散乱数据拟合的自适应分片逆尺度空间算法

散乱数据拟合(逼近)是在信号处理、计算机图形学等领域中被广泛研究的问题， 近些年，利用优化方法获得散乱数据的稀疏表示逼近解也成为了优化和曲面重构交叉领域的热 点。基于由B 样条生成的PSI 空间中的散乱点曲面拟合问题和分片稀疏的联系，将分片稀疏性 引入到Bregman 逆尺度空间算法(ISS)中，提出一种自适应的分片逆尺度空间(aP_ISS)算法，处 理散乱数据的曲面拟合问题。通过对逆尺度空间系统分片符号一致性分析，得到了自适应分片 逆尺度空间系统的性能保证定理和避免了aP_ISS 参数的选取。应用到散乱点曲面重构问题上 的数值实验结果表明，该算法不仅可以有效拟合曲面，还能够较好保护分片稀疏性。

Adaptive piecewise inverse scale space algorithm for scattered data fitting

Scattered data reconstruction has been widely studied in the fields of signal processing and computer graphics. Moreover, in recent years, to obtain sparse representation approximations of scattered data by means of sparse optimization method has also become a hot spot in the cross-domain of optimization and surface reconstruction. In this paper, we establish the connection between surface fitting of scattered data and the piecewise sparseness in PSI space generated by a B-spline, and introduce the piecewise sparsity to the Bregman inverse scale space (ISS) algorithm. In addition, an adaptive piecewise ISS algorithm is established to solve the scattered data reconstruction problem. Through the analysis of the piecewise symbolic consistency, the performance guarantee of adaptive piecewise ISS system is obtained in this paper and the selection of aP_ISS parameters can be avoided. Numerical experimental results applied to the surface reconstruction of scattered data show that, this algorithm can not only effectively fit the surface, but also protect the piecewise sparsity of coefficient of the surface.