Landa Boris, Shkolnisky Yoel
Department of Applied Mathematics, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel.
SIAM J Imaging Sci. 2017;10(2):508-534. doi: 10.1137/16M1085334. Epub 2017 Apr 13.
As modern scientific image datasets typically consist of a large number of images of high resolution, devising methods for their accurate and efficient processing is a central research task. In this paper, we consider the problem of obtaining the steerable principal components of a dataset, a procedure termed "steerable PCA" (steerable principal component analysis). The output of the procedure is the set of orthonormal basis functions which best approximate the images in the dataset and all of their planar rotations. To derive such basis functions, we first expand the images in an appropriate basis, for which the steerable PCA reduces to the eigen-decomposition of a block-diagonal matrix. If we assume that the images are well localized in space and frequency, then such an appropriate basis is the prolate spheroidal wave functions (PSWFs). We derive a fast method for computing the PSWFs expansion coefficients from the images' equally spaced samples, via a specialized quadrature integration scheme, and show that the number of required quadrature nodes is similar to the number of pixels in each image. We then establish that our PSWF-based steerable PCA is both faster and more accurate then existing methods, and more importantly, provides us with rigorous error bounds on the entire procedure.
由于现代科学图像数据集通常由大量高分辨率图像组成，设计用于精确高效处理这些数据集的方法是一项核心研究任务。在本文中，我们考虑获取数据集的可控主成分的问题，这一过程称为“可控主成分分析”（steerable principal component analysis）。该过程的输出是一组正交基函数，它们能最佳逼近数据集中的图像及其所有平面旋转。为了导出这样的基函数，我们首先在一个合适的基中展开图像，在此基下，可控主成分分析简化为一个块对角矩阵的特征分解。如果我们假设图像在空间和频率上具有良好的局部性，那么这样一个合适的基就是长球波函数（PSWFs）。我们通过一种专门的求积积分方案，推导出一种从图像的等间距样本计算PSWFs展开系数的快速方法，并表明所需求积节点的数量与每个图像中的像素数量相似。然后我们证明，基于PSWF的可控主成分分析比现有方法更快、更准确，更重要的是，为整个过程提供了严格的误差界。
