Shape descriptors have recently gained popularity in shape matching, statistical shape modeling, etc. Their discriminative ability and efficiency play a decisive role in these tasks. In this paper, we first propose a novel handcrafted anisotropic spectral descriptor using Chebyshev polynomials, called the anisotropic Chebyshev descriptor (ACD); it can effectively capture shape features in multiple directions. The ACD inherits many good characteristics of spectral descriptors, such as being intrinsic, robust to changes in surface discretization, etc. Furthermore, due to the orthogonality of Chebyshev polynomials, the ACD is compact and can disambiguate intrinsic symmetry since several directions are considered. To improve the ACD’s discrimination ability, we construct a Chebyshev spectral manifold convolutional neural network (CSMCNN) that optimizes the ACD and produces a learned ACD. Our experimental results show that the ACD outperforms existing state-of-the-art handcrafted descriptors. The combination of the ACD and the CSMCNN is better than other state-of-the-art learned descriptors in terms of discrimination, efficiency, and robustness to changes in shape resolution and discretization.

Shape descriptors are extensively used in shape analysis tasks such as shape correspondence, segmentation and retrieval, just to name a few. Their performances significantly determine the efficiency and effectiveness of subsequent applications. For this problem, we propose a novel powerful descriptor called Anisotropic Fractional Spectral Manifold Wavelet Descriptor (AFSMWD), built upon an extended manifold signal processing tool named Anisotropic Fractional Spectral Manifold Wavelet (AFSMW), which is also presented for the first time in this paper. The novelty of AFSMW is integrating the fractional theory into the common anisotropic spectral manifold wavelet. Compared to the existing wavelets, it provides one more new parameter, namely, the fractional order, to balance or enhance the transform coefficients among different shape vertices, enabling more flexible local shape analysis and more hidden shape structural information explored. Due to the advantages of this added parameter and the capability of analyzing shape features from multiple scales and orientations, the AFSMW allows us to construct the powerful descriptor AFSMWD just using the AFSMW transform coefficients of a very simple function. The proposed descriptor appears to be especially localizable, discriminative, and robust to noises. Extensive experiments have demonstrated that our descriptor has outperformed the state-of-the-art descriptors, nearly achieving 22% improvements to the most related work ASMWD and 69% to the recent popular work WEDS on the FAUST dataset. Its superiorities are also announced in some challenging occasions such as shapes with large deformation or topological partiality.