# Discrete Shearlets as a Sparsifying Transform in Low-Rank Plus Sparse Decomposition for Undersampled (k, t)-Space MR Data

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Focal Underdetermined System Solver (k-t FOCUSS)

#### 2.2. Robust Principal Component Analysis (RPCA)

#### 2.3. Sparsifying Transforms

#### 2.4. Image Registration—Motility Metric

## 3. Materials and Methods

#### 3.1. Small Bowel MR Acquisition

#### 3.2. Simulated Abdominal DCE Data

#### 3.3. Simulated Undersampled Acquisition

#### 3.4. Quantitative Evaluation

^{®}R2019b (The Mathworks Inc., Natick, MA, USA). The proposed image processing procedures were conducted on a desktop personal computer with a 2.50 GHz Intel

^{®}Core

^{TM}i7-4710HQ CPU processor and 16 GB RAM operating memory, running Windows 10 Professional Edition.

#### 3.5. Implementation Details

## 4. Results

#### 4.1. Low-Rank Plus Sparse Image Decomposition

#### 4.2. Low-Rank Plus Sparse Image Decomposition from Undersampled $(k,t)$-Space Data

#### 4.3. Quantification of Motility in Low-Rank Plus Sparse Decomposition Using Discrete Shearlets as Sparsifying Transforms

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**$L+S$ decomposition of: (

**a**) 2D abdominal DCE images, and (

**b**) 2D small bowel images. Temporal $(x-t)$ profiles along the dotted direction aim to capture the dynamic intensity changes. The low-rank (L) in both (

**a**) and (

**b**) includes the background (and periodic motion), whereas the sparse component captures rapid intensity changes either because of contrast enhancement (

**a**), or because of bowel motility (

**b**). Increased sparsity can be achieved using Fourier transformation along time (TF) and discrete shearlets. Histograms of the transformed S components with TF and DS are shown to illustrate the increased sparsity.

**Figure 3.**$L+S$ using DS as sparsifying transform decomposition using shearlets as sparsifying transform on small bowel images. Following decomposition, the rank of L was 2, whereas the rank of S equalled 50. It is clear from the L, S images and histograms that S is sparser than L. Consequently, L includes background, periodic motion, while S captures bowel motility (rapid intensity changes).

**Figure 4.**$L+S$ using DS as sparsifying transform decomposition using shearlets as sparsifying transform on abdominal DCE images. Following decomposition, the rank of L was 1, whereas S is not low-rank. It is clear from the L and S images and histograms that S is far more sparse than L. Consequently, L includes static background, while S captures contrast enhancement (rapid intensity changes).

**Figure 5.**$L+S$ reconstruction using discrete shearlets as sparsifying transforms from small bowel 4-fold and 8-fold undersampled $(k,t)$-data. L, S, $L+S$ recovered images, and time-cut representations, as in Figure 2, are illustrated for each undersampling.

**Figure 6.**Ground truth deformation fields of the original T1-weighted images (without enhancement) and deformation fields obtained after registration of the low-rank L, and $M=L+S$, following $L+S$, using DS as sparsifying transform decomposition of the simulated DCE images.

**Figure 7.**Boxplots of the motility scores across the 8 subjects with range (line), interquartile range (box), and median (horizontal line) for breath-holding (BH) and free-breathing (FB) data. Motility scores were derived with $L+S$ using DS as sparsifying transform decomposition/reconstruction of scanner images/undersampled data. The L, S, $M=L+S$ components derived from the BH data were compared with the ones from the FB data.

$\mathit{L}+\mathbf{\Phi}\left(\mathit{S}\right)$ decomposition and reconstruction algorithm |
---|

Input: $(k,t)$-space samples y, decomposition parameters ${\lambda}_{L}$, ${\lambda}_{S}$ |

Initialize: ${M}^{1}={F}_{u}^{T}y$ and ${S}^{1}=0$, iteration k=1; |

while stopping criterion is not met, do |

Singular value thresholding |

${L}^{k+1}={D}_{{\lambda}_{L}}({M}^{k}-{S}^{k})$ |

Shrinkage operator |

${S}^{k+1}={\Phi}^{-1}\left(Shrin{k}_{{\lambda}_{S}}(\Phi ({M}^{k}-{L}^{k}))\right)$ |

Subtract aliasing artifacts from $M=L+S$ |

${M}^{k+1}={L}^{k+1}+{S}^{k+1}-{F}_{u}^{T}({F}_{u}({L}^{k+1}+{S}^{k+1})-y)$ |

Stopping criterion |

$k>50$ or $\frac{{\u2225{M}^{k+1}-{M}^{k}\u2225}_{2}}{{\u2225{M}^{k}\u2225}_{2}}}<{10}^{-5$ |

end while |

Output: $L={L}^{k+1}$, $S={S}^{k+1}$ |

**Table 2.**Relative reconstruction error ($re$) of the $L+S$ reconstructions, using the identity matrix I, TF, WT, and DS as sparsifying transforms and k-t FOCUSS for the simulated DCE datasets.

