The Nyquist sampling theorem states that in order to be able to completely recover and reconstruct an analog signal we can use uniform sampling with a rate of twice the bandwidth of the band-limited signal. Miscellaneous types of natural or man made signals exists that have large bandwidth but their information content is relatively small. Following the Nyquist sampling theorem, one has to acquire many samples from the original signal before compressing techniques are deployed to store or transmit the reduced version of it. Compressed sensing (CS) tells us that instead of putting so much effort into sampling the signal with high rate and then discarding a considerable part of it we can specialize the Nyquist theorem to sparse signals and do a considerably smaller amount of sampling and afterwards hope to recover the original signal using sophisticated recovery schemes which is the basic idea of compressed sensing. Among many applications of compressed sensing, this work narrows down its scope to CS-based digital imaging, namely single-pixel imaging. Thanks to single-pixel imaging contrary to conventional cameras, we do not to take a large number of samples from the underlying scene that equals the large number N of pixels on the cameras CCD array. Instead we do M By deploying CS-based fovetaed imaging, one can acquire M Chapter 3 represents foveated single-pixel imaging and underlying sampling and recovery techniques. Finally chapter 4 presents an implementation of a typical single-pixel imaging in Matlab which employs several foveated and unfoveated sampling methods as well as underlying recovery techniques. In terms of novelty, it introduces patch-based focusing technique which falls into the category of foveated single-pixel imaging and conducts measurements to evaluate and compare its performance against its counterpart, namely superpixel-based imaging technique. In general the thesis seeks to answer three critical questions in the area of the single-pixel imaging based on the numerical analysis and simulations.