Joint Longitudinal Modeling of Brain Appearance in Multimodal MRI for the Characterization of Early Brain Developmental Processes - Longitudinal Registration and Shape Modeling - page 50

based algorithms [ 12 ] for unbiased atlas building. The result of these registrations
is that all images in the dataset share the common coordinate space of the atlas.
Mapping of a parcellation atlas onto the constructed atlas enables extraction
of major cortical regions. An expectation-maximization (EM) based procedure
for segmenting the brain into major tissue classes is implemented [ 13 ]. Improved
segmentation of early time point scans is obtained using subject-specific pri-
ors in the EM procedure. These priors are obtained from probabilistic tissue
segmentation maps of the latest time-point scan of each subject [ 14 ].

2.2 Quantification of Contrast

Contrast between two tissue classes with intensities represented by probability
density distributions requires a measure of similarity. Statistics suggest the use of
the Hellinger distance ( HD ) which knows solutions for discrete and parametric
distributions and satisfies the properties of a metric. The HD can be expressed in
terms of the Bhattacharyya coecient ( BC ) given any two intensity distributions
P 1and P 2as:

BC ( P 1 ,P 2) =

P 1( y ) P 2( y ) dy.

(1)

y

HD ( P 1 ,P 2) = 2(1 − BC ( P 1 ,P 2)) . (2)

T1W T2W

x 10 −4 x 10 −4 x 10 −4 x 10 −4 x 10 −4 x 10 −4

8 8 8 8 8 8

6 6 6 6 6 6

4 4 4 4 4 4

2 2 2 2 2 2

0 0 0 0 0 0

0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300

Intensity Intensity Intensity Intensity Intensity Intensity

a) 6 months b) 12 months c) 24 months (d) 6 months (e) 12 months (f) 24 months

Fig. 3. Intensity Distributions of gray matter (blue), white matter (red), and cere-
brospinal fluid or csf (black) in the left frontal lobe for T1W (a-c) and T2W (d-f)
images of a single subject scanned at approximately 6 months (a and d), 12 months (b
and e) and 24 months (c and f) of age

Kernel Density Estimation (KDE) with a Gaussian kernel G is used to obtain
smooth and continuous distributions for the intensities of each tissue class c k .
For an image I i,m belonging to subject i and of modality m , the intensity distri-
bution over the entire intensity range Int is denoted by P i,m ( Int|c k ). Intensity
distributions for the multimodal image set of a single subject are shown in Fig. 3 .
In the following, contrast measured in terms of HD between white and gray
matter intensity distributions is named CONT , and defined according to the
notation given above becomes :

CONT i,m = HD ( P i,m ( Int|c = WM ) ,P i,m ( Int|c = GM )) . (3)