# Calibration-Less Multi-Coil Compressed Sensing Magnetic Resonance Image Reconstruction Based on OSCAR Regularization

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works

#### 1.2. Our Contributions

#### 1.3. Outline of the Paper

## 2. Problem Statement

#### 2.1. Notation and Definitions

**A**). The transpose of a matrix

**A**is denoted by ${\mathit{A}}^{\top}$, its Hermitian transpose by ${\mathit{A}}^{*}$, its spectral norm by $\Vert \phantom{\rule{-1.0625pt}{0ex}}\left|\mathit{A}\right|\phantom{\rule{-1.0625pt}{0ex}}\Vert $, and its Frobenius norm by ${\Vert \mathit{A}\Vert}_{2}$.

#### 2.2. General Problem Formulation

**F**the 2D fast Fourier transform (FFT) and

**M**the binary under-sampling mask defined over the discrete grid where each non-zero entry in

**M**selects a row in

**F**. In non-Cartesian settings,

**F**refers to non-equispaced or nonuniform FFT [35,36] and

**M**stands for the discrete support of the k-space location measurements. Each coil measurement ${\mathit{y}}_{\ell}$, with $\ell \in \{1,\dots ,L\}$, is furthermore affected by an additive circular complex i.i.d. zero-mean Gaussian noise of variance ${\sigma}_{\ell}^{2}$, which can be characterized by a separate scan (without RF pulse) considering the same bandwidth ${\mathrm{BW}}_{\mathrm{read}}$ as the prospectively accelerated acquisition. For the sake of simplicity, we do not model here any potential between-coil covariance structure

**∑**, which is thus assumed diagonal, $\mathbf{\sum}={\mathit{I}}_{L}$.

**Ψ**decomposes the stack of L images $\mathit{X}\in {\mathbb{C}}^{N\times L}$ into a stack of coefficients $\mathbf{\Psi}\mathit{X}\in {\mathbb{C}}^{{N}_{\Psi}\times L}$ with C scales. Each scale $c\in \{1,\cdots ,C\}$ is composed of ${S}_{c}$ sub-bands. Each sub-band $s\in \{1,\cdots ,{S}_{c}\}$ has ${K}_{s\left(c\right)}$ coefficients, so that finally ${N}_{\Psi}={\sum}_{c=1}^{C}{\sum}_{s=1}^{{S}_{c}}{K}_{s\left(c\right)}$. For the sake of simplicity, in what follows we assume that ${S}_{c}=S,\forall c$ and ${K}_{s\left(c\right)}={K}_{c},\forall s$. As an example, for $n\times n$ images using decimated wavelet transform, we would have ${K}_{s\left(c\right)}=n/{2}^{c}\times n/{2}^{c}$ and ${S}_{c}=3$ for all scales except for the last one where ${S}_{c}=4$. Moreover, the kth-coefficient in the sth-sub-band of the cth-scale for the ℓth-coil will be denoted as ${z}_{csk\ell}$. Vector ${\mathit{z}}_{csk,:}\in {\mathbb{C}}^{L}$ gathers the multi-channel coefficients ${\left({z}_{csk\ell}\right)}_{1\le \ell \le L}$ at position k, sub-band s and scale c. Similarly, the larger vector ${\mathit{z}}_{cs,:}$ stacks the multi-position and multi-coil coefficients ${\left({z}_{csk\ell}\right)}_{1\le k\le {K}_{c},1\le \ell \le L}$ at a given sub-band s of scale c. Last, vector ${}_{c,:}$ stacks the multi-band multi-position and multi-coil coefficients ${\left({z}_{csk\ell}\right)}_{1\le s\le S,1\le k\le {K}_{c},1\le \ell \le L}$ at a given scale c.

