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Images of static scenes submerged beneath a wavy water surface exhibit severe non-rigid distortions. The physics of water flow suggests that water surfaces possess spatio-temporal smoothness and temporal periodicity. Hence they possess a sparse representation in the 3D discrete Fourier (DFT) basis. Motivated by this, we pose the task of restoration of such video sequences as a compressed sensing (CS) problem. We begin by tracking a few salient feature points across the frames of a video sequence of the submerged scene. Using these point trajectories, we show that the motion fields at all other (non-tracked) points can be effectively estimated using a typical CS solver. This by itself is a novel contribution in the field of non-rigid motion estimation. We show that this method outperforms state of the art algorithms for underwater image restoration. We further consider a simple optical flow algorithm based on local polynomial expansion of the image frames (PEOF). Surprisingly, we demonstrate that PEOF is more efficient and often outperforms all the state of the art methods in terms of numerical measures. Finally, we demonstrate that a two-stage approach consisting of the CS step followed by PEOF much more accurately preserves the image structure and improves the (visual as well as numerical) video quality as compared to just the PEOF stage. The source code, datasets and supplemental material can be accessed at \cite{GitRepo}, \cite{ProjectPage}.

Restoration of Non-Rigidly Distorted Underwater Images Using a Combination of Compressive Sensing and Local Polynomial Image Representations