Propuesta de valor para la industria nacional

We aim to localize the brain in every stack of slices based on the technique preliminary presented in [25] where robustness to outlier slices has been improved. In contrast to the approach presented in [29], we use an age-matched template [30] as prior and match it directly to each slice through an accelerated block matching approach. The main benefit of this approach is not only it can retrieve the location of the brain in the image but it can also retrieve its global alignment in 3D. The localization problem is formulated as a block matching algorithm in which the similarity between each block, i.e. a whole 2D slice, extracted from the template and a query image (a fetal MRI slice) is maximized. However, the search space is large and the problem is computationally heavy as we do not know a priori the position of the fetal head. We decide to limit the search space by breaking the transformation model to rotation (θ) and translation (T) parts for which parameters are estimated separately. We estimate the translation parameters for each rotation angle through the proposed block matching technique. The algorithm involves three steps: 1) block extraction and dimension reduction, 2) block matching using expectation maximization (EM), and 3) calculation of final transform by maximizing similarity.

Block extraction and dimension reduction A similarity matrix SM is generated by comput- ing the Sum of Square Distances (SSD) between a 2D block i in a template image and a 2D block j in the query image. SMi jis defined as:

SMi j=

e−SSDi j

P

ke−SSDi k

(2.1) Since SSD is not rotation invariant, the template image needs to be initially rotated using a rotation matrix (θ). Two-dimensional blocks are then extracted from the template and the similarity matrix SMθis computed for each rotation matrix. In addition, computation of SMθ is accelerated by projection of each block to a lower dimensional space though a random

Chapter 2. Automated Template-based Brain Localization and Extraction for Fetal Brain MRI Reconstruction

matrix based on Johnson-Lindenstrauss Lemma [32].

Expectation maximization for match detection Let R be the template image, S be the query image, and M be a binary matrix representing the match between blocks in the two images [33], thus EM[Mi j] = P(Mi j= 1). Matches between blocks and the translation are then iteratively

updated within an EM framework. The expectation step corresponds to calculating

EM[Mi j|R, S, Tθ] = P(Mi j= 1|R, S, Tθ), (2.2)

where Tθis the translation transferring the template to the fetal brain in the image after initial rotationθ. Then, the translation T is initially set to zero, and iteratively updated to maximize the probability of the block riin R being matched to the blocks sjin S

arg max Tθ EM[l og (P(R, M|S,T θ_{))] = (2.3)} arg min Tθ X i j P(Mi j= 1|R, S, Tθ)||Tθ◦ ri− sj||p. (2.4)

where ||.||pis the Lpnorm. If P(sj|ri, Tθ)) follows a normal Gaussian distribution, the optimal

solution of Eq. (2.4) is obtained by p = 2, i.e. by least squares optimization. Nonetheless, because of outliers P(sj|ri, Tθ)) does not follow a Gaussian distribution. Robust estimation

of the conditional probabilities in Eq. (2.4) is thus desired and achieved through L1norm

optimization, i.e. p = 1.

Matches are iteratively updated through Eq. (2.2) and Eq. (2.4). Through iterations of the EM steps, weights are assigned to a set of blocks with highest similarity and are updated to find the best match.

Final transformation Once the translation T is estimated, we apply T and the initial rotation

θ to the template, and calculate the similarity between the transformed template and the

fetal image. Finally, we select the transformation maximizing the similarity value as the most probable transformation, that is,

arg max

θ,TθNCC ((T

θ_{◦ θ)R, S) (2.5)}

where NCC corresponds to the Normalized Cross Correlation as image intensities may not be the same between the query and template images ( see Figure 2.2).

Throughout the rest of this chapter, we will consider ˆT1_s = ˆTθˆ◦ ˆθ as the global rigid transforma- tion estimated between the s-th query stack S and the template.

2.4. Novel template-based brain localization and extraction for fetal MRI

Target stack with brain mask

extracted Template brain mask matched in target stack s Template + brain mask Slice j Template blocks

Automatic Brain Extraction

Automatic Brain Localization

Estimate 2D free-form deformation

using 3rd _{order B-splines}

Relocate and reorient target brain by applying

Estimate rigid slice-to-volume transformation for all slices

Estimate global rigid transformation of stack s using block matching

Stacks of thick slices (Clinical MR Scans)

Stack s Slice j ˆ T1 s= ˆT ˆ θ_{◦ ˆθ} ˆ T2 j ˆ T1 s ˆ Ds

Figure 2.2 – Template-based brain localization and extraction from clinical MR scans. In each scan s, the initial global alignment and position of the fetal brain (global rigid transformation

ˆ

T1_s) is estimated using a block matching approach: 2D blocks of the age-matched template image are matched to slices in the fetal brain MRI scan. It corresponds to the brain localization step. The contour of the best match is indicated in red. Brain masks are then obtained through the brain extraction step. It consists of (i) cropping and reorienting the scan to the template space using ˆT1s, (ii) performing rigid slice-to-template registration to refine the

brain localization within the slice and to correct for inter-slice motion (rigid slice-to-template transformation T2_j), (iii) performing a 2D B-Spline deformable registration to take into account anatomical variability between the processed brain and the template brain (deformation field ˆDs j), and (iv) propagating the template brain mask to each slice, using the estimated

Chapter 2. Automated Template-based Brain Localization and Extraction for Fetal Brain MRI Reconstruction