**x**as ${\mathit{x}}_{\ell}={\mathit{S}}_{\ell}\mathit{x}$.

#### 2.3. Primal-Dual Optimization Algorithm

Algorithm 1: Condat-Vú algorithm |

**Ψ**is orthogonal, we get $\Vert \phantom{\rule{-1.0625pt}{0ex}}|\mathbf{\Psi}|\phantom{\rule{-1.0625pt}{0ex}}\Vert =1$. The main advantage of Algorithm 1 is that it does not involve the computation of ${\mathrm{prox}}_{g\circ \mathbf{\Psi}}$. The latter does not usually have closed form, in particular when

**Ψ**is overcomplete (e.g., undecimated wavelet transform), and would require the use of an inner iterative solver [44].

## 3. Octagonal Shrinkage and Clustering Algorithm for Regression

#### 3.1. OSCAR Regularizer

#### 3.1.1. Definition

**z**while the second term, corresponding to a pairwise ℓ

_{∞}-norm, encourages the equality of each entry pair in

**z**.

**z**sorted in decreasing order in magnitude, that is, such that

_{1}(OWL) norm defined below:

#### 3.1.2. Proximity Operator

**z**is equal to zero, then the proximity operator of the OWL norm at

**z**is also equal to zero. Otherwise, it can be efficiently computed thanks to the following Algorithm 2 as shown in ([45], SectionIII A):

Algorithm 2: Proximity operator of the OWL norm. |

_{1} Input: $\mathit{z}\in {\mathbb{C}}^{p}/\left\{0\right\}$, $\mathit{w}\in {\mathbb{R}}^{p}$; |

_{2} $\mathit{n}=\left|\mathit{z}\right|/\mathit{z}$; |

_{3} Let $\mathit{P}\in {\mathbb{R}}^{p\times p}$ s.t. ${S}_{p}\left(\mathit{n}\right)=\mathit{P}\mathit{n}$; |

_{4} Return ${\mathrm{prox}}_{{\Theta}_{\mathit{w}}}\left(\mathit{z}\right)=\mathit{n}\odot {\mathit{P}}^{\top}\mathrm{PAV}({S}_{p}\left(\mathit{n}\right)-\mathit{w})$; |

#### 3.2. OSCAR-Based Image Reconstruction

#### 3.2.1. Global OSCAR Regularization

**Z**. For that reason, we call this version global OSCAR (g-OSCAR) regularization. The wavelet coefficients are stacked together, leading to a single but large vector with entries ${\left({z}_{j}\right)}_{1\le j\le L{N}_{\Psi}}$, where we remind that ${N}_{\Psi}=S{\sum}_{c=1}^{C}{K}_{c}$ and ${N}_{\Psi}=N$ when

**Ψ**is orthogonal. The g-OSCAR regularizer then reads:

#### 3.2.2. Scalewise OSCAR Regularization

#### 3.2.3. Subbandwise OSCAR Regularization

#### 3.2.4. Coefficientwise OSCAR Regularization

## 4. Materials and Methods

#### 4.1. Reconstruction Parameters and Computational Complexity

**Ψ**a Daubechies 4 orthogonal wavelet transform (OWT) with $C=4$ decomposition scales (i.e., ${N}_{\Psi}=N$). Note that MR image quality can be improved using redundant multiscale transforms (e.g., undecimated bi-orthogonal wavelet transforms or curvelets as shown in [50]) but this kind of decomposition significantly increases the memory load and computation time of the overall algorithm. Moreover, it does not change the actual comparisons of the four versions of OSCAR-norm regularization.

`joblib`, a Python package that allows embarrassingly parallel computations (https://pypi.org/project/joblib/ (accessed on 10 February 2021)). The number of parallel threads that were used is indicated in Table 1. All experiments were run on a machine with 128 GB of RAM and an 8-core (2.40 GHz) Intel Xeon E5-2630 v3 Processor.

#### 4.2. Retrospective Study

#### 4.3. Prospective Study

#### 4.4. Hyper-Parameters Search and Sensitivity

#### 4.5. Phase Processing

`scikit-image`(https://scikit-image.org/docs/dev/auto_examples/color_exposure/plot_equalize.html (accessed on 10 February 2021)). The result is depicted in Figure 2d.

