I have read Wavelets are better than Fourier in dealing with non-stationary signals such as images, but I don't understand how images are considered stationary??
Stationarity is a statitical concept defined through stochastic processes. However the following is a practical account of why most typical images are nonstationary :
Throughout the image window, consider a small rectangular test block, say 16x16, and observe the frequency spectrum estimation of that block. Now move this test block accross the image and re-compute the resulting spectrum estimation each time. Now, if the spectrum does not change while the test block is moving accross the image, then you can loosely say that the image under concern is stationary.
Since for most natural images, such a moving block will be crossing unrelated distinct objects during its travesral of the image, the frequency spectrum will also be changing, i.e., image being nonstatinary
Basic Fourier type transforms are not very efficient and proper for this kind of local variations in spectra whereas wavelet or packets or short-space Fourier methods will be more efficiently indicating those local features.
stationarity comes from the fact that if we know state of signal at time t at time t+1 probability distribution of the signal does not change. Images are generally still hence they are stationary signal. But videos on the other hand can be non-stationary.
Some types of images might be considered stationary. However, the standard ones are patched with potentially occluding objects. Imagine the picture of a grass ground, with in the middle a Vichy diamond patterned tablecloth.
The grass can be considered somehow random stationary (in statistical sense), the Vichy red/white tablecloth has almost periodic patterns (hence stationary in a Fourier sense). But both differ in statistical law, hence non-stationary.
Note that wavelets are:
- not too bad for stationary fields; indeed, wavelets can stationarize some distributions, and yield good stationarity tests, for instance: Practical powerful wavelet packet tests for second-order stationarity
- not too bad for localized periodicity (depending on their oscillations),
- not too bad at transitions (the edge of the cloth) since then can act as derivatives or Laplacian,
- (not needed here: not too bad at slow light variations or shadow casting, because of vanishing moments).
This model is described as piece-wise regular + edges + textures + noise. We described it in this overview of 2D wavelet use in image processing: A panorama on Multiscale Geometric Representations, intertwining spatial, directional and frequency selectivity.
So they are pretty good for a large class of natural images. But if you have a perfect 2D sinusoidal image, or constant stationary fields, Fourier would be just fine, and wavelets only show-off.