Noise amplification vs. resolution tradeoff in the SLIDER technique
Steen Moeller1, Sebastian Schmitter1, and Mehmet Akcakaya1,2

1Center for Magnetic Resonance Research, University of Minnesota, Minneapolis, MN, United States, 2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, United States

### Synopsis

An investigation into the noise and resolution performance of SLIDER using a series of shifted low-resolution images to obtain a single high-resolution. The evaluation is performed both theoretical and experimentally using a FLASH acquisition, to remove experimental limitations from understanding the merits of the technique.

### Purpose

Recently, a novel reconstruction technique has been proposed for diffusion weighted imaging, termed SLIDER [1], which allows the reconstruction of a single high resolution 3D dataset from multiple low resolution 3D datasets. The technique shares similarities with the original multiband technique [2] (not SMS), which has a sqrt(M) SNR gain for M acquisitions of M slices, and also has parallels to the works [3,4]. In this work we investigate the performance of SLIDER with respect to resulting SNR and resolution in a FLASH acquisitions obtained in-vivo and in phantoms.

### Methods

A SLIDER acquisition consisting of three 3D low-resolution FLASH datasets for a FOV of 256x232x90 mm^3, with a resolution of 1x1x3 (ROxPExSL), and each shifted 1mm relative to each other along the slice-direction was acquired on a phantom and on a human subject. A matching high resolution acquisition with 1x1x1 (ROxPExSL) was acquired. For the phantom the high resolution was a 2D multi-slice and for the in-vivo a 3D acquisition was used. The phantom was head shaped with an embedded rectilinear grid. Imaging was performed in accordance with the IRB of the University of Minnesota and was acquired on Siemens Magnetom 7T system equipped with a nova medical 32 channel head coil. A 2.56 ms sinc RF pulse with a BWTP=8 was used for all experiments, which for SLIDER with 3X slice-thickness yields a subslice weigth of [0.89 1 0.89]. The in-vivo slice orientation was chosen as coronal/sagittal to null signal in the outer slices. The encoding system, denoted by $f=Ax$, for resolving the higher resolution signal $x$ was chosen as a circulant matrix. The low-resolution SLIDER images was concatenated to a single image prior to reconstruction. Using the forward mapping in SLIDER, a set of low-resolution images was generated and then reconstructed using a regularized reconstruction. The obtained simulated results where visually compared with the high resolution reconstructions from acquired and reconstructed SLIDER data. Similarly an in-vivo experiment was performed, and the acquired and reconstructed SLIDER data was compared qualitatively with high resolution FLASH acquisition of matched duration.

### Results

The impact of noise from the encoding matrix $A$ can for the $k^{th}$ element (reconstructed slice) be expressed as $$\begin{bmatrix} \left(\begin{array}{c} A^{*}A +\lambda I \end{array}\right)^{-1} A^{*}A \left(\begin{array}{c} A^{*}A +\lambda I \end{array}\right)^{-1} \end{bmatrix}_{k,k}$$ When $A$ is a circulant matrix, the noise scaling is identical for all slices and can for each estimated value be expressed as $$std(f)=\sigma \sqrt{\frac{1}{N}\sum_{i}\frac{|h_i|^2}{(|h_i|^2+\lambda)^2}}=\sigma \eta,$$ where $h_i$ are the singular values of the encoding system $\sigma^2$ the noise variance, $N$ total number of slices. Noise amplification is plotted in figure 1A. Using a value of $\lambda$ which corresponds to a noise amplification of 1 (one), the simulation for reconstructing a square profile is plotted in figure 1B, based on an RF slice-profile as shown in figure 1C. The simulation of a SLIDER encoding reconstruction is shown in figure 2. The simulated data (Figure 2A) is based on the image in figure 2B. Reconstructions based on simulated SLIDER data are shown in figure 2C and 2D for different values of $\lambda$. A matching view of the same phantom acquired with a SLIDER technique are shown in figure 3A-D, where the concatenation of the low-resolution images is displayed in figure 3A, and can be compared with the directly acquired high resolution image 3B. The reconstruction for different values of $\lambda$ are shown in figures 3C and 3D. The results from the In-vivo data are shown in figure 4. A matched slice from the high resolution acquisition is shown in figure 4B, and the anatomical resolution can be compared with the acquired SLIDER data in 4A. Reconstructed data are shown in figures 4C-4E for different values of $\lambda$.The thermal noise is the same in the low-resolution and high-resolution acquisitions, and the SNR in the low-resolution images are larger than in the high resolution, since the signal is averaged. The noise in the SLIDER reconstructed images are the thermal noise time multiplied with the $\eta$ from the circulant encoding matrix.

### Discussion

Noise amplification have been analyzed for the SLIDER technique and it has been demonstrated how for low values of lambda, there is noise amplification, which can be reduced with increasing lambda at the cost of through-slice blurring. These theoretical observations have been validated both in phantom and in-vivo, and evaluated using a conventional FLASH sequence. The theoretical and experimental observations demonstrated here, shows that if the high resolution acquisition can be performed directly, then that that will have both improved SNR and resolution compared to SLIDER.

### Acknowledgements

P41 EB015894. Kawin Setsompop for discussion on noise properties in SLIDER and Xiaoping Wu for alternate denoising techniques.

### References

[1] SLIce Dithered Enhanced Resolution Simultaneous MultiSlice (SLIDER-SMS) for high resolution (700 um) diffusion imaging of the human brain. Kawin Setsompop et al. page 339, ISMRM 2015.

[2] SIMA: Simultaneous Multislice Acquisition of MR Images by Hadamard-encoded Excitation. S.P. Souza, J. Szumowski, C.L. Dumoulin, D.P. Plewes, G.H. Glover, J. Comput. Assist. Tomogr. 12 (6) pp. 1026-30 (Nov-Dec. 1988).

[3] A Spectral Approach to Analyzing Slice Selection inPlanar Imaging: Optimization for Through-Plane Interpolation. Douglas Noll, MRM 38:151-160 (1997)

[4] Super-resolution reconstruction of diffusion parameters from diffusion-weighted images with different slice orientations. Van Steenkiste G, Jeurissen B, Veraart J, den Dekker AJ, Parizel PM, Poot DHJ, Sijbers J. Magnetic Resonance in Medicine 2015

### Figures

Figure 1. Figure A, show the noise-amplification from the SLIDER reconstruction for a square profile with [0.89 1 0.89]. Figure B shows a simulated square profile and the regularized SLIDER reconstruction. Figure C shows the slice profile used for acquisition and reconstruction.

Figure 2. Figure A, shows a coronal slice from a resolution phantom. Figure B shows the concatenated low-resolution images for a SLIDER acquisition. The nominal high resolution reconstruction from the simulated SLIDER data is in (C)+(D) for different values of $\lambda$.

Figure 3. Figure A, shows a coronal slice from a resolution phantom. Figure B shows the concatenated low-resolution images from a SLIDER acquisition. The nominal high resolution reconstruction from the acquired SLIDER data is in (C)+(D) for different values of $\lambda$.

Figure 4. Figure A, shows a coronal slice from a high resolution 3D FLASH acquisition. Figure B shows the concatenated low-resolution images with a SLIDER acquisition. The nominal high resolution reconstruction from the simulated SLIDER data is in (C)-(E) for different values of $\lambda$.

A SLIDER acquisition (B) and reconstruction (C)-(E), with a high resolution reference shown in (A). The SLIDER acquisitions are multi-slice acquisitions and the reference is a 3D acquistion

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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