## 5. Results

#### 5.1. Retrospective Studies

#### 5.2. Prospective Studies

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Donoho, D. Compressed sensing. IEEE Trans. Inf. Theory
**2006**, 52, 1289–1306. [Google Scholar] [CrossRef] - Candès, E.; Romberg, J.; Tao, T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory
**2006**, 52, 489–509. [Google Scholar] [CrossRef] [Green Version] - Lustig, M.; Donoho, D.; Pauly, J. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn. Reson. Med.
**2007**, 58, 1182–1195. [Google Scholar] [CrossRef] - Lustig, M.; Donoho, D.L.; Santos, J.M.; Pauly, J.M. Compressed Sensing MRI. IEEE Signal Process. Mag.
**2008**, 25, 72–82. [Google Scholar] [CrossRef] - FDA Clears Compressed Sensing MRI Acceleration Technology from Siemens Healthineers. Available online: https://www.siemens-healthineers.com/en-us/news/fda-clears-mri-technology-02-21-2017.html (accessed on 15 April 2019).
- Pipe, J.G. Motion correction with PROPELLER MRI: Application to head motion and free-breathing cardiac imaging. Magn. Reson. Med.
**1999**, 42, 963–969. [Google Scholar] [CrossRef] - Lee, J.H.; Hargreaves, B.A.; Hu, B.S.; Nishimura, D.G. Fast 3D imaging using variable-density spiral trajectories with applications to limb perfusion. Magn. Reson. Med.
**2003**, 50, 1276–1285. [Google Scholar] [CrossRef] [PubMed] - Feng, L.; Axel, L.; Chandarana, H.; Block, K.T.; Sodickson, D.K.; Otazo, R. XD-GRASP: Golden-angle radial MRI with reconstruction of extra motion-state dimensions using compressed sensing. Magn. Reson. Med.
**2016**, 75, 775–788. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Boyer, C.; Chauffert, N.; Ciuciu, P.; Kahn, J.; Weiss, P. On the generation of sampling schemes for magnetic resonance imaging. SIAM J. Imaging Sci.
**2016**, 9, 2039–2072. [Google Scholar] [CrossRef] [Green Version] - Kasper, L.; Engel, M.; Barmet, C.; Haeberlin, M.; Wilm, B.; Dietrich, B.; Schmid, T.; Gross, S.; Brunner, D.; Stephan, K.; et al. Rapid anatomical brain imaging using spiral acquisition and an expanded signal model. Neuroimage
**2018**, 168, 88–100. [Google Scholar] [CrossRef] [Green Version] - Lazarus, C.; Weiss, P.; Chauffert, N.; Mauconduit, F.; Gueddari, L.E.; Destrieux, C.; Zemmoura, I.; Vignaud, A.; Ciuciu, P. SPARKLING: Variable-density k-space filling curves for accelerated T2*-weighted MRI. Magn. Reson. Med.
**2019**, 81, 3643–3661. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chaithya, G.R.; Weiss, P.; Massire, A.; Vignaud, A.; Ciuciu, P. Globally optimized 3D SPARKLING trajectories for high-resolution T2*-weighted Magnetic Resonance imaging. IEEE Trans. Med. Imaging
**2020**. under review. [Google Scholar] - Roemer, P.; Edelstein, W.; Hayes, C.; Souza, S.; Mueller, O. The NMR phased array. Magn. Reson. Med.
**1990**, 16, 192–225. [Google Scholar] [CrossRef] [PubMed] - Shin, P.J.; Larson, P.E.; Ohliger, M.A.; Elad, M.; Pauly, J.M.; Vigneron, D.B.; Lustig, M. Calibrationless parallel imaging reconstruction based on structured low-rank matrix completion. Magn. Reson. Med.
**2014**, 72, 959–970. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Haldar, J.P.; Zhuo, J. P-LORAKS: Low-rank modeling of local k-space neighborhoods with parallel imaging data. Magn. Reson. Med.
**2016**, 75, 1499–1514. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Guerquin-Kern, M.; Haberlin, M.; Pruessmann, K.P.; Unser, M. A fast wavelet-based reconstruction method for magnetic resonance imaging. IEEE Trans. Med. Imaging
**2011**, 30, 1649–1660. [Google Scholar] [CrossRef] [Green Version] - Chaâri, L.; Pesquet, J.C.; Benazza-Benyahia, A.; Ciuciu, P. A wavelet-based regularized reconstruction algorithm for SENSE parallel MRI with applications to neuroimaging. Med. Image Anal.
