Longitudinal Guided Super-Resolution Reconstruction of Neonatal Brain MR Images - Longitudinal Registration and Shape Modeling - page 64

proposed non-local guidance from longitudinal prior. We then introduce the regulari-
zation terms as well as optimization steps for solving the cost function. The input will
include a neonatal LR image and a longitudinal HR follow-up image, and the output
will be the estimated neonatal HR image.

2.1 Super-Resolution Problem

We employ a physical model for capturing the degradation processes involved in
reducing a high-resolution image to a low-resolution image [1]:

(1)
where T denotes the observed LR image, D is a downsampling operator, S is a blur-
ring operator, X is the to-be-recovered HR image, and n represents the observation
noise. The HR image can be estimated using this model by minimizing the following
cost function:

min (2)

where the first term is a data fidelity term used for penalizing the differences between
the degraded HR image X and the observed LR image T . The second term is a regula-
rization term often defined based on prior knowledge. Weight λ is introduced to bal-
ance the contributions of the fidelity term and regularization term.

2.2 Non-local Guidance from Longitudinal Prior

We use a non-local approach to learn the spatial relationships of structures in high-
resolution longitudinal images and then apply this information to the reconstruction of
high-resolution neonatal image. In other words, the recurring patterns throughout in
the longitudinal scans are leveraged for reconstructing the neonatal image.
The non-local strategy has been proposed for image denoising [10]. For each voxel
v in image X , the similarity between v and each voxel k in a non-local search domain
Ω is measured using their local patches. A weighted graph w can thus be obtained
to represent the non-local relationships between v and other voxels in the large non-
local search domain. Then, a non-local mean (NLM) image is obtained by updating
each voxel in the image using this strategy:

,
Ω

,

(3)

In our case, X is the neonatal HR image that needs to be recovered. To utilize the
longitudinal prior, we propose to learn the weighted graph w for each voxel using
both the longitudinal HR image L and the pre-estimated X :
, 1

/ /

(4)

where , is the weight associating the center voxel to a voxel in its search
domain Ω , and are 3D patches of longitudinal HR image centered