  The importance of regularization has been well established in image
reconstruction -- which is the computational inversion of imaging forward model
-- with applications including deconvolution for microscopy, tomographic
reconstruction, magnetic resonance imaging, and so on. Originally, the primary
role of the regularization was to stabilize the computational inversion of the
imaging forward model against noise. However, a recent framework pioneered by
Donoho and others, known as compressive sensing, brought the role of
regularization beyond the stabilization of inversion. It established a
possibility that regularization can recover full images from highly
undersampled measurements. However, it was observed that the quality of
reconstruction yielded by compressive sensing methods falls abruptly when the
under-sampling and/or measurement noise goes beyond a certain threshold.
Recently developed learning-based methods are believed to outperform the
compressive sensing methods without a steep drop in the reconstruction quality
under such imaging conditions. However, the need for training data limits their
applicability. In this paper, we develop a regularization method that
outperforms compressive sensing methods as well as selected learning-based
methods, without any need for training data. The regularization is constructed
as a spatially varying weighted sum of first- and canonical second-order
derivatives, with the weights determined to be adaptive to the image structure;
the weights are determined such that the attenuation of sharp image features --
which is inevitable with the use of any regularization -- is significantly
reduced. We demonstrate the effectiveness of the proposed method by performing
reconstruction on sparse Fourier samples simulated from a variety of MRI
images.
