I am trying to understand the motivation for the above question for signal processing and image processing? Can anyone answer this question, or provide me with resources for this?
The fourier transform gives you very fine resolution in the frequency domain, but during the transformation, you loose all the information about when (for time signals) or where (for images) these frequencies occur in your input signal.
The Gabor transform alleviates this problem by windowing the base functions of the fourier transform with a Gaussian impulse. The transform gives the similarity between the signal and the time- and frequency-shifted window, so you may get a sense when/where certain frequencies occur in your input signal.
However, the windowing comes at the cost of losing some of the frequency resolution. The Gabor transform is optimum in the sense that it's windowing function yields the minimum product between time and frequency resolution.