Efficient 3D Geometric and Zernike Moments Computation from Unstructured Surface Meshes

Abstract

This paper introduces and evaluates a fast exact algorithm and a series of faster approximate algorithms for the computation of 3D geometric moments from an unstructured surface mesh of triangles. Being based on the object surface reduces the computational complexity of these algorithms with respect to volumetric grid-based algorithms. In contrast, it can only be applied for the computation of geometric moments of homogeneous objects. This advantage and restriction is shared with other proposed algorithms based on the object boundary. The proposed exact algorithm reduces the computational complexity for computing geometric moments up to order N with respect to previously proposed exact algorithms, from N(9) to N(6). The approximate series algorithm appears as a power series on the rate between triangle size and object size, which can be truncated at any desired degree. The higher the number and quality of the triangles, the better the approximation. This approximate algorithm reduces the computational complexity to N(3). In addition, the paper introduces a fast algorithm for the computation of 3D Zernike moments from the computed geometric moments, with a computational complexity N(4), while the previously proposed algorithm is of order N(6). The error introduced by the proposed approximate algorithms is evaluated in different shapes and the cost-benefit ratio in terms of error, and computational time is analyzed for different moment orders.

Correspondence-Less Non-rigid Registration of Triangular Surface Meshes

“…Note that the integrands are simply various geometric moments of tetrahedrons and , which can be efficiently computed by applying the methods proposed in [13,20].…”

“…The computational complexity of the ℎ integral would be ( 9 ), where = + + is the order of the moments. However, the summations in (16) can be rearranged using the following recursive formulas [20]:…”

“…As a result, the complexity reduces to ( 6 ) [20]. A further reduction to ( 3 ) can be achieved by applying the results of [13] for rearranging (17), (18), and (19) as…”

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“…We utilize the efficient algorithm proposed in [11] for unstructured surface meshes of triangles. As we have a triangle mesh, it allows us to decompose the integral, involving geometric moments, in different tetrahedra.…”

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“…11 New algorithms to compute both exact and approximate moments from triangle meshes have been presented very recently. 12 The implementation we present in this paper is an adaptation of Novotni and Klein's algorithm for GPUs which makes it suitable for interactive applications.…”