**2011**, 15, 185–201. [Google Scholar] [CrossRef] - Chauffert, N.; Ciuciu, P.; Weiss, P. Variable density compressed sensing in MRI. Theoretical vs heuristic sampling strategies. In Proceedings of the 10th IEEE International Symposium on Biomedical Imaging (ISBI 2013), San Francisco, CA, USA, 7–11 April 2013; pp. 298–301. [Google Scholar]
- Chauffert, N.; Ciuciu, P.; Kahn, J.; Weiss, P. Variable density sampling with continuous trajectories. Application to MRI. SIAM J. Imaging Sci.
**2014**, 7, 1962–1992. [Google Scholar] [CrossRef] [Green Version] - McKenzie, C.A.; Yeh, E.N.; Ohliger, M.A.; Price, M.D.; Sodickson, D.K. Self-calibrating parallel imaging with automatic coil sensitivity extraction. Magn. Reson. Med.
**2002**, 47, 529–538. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Uecker, M.; Lai, P.; Murphy, M.; Virtue, P.; Elad, M.; Pauly, J.; Vasanawala, S.; Lustig, M. ESPIRiT—An eigenvalue approach to autocalibrating parallel MRI: Where SENSE meets GRAPPA. Magn. Reson. Med.
**2014**, 71, 990–1001. [Google Scholar] [CrossRef] [Green Version] - Gueddari, L.; Lazarus, C.; Carrié, H.; Vignaud, A.; Ciuciu, P. Self-calibrating nonlinear reconstruction algorithms for variable density sampling and parallel reception MRI. In Proceedings of the IEEE 10th Sensor Array and Multichannel Signal Processing Workshop (SAM 2018), Sheffield, UK, 8–11 July 2018; pp. 415–419. [Google Scholar]
- Ying, L.; Sheng, J. Joint image reconstruction and sensitivity estimation in SENSE (JSENSE). Magn. Reson. Med.
**2007**, 57, 1196–1202. [Google Scholar] [CrossRef] - Uecker, M.; Hohage, T.; Block, K.; Frahm, J. Image reconstruction by regularized nonlinear inversion—Joint estimation of coil sensitivities and image content. Magn. Reson. Med.
**2008**, 60, 674–682. [Google Scholar] [CrossRef] [Green Version] - Majumdar, A.; Ward, R.K. Iterative estimation of MRI sensitivity maps and image based on sense reconstruction method (isense). Concepts Magn. Reson. Part A
**2012**, 40, 269–280. [Google Scholar] [CrossRef] - Dwork, N.; Johnson, E.M.; O’Connor, D.; Gordon, J.W.; Kerr, A.B.; Baron, C.A.; Pauly, J.M.; Larson, P.E. Calibrationless Multi-coil Magnetic Resonance Imaging with Compressed Sensing. arXiv
**2020**, arXiv:2007.00165. [Google Scholar] - Majumdar, A.; Ward, R.K. Calibration-less multi-coil MR image reconstruction. Magn. Reson. Imaging
**2012**, 30, 1032–1045. [Google Scholar] [CrossRef] - Chun, I.; Adcock, B.; Talavage, T. Efficient compressed sensing SENSE pMRI reconstruction with joint sparsity promotion. IEEE Trans. Med Imaging
**2016**, 35, 354–368. [Google Scholar] [CrossRef] - Trzasko, J.; Manduca, A. Calibrationless parallel MRI using CLEAR. In Proceedings of the 45th Asilomar Conference on Signals, Systems and Computers (ASILOMAR 2011), Pacific Grove, CA, USA, 6–9 November 2011; IEEE: Piscataway, NJ, USA, 2011; pp. 75–79. [Google Scholar]
- Bondell, H.; Reich, B. Simultaneous regression shrinkage, variable selection, and supervised clustering of predictors with OSCAR. Biometrics
