# Sparse image reconstruction for molecular imaging

Michael Ting, Raviv RaichAlfred O. Hero III

## Abstract

The application that motivates this paper is molecular imaging at the atomic level. When discretized at sub-atomic distances, the volume is inherently sparse. Noiseless measurements from an imaging technology can be modeled by convolution of the image with the system point spread function (psf). Such is the case with magnetic resonance force microscopy (MRFM), an emerging technology where imaging of an individual tobacco mosaic virus was recently demonstrated with nanometer resolution. We also consider additive white Gaussian noise (AWGN) in the measurements. Many prior works of sparse estimators have focused on the case when H has low coherence; however, the system matrix H in our application is the convolution matrix for the system psf. A typical convolution matrix has high coherence. The paper therefore does not assume a low coherence H. A discrete-continuous form of the Laplacian and atom at zero (LAZE) p.d.f. used by Johnstone and Silverman is formulated, and two sparse estimators derived by maximizing the joint p.d.f. of the observation and image conditioned on the hyperparameters. A thresholding rule that generalizes the hard and soft thresholding rule appears in the course of the derivation. This so-called hybrid thresholding rule, when used in the iterative thresholding framework, gives rise to the hybrid estimator, a generalization of the lasso. Estimates of the hyperparameters for the lasso and hybrid estimator are obtained via Stein’s unbiased risk estimate (SURE). A numerical study with a Gaussian psf and two sparse images shows that the hybrid estimator outperforms the lasso.

## Paper

M. Ting, R. Raich and A.O. Hero, “Sparse image reconstruction for molecular imaging,” IEEE Trans. on Image Processing, vol. 18, no. 6, pp. 1215-1227, June 2009. (.pdf)

## Matlab Code

Associated sparse image toolbox (written by M. Ting in 2014. The source code was used to generate the results of the work Molecular Imaging in Nano MRI by Michael Ting, Wiley 2014. The latest version is 0.0.1.

https://github.com/mt94/Sparse-Image-Reconstruction

## Tested Configurations

The Matlab code has been tested on Matlab 7.1 (R14) on Linux. For instructions on how to generate the results of the paper, refer to the following file. (.pdf)

## Figures

• Figure 1. Hybrid thresholding rule.
• Figure 2. Illustration of the two types of θ used in the simulations; as well, the Gaussian blur psf is shown.
• Figure 3. Reconstructed images for the binary-valued θ under an SNR of 1.76 dB for SBL, StOMP (CFAR), MAP2 (g*=2-1/2), and lasso-SURE.
• Figure 4. Performance vs. SNR for Landweber iterations, MAP1, MAP2, LASSO-SURE, and H-SURE when applied to the binary-valued θ.
• Figure 5. Image θ and noiseless projection Hθ used in the MRFM reconstruction example.
• Figure 6. Landweber reconstruction of the MRFM example under a SNR of 4.77 dB.
• Figure 7. LASSO-SURE reconstruction of the MRFM example under a SNR of 4.77 dB.
• Figure 8. Histogram of <latex>\hat{\theta}_i</latex> for the Landweber and LASSO-SURE estimator.

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