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Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs cation of edges in a way that is robust with respect to parameter changes. In particular, within the four regions marked in Fig. 6.11 the expectation of the phase field is highly influenced by the choice of µ and ν as visible in Fig. 6.15. The blurred edge at position 1 can be seen in the expectation of the phase field only when we use a narrow phase field. In region 2, we have a different situation in which the edge can be identified using a widish phase field. In addition, the edges at positions 3 and 4 can be identified using adjusted parameters. However, note that one of these edges is not seen in the expectation of φ because of the particular choice of parameters; a high variance of φ indicates the possible existence of an edge. In particular, this is true for the regions 1 and 2. Moreover, the algorithm estimates the reliability of detected edges: A low expected value of the phase field and a low variance indicate that the edge is robust and not influenced by the noise and uncertainty of the acquisition process. This is true for the upper edges of the structure. In contrast, a high variance in regions with a high or low expected value of the phase field, e.g. the labeled regions 1–4, indicates regions where the detected edge is sensitive to the noise. In addition, we can easily extract the distribution of the gray values for any pixel location inside the image and the phase field from the polynomial chaos expansion obtained via the GSD method. Fig. 6.12 shows the PDF of a pixel from the phase field computed via the GSD. Adaptive Grids With a combination of the stochastic method and the adaptive grid approach described in Section 4.5, we decrease the memory requirements and increase the performance further. Fig. 6.17 shows the results of the adaptive method and compares them with the results from the uniform grid. For the computations, we used a threshold for the error indicator of ι = 0.005. Thus, elements where the error indicator S(x) is smaller than ι at every node are not further refined. The choice of a suitable value for the error indicator threshold is important, because too small values lead to unnecessary fine grids, whereas too high values for the threshold lead to coarse grids even close to edges. This causes over-shootings in the numerical computation of the phase field, i.e. the phase field has no longer values between zero and one. The results shown in Fig. 6.17 use a maximal level of 7, leading to a uniform grid of size 129 × 129 at the beginning. The final adaptive grid depicted on the right of Fig. 6.17 has about 70% fewer degrees of freedom, but yields nearly the same results. Fig. 6.17 shows no visible difference between the uniform grid and the adaptive grid solution. From Fig. 6.17 it is visible that the saturation condition, required to build admissible grids, leads to an area of increasing element size around detected edges. In flat regions, i.e. where the image gradient magnitude |∇u| is small, the elements are coarser compared to regions close to edges. Edge Continuity and Edge Consistency The combination of the stochastic Ambrosio-Tortorelli segmentation with an edge linking step is a great advance on the way to a detection and volumetry of objects in stochastic images. Fig. 6.16 shows the results of the stochastic extension of the edge linking step. We used α = 10, which turns out to be a good weighting between smoothing out unwanted edges and sharpening regions to get closed contours, respectively linked edges. We used s = 4, i.e. we used the two neighboring pixels in the directions perpendicular to the image gradient to compute the feedback measure ζ dc (ξ ). The figures indicate that the use of the modified diffusivity in the image equation of (6.16) leads to a better detection of closed contours in the stochastic images. These closed contours lead to cartoon-like smoothed images, because they avoid the smoothing over undetected edges. It is possible to combine the edge linking step with the adaptive grid approach, leading to a fast and accurate extension of the initial stochastic Ambrosio-Tortorelli model. The last column of Fig. 6.16 shows the result of such a combination. 76
6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images full grid adaptive grid full vs. adaptive grid E Image Var E Phase Field Var Figure 6.17: Comparison of the full grid and adaptive grid solution. The full grid and adaptive grid solution are visually identical, but the computation of the adaptive grid solution needs significantly less DOFs. Thus, it can be applied on high-resolution images. Conclusion We presented extensions of the random walker and the Ambrosio-Tortorelli model to stochastic images and applied the methods on different data sets. Especially, the intuitive visualization of the stochastic random walker method via the visualization of contour realizations and the objects volume PDF can be useful to convince the image processing community of stochastic modeling. Furthermore, we presented a detailed theoretical foundation of the stochastic Ambrosio-Tortorelli extension. The availability of the theoretical foundation along with the intuitive visualization of the results is the key to a widely accepted method in image processing. The acceleration of the algorithm via an adaptive grid approach and the integration of the edge linking step shows the potential of the proposed methods to be extended to the users’ needs easily. 77
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Segmentation of Stochastic Images u
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Abstract The task of segmentation,
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7.5 Segmentation of Stochastic Imag
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Chapter 1 Introduction The developm
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Figure 1.3: This thesis combines fi
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the presentation of the stochastic
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Chapter 2 Image Segmentation and Li
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Bibliography [147] J. Tryoen, O. Le
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