**2008**, 64, 115–123. [Google Scholar] [CrossRef] - Bogdan, M.; Van Den Berg, E.; Sabatti, C.; Su, W.; Candès, E.J. SLOPE-adaptive variable selection via convex optimization. Ann. Appl. Stat.
**2015**, 9, 1103. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bauschke, H.H.; Combettes, P.L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd ed.; Springer International Publishing: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Moreau, J.J. Proximité et dualité dans un espace hilbertien. Bull. De La Société Mathématique De Fr.
**1965**, 93, 273–299. [Google Scholar] [CrossRef] - Man, L.C.; Pauly, J.M.; Macovski, A. Multifrequency interpolation for fast off-resonance correction. Magn. Reson. Med.
**1997**, 37, 785–792. [Google Scholar] [CrossRef] - Keiner, J.; Kunis, S.; Potts, D. Using NFFT 3—a software library for various nonequispaced fast Fourier transforms. ACM Trans. Math. Softw. (TOMS)
**2009**, 36, 19. [Google Scholar] [CrossRef] - Fessler, J.; Sutton, B. Nonuniform fast Fourier transforms using min-max interpolation. IEEE Trans. Signal Process.
**2003**, 51, 560–574. [Google Scholar] [CrossRef] [Green Version] - Elad, M.; Milanfar, P.; Rubinstein, R. Analysis versus synthesis in signal priors. Inverse Probl.
**2007**, 23, 947. [Google Scholar] [CrossRef] [Green Version] - Florescu, A.; Chouzenoux, E.; Pesquet, J.C.; Ciuciu, P.; Ciochina, S. A majorize-minimize memory gradient method for complex-valued inverse problems. Signal Process.
**2014**, 103, 285–295. [Google Scholar] [CrossRef] [Green Version] - Parker, D.L.; Payne, A.; Todd, N.; Hadley, J.R. Phase reconstruction from multiple coil data using a virtual reference coil. Magn. Reson. Med.
**2014**, 72, 563–569. [Google Scholar] [CrossRef] [Green Version] - Komodakis, N.; Pesquet, J. Playing with Duality: An overview of recent primal-dual approaches for solving large-scale optimization problems. IEEE Signal Process. Mag.
**2015**, 32, 31–54. [Google Scholar] [CrossRef] [Green Version] - Combettes, P.; Pesquet, J. Fixed Point Strategies in Data Science. arXiv
**2021**, arXiv:2008.02260. [Google Scholar] - Condat, L. A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms. J. Optim. Theory Appl.
**2013**, 158, 460–479. [Google Scholar] [CrossRef] [Green Version] - Vũ, B. A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math.
**2013**, 38, 667–681. [Google Scholar] [CrossRef] - Combettes, P.L.; Pesquet, J.C. Proximal splitting methods in signal processing. In Fixed-Point Algorithms for Inverse Problems in Science and Engineering; Springer: Berlin/Heidelberg, Germany, 2011; pp. 185–212. [Google Scholar]
- Zeng, X.; Figueiredo, M.A. The Ordered Weighted l1 Norm: Atomic Formulation, Projections, and Algorithms. arXiv
**2014**, arXiv:1409.4271. [Google Scholar] - Mair, P.; Hornik, K.; de Leeuw, J. Isotone optimization in R: Pool-adjacent-violators algorithm (PAVA) and active set methods. J. Stat. Softw.
**2009**, 32, 1–24. [Google Scholar] - Zou, H.; Hastie, T. Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B (Stat. Methodol.)
**2005**, 67, 301–320. [Google Scholar] [CrossRef] [Green Version] - Argyriou, A.; Foygel, R.; Srebro, N. Sparse Prediction with the k-Support Norm. arXiv
**2012**, arXiv:1204.5043. [Google Scholar] - Haldar, J.P. Autocalibrated LORAKS for fast constrained MRI reconstruction. In Proceedings of the IEEE 12th International Symposium on Biomedical Imaging (ISBI 2015), Brooklyn, NY, USA, 16–19 April 2015; IEEE: Piscataway, NJ, USA, 2015; pp. 910–913. [Google Scholar]
- Cherkaoui, H.; Gueddari, L.; Lazarus, C.; Grigis, A.; Poupon, F.; Vignaud, A.; Farrens, S.; Starck, J.; Ciuciu, P. Analysis vs Synthesis-based Regularization for combined Compressed Sensing and Parallel MRI Reconstruction at 7 Tesla. In Proceedings of the 26th European Signal Processing Conference (EUSIPCO 2018), Rome, Italy, 3–7 September 2018. [Google Scholar]
