IBM Almaden Res. Center, San Jose, CA.
IEEE Trans Image Process. 1996;5(1):71-88. doi: 10.1109/83.481672.
Presents a new covariant basis, dubbed the quasi-orthogonal Q-spline basis, for the space of n-degree periodic uniform splines with k knots. This basis is obtained analogously to the B-spline basis by scaling and periodically translating a single spline function of bounded support. The construction hinges on an important theorem involving the asymptotic behavior (in the dimension) of the inverse of banded Toeplitz matrices. The authors show that the Gram matrix for this basis is nearly diagonal, hence, the name "quasi-orthogonal". The new basis is applied to the problem of approximating closed digital curves in 2D images by least-squares fitting. Since the new spline basis is almost orthogonal, the least-squares solution can be approximated by decimating a convolution between a resolution-dependent kernel and the given data. The approximating curve is expressed as a linear combination of the new spline functions and new "control points". Another convolution maps these control points to the classical B-spline control points. A generalization of the result has relevance to the solution of regularized fitting problems.
提出了一种新的协变基，称为拟正交 Q-样条基，用于具有 k 个结的 n 次周期均匀样条函数的空间。该基类似于 B-样条基，通过缩放和周期性地平移具有有界支撑的单个样条函数来获得。该构造取决于一个涉及带限 Toeplitz 矩阵逆的渐近行为（在维度上）的重要定理。作者表明，该基的 Gram 矩阵几乎是对角的，因此得名“拟正交”。新基应用于通过最小二乘拟合逼近 2D 图像中封闭数字曲线的问题。由于新的样条基几乎是正交的，因此可以通过对分辨率相关核与给定数据的卷积进行抽取来近似最小二乘解。逼近曲线表示为新样条函数和新的“控制点”的线性组合。该结果的推广与正则拟合问题的解有关。
