In order to illustrate the robustness of our new proposed model further numerical results will be applied in a range of artificial, synthetic and real images, with different types of contours and shapes. Our work will also be compared with Badshah-Chen [12] model. We will see that for problems which can be solved by both models, our method is less dependent on the choice of regularized Heaviside functions while there exist some cases where the latter method does not work.
In our numerical experiments, we generally choose two image sizesn= 128,256 and the parameters as follows:
λ1G= 1, λ2G= 1, λ1 = 1, λ2= 1, λ3 = 1, τ = 4, h= 1 (the step space),
△t= 0.1 or 0.01 (the time step),α= 0.001,
µL= 0.4, µG = 0.4,µ1=n2/10,µ2 =n2/10
(if a given image has no noise, then allµparameters can be chosen smaller). The initial global level set, placed as a circle, has the form
ϕ0G= √ (x−x0 G)2+ (y−yG0)2−r 0 G,
where (x0G, yG0) is the centre of the circle, usually at the center of Ω and rG0 =n/5 the radius, and the initial local level set is placed similarly as
ϕ0L= √ (x−x0 L)2+ (y−yL0)2−r 0 L,
where (x0L, yL0) is the centre of the markers in set A and the radius r0L is the min- imum distance of the markers∑ rL0 = mina̸=b||pa−pb||, where pa, pb ∈ A; here x0L =
x-comp of markers no. of markers ,y 0 L= ∑ y-comp of markers
no. of markers . Since the approximated Heaviside functions can be grouped into two: big or small support in the interval [−ϵ, ϵ], see Chapter 3, we consider one for each group here.
H1ϵ(z) = 0 z <−ϵ 1 2 [ 1 +zϵ +π1sin(πzϵ )] |z| ≤ϵ 1 z > ϵ , δ1ϵ(z) = 0 z <−ϵ 1 2 [1 ϵ + 1 ϵ cos( πz ϵ ) ] |z| ≤ϵ 1 z > ϵ , H2ϵ(z) = 1 2(1 + 2 πarctan( z ϵ)), δ2ϵ(z) = 1 ϵπ(1 +x2/ϵ2),
As explained in §2.2.3, H2ϵ has a bigger support in the interval [−ϵ, ϵ], which means
that a moderately largeϵ may lead to spurious results.
In test comparisons, the initial local level set initialization and the choices of pa- rameters are the same for the Badshah-Chen method [12].
Note: As far as selective segmentation is concerned, easier problems refer to those images where the selective target is well separated from all other nearby features; in the extreme case where the separation distance is extremely large and the target feature is of a simple convex shape, one may even use the non-selective models such as [39, 92] by starting evolving contours near the geometric markers.
5.5.1 Test Set 1 — robustness of the new model
First we show some easier problems referring to those images where the selective target is well separated from all other nearby features. Numerical results of our new method for segmenting 8 different images will be shown.
The top left image in Figure 5.4 shows an image with many features where the spiral was the aim of detection. The top right image shows results of all features captured by our global level set, and the last images show the segmentation result of the spiral using 3 geometric markers.
In Figure 5.5, we test the model on a real CT image where the right kidney is to be selected; again the bottom two images show the correctly segmented organ, using 3 geometric markers.
Figure 5.6, shows three test results (of an artificial flower, the cameraman and a cell image) by our model; clearly our selection model delivers good results.
Finally Figure 5.7 shows three more results obtained from segmentation of images with strong noise or smooth contours. Again our model gives the correctly segmented results satisfying the expected selection requirement.
5.5.2 Test Set 2 — comparison of segmentation of easier problems For easy problems, cases where the separation distance is of more than 3 pixels away and the target feature is of a simple convex shape, one may rely on the Badshah-Chen [12] or Gout-Guyader [67, 69] models by starting evolving contours near the geometric markers. Here we compare our model with Badshah-Chen [12] for three easier problems as shown in Figures 5.8, 5.9, 5.10. Comparative results between [12] and [67, 69] can be found in [12]. For the test results in Figure 5.10, although both models give almost identical segmentation usingH1ϵ, the Badshah-Chen method is more sensitive to the ϵ
parameter choice for the regularized HeavisideH2ϵ; specifically our model would work
forϵ= 0.01, orϵ= 1 while the Badshah-Chen method must use the smaller parameter (otherwise redundant features are captured).
5.5.3 Test Set 3 — comparison of segmentation of harder problems In this set of 4 test problems we consider harder and more challenging cases. In this set the separation distance between features is small or the intensity difference between features and background is small. In these difficult cases, the previous models from [12, 67, 69] will not work. The experimental results from these methods will be shown in each of these examples. Figures 5.11, 5.12, 5.13 and 5.14 show four respective images and their segmented results of one feature; in each case, the top line of images shows the results of [12] which are not correct due to inclusion of redundant features in the selective segmentation and the bottom line shows the correctly segmented results by our new model. Clearly our model is robust.
5.5.4 Test Set 4 — necessity of a selection model
Here we show one final experiment of a selective segmentation model in clear contrast to other widely known methods for global segmentation.
In Figure 5.15, we compare three sets of usual segmentation results with our selective segmentation result. Here the image in Figure 5.15 (a) is the original image, given with the markers indicating where the feature is to be extracted. First (a) is segmented by
the Chan-Vese [39] algorithm to obtain the segmented image in Figure 5.15 (b). Then two cropped and smaller images (c)-(d) of Figure 5.15 (a) are respectively segmented to give the results in (e)-(f). Finally our proposed method gives the correctly segmented result in Figure 5.15 (g)-(h).
Clearly one observes the correct segmentation in such situations where selection is required and which only can be delivered by a selective model such as ours.