Location constrained approximate message passing for compressed sensing MRI

Abstract

Iterative thresholding methods have been extensively studied as faster alternatives to convex optimization methods for solving large-sized problems in compressed sensing. A novel iterative thresholding method called LCAMP (Location Constrained Approximate Message Passing) is presented for reducing computational complexity and improving reconstruction accuracy when a nonzero location (or sparse support) constraint can be obtained from view shared images. LCAMP modifies the existing approximate message passing algorithm by replacing the thresholding stage with a location constraint, which avoids adjusting regularization parameters or thresholding levels. This work is first compared with other conventional reconstruction methods using random one-dimention signals and then applied to dynamic contrast-enhanced breast magnetic resonance imaging to demonstrate the excellent reconstruction accuracy (less than 2% absolute difference) and low computation time (5-10 s using Matlab) with highly undersampled three-dimentional data (244 × 128 × 48; overall reduction factor = 10).

Keywords: compressed sensing; image reconstruction; iterative reconstruction; quantitative DCE MRI; wavelet transformation.

© 2012 Wiley Periodicals, Inc.

Figures

Figure 1

Illustration of three iterative thresholding-based methods (IST/IHT, AMP and LCAMP) for one iteration. Note that all three methods share the same matrix-vector operations (Φψ* and ψΦ*), which are the main computational load, and the differences between each methods are highlighted in the gray box.

Figure 2

Numerical simulation of dynamic signal behaviors in DCE MRI. (a) The image domain contains different signal dynamics in different regions where each signal dynamic follows an actual DCE signal. Temporal behaviors for two selected regions (1: blue and 2: red) are plotted in all time frames. (b) The corresponding wavelet domain also contains different amplitudes of wavelet coefficients at different time frames, but (c) the location of the non-zero wavelet coefficients (white for non-zeros and black for zeros) does not change for all time frames.

Figure 3

Illustration of k-space undersampling patterns (ky - kz - t) and non-zero location masks. (a) The undersampling patterns consist of three regions with different reduction factors (R = 3, 6 and 12), and each region becomes fully sampled when averaged over R frames. The non-zero location masks estimated by (b) original images and (c) composite images. The composite images are formed by temporal sharing undersampled data to generate a full k-space data set.

Figure 4

Reconstruction examples of (a) L1 LSP, (b) AMP and (c) LCAMP with added noise (σ = 0.01 and maximum SNR = 100). Both L1 LSP and AMP have similar nRMSE (nRMSE = 0.0171 and 0.0175) while LCAMP shows the lowest nRMSE (0.0113). One of the reconstruction failures is zoomed-in in the image and wavelet domains. Note that the location constraint is marked as gray regions in (c).

Figure 5

Reconstruction comparison of L1 LSP, AMP and LCAMP using 1D random signals with different noise standard deviations of (a) σ = 0.005, (b) σ = 0.001 and (c) σ = 0.05. Mean and standard deviation of nRMSE and average computation time were computed over 300 repetitions for each reduction factor.

Figure 6

Reconstruction comparison of AMP and LCAMP with different overall reduction factors: (a) Original, (b) Rnet = 2.5, (c) Rnet = 4.4 and (d) Rnet = 7. The absolute difference shows the reconstruction performance of AMP and LCAMP as the overall reduction factors increase. Two regions of interest (ROI A and B) are shown in the original image.

Figure 7

DCE breast images at selected slice locations: (a) original, (b) LCAMP and (c) the absolute difference between original and LCAMP. The absolute difference is scaled to be 10% of the maximum signal and was typically less than 2% of the maximum.

Figure 8

DCE breast images at selected time frames: (a) original, (b) LCAMP and (c) the absolute difference between original and LCAMP. The absolute difference is scaled to be 10% of the maximum signal, and the absolute difference was typically less than 2% of the maximum.

Figure 9

Comparison of temporal behavior in x-f space. (a) The original signal in the x-f domain is generated by applying the temporal Fourier transform to DCE breast images and by selecting the x-f plane that includes a tumor region (see the arrow). The x-f signals reconstructed by (b) view sharing, (c) modified CS (MCS), and (d) LCAMP are shown with the absolute difference between original and each method. Note that different types of temporal blurring in the tumor region can be observed in view sharing and MCS, indicated by the arrows, and LCAMP dramatically reduces this temporal blurring.

Figure 10

The frame-by-frame plots of nRMSE for view sharing, MCS, and LCAMP with all DCE breast cases. (a - c) The nRMSE plots computed by using all slices, and (d - f) the nRMSE plots computed by using the slices containing a tumor region.

Figure 11

Maps of initial slope of contrast enhancement generated using (a) original, (b) view sharing, (c) MCS and (d) LCAMP. Magnitude images are shown for the anatomical reference, and a region containing a tumor, indicated by the arrows, is zoomed-in to show the difference in the initial slopes. Different colors in the initial slope map indicate various degrees of line slope, computed by a linear regression of initial enhancement signals in arbitrary units and typically expresses the steepness of initial signal uptake.