Multidimensional Signal and Color Image Processing Using Lattices

6Sampling, Reconstruction and Sampling Theorems for Multidimensional Signals

6.1 Introduction

Physical real‐world images are continuous in space and time, while we must use sampled signals for digital image processing. Thus we must be able to convert continuous‐domain signals to discrete‐domain signals, a process known as sampling. Likewise, discrete‐domain signals may need to be converted to the continuous domain for viewing, a process known as reconstruction. This chapter presents the theory of ideal sampling and reconstruction, which is strongly based on Fourier analysis. Some of the issues involved in practical sampling and reconstruction are then discussed.

Sampling of an aperiodic continuous‐domain signal produces an aperiodic discrete‐domain signal. For periodic continuous‐domain signals, if the periodicity lattice is a sublattice of the sampling lattice, sampling produces a periodic discrete‐domain signal with the same periodicity lattice. We can also sample the Fourier transform of an aperiodic continuous‐domain signal, yielding a periodic continuous‐domain signal. Finally, we can sample the Fourier transform of an aperiodic discrete‐domain signal (with conditions on the lattices), which yields a periodic discrete‐domain signal. Thus, all the categories of signals we have studied can be related though sampling in the signal domain or the frequency domain. This will prove very useful in evaluating Fourier transforms and carrying out processing in the frequency domain. ...