The downside of increasing the resolution of surface scanning devices is that the amount of acquired data makes the morphological analysis of the scanned surfaces computationally challenging. This limitation is circumvented by using scalable and non–memory–intensive harmonic expansions. In this paper, the projection of parametric surfaces onto disk and spherical harmonics bases is investigated, and three novel computationally efficient algorithms are proposed based on the randomised Kaczmarz (RK). To boost the computational performance and convergence of the root mean square error (RMSE) of the reconstructed surfaces we exploited the conjugate symmetry property of the harmonic basis functions. Further, the sparsity of the signals is used for estimating the projection coefficients from an undersampled surface. The first algorithm only takes into consideration the conjugate symmetry property for enhancing the convergence of the RMSE. The second algorithm endows the sparse version of the RK algorithm with the conjugate symmetry property. The third algorithm combines the previous two to further accelerate the convergence of the RMSE. The performance of the developed algorithms is tested on three surfaces where we demonstrate that they outperform conventional reconstruction techniques in terms of processing time with comparable precision.

中文翻译：

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随机Kaczmarz方法论参数曲面调和分解的共轭对称性和稀疏性

提高表面扫描设备分辨率的缺点是，获取的数据量使得扫描表面的形态分析在计算上具有挑战性。通过使用可扩展且非内存密集型谐波展开可以规避此限制。本文研究了参数曲面在圆盘和球谐函数基上的投影，并基于随机 Kaczmarz (RK) 提出了三种新颖的计算高效算法。为了提高重构表面的计算性能和均方根误差（RMSE）的收敛性，我们利用了调和基函数的共轭对称性。此外，信号的稀疏性用于估计来自欠采样表面的投影系数。第一种算法仅考虑共轭对称性以增强RMSE的收敛性。第二种算法赋予稀疏版本的RK算法共轭对称性。第三种算法结合了前两种算法，进一步加速了 RMSE 的收敛。所开发算法的性能在三个表面上进行了测试，我们证明它们在处理时间和精度方面优于传统的重建技术。