- Gueddari, L.E.; Chaithya, G.R.; Ramzi, Z.; Farrens, S.; Starck, S.; Grigis, A.; Starck, J.L.; Ciuciu, P. PySAP-MRI: A Python Package for MR Image Reconstruction. In Proceedings of the ISMRM Workshop on Data Sampling and Image Reconstruction, Sedona, AZ, USA, 26–29 January 2020. [Google Scholar]
- Farrens, S.; Grigis, A.; El Gueddari, L.; Ramzi, Z.; Chaithya, G.R.; Starck, S.; Sarthou, B.; Cherkaoui, H.; Ciuciu, P.; Starck, J.L. PySAP: Python Sparse Data Analysis Package for multidisciplinary image processing. Astron. Comput.
**2020**, 32, 100402. [Google Scholar] [CrossRef] - Knoll, F.; Schwarzl, A.; Diwoky, C.; Sodickson, D. gpuNUFFT—An Open Source GPU Library for 3D Regridding with Direct Matlab Interface. In Proceedings of the 22nd Annual Meeting of ISMRM, Milan, Italy, 10–16 May 2014. [Google Scholar]
- Zbontar, J.; Knoll, F.; Sriram, A.; Muckley, M.J.; Bruno, M.; Defazio, A.; Parente, M.; Geras, K.J.; Katsnelson, J.; Chandarana, H.; et al. fastMRI: An Open Dataset and Benchmarks for Accelerated MRI. arXiv
**2018**, arXiv:1811.08839. [Google Scholar] - Knoll, F.; Zbontar, J.; Sriram, A.; Muckley, M.J.; Bruno, M.; Defazio, A.; Parente, M.; Geras, K.J.; Katsnelson, J.; Chandarana, H.; et al. fastMRI: A publicly available raw k-space and DICOM dataset of knee images for accelerated MR image reconstruction using machine learning. Radiol. Artif. Intell.
**2020**, 2, e190007. [Google Scholar] [CrossRef] [PubMed] - Wang, Z.; Bovik, A.; Sheikh, H.; Simoncelli, E. Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Process.
**2004**, 13, 600–612. [Google Scholar] [CrossRef] [Green Version] - Haacke, E.M.; Xu, Y.; Cheng, Y.C.N.; Reichenbach, J.R. Susceptibility weighted imaging (SWI). Magn. Reson. Med.
**2004**, 52, 612–618. [Google Scholar] [CrossRef] [PubMed] - Ramani, S.; Fessler, J.A. Parallel MR image reconstruction using augmented Lagrangian methods. IEEE Trans. Med Imaging
**2010**, 30, 694–706. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhu, B.; Liu, J.Z.; Cauley, S.F.; Rosen, B.R.; Rosen, M.S. Image reconstruction by domain-transform manifold learning. Nature
**2018**, 555, 487. [Google Scholar] [CrossRef] [Green Version] - Mardani, M.; Gong, E.; Cheng, J.Y.; Vasanawala, S.S.; Zaharchuk, G.; Xing, L.; Pauly, J.M. Deep generative adversarial neural networks for compressive sensing MRI. IEEE Trans. Med. Imaging
**2018**, 38, 167–179. [Google Scholar] [CrossRef] - Ramzi, Z.; Ciuciu, P.; Starck, J.L. Benchmarking MRI reconstruction neural networks on large public datasets. Appl. Sci.
**2020**, 10, 1816. [Google Scholar] [CrossRef] [Green Version] - Antun, V.; Renna, F.; Poon, C.; Adcock, B.; Hansen, A.C. On instabilities of deep learning in image reconstruction-Does AI come at a cost? arXiv
**2019**, arXiv:1902.05300. [Google Scholar] - Ramani, S.; Liu, Z.; Rosen, J.; Nielsen, J.F.; Fessler, J.A. Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods. IEEE Trans. Image Process.
**2012**, 21, 3659–3672. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ma, J. Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets. Appl. Comput. Harmon. Anal.
**2017**, 42, 294–318. [Google Scholar] [CrossRef] [Green Version] - El Gueddari, L.; Chouzenoux, E.; Vignaud, A.; Pesquet, J.C.; Ciuciu, P. Online MR image reconstruction for compressed sensing acquisition in T2* imaging. In Proceedings of the Wavelets and Sparsity XVIII. International Society for Optics and Photonics, San Diego, CA, USA, 13–15 August 2019; Volume 11138, p. 1113819. [Google Scholar]

