Test Set 1 — robustness of the 3-D dual level set selective method

6.4 A 3-D Dual Level Set Variational Model

6.5.1 Test Set 1 — robustness of the 3-D dual level set selective method

The ﬁrst set of tests respectively in Figures 6.10, 6.11, 6.12 and 6.13 all consist of artiﬁcial geometrical objects, similarly to the Badshah-Chen model, we can see that the new dual level set model gives the same results. However we can notice that the new model is less sensitive to the choice of parameters, especially forϵ.

6.5.2 Test Set 2 — comparison of 3-D segmentation of harder prob- lems

In this set of 5 test problems we consider harder and more challenging 3-D cases. In the above set of examples we observe that the target feature/object is well separated from the other nearby ones. A challenging test would be when the distance of the objects or the intensity difference from others is small (see failed Badshah-Chen and Gout-Guyader models, Figure 6.8). The first example shown in Figure 6.14 presents a correctly segmented result by the 3-D dual level set model for the object presented in Figure 6.8, that failed with the Badshah-Chen model. In medical imaging such as CT and MRI, often, the whole image has low contrast so this small intensity difference poses a great challenge to selective segmentation models. The Badshah-Chen and Gout-Guyader models give reasonable segmentation at early iterations, such as Figure 6.15, and totally fail in other cases, such as Figure 6.14, 6.16 and 6.17. All of these 4 examples are correctly segmented in Figure 6.14, 6.15, 6.16, and 6.17 respectively, using the 3-D dual level set selective segmentation. One more example showing the practical importance of accurate selective segmentation in 3-D measuring of medical data is shown in Figure 6.18 where the cancer detection in the brain has been successfully segmented.

(a) Various slices of the segmented object fromxy,yz,and

xzdirection

(b) Selected object with the new method

Figure 6.10: Set 1 Example 1: Successful detection of the pyramid in a clean and synthetic 3D image with 4 markers with 3-D dual level set method. Initial level set is a sphere within the center of the markers and n= 128, dt= 0.1, ϵ= 10−2 orϵ= 100 after 100 iterations.

slice number42x−y plane slice slice number42x−z plane slice

slice number42y−z plane image

(a) Various slices of the segmented object from

xy, yz, and xz direction after 100 iterations

slice number42x−y plane slice slice number42x−z plane slice

slice number42y−z plane image

(b) Various slices of the segmented object from

xy, yz, and xz direction after 300 iterations

Figure 6.11: Set 1 Example 2: Successful detection of one sphere out of two in a clean and synthetic 3-D image with 4 markers with 3-D dual level set method. Initial level set is a sphere within the centre of the markers and n = 128, dt = 0.1, ϵ = 10−2 or

(a) Tested object (b) Various slices of the segmented object fromxy,yz,and

xzdirection

Figure 6.12: Set 1 Example 3: Successful detection of the small cuboid in a clean and synthetic 3-D image with 4 markers with 3-D dual level set method. Initial level set is a sphere within the centre of the markers and n= 128, dt= 0.1, ϵ= 10−2 orϵ= 100 after 100 iterations.

(a) Given 3-D volumetric with 4 spheres

slice number64x−y plane slice slice number42x−z plane slice

slice number42y−z plane image

(b) Various slices of the segmented object fromxy,yz,and

xzdirection

Figure 6.13: Set 1 Example 4: Successful detection of one sphere out of 4 in a clean and synthetic 3-D image with 4 markers with 3-D dual level set method. Initial level set is a sphere within the centre of the markers and n = 128, dt = 0.1, ϵ = 10−2 or

(a) The given image (b) Various slices of the segmented object inxydirection with the new method

Figure 6.14: Set 2 Example 1: Successful detection of the prism in a clean and synthetic 3-D image with 6 markers with 3-D dual level set method while the Badshah-Chen [12] fails. Parameters aren= 128, dt= 0.1,ϵ= 10−2 _(or_ϵ_{= 10}0_).

6.5.3 Test Set 3 — useful applications of a 3-D selection model After a 3-D organ or object has been segmented with our selective segmentation method, we can look inside of this organ for a deeper segmentation. For example in the case of the kidney we can segment its inside structures and see what the veins looks like. We can do this by either the Chan-Vese model or Li-Kao-Gore-Ding model [91, 90]. The main diﬀerence is that the two piecewise constants c1 and c2 in Chan-Vese have been replaced by two piecewise smooth functions f1(x) and f2(x), ﬁtting the image intensities near each pointx in a local way. The experimental results show that both models succeed when the background intensity is equal to the mean of the object intensity, but if the background intensity is equal to 0, only Li-Kao-Gore-Ding model can succeed in both 2-D and 3-D cases, see Figure 6.19(d). In Figure 6.19 we show the results in a 2-D slice after the image has been processed with a selective segmentation model and the kidney has been captured. A full view of the 3-D case is shown in Figure 6.20 and Figure 6.21. Clearly our new selective model provides useful input to the deep segmentation models.

