High-frequency subband compressed sensing MRI using quadruplet sampling

Abstract

Purpose: To present and validate a new method that formalizes a direct link between k-space and wavelet domains to apply separate undersampling and reconstruction for high- and low-spatial-frequency k-space data.

Theory and methods: High- and low-spatial-frequency regions are defined in k-space based on the separation of wavelet subbands, and the conventional compressed sensing problem is transformed into one of localized k-space estimation. To better exploit wavelet-domain sparsity, compressed sensing can be used for high-spatial-frequency regions, whereas parallel imaging can be used for low-spatial-frequency regions. Fourier undersampling is also customized to better accommodate each reconstruction method: random undersampling for compressed sensing and regular undersampling for parallel imaging.

Results: Examples using the proposed method demonstrate successful reconstruction of both low-spatial-frequency content and fine structures in high-resolution three-dimensional breast imaging with a net acceleration of 11-12.

Conclusion: The proposed method improves the reconstruction accuracy of high-spatial-frequency signal content and avoids incoherent artifacts in low-spatial-frequency regions. This new formulation also reduces the reconstruction time due to the smaller problem size.

Keywords: compressed sensing; image reconstruction; iterative reconstruction; parallel imaging; wavelet transformation.

Copyright © 2012 Wiley Periodicals, Inc.

Figures

Figure 1

1D discrete wavelet transform using an orthogonal basis: (a) the schematic diagram to realize the fast wavelet transform with a single scale, and (b) its behavior in frequency domain.

Figure 2

Illustration of the domains used in CS MRI and the property of wavelet transforms. (a) The relationship among three domains (x: image, y: k-space, and w: wavelet). High-frequency subbands and the corresponding k-space data are colored (LH: green, HL: red, and HH: blue). (b) High-frequency subband spectral weightings. The Daubechies-6 wavelet has been used here and an ideal set of the zero-transition-band filters is indicated by the dotted lines.

Figure 3

Illustration of the 2D Fourier-Wavelet transform, ФΨ−1, consisting of (a) 2D upsampling and convolution, and (b) the sum of the localized k-space data. The upsampling in 2D space is the same as the replication in frequency domain, and the 2D convolution is the same as the multiplication of the spectral weighting Dn in frequency domain. The sum of all the localized k-space data regions (yn) becomes a full k-space data set (y).

Figure 4

Illustration of (a) the wavelet tree structure and (b) the non-zero location estimation. Wavelet coefficients naturally form a tree structure flowing from the coarsest scale to the finest scale, described by the arrows, and are typically non-increasing along the branches of the tree. This results in two main observations: the high-frequency subbands are the most sparse, and possible non-zero locations of the high-frequency subbands are limited by the coarser scale wavelet subbands.

Figure 5

HiSub CS: quadruplet sampling and reconstruction. (a) The generation of k-space under-sampling consists of the duplication of Фs, the addition of regular undersampling, and the addition of a fully sampled region. The proposed undersampling pattern is customized to accommodate the separate reconstruction methods, CS and parallel imaging. (b) The serial reconstruction of ARC and HiSub CS shows how each reconstruction method recovers two different k-space regions (low- and high-spatial-frequency contents).

Figure 6

Comparison of ARC, Standard CS and ARC+HiSub CS: (a) illustration of each method with associated k-space undersampling patterns and (b) nRMSE (mean ± SD) plots with different reduction factors. For all reduction factors, ARC+HiSub CS shows the best performance based on this metric.

Figure 7

Reconstruction examples of the two methods (Standard CS and ARC+HiSub CS). The original image and ARC (2 × 2) are shown on the left as references. Different regions of reconstructed images are zoomed in to better depict high-frequency (comb shape) and low-frequency (slice thickness bar) contents. For R = 6 and R = 10, Standard CS contains residual artifacts, indicated by the arrows, while ARC+HiSub CS shows excellent reconstruction.

Figure 8

High-resolution 3D grid phantom images using the three reconstruction methods: ARC, ARC+HiSub CS, ARC+HiSub CS with cutting the k-space corners. (a) Reconstructed images, (b) zoomed-in versions of reconstructed images in a different slice location and (c) associated k-space undersampling schemes. Note that an example of high-resolution features, indicated by the arrows, is well maintained.

Figure 9

High-resolution 3D breast imaging with fat saturation using the different reconstruction methods: (a) fully sampled, (b) ARC (R = 5.8), (c) L1-SPIRiT (R = 10.7), and (d) ARC+HiSub CS (R = 10.7). The k-space undersampling patterns are placed on the left to describe different ky − kz sampling schemes for different reconstruction methods. Note that L1-SPIRiT and HiSub CS have the different sharpness of the breast images (see the arrows).

Figure 10

High-resolution 3D T1-weighted breast imaging without fat saturation using: (a) fully sampled, (b) ARC (R = 5.8), (c) L1-SPIRiT (R = 12.2), and (d) ARC+HiSub CS (R = 12.2). Images are shown in the coronal plane, and fine structures are magnified to show the differences (see the arrows).