Statistical Signal Processing in Engineering

182D Signals and Physical Filters

A 2D signal is any function that depends on two variables: these could be either space‐space, or space‐time, or any other. 2D signals are very common in statistical signal processing and this is a situation where multidimensionality offers remarkable benefits, but specific processing tools need to be designed. Properties of 2D signals are very specific—see [77] for a comprehensive treatment. Most of the properties are just an extension of 1D signals, but there are specific properties that are a consequence of the augmented dimension. Appendix A covers the main properties useful here. Images are the most common 2D signals, and these have specific processing tools for image manipulation and enhancement that are not considered extensively here as they are too specific, but [78,80] are excellent references on the topic. Statistical signal processing is known to be far more effective when considering parametric models for the data, and this is the subject of this chapter where 2D signals are generated and manipulated according to their physical generation models based on partial differential equations (PDEs). Furthermore, there is strong interest in revisiting PDEs as some 2D filtering in image processing can be reduced to smoothing PDE and smoothing‐enhancing PDE. The reader is encouraged to read the reference by G. Aubert and P. Kornprobst [81].

The 2D Fourier transform of any 2D signal (Appendix B) is

where the angular frequency¹ ω_x, ω_y (rad/m) ...