**Figure 1.**Map of (

**a**) structural similarity (SSIM) (

**b**) peak signal-to-noise ratio (pSNR) score as a function of hyperparameters $(\lambda ,\gamma )$ involved in OSCAR-band (b-OSCAR) regularization using 20-fold prospectively accelerated Sparkling sampling scheme.

**Figure 2.**Phase processing for better visualization of the brain structure and hence easier comparisons. (

**a**) Raw wrapped phase image. (

**b**) Unwrapped phase image. (

**c**) High pass filtered phase image, notice that the structures in the brain are more visible here. (

**d**) Contrast stretching of 2nd and 98th percentiles of intensity values that permits contrast enhancement and improved visualization.

**Figure 3.**Retrospective results for Cartesian mask (

**A**) and non-Cartesian undersampling pattern (

**B**) on T1 weighted (

**i**) and FLAIR (

**ii**) images of the brain fastMRI data set. The results are presented in the form of box plots, computed over $S=5$ slices. From left to right in each boxplot, we compared subbandwise OSCAR (b-OSCAR), scalewise OSCAR (s-OSCAR), global OSCAR (g-OSCAR), CaLM, L1-ESPiRIT and AC-LORAKS, reconstruction methods.

**Figure 4.**Retrospective results for a single slice of FLAIR (

**top row**) and T1 weighted (

**bottom row**) images of the brain fastMRI data set obtained using the Cartesian mask shown in Figure 3A with $\mathrm{UF}=4$, corresponding to $\mathrm{AF}\simeq 4$. The fully sampled Cartesian reference and the different methods (Zero filled Inverse, g-OSCAR, CaLM, L1-ESPiRIT and AC-LORAKS) are shown from left to right and the SSIM scores are indicated to reflect the performance of each method.

**Figure 5.**Retrospective results for a single slice of FLAIR (

**top row**) and T1 weighted (

**bottom row**) images of the brain fastMRI data set obtained using the Non-Cartesian sampling pattern shown in Figure 3B with $\mathrm{AF}=16$ and $\mathrm{UF}=1.66$. The fully sampled Cartesian reference and the different methods (Density Compensated (DC) adjoint NUFFT, s-OSCAR, CaLM, L1-ESPiRIT and AC-LORAKS) are shown from left to right and the SSIM scores are indicated to reflect the performance of each method.

**Figure 6.**(

**Top Row**) Reconstructed MR images (magnitude) from 20-fold accelerated SPARKLING acquisitions using different methods. The scan time for Cartesian reference was 4min41s while the scan time of accelerated SPARKLING was 15s. (

**a**) Cartesian reference. (

**b**) The density compensated adjoint of raw k-space data (DC adj-NUFFT). (

**c**) Reconstruction based on the b-OSCAR formulation. (

**d**) calibration-less reconstruction based on CaLM or group-LASSO regularization. (

**e**) Self-calibrating ${\ell}_{1}$-ESPIRiT reconstruction. (

**f**) Auto-calibrated (AC) LORAKS reconstruction. (

**Second Row**) Respective zooms in the red frame. Reconstructed MR images. (

**Third Row**) Enhanced structures in phase images obtained by the method described in Section 4.5 on the virtual coil reconstructions of each method. The respective MSE of the phase images with respect to Cartesian reference are also reported. (

**Bottom Row**) Respective zooms to highlight details.

**Table 1.**Numerical complexity and parallelization capacity of OSCAR-norm regularizations using Daub. 4 OWT and $(N,C,L)=(640\times 320,4,16)$ for MRI reconstruction. The computation times of other state of art methods (CaLM, ${\ell}_{1}$-ESPIRiT and AC-LORAKS) are also mentioned below.

