration, classification, and detection of features of interest