A parallel implementation of 3D Zernike moment analysis

Abstract

Zernike polynomials are a well known set of functions that find many applications in image or pattern characterization because they allow to construct shape descriptors that are invariant against translations, rotations or scale changes. The concepts behind them can be extended to higher dimension spaces, making them also fit to describe volumetric data. They have been less used than their properties might suggest due to their high computational cost.We present a parallel implementation of 3D Zernike moments analysis, written in C with CUDA extensions, which makes it practical to employ Zernike descriptors in interactive applications, yielding a performance of several frames per second in voxel datasets about 200 3 in size.In our contribution, we describe the challenges of implementing 3D Zernike analysis in a general-purpose GPU. These include how to deal with numerical inaccuracies, due to the high precision demands of the algorithm, or how to deal with the high volume of input data so that it does not become a bottleneck for the system.

Efficient and Stable Voxel-Based Algorithm for Computing 3D Zernike Moments and Shape Reconstruction

“…We also used algorithm provided by Novotni and Klein in [1,2] and applied it to the image Michelangelo's David. As reported in [12], when the maximum order is 20, the absolute value of the geometric moment coefficient ranges from 10 0 to 10 9 , and the geometric moments also have a great dynamic range. The calculation error accumulates very quickly and is directly reflected in the shape reconstruction at 𝑛 𝑚𝑎𝑥 = 20, as shown in Figure 3.…”

“…Theorem 3.2.1 in [31]) can be used to derive the three-term recurrence formulas of the associated Legendre polynomials and the one for 3D Zernike radial polynomials described in Theorem A and Theorem D, respectively. The definition of the radial polynomial given in (12) looks different from that of Canterakis [9,10], but these two definitions are equivalent [34].…”

“…The 3D moments are expressed by a complex number linear combination of geometric moments, which are widely used in image processing and pattern recognition. In [11][12][13], the algorithms for calculating geometric moments was proposed by using exact integration in each voxel. Based on the theory in [1,2], Novotni and Klein developed the algorithm for calculating Zernike-Canterakis moments.…”

“…A reformulation of Zernike-Canterakis moments and its implementation were provided in the article by Callahan and De Graef [11]. The coefficients of geometric moments range from 10 0 to10 9 for the maximal order 20 and the geometric moments also have a great dynamic range [12]. This method of computing 3D Zernike moments is subject to severe instability of the geometric moment in the case of higher orders [13].…”

Stable Evaluation of 3D Zernike Moments for Surface Meshes

“…It is easy to see that this discretization error increases as the order of the moment increases. Finally, the relationships between geometric moments and Zernike moments lead to numerical instabilities, limiting the order N of the moments that can be computed accurately, usually with N < 50 (see [22,46]).…”