Structurally Adaptive Multi-Derivative Regularization for Image Recovery from Sparse Fourier Samples

Abstract: 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.