Unders. Factor 4 | Unders. Factor 8 | |
---|---|---|

k-t FOCUSS | 0.162 | 0.203 |

$L+S$ using I as sparsifying transform | 0.149 | 0.194 |

$L+S$ using TF as sparsifying transform | 0.147 | 0.189 |

$L+S$ using WT as sparsifying transform | 0.147 | 0.188 |

$L+S$ using DS as sparsifying transform | 0.144 | 0.187 |

**Table 3.**Median and interquartile range (iQR) of relative reconstruction error ($re$) of the reconstructions k-t FOCUSS, $L+S$ reconstructions (using identity matrix I, TF, WT, and DS as sparsifying transforms), across the 8 small bowel datasets.

Undersampling Factor 4 | Median | iQR |

k-t FOCUSS | 0.077 | 0.013 |

$L+S$ using I as sparsifying transform | 0.080 | 0.012 |

$L+S$ using TF as sparsifying transform | 0.079 | 0.012 |

$L+S$ using WT as sparsifying transform | 0.075 | 0.011 |

$L+S$ using DS as sparsifying transform | 0.064 * | 0.006 |

Undersampling Factor 8 | Median | iQR |

k-t FOCUSS | 0.120 | 0.025 |

$L+S$ using I as sparsifying transform | 0.119 | 0.018 |

$L+S$ using TF as sparsifying transform | 0.115 | 0.017 |

$L+S$ using WT as sparsifying transform | 0.112 | 0.021 |

$L+S$ using DS as sparsifying transform | 0.106 | 0.029 |

**Table 4.**Relative error between the ground truth deformation fields and the ones obtained after registration of the low-rank (L), and $M=L+S$ following $L+S$ using DS as sparsifying transform decomposition of the simulated DCE images and reconstruction from the 4/8-fold undersampled DCE $(k,t)$-space data (US4/US8).

L | $\mathit{M}=\mathit{L}+\mathit{S}$ | |
---|---|---|

DCE images | 0.016 | 0.031 |

US4 | 0.021 | 0.025 |

US8 | 0.024 | 0.029 |

**Table 5.**Median motility scores $\mu \left({\sigma}_{J}\right)$ for S and $L+S$ in the breath-holding and free-breathing data.

Scanner images | BH | FB | p-value |

S | 0.044 | 0.049 | 0.20 |

$L+S$ | 0.036 | 0.047 | 0.02 |

Four-fold | BH | FB | p-value |

S | 0.039 | 0.045 | 0.08 |

$L+S$ | 0.034 | 0.043 | 0.02 |

Eight-fold | BH | FB | p-value |

S | 0.039 | 0.047 | 0.04 |

$L+S$ | 0.035 | 0.043 | 0.01 |

**Table 6.**p-values measuring the effect of undersampling in the value of the motility score: median motility scores calculated from L, S, and M of the free-breathing scanner images, compared to the ones from the four-fold and eight-fold undersampled data, respectively.

L | S | $\mathit{M}=\mathit{L}+\mathit{S}$ | |
---|---|---|---|

Four-fold | 0.10 | 0.27 | 0.35 |

Eight-fold | 0.01 | 0.45 | 0.49 |

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**MDPI and ACS Style**

Protonotarios, N.E.; Tzampazidou, E.; Kastis, G.A.; Dikaios, N.
Discrete Shearlets as a Sparsifying Transform in Low-Rank Plus Sparse Decomposition for Undersampled (*k*, *t*)-Space MR Data. *J. Imaging* **2022**, *8*, 29.
https://doi.org/10.3390/jimaging8020029

**AMA Style**

Protonotarios NE, Tzampazidou E, Kastis GA, Dikaios N.
Discrete Shearlets as a Sparsifying Transform in Low-Rank Plus Sparse Decomposition for Undersampled (*k*, *t*)-Space MR Data. *Journal of Imaging*. 2022; 8(2):29.
https://doi.org/10.3390/jimaging8020029

**Chicago/Turabian Style**

Protonotarios, Nicholas E., Evangelia Tzampazidou, George A. Kastis, and Nikolaos Dikaios.
2022. "Discrete Shearlets as a Sparsifying Transform in Low-Rank Plus Sparse Decomposition for Undersampled (*k*, *t*)-Space MR Data" *Journal of Imaging* 8, no. 2: 29.
https://doi.org/10.3390/jimaging8020029