6.6 Conclusions

We have developed two new selective segmentation 3-D models for selective segmentation. The first model is an extension of the 2-D Badshah-Chen model based on a single level set function and the second new model is based on dual level set selective segmentation functions. The results indicate that the latter model is more robust or images with nearby objects having similar intensity values. The issues of refinements to the model to have more flexibility in geometric constraints and fast solution will be considered in Chapter 8.

slice number70x−y plane slice slice number79x−z plane slice

slice number84y−z plane image

(a) Various slices in xy, yz and xz direction after segmentation with Badshah- Chen [12]

(b) Badshah-Chen model [12] accurate segmentation but failed due to spurious detections

slice number70x−y plane slice slice number79x−z plane slice

slice number84y−z plane image

(d) Reasonable but inaccurate segmentation of Gout-Guyader model [69] due to noise

slice number70x−y plane slice slice number79x−z plane slice

slice number84y−z plane image

(e) Various slice in xy, yz and xz direction after segmentation with the new method

(f) Detected object with the new method

slice number140x−y plane slice slice number158x−z plane slice

slice number168y−z plane image

(a) Various slices inxy,yzandxzdirec- tion after segmentation with B-Ch [12]

(b) B-Ch model [12] failed to detect the object

slice number140x−y plane slice slice number158x−z plane slice

slice number168y−z plane image

(d) G-G model [69] failed to detect the object

slice number140x−y plane slice slice number158x−z plane slice

slice number168y−z plane image

(e) Various slice inxydirection (f) Detected object

Figure 6.16: Set 2 Example 3: Successful detection of the right kidney in a 3-D CT volume data with 3-D dual level set method. Initial level set is a polyhedron constructed from 5 given markers in 3 diﬀerent layers and n = 128, dt = 0.1, ϵ = 10−2 after 200 iterations.

slice number45x−y plane slice slice number50x−z plane slice

slice number60y−z plane image

(a) Failed segmentation with B-C model [12]

slice number45x−y plane slice slice number50x−z plane slice

slice number60y−z plane image

(b) Failed segmentation with G-G model [69]

(d) Successful detected object with dual level set

Figure 6.17: Set 2 Example 4: Successful detection of liver in a 3-D CT volume data with 3-D dual level set method, while the old models fail. Initial level set is a polyhedron constructed from 5 given markers in 4 diﬀerent layers and n= 256, dt= 0.1,ϵ= 10−2 after 200 iterations.

Figure 6.18: Set 2 Example 5: Slice in xy direction after segmentation of brain CT image with low glioma with 3-D dual level set. Successful detection of glioma with a polyhedron level set constructed with 4 given markers in each of the three diﬀerent layers using the 3-D Dual Level Set method. The parameters are n= 128, dt = 0.01,

ϵ= 10−2 after 100 iterations.

(a) Given selective image with background intensity equal to the mean of the object intensity

(b) Deep segmentation with Chan-Vese model

(d) Deep segmentation with Li- Kao-Gore-Ding model of the selected object with background intensity equal to 0

Figure 6.19: Set 3 Example 1: CT image post processed with Chan-Vese and Li-Kao- Gore-Ding model.

20 40 60 80 100 120 20 40 60 80 100 120

(a) 2-D slice of 3-D deep segmentation processed with C-V model

(b) 3-D shape of the deep segmentation with C-V model

(d) 3-D shape of the deep segmentation with Li-Kao-Gore- Ding model

Figure 6.20: Set 3 Example 1: CT 3-D volume image post processed with Chan-Vese and Li-Kao-Gore-Ding model. Blood ﬂow into veins has been distinguished.

(a) 3-D view of blood veins ﬂow segmentation of the kidney

Figure 6.21: Set 3 Example 2: 3-D volume image post processed with Chan-Vese and Li-Kao-Gore-Ding models. In red we can see the segmentation obtained which shows the blood ﬂow of the selected kidney.

Chapter 7

On a Variational Model for

Selective Image Segmentation of

Features with Inﬁnite Perimeter

The variational models already detailed in the previous chapters are based on the min- imization of the length of a feature’s boundaries (i.e. H1Hausdorﬀ measure). However there exist problems with irregular and oscillatory object boundaries, where minimiz- ing such a length is not appropriate, as noted by Barchiesi et al. [14] who proposed to minimize theL2 Lebesgue measure of the γ-neighbourhood of the boundaries instead. This chapter presents a dual level set selective segmentation model which improves the selective model in cases of real life images with oscillatory boundaries. We will show qualitative results demonstrating the eﬀectiveness of the proposed method for these cases.

7.1 Introduction

In this chapter we will show a way of improving the 2-D selective segmentation work already introduced in Chapter 5. The model deals with the challenging task of se- lecting one feature/object. For the dual level set model it has been noticed that for proper incorporation of some geometric prior information (markers) into the image, edge detection techniques and a proper ﬁtting term for the local level set could lead to good convergence to the target object. However, these models might not give good results in cases where the image has oscillatory boundaries since all techniques based on the weighted penalization term of the length can give rise to some diﬃculties for these cases.

In this chapter, we make use of theL2 Lebesgue measure of theγ-neighbourhood of the contour [14] as a penalization term instead of theH1Hausdorﬀ measure and propose a local selective segmentation model enabling us to extract features with irregular and oscillatory boundaries. The diﬀerence between these measures can be demonstrated with a simple example:

Given Ω = (0,1)2,u0(x, y) is deﬁned as

u0(x, y) = {

1 y <1 +x₂sin(1_x),

for all (x, y)∈Ω. In this case, the contour is given by Γ = (x, y) :y= 1 +x₂sin(1_x) as shown in Fig. 7.1. It can be easily noticed that L(γ-Γ) ≤ 1,∀γ > 0, while a simple calculation shows thatH1(Γ) =∞.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Figure 7.1: Illustration of an inﬁnite perimeter function.

The remainder of this chapter is organized as follows. In Section §7.2, we describe the Barchiesi-Kang-Le-Morini-Ponsiglione model [14], which uses theL2Lebesgue measure of theγ-neighbourhood of the contour as the regularization term of their variational model. A new dual level-set selective model, which uses the L2 Lebesgue measure of the γ-neighbourhood of the contour as a penalization term, is presented in Section

§7.3. Time marching and AOS techniques are developed for the numerical solution. In Section§7.4 the experimental results are presented based on artiﬁcial and real images. Conclusions are drawn in Section§7.5.

In document Variational models and numerical algorithms for selective image segmentation (Page 185-196)