Non-rigid shapes are generally known as objects whose three dimensional geometry may deform by internal and/or external forces. Deformable shapes are all around us, ranging from protein molecules, to natural objects such as the trees in the forest or the fruits in our gardens, and even human bodies. Two deformable shapes may be related by isometry, which means their intrinsic geometries are preserved, even though their extrinsic geometries are dissimilar. An important problem in the analysis of the deformable shapes is to identify the three-dimensional correspondence between two isometric shapes, given that the two shapes may be deviated from isometry by intrinsic distortions. A major challenge is that non-rigid shapes have large degrees of freedom on how to deform. Nevertheless, irrespective of how they are deformed, they may be aligned such that the geodesic distance between two arbitrary points on two shapes are nearly equal. Such alignment may be expressed by a permutation matrix (a matrix with binary entries) that corresponds to every paired geodesic distance in between the two shapes. The alignment involves searching the space over all possible mappings (that is all the permutations) to locate the one that minimizes the amount of deviation from isometry. A brute-force search to locate the correspondence is not computationally feasible. This thesis introduces a novel approach created to locate such correspondences, in spite of the large solution space that encompasses all possible mappings and the presence of intrinsic distortion. In order to find correspondences between two shapes, the first step is to create a suitable descriptor to accurately describe the deformable shapes. To this end, we developed deformation-invariant metric descriptors. A descriptor constitutes pair-wise geodesic distances among arbitrary number of discrete points that represent the topology of the non-rigid shape. Our descriptor provides isometric-invariant representation of the shape irrespective of its circumstantial deformation. Two isometric-invariant descriptors, representing two candidate deformable shapes, are the input parameters to our optimization algorithm. We then proceed to locate the permutation matrix that aligns the two descriptors, that minimizes the deviation from isometry. Once we have developed such a descriptor, we turn our attention to finding correspondences between non deformable shapes. In this study, we investigate the use of both classical and quantum particle swarm optimization (PSO) algorithms for this task. To explore the merits of variants of PSO, integer optimization involving test functions with large dimensions were performed, and the results and the analysis suggest that quantum PSO is more effective optimization method than its classical PSO counterpart. Further, a scheme is proposed to structure the solution space, composed of permutation matrices, in lexicographic ordering. The search in the solution space is accordingly simplified to integer optimization to find the integer rank of the targeted permutation matrix. Empirical results suggest that this scheme improves the scalability of quantum PSO across large solution spaces. Yet, quantum PSO's global search capability requires assistance in order to more effectively manoeuvre through the local extrema prevalent in the large solution spaces. A mutation based genetic algorithm (GA) is employed to augment the search diversity of quantum PSO when/if the swarm stagnates among the local extrema. The mutation based GA instantly disengages the optimization engine from the local extrema in order to reorient the optimization energy to the trajectories that steer to the global extrema, or the targeted permutation matrix. Our resultant optimization algorithm combines quantum Particle Swarm Optimization (PSO) and mutation based Genetic Algorithm (GA). Empirical results show that the optimization method presented is scalable and efficient on standard hardware across different solution space sizes. The performance of the optimization method, in simulations and on various near-isometric shapes, is discussed. In all cases investigated, the method could successfully identify the correspondence among the non-rigid deformable shapes that were related by isometry.

Non-rigid shapes are generally known as objects whose three dimensional geometry may deform by internal and/or external forces. Deformable shapes are all around us, ranging from protein molecules, to natural objects such as the trees in the forest or the fruits in our gardens, and even human bodies. Two deformable shapes may be related by isometry, which means their intrinsic geometries are preserved, even though their extrinsic geometries are dissimilar. An important problem in the analysis of the deformable shapes is to identify the three-dimensional correspondence between two isometric shapes, given that the two shapes may be deviated from isometry by intrinsic distortions. A major challenge is that non-rigid shapes have large degrees of freedom on how to deform. Nevertheless, irrespective of how they are deformed, they may be aligned such that the geodesic distance between two arbitrary points on two shapes are nearly equal. Such alignment may be expressed by a permutation matrix (a matrix with binary entries) that corresponds to every paired geodesic distance in between the two shapes. The alignment involves searching the space over all possible mappings (that is all the permutations) to locate the one that minimizes the amount of deviation from isometry. A brute-force search to locate the correspondence is not computationally feasible. This thesis introduces a novel approach created to locate such correspondences, in spite of the large solution space that encompasses all possible mappings and the presence of intrinsic distortion. In order to find correspondences between two shapes, the first step is to create a suitable descriptor to accurately describe the deformable shapes. To this end, we developed deformation-invariant metric descriptors. A descriptor constitutes pair-wise geodesic distances among arbitrary number of discrete points that represent the topology of the non-rigid shape. Our descriptor provides isometric-invariant representation of the shape irrespective of its circumstantial deformation. Two isometric-invariant descriptors, representing two candidate deformable shapes, are the input parameters to our optimization algorithm. We then proceed to locate the permutation matrix that aligns the two descriptors, that minimizes the deviation from isometry. Once we have developed such a descriptor, we turn our attention to finding correspondences between non deformable shapes. In this study, we investigate the use of both classical and quantum particle swarm optimization (PSO) algorithms for this task. To explore the merits of variants of PSO, integer optimization involving test functions with large dimensions were performed, and the results and the analysis suggest that quantum PSO is more effective optimization method than its classical PSO counterpart. Further, a scheme is proposed to structure the solution space, composed of permutation matrices, in lexicographic ordering. The search in the solution space is accordingly simplified to integer optimization to find the integer rank of the targeted permutation matrix. Empirical results suggest that this scheme improves the scalability of quantum PSO across large solution spaces. Yet, quantum PSO's global search capability requires assistance in order to more effectively manoeuvre through the local extrema prevalent in the large solution spaces. A mutation based genetic algorithm (GA) is employed to augment the search diversity of quantum PSO when/if the swarm stagnates among the local extrema. The mutation based GA instantly disengages the optimization engine from the local extrema in order to reorient the optimization energy to the trajectories that steer to the global extrema, or the targeted permutation matrix. Our resultant optimization algorithm combines quantum Particle Swarm Optimization (PSO) and mutation based Genetic Algorithm (GA). Empirical results show that the optimization method presented is scalable and efficient on standard hardware across different solution space sizes. The performance of the optimization method, in simulations and on various near-isometric shapes, is discussed. In all cases investigated, the method could successfully identify the correspondence among the non-rigid deformable shapes that were related by isometry.