Monday, March 14th, 2005
The Importance of Phase in Images
Many papers have suggested that phase information in an image is very important. A report from Alan Oppenheim in 1979 entitled Phase in Speech and Pictures demonstrated that much of the structural information in an image is preserved even when it is represented by phase alone.
He describes an experiment in which an image is decomposed into phase and magnitude parts using a Fourier transform, then the magnitude is set to unity, and an image is reconstructed from the remaining phase information.
The idea is that Fourier phase includes important information about the features and details in an image. The following figures show an original and the phase-only reconstruction of an example image. These were produced by the following MATLAB commands:
% start with an image stored in variable "im"
im_fourier = fft2(im);
im_phase = angle(im_fourier);
im_reconstruct_from_phase = abs(ifft2(exp(i*im_phase)));
im_reconstruct_from_phase
% display original & reconstructed image
% (scaled for visibility)
imshow(im,[])
imshow(im_reconstruct_from_phase.^.4),[])
Many of the high-frequency structures have been preserved in the phase-only image. Indeed, the transformation into a phase-only image can be approximately interpreted as a high pass filtering operation.
It turns out that the intelligibility of the phase-only representation depends on the magnitude “smoothness” of the signal being looked at. Since most natural images contain mostly low frequency content, their magnitude rolls off quickly at high frequency and this leads to the situation where the “high pass” interpretation of the phase-only transform holds.
