Improved robust tensor principal component analysis for accelerating dynamic MR imaging reconstruction

Abstract

Dynamic magnetic resonance imaging (dMRI) strikes a balance between reconstruction speed and image accuracy in medical imaging field. In this paper, an improved robust tensor principal component analysis (RTPCA) method is proposed to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data. The MR reconstruction problem is formulated as a high-order low-rank tenor plus sparse tensor recovery problem, which is solved by robust tensor principal component analysis (RTPCA) with a new tensor nuclear norm (TNN). To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into TNN for enforcing the low-rank structure in MRI datasets. The experimental results show that the proposed method outperforms state-of-the-art methods in terms of both MR image reconstruction accuracy and computational efficiency on 3D and 4D experiment datasets, especially for 4D MR image reconstruction.

Appendices

Appendix I. Robust principal component analysis model based on matrix and tensor

The RPCA decomposes a matrix $X\in {\mathrm{R}}^{N_1\times {N}_2}$ into a low-rank matrix L and a sparse component S, and forms the following convex optimization problem:

$$ \underset{L,S}{\min }{\left\Vert L\right\Vert}_{\ast }+\lambda {\left\Vert S\right\Vert}_1,\kern0.5em s.t.X=L+S $$

where ‖L‖_∗ is the matrix nuclear norm, which is the sum of the singular values of matrix L, and ‖S‖₁ denotes the l₁ norm of matrix S.

The RTPCA decomposes 3-order tensor $\chi \in {\mathrm{R}}^{N_1\times {N}_2\times {N}_3}$ into a low-rank tensor and sparse tensor as follows: χ = ℓ + ξ, where ℓ and ξ denote the low-rank component and sparse component respectively. Thus,

$$ \underset{\ell, \xi }{\min }{\left\Vert \ell \right\Vert}_{\ast }+{\left\Vert \xi \right\Vert}_1,\kern1em s.t.\kern0.5em \ell +\xi =\chi $$

RPCA is classical principal component analysis that has been widely used for data processing problems. However, it is based on matrix mode. It is natural to consider extending RPCA to manipulate the tensor data by taking advantages of its multi-dimensional structure. Robust tensor principal component analysis extracts the low-rank and sparse component of multi-dimensional data by t-SVD, which can be used for many data analysis problems. RTPCA has been applied to three groups of numerical experiments on image denoising, face images, and motion separation in videos.

Appendix II The tensor singular value decomposition operator

Given a 3-order tensor $\chi \in {\mathrm{R}}^{N_1\times {N}_2\times {N}_3}$, N₁, N₂, and N₃ are the 1^th, 2^th, and 3^thdimensions of χ. The t-SVD is given by χ = U ∗ S ∗ V^H, where U and V are orthogonal tensors of size N₁ × N₁ × N₃ and N₂ × N₂ × N₃, respectively, and S is an f-diagonal tensor. This t-SVD is constructed as follows: tensor χ is transformed into Fourier domain, and let $\overline{\chi}\left(:,:,i\right)={\overline{U}}^{(i)}{\overline{S}}^{(i)}{\left({\overline{V}}^{(i)}\right)}^H$ be matrix singular value decomposition for frontal slices $i=1,\dots, \left\lceil \frac{N_3+1}{2}\right\rceil$. For frontal slices $i=\left\lceil \frac{N_3+1}{2}\right\rceil +1,\dots, {N}_3,$let${\overline{U}}^{(i)}=\mathrm{conj}\left({\overline{U}}^{\left({N}_3-i+2\right)}\right)$,${\overline{S}}^{(i)}={\overline{S}}^{\left({N}_3-i+2\right)}$, and ${\overline{V}}^{(i)}==\mathrm{conj}\left({\overline{V}}^{\left({N}_3-i+2\right)}\right).$ This processing scheme on all frontal slices is consistent with tensor-tensor product mode. Notice that the work reduces the number of singular value decomposition (SVD) to $\left\lceil \frac{N_3+1}{2}\right\rceil$, while the previous works require computing SVD N₃times; therefore, computation efficiency of this novel tensor nuclear norm is improved significantly when N₃ is large [37].

T-SVD method for the 3-order tensor

Input: $\chi \in {\mathrm{R}}^{N_1\times {N}_2\times {N}_3}$ 1. Compute $\overline{\chi}\leftarrow \mathrm{fft}\left(\chi, \left[\right],3\right),$ 2. Compute each frontal slice of $\overline{U},\kern0.5em \overline{S}$ and $\overline{V}$ from the $\overline{\chi}$ by for i = 1, ... , to $\left\lceil \frac{N_3+1}{2}\right\rceil$ do [${\overline{\mathrm{U}}}^{\left(\mathrm{i}\right)},{\overline{\mathrm{S}}}^{\left(\mathrm{i}\right)},{\overline{\mathrm{V}}}^{\left(\mathrm{i}\right)}$] = SVD(χ(:,  : , i)), end for for $i=\left\lceil \frac{N_3+1}{2}\right\rceil +1,...,{N}_3$ do ${\overline{\mathrm{U}}}^{\left(\mathrm{i}\right)}=\mathrm{conj}\left({\overline{\mathrm{U}}}^{\left({N}_3-\mathrm{i}+2\right)}\right);$ ${\overline{\mathrm{S}}}^{\left(\mathrm{i}\right)}={\overline{\mathrm{S}}}^{\left({N}_3-\mathrm{i}+2\right)};$ ${\overline{\mathrm{V}}}^{\left(\mathrm{i}\right)}=\mathrm{conj}\left({\overline{\mathrm{V}}}^{\left({N}_3-\mathrm{i}+2\right)}\right);$ end for 3. $U\leftarrow \mathrm{ifft}\left(\overline{U},\left[\right],3\right),S\leftarrow \mathrm{ifft}\left(\overline{S},\left[\right],3\right),V\leftarrow \mathrm{ifft}\left(\overline{V},\left[\right],3\right),$ Output: U, S, V