Image Processing Reference
In-Depth Information
Phantom Data
Right Horizontal Asymptote
Inflection point
2
2
12
Variable 1
Variable 2
1.9
1.8
11.5
1.6
1.8
11
1.7
1.4
10.5
1.2
1.6
1.5
1
10
0.8
1.4
9.5
0.6
1.3
9
0.4
1.2
8.5
0.2
1.1
0
1
8
0
5
10
15
20
25
30
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
3
3.5
4
4.5
5
5.5
6
6.5
7
Covariate
Asymptote (Variable 1)
Inflection Point (Variable 1)
2
2
12
1.9
1.8
11.5
1.6
1.8
11
1.7
1.4
10.5
1.2
1.6
1
1.5
10
0.8
1.4
9.5
0.6
1.3
9
0.4
1.2
8.5
0.2
1.1
0
1
8
0
5
10
15
20
25
30
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
3
3.5
4
4.5
5
5.5
6
6.5
7
Covariate
Asymptote (Variable 1)
Inflection point (Variable 1)
Fig. 5. (Leftmost column) Trajectories of change plotted with respect to the covari-
ate for two synthetically generated variables. These variables (1 and 2) have strongly
correlated (top row) and uncorrelated (bottom row) random effects parameters for
the underlying logistic growth function. (Middle and right columns) Experimentally
designed values of mixed effects parameters (fixed + random effects) for variable 2 vs.
variable 1 are shown in 2D scatter plots : (middle column) right horizontal asymp-
tote
φ 3 . The parameters in the top row show
strong correlation between the two variables while the parameters in the bottom row
show no correlation. The inflection point assumes same unit as the covariate while the
asymptote assumes the unit of the variable modeled.
φ 2
and (right column) inflection point
parietal lobes while temporal and frontal lobes undergo contrast change over a
longer time period.
Analysis of Logistic Growth Function Parameters Describing Con-
trast Change. Statistical inferences based on the non-zero parameters (right-
asymptote, inflection point, and rate) of the logistic function of the form defined
earlier, quantify the established qualitative knowledge that regional contrast
change takes place with varying rates in different modalities. Statistical hypoth-
esis testing using the Student's t-test confirms significant differences in both the
inflection point and rate parameters between T1W and T2W modalities for all
lobes (results not shown). These timing parameters plotted in Fig. 6 numeri-
cally exemplify differences seen in the appearance of white-gray matter contrast
between T1W and T2W modalities. Although absolute values of these parame-
ters might vary depending on the type of pulse sequence used, they can still be
compared to assess delay in appearance of adult-like white-gray contrast for cer-
tain modalities. The inflection point is particularly crucial to this finding since it
indicates the time taken to reach half the right asymptotic value (assuming that
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