Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

Nonlinear Filtering for Image Denoising 27-927.2.3 Radon-Based Digital ImplementationEquations 27.10 and 27.11 suggest that the discrete PCT can be implemented by a Radon transform, thatis, taking the 1-D DCT on Radon slices:Cf [k, u] ¼ XN 1 Rf [t, u] cos p N t þ 1 k : (27:23)2t¼0As the Fourier Slice Theorem shows, the Radon transform can be implemented by taking the centralslice in the Fourier spectrum and then performing a 1-D inverse Fourier transform on it. Applying DCTon the Radon slices, we can obtain the discrete PCT.This gives a means for a digital implementation of the discrete polar cosine transform. For the Radontransform, various ways to implement it in a discrete fashion have been attempted, among which theFast Slant Stack [25] was chosen, which is based on a pseudo-polar Fast Fourier Transform (FFT) [26].According to Ref. [25], the transform is computationally efficient, algebraically exact, geometricallyfaithful, and its inversion is numerically stable.With the implementation described above, the PCT basis vectors for an 8 8 image block can be seen inFigure 27.1. The basis vectors are indexed by frequency k and orientation u. As a result of the Cartesianto-polarconversion in the frequency domain, the low-frequency basis vectors are clearly being over-sampled,resulting in a redundant frame where the number of coefficients is four times the original data size.27.2.4 Butterfly-Based Digital ImplementationThe above digital implementation of the PCT based on the Radon transform is akin to the digital ridgelettransform [27]. Such a Radon-based approach is flexible in constructing various directional ridge-typetransforms, but it suffers from several drawbacks of the underlying Radon transform. First is the highcomputational requirement both for the forward and inverse transforms. The forward transform includesa pseudo-polar Fourier transform, 1-D inverse Fourier transforms on polar slices, and forward cosineFIGURE 27.18 8 discrete polar cosine basis vectors.