Segmentation of Stochastic Images using ... - Jacobs University
Segmentation of Stochastic Images using ... - Jacobs University
Segmentation of Stochastic Images using ... - Jacobs University
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List <strong>of</strong> Figures<br />
6.16 Comparison <strong>of</strong> the stochastic Ambrosio-Tortorelli model (left column) with the extended<br />
model <strong>using</strong> the edge linking procedure described in Section 2.3.3 (middle<br />
column) and a combination <strong>of</strong> the edge linking and adaptive grid approach (right<br />
column). Note that these results are computed with the same parameter set. The<br />
differences in the results are due to the additional edge linking parameter c only. . . . 75<br />
6.17 Comparison <strong>of</strong> the full grid and adaptive grid solution. The full grid and adaptive<br />
grid solution are visually identical, but the computation <strong>of</strong> the adaptive grid solution<br />
needs significantly less DOFs. Thus, it can be applied on high-resolution images. . . 77<br />
7.1 <strong>Stochastic</strong> level sets do not have a fixed position where φ(x) = 0. Instead, there is<br />
a band with positive probability that the level set is equal to zero, i.e. the position <strong>of</strong><br />
the zero level set is random and it is possible to estimate the PDF <strong>of</strong> the interface<br />
location in the normal direction <strong>of</strong> the expected value <strong>of</strong> the interface (lower right<br />
corner). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82<br />
7.2 Comparison <strong>of</strong> expected value and variance <strong>of</strong> the resulting phase field for the cosine<br />
test <strong>of</strong> (7.18) <strong>using</strong> the polynomial chaos (PC), stochastic collocation (SC), Monte<br />
Carlo simulation (MC), and Monte Carlo simulation <strong>of</strong> the original level set equation<br />
(MCL). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85<br />
7.3 Comparison <strong>of</strong> the expected value and variance <strong>of</strong> the resulting phase field for the<br />
rarefaction fan and the shock, two classical tests for level set propagation. The figure<br />
shows the comparison <strong>of</strong> the four discretizations <strong>of</strong> the stochastic phase field equation. 86<br />
7.4 Expected value color-coded by the variance for the Stanford bunny after shrinkage<br />
under an uncertain speed in the normal direction. Red indicates regions with a high<br />
variance and green regions with low variance. In addition, we show one slice <strong>of</strong> the<br />
variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87<br />
7.5 Mean <strong>of</strong> the CT data set (left) and the liver data set (right) for the segmentation test. . 88<br />
7.6 Left: Mean contour during the evolution <strong>of</strong> the stochastic level set. The iso-contours<br />
are drawn on the variance image <strong>of</strong> the final, magenta contour. The contour detection<br />
is influenced by the image noise on the bottom and the right <strong>of</strong> the object (high<br />
variance). Right: Contour realizations <strong>of</strong> the stochastic gradient-based segmentation<br />
<strong>of</strong> the CT data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89<br />
7.7 Resulting image with the expected value <strong>of</strong> the contour (red) <strong>of</strong> the segmented object<br />
and the phase field variance with the expected value <strong>of</strong> the contour for gradientbased<br />
segmentation <strong>of</strong> a stochastic CT image. The variance is constant in the normal<br />
direction <strong>of</strong> the expected value <strong>of</strong> zero level set. . . . . . . . . . . . . . . . . . . . . 90<br />
7.8 Mean and variance <strong>of</strong> the stochastic geodesic active contour segmentation <strong>of</strong> the<br />
stochastic CT data set. The variance is constant in the normal direction <strong>of</strong> the zero<br />
level set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91<br />
7.9 Left: Evolution <strong>of</strong> the expected value contour <strong>of</strong> the stochastic geodesic active contour<br />
method. The shown variance corresponds to the contour after 240 iterations (the<br />
magenta contour). Right: Mean value <strong>of</strong> the stochastic image to be segmented and<br />
the contours at time points <strong>of</strong> the level set evolution. The final contour matches the<br />
object boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92<br />
7.10 Mean (left) <strong>of</strong> the stochastic CT and the variance (right) <strong>of</strong> the stochastic Chan-Vese<br />
solution. Additionally, we show the expected value contour at different time steps. . . 93<br />
7.11 Left: MC-realizations <strong>of</strong> the stochastic contour from the stochastic Chan-Vese segmentation<br />
applied on the CT data set. Right: Realizations <strong>of</strong> the stochastic contour<br />
from the stochastic geodesic active contour approach applied on the CT data set. . . . 94<br />
114