Five good reasons for complex-valued transforms in image processing

In 1946, Dennis Gabor introduced the analytic signal for one-dimensional signals. This complexification of functions gives access to their amplitude and phase information and has since then given well-interpretable insight into the properties of the signals over time. However, complex-valued transforms still have not found their place in image processing, except for the Fourier transform and the Gabor transform, which both have proven their performance in many contexts. In this chapter, we give five reasons to consider more general complex transforms for image analysis. We discuss the challenges and advantages of those transforms.