  The problem of minimization of the number of measurements needed for digital
image acquisition and reconstruction with a given accuracy is addressed. Basics
of the sampling theory are outlined to show that the lower bound of signal
sampling rate sufficient for signal reconstruction with a given accuracy is
equal to the spectrum sparsity of the signal sparse approximation that has this
accuracy. It is revealed that the compressed sensing approach, which was
advanced as a solution to the sampling rate minimization problem, is far from
reaching the sampling rate theoretical minimum. Potentials and limitations of
compressed sensing are demystified using a simple and intutive model, A method
of image Arbitrary Sampling and Bounded Spectrum Reconstruction (ASBSR-method)
is described that allows to draw near the image sampling rate theoretical
minimum. Presented and discussed are also results of experimental verification
of the ASBSR-method and its possible applicability extensions to solving
various underdetermined inverse problems such as color image demosaicing, image
in-painting, image reconstruction from their sparsely sampled or decimated
projections, image reconstruction from the modulus of its Fourier spectrum, and
image reconstruction from its sparse samples in Fourier domain
