Hi

Vector Diffusion

Segmentation Map

Region Map

Region Gradient Map

Region Force Vector Field

Figure 10.14: RAGS processing schema (color slide).

Initial Snake

Signed Distance Transfoim

Initial Level Sets

Evolve Level Sets r

Initial Snake

Signed Distance Transfoim

Initial Level Sets

Evolve Level Sets

Final Level Sets

Extract Zero Level Set

Converged RAGS

images where there are weak edges or noisy regions preventing the aforementioned snakes to perform at their best. Although GGVFs have been reported only using gray level image gradients, we can also apply them to "color" gradients (obtained as described in Section 10.6), which allows direct comparison with the color RAGS. It must also be noted that the GGVF can sometimes perform better than we have shown in some of the following examples as long as it is initialized differently, i.e. much closer to the desired boundary. In all the experiments, we have initiated the geometric, GGVF, and RAGS snakes at the same starting position, unless specifically stated.

10.9.1 Preventing Weak-Edge Leakage

We first illustrate the way weak-edge leakage is handled on a synthetic image. The test object is a circular shape with a small blurred area on the upper right boundary as shown in Fig. 10.15.

The standard geometric snake steps through the weak edge because the intensity changes so gradually that there is no clear boundary indication in the edge map. The RAGS snake converges to the correct boundary since the extra diffused region force delivers useful global information about the object boundary and helps prevent the snake from stepping through. Figure 10.16 shows, for the test object in Fig. 10.15, the edge map, the stopping function g( ), its gradient magnitude |Vg( )|, the region segmentation map S, and the vector map of the diffused region force R.

Figure 10.15: Weak-edge leakage testing on a synthetic image. Top row: geodesic snake steps through. Bottom row: RAGS snake converges properly using its extra region force.
Figure 10.16: Diffused region force on weak edge. From left: the edge map, the stopping function g( ) of edge map, the magnitude of its gradient Vg( ), the region segmentation map, and the vector map of the diffused region force R.

10.9.2 Neighboring Weak/Strong Edges

The next experiment is designed to demonstrate that both the standard geometric snake and the GGVF snake readily step through a weak edge to reach a neighboring strong edge. The test object in Fig. 10.17 contains a prominent circle inside a faint one. The presence of the weaker edge at the outer boundary is detected only by the RAGS snake. The geodesic snake fails because the weaker outer boundary allows the whole snake to leak through (similar to but in the opposite direction of propagation in Fig. 10.15). The GGVF snake fails due to the strong gradient vector force caused by the inner object boundary. Practical examples of this can also be observed in most of the real images shown later, such as Figs 10.20 and 10.26.

10.9.3 Testing on Noisy Images

We also performed comparative tests to examine and quantify the tolerance to noise for the standard geometric, the geometric GGVF, and the RAGS snakes. For this a harmonic shape was used as shown in Fig. 10.18. It was generated

Figure 10.17: Strong neighboring edge leakage. From left: initial snake, geodesic snake steps through weak edge in top right of outer boundary, GGVF is attracted by the stronger inner edge, and RAGS snake converges properly using extra region force.

Figure 10.18: A shape and its boundary (a harmonic curve).

where r is the length from any edge point to the center of the shape, a, b, and c remain constant, and m can be used to produce different numbers of 'bumps'; in this case m = 6. We added varying amounts of noise and measured the accuracy of fit (i.e. boundary description) after convergence. The accuracy was computed using maximum radial error (MRE), i.e. the maximum distance in the radial direction between the true boundary and each active contour.

Impulse noise was added to the original image from 10% to 60% as shown in the first column of Fig. 10.19. The region segmentation data used for RAGS is in the second column (without any post-processing to close gaps, etc.). The third, fourth, and fifth columns show the converged snake for the standard geometric, the GGVF, and RAGS snakes respectively. A simple subjective examination clearly demonstrates the superior segmentation quality of the proposed snake. The initial state for the standard geometric and RAGS snakes is a square at the edge of the image, while for the GGVF it is set close to the true boundary to ensure better convergence. At low percentages of noise, all snakes could find the boundary accurately enough. However, at increasing noise levels (>20%), more and more local maxima appear in the gradient flow force field, which prevent the standard geometric and GGVF snakes from converging to the true boundaries. The RAGS snake has a global view of the noisy image and the underlying region force pushes it toward the boundary. The MRE results are shown in Table 10.1. These verify RAGS error values to be consistently and significantly lower than the other two snake types for noise levels > 10%.

10.9.4 Results on gray level images

Figures 10.20-10.22 demonstrate RAGS in comparison to the standard geometric and GGVF snakes on various gray level images. Figure (10.20) shows a good example of weak-edge leakage on the lower side of the object of interest. While

Figure 10.19: Shape recovery in noisy images. (Column 1) original image with various levels of added Gaussian noise [0%, 10%,..., 60%], (column 2) the region maps later diffused by RAGS, (column 3) standard geometric snake results, (column 4) GGVF snake results, and (column 5) RAGS results.

