Yeo Boon Thye Thomas, Ou Wanmei, Golland Polina
Department of Electrical Engineering and Computer Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
IEEE Trans Image Process. 2008 Mar;17(3):283-300. doi: 10.1109/TIP.2007.915550.
The theories of signal sampling, filter banks, wavelets, and "overcomplete wavelets" are well established for the Euclidean spaces and are widely used in the processing and analysis of images. While recent advances have extended some filtering methods to spherical images, many key challenges remain. In this paper, we develop theoretical conditions for the invertibility of filter banks under continuous spherical convolution. Furthermore, we present an analogue of the Papoulis generalized sampling theorem on the 2-Sphere. We use the theoretical results to establish a general framework for the design of invertible filter banks on the sphere and demonstrate the approach with examples of self-invertible spherical wavelets and steerable pyramids. We conclude by examining the use of a self-invertible spherical steerable pyramid in a denoising experiment and discussing the computational complexity of the filtering framework.
信号采样、滤波器组、小波和“超完备小波”的理论在欧几里得空间中已得到充分确立，并广泛应用于图像的处理和分析。虽然最近的进展已将一些滤波方法扩展到球面图像，但仍存在许多关键挑战。在本文中，我们给出了连续球面卷积下滤波器组可逆性的理论条件。此外，我们给出了二维球面上帕普利斯广义采样定理的类似结果。我们利用这些理论结果建立了球面上可逆滤波器组设计的通用框架，并通过自可逆球面小波和可控金字塔的例子展示了该方法。我们通过在去噪实验中检验自可逆球面可控金字塔的使用并讨论滤波框架的计算复杂性来结束本文。
