Compressed Sensing Image Reconstruction via Recursive Spatially Adaptive Filtering

nonparametric stochastic approximation approach


Introduction

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Demonstration software for Matlab
ver. 1.1, released July 23, 2015

Supports Windows, Linux, and Mac OS X (32-and 64-bit versions)

Results

People

Publications






Introduction


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.



Results


Simulation of Radon 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. As the recursive algorithm progresses, the reconstruction error improves steadily until numerical convergence; for all three cases the reconstruction is exact, with the estimate converging to the original unobserved phantom.

- Sparse projections: 22 radial lines

available portion of the spectrumBack-projection estimaterecursive algorithmFinal estimate
available portion of the spectrum
(22 radial lines)
Back-projection estimateEstimate after convergence
(exact reconstruction)

download animationExact reconstruction from 22 radial lines (ANIMATION)

- Sparse projections: 11 radial lines

available portion of the spectrumBack-projection estimaterecursive algorithmFinal estimate
available portion of the spectrum
(11 radial lines)
Back-projection estimateEstimate after convergence
(exact reconstruction)

download animationExact reconstruction from 11 radial lines (ANIMATION)

- Limited-angle

available portion of the spectrumBack-projection estimaterecursive algorithmFinal estimate
available portion of the spectrum
(90 degrees aperture, 61 radial lines)
Back-projection estimateEstimate after convergence
(exact reconstruction)

download animationExact reconstruction from 90 degrees limited-angle with 61 radial lines (ANIMATION)

Notes:
The animation files are rather big (about 6 Mbytes). They are in .GIF format.
Because of their length, it is recommended to download the files (right click, Save target as...) and open them with an external viewer (e.g., Irfanview).
In the animations, the three numbers on the top of the frames are the PSNR(dB), the iteration count, and the maximum absolute error (l-infinity norm).
The animations show the image estimate and its difference from the original (unknown) image. This difference is normalized with respect to the maximum absolute error.


Reconstruction from low-frequency portion of Fourier spectrum

We present the result of reconstruction of the Cameraman image (256x256 pixels) from the low-frequency portion of its Fourier spectrum (a 128128 square centered at the DC).

available portion of the spectrumBack-projection estimaterecursive algorithmEstimate after 62 iterations
available portion of the spectrum
(low frequencies)
Back-projection estimate
(zero-padding the high frequencies)
Estimate after 62 iterations


For all the above experiments, the block-matching and 3D filtering algorithm (BM3D) served 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.


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Demonstration software
for Matlab (ver. 7.4 or later)

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233-kbyte zip-file
includes scripts reproducing the above experiments

v1.1, released July 23, 2015



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People


Aram Danielyan
Karen Egiazarian
Alessandro Foi
Vladimir Katkovnik



Publications

2010

PDFDanielyan, A., A. Foi, V. Katkovnik, and K. Egiazarian, “Spatially adaptive filtering as regularization in inverse imaging: compressive sensing, upsampling, and super-resolution”, in Super-Resolution Imaging (P. Milanfar, ed.), CRC Press / Taylor & Francis, ISBN: 978-1-4398-1930-2, September 2010     to publisherto CRC Press   to amazon.comto amazon.com   examplesExamples of super-resolution reconstruction as zipped Matlab MAT-files.

2008

PDFDanielyan, A., A. Foi, V. Katkovnik, and K. Egiazarian, “Image and video super-resolution via spatially adaptive block-matching filtering”, Proc. Int. Workshop on Local and Non-Local Approx. in Image Process., LNLA 2008, Lausanne, Switzerland, August 2008.
PDFDanielyan, A., A. Foi, V. Katkovnik, and K. Egiazarian, “Image Upsampling Via Spatially Adaptive Block-Matching Filtering”, Proc. 16th European Signal Process. Conf., EUSIPCO 2008, Lausanne, Switzerland, August 2008.

2007

PDF Egiazarian, K., A. Foi, and V. Katkovnik, “Compressed Sensing Image Reconstruction via Recursive Spatially Adaptive Filtering”, Proc. IEEE Int. Conf. Image Process., ICIP 2007, San Antonio (TX), USA, pp. 549-552, September 2007.


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