Boosting Compressed Sensing Using Local Measurements and Sliding Window Reconstruction

In the framework of compressed sensing, image data is measured using less measurements than the total number of pixels. Each measurement consists of a (random) linear combination of all pixels. Since image data is approximately sparse in an appropriate transform domain, a reasonable reconstruction is possible for many measurement matrices, especially for i.i.d. Gaussian measurement matrices. In a seemingly different field, non-regular sampling techniques such as three-quarter sampling have shown promising results to enhance the resolution of an imaging sensor by effectively sub-sampling a higher resolution image. Here, the measurements can be described as linear combinations of only three pixels, which can also be seen as a (spectral) compressed sensing measurement. Since each measurement is spatially localized, the reconstruction can be performed in overlapping sliding windows. In this work, we show that compressed sensing reconstruction algorithms can greatly benefit from such an overlapping sliding window reconstruction. Compared to conventional block-wise compressed sensing with i.i.d. Gaussian measurement matrices, the reconstruction quality in terms of the PSNR increases up to +5dB using small, local i.d.d. Gaussian measurement blocks. Additionally, we propose a local joint sparse deconvolution and extrapolation (L-JSDE) to reconstruct images from arbitrary local measurements. For several applications with local measurements we show that L-JSDE increases the PSNR by +2.2dB relative to conventional block-wise i.i.d. Gaussian measurements reconstructed with the state-of-the-art reconstruction algorithm D-AMP using the same overall sampling density.