  In compressive sensing, it is challenging to reconstruct image of high
quality from very few noisy linear projections. Existing methods mostly work
well on piecewise constant images but not so well on piecewise smooth images
such as natural images, medical images that contain a lot of details. We
propose a two-stage method called GeoCS to recover images with rich geometric
information from very limited amount of noisy measurements. The method adopts
the shearlet transform that is mathematically proven to be optimal in sparsely
representing images containing anisotropic features such as edges, corners,
spikes etc. It also uses the weighted total variation (TV) sparsity with
spatially variant weights to preserve sharp edges but to reduce the staircase
effects of TV. Geometric information extracted from the results of stage I
serves as an initial prior for stage II which alternates image reconstruction
and geometric information update in a mutually beneficial way. GeoCS has been
tested on incomplete spectral Fourier samples. It is applicable to other types
of measurements as well. Experimental results on various complicated images
show that GeoCS is efficient and generates high-quality images.
