  We consider the problem of reconstructing 2D images from randomly
under-sampled confocal microscopy samples. The well known and widely celebrated
total variation regularization, which is the L1 norm of derivatives, turns out
to be unsuitable for this problem; it is unable to handle both noise and
under-sampling together. This issue is linked with the notion of phase
transition phenomenon observed in compressive sensing research, which is
essentially the break-down of total variation methods, when sampling density
gets lower than certain threshold. The severity of this breakdown is determined
by the so-called mutual incoherence between the derivative operators and
measurement operator. In our problem, the mutual incoherence is low, and hence
the total variation regularization gives serious artifacts in the presence of
noise even when the sampling density is not very low. There has been very few
attempts in developing regularization methods that perform better than total
variation regularization for this problem. We develop a multi-resolution based
regularization method that is adaptive to image structure. In our approach, the
desired reconstruction is formulated as a series of coarse-to-fine
multi-resolution reconstructions; for reconstruction at each level, the
regularization is constructed to be adaptive to the image structure, where the
information for adaption is obtained from the reconstruction obtained at
coarser resolution level. This adaptation is achieved by using maximum entropy
principle, where the required adaptive regularization is determined as the
maximizer of entropy subject to the information extracted from the coarse
reconstruction as constraints. We demonstrate the superiority of the proposed
regularization method over existing ones using several reconstruction examples.
