| Non-rigid shape correspondence is an important problem in the fields of computer graphics and computer vision.It is the research basis of shape recognition,shape retrieval,shape registration and shape segmentation,and provides a solid theoretical basis for 3D visualization,biological computing,face recognition and other application fields.Aiming at existing problems in non-rigid shape correspondence,this dissertation has conducted in-depth research and proposed several efficient corresponding methods.The main research achievements and contributions include:(1)We propose a novel convolutional neural network(CNN)for shape correspondence,termed Anisotropic Chebyshev Spectral CNN(ACSCNN).Firstly,we present an anisotropic spectral manifold convolution operator based on anisotropic Laplace-Beltrami operator,which aggregate the local features of signals by a set of oriented kernels around each point.Rather than using fixed oriented kernels in the spatial domain in previous CNNs,the proposed anisotropic convolution realizes the data-driven convolution kernel by learning spectral filtering parameters.To reduce the computational complexity,we employ an explicit expansion of the Chebyshev polynomial basis to represent the spectral filters whose expansion coefficients are trainable.Through the benchmark experiments of shape correspondence,our architecture is demonstrated to be efficient and be able to provide better than the state-of-the-art results in several datasets even if using constant functions as inputs.(2)We propose an efficient shape correspondence via multiscale spectral manifold preservation.First,a novel,simple and efficient constraint is proposed to compute the functional maps with the core idea of requiring the multiscale spectral manifold wavelet function to preserve correspondingly at each scale.The constraint can extract the shape feature information at different scales and strongly guarantees the isometric property of the mapping.Then,we demonstrate that if the tight wavelet framework is used,the computation of the above functional maps boils down to a simple filtering process of a low-pass and a set of band-pass filters,thus effectively avoiding the problems of computational inefficiency and lack of stability brought by solving a linear system of equations in the existing functional map computation.Finally,an efficient iterative approach is proposed to compute the optimal shape correspondence by alternating the optimization of the functional maps and point-wise maps.Experiments on different datasets demonstrate that our method significantly outperforms existing methods in terms of shape correspondence quality and computational efficiency.(3)We propose an efficient unsupervised learning method for nonrigid shape correspondence.First,an initial point-wise map is obtained from the learned descriptors using the optimal transport,and then the final functional map is obtained by a simple relationship between the point-wise map and the functional map.Such a mechanism not only effectively avoids the problem of unstable network training caused using least square method in existing work,but also allows embedding the powerful geometric constraint of multiscale spectral manifold wavelets preservation into the neural network,making it possible to further optimize the functional map by a simple matrix multiplication calculation only.Finally,a simple unsupervised loss function is constructed by penalizing the corresponding distortion of the impulse function between shapes.It is demonstrated that this method achieves significant improvement in both correspondence quality and computational efficiency compared with existing methods,as well as strong generalization ability in terms of cross datasets and shape discretization.In a word,this thesis proposes several kinds of effective shape correspondence methods,which effectively solve some problems existing in the recent non-rigid shape correspondence methods.Many experimental results also demonstrate the effectiveness of our proposed method.The work has made some innovations and attempts for complex 3D shape correspondence and lays a foundation for subsequent research.There are 42 figures,9 tables,and 111 citations in this thesis. |
