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

_{1}*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 also utilize the directionally adaptive second order derivatives for constructing the regularization with directions guided by the given coarse reconstruction, which leads to an improved suppression of artifacts. We demonstrate the superiority of the proposed regularization method over existing ones using several reconstruction examples.

The problem of reconstructing an image from nonuniformly spaced, spatial point measurements is frequently encountered in bioimaging and other scientific disciplines. The most successful class of methods in handling this problem uses the regularization approach involving the minimization of a derivative-based roughness functional. It has been well demonstrated, in the presence of noise, that nonquadratic roughness functionals such as