  In this paper we study the topic of signal restoration using complexity
regularization, quantifying the compression bit-cost of the signal estimate.
While complexity-regularized restoration is an established concept, solid
practical methods were suggested only for the Gaussian denoising task, leaving
more complicated restoration problems without a generally constructive
approach. Here we present practical methods for complexity-regularized
restoration of signals, accommodating degradations caused by a known linear
operator of an arbitrary form. Our iterative procedure, obtained using the
alternating direction method of multipliers (ADMM) approach, addresses the
restoration task as a sequence of simpler problems involving L2-regularized
estimations and rate-distortion optimizations (considering the squared-error
criterion). Further, we replace the rate-distortion optimizations with an
arbitrary standardized compression technique and thereby restore the signal by
leveraging underlying models designed for compression. Additionally, we propose
a shift-invariant complexity regularizer, measuring the bit-cost of all the
shifted forms of the estimate, extending our method to use averaging of
decompressed outputs gathered from compression of shifted signals. On the
theoretical side, we present an analysis of complexity-regularized restoration
of a cyclo-stationary Gaussian signal from deterioration by a linear
shift-invariant operator and an additive white Gaussian noise. The theory shows
that optimal complexity-regularized restoration relies on an elementary
restoration filter and compression spreading reconstruction quality unevenly
based on the energy distribution of the degradation filter. Nicely, these ideas
are realized also in the proposed practical methods. The presented experiments
show good results for image deblurring and inpainting using the JPEG2000 and
HEVC compression standards.
