Thursday, March 03, 2011

Compressed Sensing Image Reconstruction via Recursive Spatially Adaptive Filtering

Alessandro Foi just sent me the following:

Dear Igor,

a demonstration software for Matlab is now online at


Thanks Alessandro. Our readers may recall that the algorithm is about Compressed Sensing Image Reconstruction via Recursive Spatially Adaptive Filtering introduced as such:

We introduce a new approach to image reconstruction from highly incomplete data. The available data are assumed to be a small collection of spectral coefficients of an arbitrary linear transform. This reconstruction problem is the subject of intensive study in the recent field of "compressed sensing" (also known as "compressive sampling"). Our approach is based on a quite specific recursive filtering procedure. At every iteration the algorithm is excited by injection of random noise in the unobserved portion of the spectrum and a spatially adaptive image denoising filter, working in the image domain, is exploited to attenuate the noise and reveal new features and details out of the incomplete and degraded observations. This recursive algorithm can be interpreted as a special type of the Robbins-Monro stochastic approximation procedure with regularization enabled by a spatially adaptive filter. Overall, we replace the conventional parametric modeling used in CS by a nonparametric one. We illustrate the effectiveness of the proposed approach for two important inverse problems from computerized tomography: Radon inversion from sparse projections and limited-angle tomography. In particular we show that the algorithm allows to achieve exact reconstruction of synthetic phantom data even from a very small number projections. The accuracy of our reconstruction is in line with the best results in the compressed sensing field.

From the description file:

This demo software reproduces the following four experiments from the paper
K. Egiazarian, A. Foi, and V. Katkovnik, "Compressed Sensing Image Reconstruction via Recursive Spatially Adaptive Filtering," Proc. IEEE ICIP 2007, San Antonio (TX), USA, pp. 549-552, Sept. 2007. DOI
Experiments 1-3
Tomographic reconstruction from sparse projections (approximating Radon projections as radial lines in FFT domain) We consider three illustrative inverse problems of compressed sensing for computerized tomography. In particular, we show reconstruction examples of the Shepp-Logan phantom from sparse Radon projections, with 22 and 11 radial lines in FFT-domain (i.e., available Radon projections), and reconstruction from limited-angle projections, with a reduced subset of 61 projections within a 90 degrees aperture. The available portions of the spectrum and the initial back-projection estimates are shown below.
experiment_case=1 - Sparse projections: 22 radial lines
experiment_case=2 - Sparse projections: 11 radial lines
experiment_case=3 - Limited-angle (90 degrees aperture, 61 radial lines)
Experiment 4
Reconstruction from low-frequency portion of Fourier spectrum. Reconstruction of the Cameraman image (256x256 pixels) from the low-frequency portion of its Fourier spectrum (a 128×128 square centered at the DC).
For all the above experiments, the block-matching and 3D filtering algorithm (BM3D) serves as the spatially adaptive filter used in the recursive stochastic approximation algorithm. The separable 3D Haar wavelet decomposition is adopted as the transform utilized internally by the BM3D algorithm. The general version of the BM3D algorithm is available at
K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian,
"Image denoising by sparse 3D transform-domain collaborative filtering,"
IEEE Trans. Image Process., vol. 16, no. 8, pp. 2080-2095, August 2007.
I'll be listing this algorithm shortly in the reconstruction section of the Big picture.

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