正则化三角逼近的Levinson-Galerkin算法
A Levinson-Galerkin algorithm for regularized trigonometric approximation
AI总结：
针对非均匀采样且含噪数据的三角逼近，提出以多项式次数为正则化参数的多级Levinson-Galerkin算法，通过求解嵌套Toeplitz系统高效获得最优光滑度最小二乘解，并成功应用于超声心动图左心室边界恢复。
摘要：
三角多项式广泛用于从一组非均匀分布的样本$\{f(x_j)\}_{j=0}^{N-1}$逼近光滑函数$f$。如果样本受到噪声干扰，控制三角逼近的光滑度成为避免数据过拟合和欠拟合的关键问题。我们以多项式次数作为正则化参数，推导出一种多级算法，该算法迭代地适应最优光滑度的最小二乘解。所提出的算法通过求解一族嵌套的Toeplitz系统，以最多$\cal{O}(NM + M^2)$次运算（其中$M$为逼近的多项式次数）计算解。文中展示了所提方法如何扩展到多元三角逼近。我们通过将该算法应用于超声心动图中左心室边界的恢复，证明了其性能。
英文摘要：
Trigonometric polynomials are widely used for the approximation of a smooth function $f$ from a set of nonuniformly spaced samples $\{f(x_j)\}_{j=0}^{N-1}$. If the samples are perturbed by noise, controlling the smoothness of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data. Using the polynomial degree as regularization parameter we derive a multi-level algorithm that iteratively adapts to the least squares solution of optimal smoothness. The proposed algorithm computes the solution in at most $\cal{O}(NM + M^2)$ operations ($M$ being the polynomial degree of the approximation) by solving a family of nested Toeplitz systems. It is shown how the presented method can be extended to multivariate trigonometric approximation. We demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the Left Ventricle.
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