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MOCCA: A Fast Algorithm for Parallel MRI Reconstruction Using Model Based Coil Calibration

Gerlind Plonka, Yannick Riebe

TL;DR

A new fast algorithm for simultaneous recovery of the coil sensitivities and of the magnetization image from incomplete Fourier measurements in parallel MRI using bivariate trigonometric polynomials of small degree and can outperform several other reconstruction methods.

Abstract

We propose a new fast algorithm for simultaneous recovery of the coil sensitivities and of the magnetization image from incomplete Fourier measurements in parallel MRI. Our approach is based on a parameter model for the coil sensitivities using bivariate trigonometric polynomials of small degree. The derived MOCCA algorithm has low computational complexity of $O(N_c N^2 \log N)$ for $N \times N$ images and $N_c$ coils and achieves very good performance for incomplete MRI data. We present a complete mathematical analysis of the proposed reconstruction method. Further, we show that MOCCA achieves similarly good reconstruction results as ESPIRiT with a considerably smaller numerical effort which is due to the employed parameter model. Our numerical examples show that MOCCA can outperform several other reconstruction methods.

MOCCA: A Fast Algorithm for Parallel MRI Reconstruction Using Model Based Coil Calibration

TL;DR

A new fast algorithm for simultaneous recovery of the coil sensitivities and of the magnetization image from incomplete Fourier measurements in parallel MRI using bivariate trigonometric polynomials of small degree and can outperform several other reconstruction methods.

Abstract

We propose a new fast algorithm for simultaneous recovery of the coil sensitivities and of the magnetization image from incomplete Fourier measurements in parallel MRI. Our approach is based on a parameter model for the coil sensitivities using bivariate trigonometric polynomials of small degree. The derived MOCCA algorithm has low computational complexity of for images and coils and achieves very good performance for incomplete MRI data. We present a complete mathematical analysis of the proposed reconstruction method. Further, we show that MOCCA achieves similarly good reconstruction results as ESPIRiT with a considerably smaller numerical effort which is due to the employed parameter model. Our numerical examples show that MOCCA can outperform several other reconstruction methods.
Paper Structure (20 sections, 7 theorems, 53 equations, 7 figures, 2 tables, 3 algorithms)

This paper contains 20 sections, 7 theorems, 53 equations, 7 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Modeling of Coil Sensitivities and the Discrete Problem in Parallel MRI
  3. Model for Coil Sensitivities
  4. Acquired $k$-space Measurements
  5. Problem Statement
  6. Generalized Model and Ambiguities
  7. Sum-of-Squares Condition
  8. MOCCA Algorithm for Reconstruction from Incomplete $k$-space Data
  9. MOCCA Algorithm
  10. Complexity of the MOCCA Algorithm
  11. Efficient Solution of the Least-Squares Problem
  12. Weighted Jacobi-Richardson Iteration
  13. Direct Fast Algorithm for Structured Incomplete Data
  14. Nonlinear Local Smoothing
  15. Theoretical Background of Algorithms \ref{['alg2']}-\ref{['alg4']}
  16. ...and 5 more sections

Key Result

Theorem 3.1

Assume that the vectorized coil sensitivities ${\mathbf s}^{(j)} \in {\mathbb C}^{N^2}$ satisfy $(sjk)$, i.e., ${\mathbf s}^{(j)}= {\mathbf W} {\mathbf c}^{(j)}$ for $j=0, \ldots , N_c-1$, with $\mathbf c^{(j)} \in {\mathbb C}^{L^2}$ as in $(sj)$. If the model $(discmodel)$ is satisfied then where ${\mathbf Y}_{N,L}^{(j)} := ({y}^{(j)} _{(\boldsymbol \nu-\mathbf r) \scriptsize{{\mathrm{mod}}}\, \

Figures (7)

  • Figure 1: Illustration of the grid $\Lambda_N$ (with $N=100$), and the subgrids $\Lambda_L \subset \Lambda_M \subset \Lambda_{M+L-1}$, the blue subgrid $\Lambda_L$ (with $L=5$), the green subgrid $\Lambda_M$ (with $M=9$), and the yellow subgrid $\Lambda_{M+L-1}$, which needs to be part of the ACS region.
  • Figure 2: Magnitudes of the coil images for the two used test data sets together with the full sos reconstruction at the end.
  • Figure 3: Reconstruction results for the first data set obtained from a third of the $k$-space data of 8 coils (every third column acquired) for different methods. From left to right: (1) MOCCA using Algorithm \ref{['alg2']} (taking Algorithm \ref{['alg3']} with 40 iterations), (2) MOCCA-S (with 40 iterations and smoothing), (3) L1-ESPIRiT, (4) ESPIRiT and (5) GRAPPA. Corresponding error maps are given below. All error images use the same scale with relative error in $[0,0.12]$, where $0$ corresponds to black and $0.12$ to white.
  • Figure 4: Magnitude (left) and phase (in $[-\pi,\pi]$) (right) of the $8$ coil sensitivities obtained for MRI reconstruction for $L=5$ for the first data set before and after multiplication with $\mathrm{sign}(\mathbf m)$, see step 7 of Algorithm \ref{['alg2']}. The normalized sensitivities are samples of bivariate trigonometric polynomials with $L^2=25$ nonzero coefficients, pointwisely multiplied with the normalization factors $({\mathbf d}^{+})^{\frac{1}{2}}$ to ensure the sos condition.
  • Figure 5: Illustration of the singular values of the matrix ${\mathbf A}_M$ for the first and the second data set. From left to right: (1) Singular values of ${\mathbf A}_M$ with $L=5$ and $25\cdot 8=200$ columns for the first data set. (2) Singular values of ${\mathbf A}_M$ with $L=5$ and $25\cdot 8=200$ columns, (3) with $L=7$ and $49\cdot8= 392$ columns, and (4) with $L=9$ and $81\cdot 8=648$ columns for the second data set.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Remark 2.1
  • Theorem 3.1
  • Remark 3.3
  • Remark 3.5
  • Remark 3.7
  • Theorem 4.1
  • Lemma 4.1
  • Theorem 4.2
  • Remark 4.3
  • Theorem 4.4
  • ...and 2 more