Table 10.1: MRE comparison for the harmonic shapes in

Fig. 10.19

Table 10.1: MRE comparison for the harmonic shapes in

Fig. 10.19

Standard geometric

GGVF

RAGS

% noise

snake error

snake error

snake error

0

2.00

2.00

2.00

10

2.23

2.24

2.00

20

5.00

7.07

4.03

30

10.00

16.03

3.41

40

16.16

21.31

5.22

50

15.81

21.00

5.38

60

28.17

20.10

5.83

RAGS does extremely well here, the geometric snake leaks through and the GGVF snake leaks and fails to progress at all in the narrow object. In Fig. 10.21, RAGS achieves a much better overall fit than the other snakes, particularly in the lower regions of the right-hand snake and the upper-right regions of the left-hand snake. In Fig. 10.22, again RAGS manages to segment the desired region much better than the standard geometric and the GGVF snakes. Note the stan-

Figure 10.20: Brain MRI (corpus callosum) image. Top row: initial snake, standard geometric snake. Bottom row: GGVF snake and RAGS snake (original image courtesy of GE Medical Systems).
Figure 10.21: Heart MRI image. Top row: initial snakes and standard geometric snakes. Bottom row: GGVF snakes and final RAGS snakes showing improvement on the top right of the left snake and the lower region of the right snake.

dard snake leaks out of the object, similar to the effect demonstrated with the synthetic image in Fig. 10.15.

10.9.5 Results on Color Images

We now consider the performance of the RAGS snake on color images. In Fig. 10.23 we can see a cell image with both strong and fuzzy region boundaries. Note how the fuzzy boundaries to the right of the cell "dilute" gradually into the background. So the results in the top-right image again demonstrate an example of weak-edge leakage, similar to the example in Fig. 10.22, where the standard geometric snake fails to converge on the outer boundary. The middle and bottom rows show the converged RAGS snake using the oversegmen-tation and undersegmentation color region maps produced by the mean shift algorithm.

A very similar example is demonstrated in Fig. 10.24 in application to images of the optic disk in which the blood vessels have been removed using color mathematical morphology techniques. Again, the failing performance of the standard

Figure 10.22: Heart MRI image. Top row: initial snake, and standard geometric snake. Bottom row: GGVF snake and final RAGS snake showing better convergence and no leakage (original image courtesy of GE Medical Systems).

snake is shown along with the RAGS results on both oversegmentation and undersegmentation regions.

In Fig. 10.25, a full application of RAGS is presented where the resulting regions from the RAGS snake are quantitatively evaluated against those hand-labeled by an expert ophthalmologist. The first column represents these groundtruth boundaries. The second column shows the position of the starting RAGS snakes. The boundary of the optic disk is quite fuzzy and well blended with the background. The region force helps the proposed snake stop at weak edges while the standard geometric snake leaks through (as shown in Fig. 10.24) and the accuracy of the GGVF snake is highly dependent on where the initial snake is placed (hence GGVF snake results are not provided). The last two columns illustrate the RAGS results using oversegmented and undersegmented regions of the mean shift algorithm respectively.

A simple measure of overlap is used to evaluate the performance of the RAGS snake against its corresponding groundtruth:

Figure 10.23: Weak-edge leakage testing. Top row: original image with starting contour and geodesic snake which steps through. Middle row: oversegmentation color region map and converged RAGS snake. Bottom row: undersegmentation color region map and converged RAGS snake (original image courtesy of Bristol Biomedical Image Archive, Bristol University, UK) (color slide).

Figure 10.23: Weak-edge leakage testing. Top row: original image with starting contour and geodesic snake which steps through. Middle row: oversegmentation color region map and converged RAGS snake. Bottom row: undersegmentation color region map and converged RAGS snake (original image courtesy of Bristol Biomedical Image Archive, Bristol University, UK) (color slide).

where A and B correspond to ground-truth and RAGS localized optic disk regions respectively, and n( ) is the number of pixels in a region. Table 10.2 shows the result of measurement M demonstrating a 91.7% average performances for both over/undersegmentation RAGS respectively.

The final example in Fig. 10.26 shows a darker cell center compared to the cell outer region, but more significantlythe object of interest is surrounded by

Table i0.2: Quantitative evaluation of RAGS snake on the optic disks in i0.25

Image

1

2

3

4

5

6

Average

% RAGS (over)

91.4

90.0

91.9

93.1

93.1

90.5

91.7

% RAGS (under)

90.7

89.5

93.1

91.3

93.0

92.7

91.7

other strong features. The standard geometric snake splits and converges unsatisfactorily and the GGVF snake is pulled in and out by the stronger inner cell nucleus and neighboring cells respectively, while the RAGS snake converges well to the outer cell boundary without leaking through.

Figure i0.24: Optic disk localization. Top row: initial contour and geodesic snake which steps through to the stronger central region. Middle row: overseg-mentation color region map and final RAGS snake. Bottom row: undersegmen-tation color region map and final RAGS snake (color slide).

Figure 10.25: RAGS segmentation comparison with ground-truth. (Column 1) ground-truth, (column 2) initial snakes, (column 3) RAGS results with over-segmentation, and (column 4) RAGS results with undersegmentation (color slide).

Figure 10.26: Cell with strong nucleus feature. Top row: initial snake and standard geometric snake. Bottom row: GGVF snake and RAGS snake showing how the stronger inner edge in the cell nucleus does not cause it to lose the outer weaker edge (original image courtesy of Bristol Biomedical Image Archive, Bristol University, UK) (color slide).

Figure 10.26: Cell with strong nucleus feature. Top row: initial snake and standard geometric snake. Bottom row: GGVF snake and RAGS snake showing how the stronger inner edge in the cell nucleus does not cause it to lose the outer weaker edge (original image courtesy of Bristol Biomedical Image Archive, Bristol University, UK) (color slide).

All the examples shown here illustrate the resilience of RAGS to weak edges and noise. However, the RAGS snake does suffer from some shortcomings. As with the standard geometric snake, or the geometric GGVF snake, it will not perform well in highly textured regions in which the gradient flow forces may be hampered by multitudes of texture edge information. It is also dependent on a reasonable segmentation stage, although this was shown to be quite flexible using a popular method of image segmentation.

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