Proximity Numerical Complexity | Computation Time Per Prox. (S) | Parallelization | Computation Time Per Iter. (S) | |
---|---|---|---|---|

g-OSCAR | $O(L{N}_{\Psi}log\left(L{N}_{\Psi}\right))$ | 0.334 | N.A. | 2.894 |

s-OSCAR | $O({\sum}_{c=1}^{C}L{K}_{c}{S}_{c}log\left(L{S}_{c}{K}_{c}\right))$ | 1.005 | C | 6.711 |

b-OSCAR | $O({\sum}_{c=1}^{C}L{K}_{c}{S}_{c}log\left(L{K}_{c}\right))$ | 3.094 | $CS$ | 4.418 |

c-OSCAR | $O({N}_{\Psi}LlogL)$ | 159.75 | ${N}_{\Psi}$ | 161.13 |

CaLM | 1.944 | |||

${\ell}_{1}$-ESPIRiT | 4.360 | |||

AC-LORAKS | 2.516 |

**Table 2.**Comparison of different OSCAR-norm regularizations with ${\ell}_{1}$-ESPIRiT, CaLM and AC-LORAKS. The hyper-parameters were set to maximize the SSIM score. Best image quality metrics computed per row appear in bold font. At most, three scores are outlined on each row.

AF | g-OSCAR | s-OSCAR | b-OSCAR | CaLM | ${\mathit{\ell}}_{1}$-ESPIRiT | AC-LORAKS | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

SSIM | pSNR | SSIM | pSNR | SSIM | pSNR | SSIM | pSNR | SSIM | pSNR | SSIM | pSNR | |

8 | 0.923 | 30.52 | 0.925 | 31.66 | 0.926 | 31.68 | 0.921 | 30.51 | 0.911 | 27.82 | 0.894 | 26.09 |

10 | 0.920 | 29.21 | 0.921 | 29.62 | 0.922 | 30.28 | 0.921 | 29.54 | 0.906 | 26.58 | 0.897 | 26.23 |

12 | 0.916 | 28.81 | 0.918 | 28.40 | 0.918 | 29.78 | 0.917 | 29.05 | 0.904 | 27.17 | 0.893 | 26.25 |

15 | 0.912 | 29.28 | 0.912 | 29.05 | 0.913 | 29.52 | 0.912 | 28.87 | 0.900 | 26.29 | 0.884 | 25.94 |

20 | 0.899 | 29.12 | 0.896 | 28.35 | 0.899 | 29.52 | 0.897 | 28.59 | 0.885 | 26.48 | 0.753 | 25.52 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

El Gueddari, L.; Giliyar Radhakrishna, C.; Chouzenoux, E.; Ciuciu, P.
Calibration-Less Multi-Coil Compressed Sensing Magnetic Resonance Image Reconstruction Based on OSCAR Regularization. *J. Imaging* **2021**, *7*, 58.
https://doi.org/10.3390/jimaging7030058

**AMA Style**

El Gueddari L, Giliyar Radhakrishna C, Chouzenoux E, Ciuciu P.
Calibration-Less Multi-Coil Compressed Sensing Magnetic Resonance Image Reconstruction Based on OSCAR Regularization. *Journal of Imaging*. 2021; 7(3):58.
https://doi.org/10.3390/jimaging7030058

**Chicago/Turabian Style**

El Gueddari, Loubna, Chaithya Giliyar Radhakrishna, Emilie Chouzenoux, and Philippe Ciuciu.
2021. "Calibration-Less Multi-Coil Compressed Sensing Magnetic Resonance Image Reconstruction Based on OSCAR Regularization" *Journal of Imaging* 7, no. 3: 58.
https://doi.org/10.3390/jimaging7030058