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.

中文翻译：

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频域本地化图像的可控主要组件。

由于现代科学图像数据集通常由大量高分辨率图像组成，因此设计准确有效的处理方法是一项中心研究任务。在本文中，我们考虑了获取数据集的可控主成分的问题，该过程称为“可控PCA”（可控主成分分析）。该过程的输出是一组正交基函数，这些函数最接近数据集中的图像及其所有平面旋转。为了推导这样的基函数，我们首先在适当的基础上扩展图像，为此，可转向的PCA减少为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都很好地定位，则这样的适当基础就是扁长球面波函数（PSWF）。通过特殊的正交积分方案，我们得出了一种从图像的等距样本中计算PSWFs膨胀系数的快速方法，并表明所需的正交节点数与每个图像中的像素数相似。然后，我们确定基于PSWF的可控PCA比现有方法更快，更准确，更重要的是，在整个过程中为我们提供了严格的误差范围。