Exploiting the separation

In this section, the utility of the intrinsic separation of the reconstructed data into low-rank and sparse components is demonstrated in the context of motion estimation in dynamic contrast enhanced (DCE) MRI.

In DCE MRI, acquisition of multiple MR images is taken continuously before, during, and after the administration of a contrast agent. The uptake and washout of the contrast agent concentration over time in the body corresponds to local changes of intensity in the MR images. Pharmacokinetic analysis can then be used to relate to tissue characteristics [147]. However, patient motion during acquisition (such as heartbeats, breathing or involuntary movements) produces inter-frame misalignment and complicates the estimation of the rate of the uptake by the tissue. Image registration can be used to solve this problem, but the presence of the contrast enhanced images interferes with the registration procedure because conventional algorithms can interpret local intensity changes as motion.

As it has been demonstrated for RPCA in figure 6.2, the proposed k-t RPCA approach is also expected to separate slow time-varying elements (background) from more abrupt changes due to the embedded separation. We show in figure 6.10 the different decompositions obtained with various ρ using the numerical phantom of the previous section that includes a combination of local intensity changes and slow varying motion. This figure demonstrates that it is possible to some degree to separate local changes of intensity in the sparse component from the background for an appropriate ρ parameter. To help in motion estimation, the idea is then to register low-rank images that will include most of the slow varying motion and less local intensity changes provoked by the contrast agent. The displacement field obtained from registering only the low-rank images (without interference from local intensity changes) is likely to be closer to the ”ground truth displacement field”,

Numerical simulations ρ = 0.5 ρ = 1.5 ρ = 2 ρ = 3.5

a

b

c

d

Figure 6.10: Different types of separation into low-rank and sparse components using k-t RPCA with different decomposition parameters ρ. It can be observed that this parameter acts as a trade-off between the two components. The undersampling rate is 0.25. (a) Low-rank time frames (b) Sparse time frames (c) x-t temporal profiles of low-rank component (d) x-t temporal profiles of sparse component.

Joint reconstruction–separation via matrix decomposition

Ground truth Noisy with CE k-t FOCUSS k-t SLR k-t RPCA k-t RPCA (L) k-t RPCA (S)

Figure 6.11: x-t temporal profiles used in the registration procedure. For the reg- istration of k-t RPCA, only the low-rank part is used which mostly contains images without contrast enhancement (CE) thanks to the separation process.

that is the displacement field from the same MR signal without contrast enhanced images. This displacement field can then be employed for more accurate motion correction as it has been shown in Ref. [148].

As a proof of concept, the numerical phantom of the previous section (combina- tion of local intensity changes and motion) is used for the purpose of demonstration with an acceleration factor of 4 using the pseudo-radial sampling. k-t FOCUSS and k-t SLR are reconstructed as previously, using the best regularisation parameters. However, k-t RPCA reconstruction is now explicitly selected with ρ = 1.5 to end up with mostly the local intensity changes in the sparse part, and motion in the low-rank part. Reconstruction errors for k-t FOCUSS, k-t SLR and k-t RPCA were respectively 25.3dB, 31.1dB and 26.3dB.

A sequential registration of each frame of k-t FOCUSS, k-t SLR and the low-rank part of k-t RPCA is performed with NiftyReg [149], an efficient C++ implementation of a parallel formulation of the free-form deformation (FFD) algorithm [150] based on cubic B-splines. Local normalised cross correlation is used as measure of similarity (standard deviation of the Gaussian kernel set to 5 pixels for all time points) and a control point spacing of 2 pixels in all directions. The time profiles of the different reconstruction methods along the ground truth are shown in figure 6.11. Note the ground truth was obtained with the noiseless phantom created without intensity changes but with the same motion. The reference images taken for registration were the last time frame images in the respective dynamic reconstructed sequences.

In the registration procedure, we also compute the displacement fields of the image sequence (which is called optical flow in computer vision) via NiftyReg. This results in displacement fields Dx and Dy, respectively along the x direction and y

direction. Displacement vector fields from one time frame to the next time frame are shown (in blue) in figure 6.12 with source images used for registration in the back- ground. The definition of the Jacobian J (x, y) in 2D depends on the displacement fields Dx and Dy, J (x, y) = "_∂D x ∂x ∂Dy ∂x ∂Dx ∂y ∂Dy ∂y , # (6.19)

Discussion

a b c d e

Figure 6.12: Displacement fields (zoom-in) over source images used for registration. Table 6.5 provides the associated quantitative results. (a) Ground truth noiseless phan- tom (b) Noisy phantom with local intensity changes (c) k-t FOCUSS (d) k-t SLR (e) k-t RPCA, low-rank part. It can be seen that the displacement field is better estimated in the region with local changes of intensity in k-t RPCA.

Noisy phantom with local intensity changes 11.0

k-t FOCUSS 10.4

k-t SLR 14.6

k-t RPCA – Low-rank component 15.2

Table 6.5: Displacement fields results in the region of interest with local inten- sity changes. Quantities are in dB and have been computed using the Jacobian and Eq. (B.2).

and in particular its determinant is given by |J (x, y)| = ∂Dx ∂x ∂Dy ∂y − ∂Dx ∂y ∂Dy ∂x . (6.20)

The metric defined in Eq. (B.2) can then be used to compute the displacement fields errors based on the Jacobian. Results are reported in table 6.5 and show a slight improvement for k-t RPCA over other methods.

6.4

Discussion