Fourier transform of Moisan periodic image component

The function perfft2() implements Moisan's "Periodic plus Smooth Image Decomposition" which decomposes an image into two components

    img = p + s

where s is the 'smooth' component with mean 0 and p is the 'periodic' component which has no sharp discontinuities when one moves cyclically across the image boundaries.

This decomposition is very useful when one wants to obtain an FFT of an image with minimal artifacts introduced from the boundary discontinuities. The image p gathers most of the image information but avoids periodization artifacts.

Reference: L. Moisan, "Periodic plus Smooth Image Decomposition", Journal of Mathematical Imaging and Vision, vol 39:2, pp. 161-179, 2011.

using Images
using FFTW
using ImagePhaseCongruency
using ImageContrastAdjustment
using TestImages

img = Float64.(Gray.(testimage("lena")))

IMG = fft(img)               # 'Standard' fft
(P, S, p, s) = perfft2(img)  # 'Periodic' fft

mosaic(
    adjust_histogram(Gray.(p), LinearStretching()),
    adjust_histogram(s, LinearStretching()),
    # Note the vertical and horizontal cross in
    # the spectrum induced by the non-periodic edges.
    adjust_histogram(log.(abs.(fftshift(IMG)) .+ 1), LinearStretching()),
    # Note the clean spectrum because p is periodic.
    adjust_histogram(log.(abs.(fftshift(P)) .+ 1), LinearStretching());
    nrow=2, rowmajor=true
)

Top 1) left: periodic component 2) right: smooth component

Bottom 3) left: spectrum of standard FFT 4) right: spectrum of periodic component

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