Segmentation of Stochastic Images using ... - Jacobs University

List of Figures
6.2 Mean and variance of the probabilities for pixels to belong to the object. Furthermore,
we show in red Monte Carlo realizations of the object boundary sampled from
the stochastic result. A high variance indicates pixels where the gray value uncertainty
highly influences the result. For comparison we added a classical random
walker segmentation result in the last column. There the variance image is not available,
because the method acts on a classical image. . . . . . . . . . . . . . . . . . . 61
6.3 MC-realizations of the stochastic object boundary for the stochastic liver image segmented
with the stochastic random walker approach with β = 10. On the right we
highlight a region of the image, where the noise in the input image influences the
result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.4 PDF of the area of the segmented person from the street image for β = 25 (black)
and β = 50 (gray). From the PDF we judge the reliability of the segmentation, a
narrow PDF indicates that the image noise influences the segmentation marginally. . 63
6.5 Comparison of the discretization methods for the computation of the stochastic random
walker result to verify the intrusive discretization. The small difference between
the intrusive discretization via the GSD method and the two other sampling based approaches
might be due to the projection of the Laplacian matrix on the polynomial
chaos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.6 Input “doughnut” without noise (left) and noisy input image treated as expected value
of the stochastic image (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.7 Left: The object seed points (yellow) and background seed points (red) used as initialization
of the stochastic random walker method. Right: The MC-realizations of
the stochastic segmentation result differ significantly for different noise realizations. 66
6.8 The PDF for both possibilities of the volume computation, the summation of the
random variables (gray) and the thresholding (black). The true volume is 60 pixels. . 66
6.9 Structure of the block system of an SPDE. Every block has the sparsity structure
of a classical finite element matrix and the block structure of the matrix is sparse,
meaning that some of the blocks are zero. The sparsity structure on the block level
depends on the number of random variables and the polynomial chaos degree used
in the discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.10 Nonzero pattern of the SFEM matrix for the smoothed stochastic image using n = 5
random variables and a polynomial degree p = 3. A black dot denotes a block that
has a nonzero stochastic part, thus having the sparsity structure of a classical FEM
matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.11 Mean value of the three data sets used to demonstrate the stochastic Ambrosio-
Tortorelli method. For the second data set, we denoted image regions the text refers
to. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.12 PDF of a pixel from the phase field computed from the polynomial chaos expansion
of the pixel via a sampling approach. Although we use uniform basic random
variables for the polynomial chaos, the resulting random variables have skewed and
Gaussian like distributions due to the use of higher order polynomials in the basic
random variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.13 Segmentation result of the street scene. On the left we show the five samples the
stochastic input image is computed from. On the right we compare the results computed
via the GSD method and a Monte Carlo sampling. . . . . . . . . . . . . . . . 73
6.14 Expected value and variance of the stochastic input image of the street scene. . . . . 73
6.15 Mean and variance of the image and phase field for varying ε and µ using the US
data. For comparison, we added the result from the deterministic method applied on
the mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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