Numerical Implementation based on the Anisotropic Fast Marching Method
3.5.3 Region-Constrained Minimal Path Model
The goal in this section is to find a minimal cost path inside a given prior region instead of the whole image domain Ω. For this purpose, we introduce some no-tations. Given an open subset U ✓ Ω ⇢ R2, bounded and connected, one has a new radius-lifted domain ˆU = U ⇥[Rmin, Rmax] ( [Rmin, Rmax] is the admissi-ble radius space ). Let AUˆ be the collection of all the regular radius-lifted paths γc : [0,1]!Uˆ:
AUˆ :=7
γc; γc : [0,1]!U , γˆ c(0) = ˆs, γ(1) = ˆx .
• Minimal action mapUd and geodesic Cs,p.
5: Recover the minimal pathCs,p.
6: Set IfStop True.
7: end if
8: Find the back-tracked point z of xmin by (3.29).
9: Compute Dc(xmin) and Dd(xmin) by (3.30) and (3.31) respectively.
15: Calculate the dynamic tensorMd(y) by (3.32).
16: Compute Unew(y) by solving the Hopf-Lax operator (2.86).
17: if Unew(y)<U(y) then
18: U(y) Unew(y).
19: end if
20: end for
21: end while
The geodesic energy Lc with respect to the radius-lifted metric RAR (2.61) and the constrained domain ˆU can be defined as
Lc(γ) :=
Algorithm 5 Anisotropic Fast Marching method with Constrained Region Input:
• Radius-lifted tensor field Mr.
• Constrained domain ˆU.
• Initial source point ˆs.
• Local stencil S.
1: while at least one grid point is tagged asTrial do
2: Find ˆxmin, the Trial point which minimizes Uc.
3: Set V(ˆxmin) Accepted.
4: if xˆmin 2Uˆ then
5: for All ˆysuch that ˆxmin 2S(ˆy) and V(ˆy)6=Accepted do
6: Compute Unew(ˆy) using Hopf-Lax update in (2.86).
7: if V(ˆy)6=Trial then
Therefore, the minimal action map Uc with respect to the initial source point ˆ
s2Uˆ, defined over the whole radius-lifted domain ˆΩ can be expressed as:
Uc(ˆx) :=
Numerically, the minimal action map Uc can be naturally computed by the fast marching method with a freezing scheme as described in Algorithm5. Comparing against the regular fast marching method, the freezing scheme-based method set the value of the latestAccepted point as1if this point is outside the radius-lifted constrained region ˆU.
Figure 3.12: Steps for the proposed Retinal vessel extraction method. (a) Minimal path (red curve) by using the dynamic Riemannian metric. (b)Tubular neighbourhood region of the minimal path shown in (a). (c) Minimal path extraction result obtained using the region-constrained minimal path model, where the cyan curve denotes the
centreline and red curve denotes the vessel boundary.
(a) (b) (c)
Figure 3.13: Centreline bias correction. (a)Minimal paths extracted by the proposed region-constrained minimal path model and the dynamic metric model, where the results are indicated by cyan solid line and red dash line respectively. In this figure, we only demonstrate the physical path of the obtain radius-lifted minimal path. (b) and (c)
Details for both the minimal paths in (a).
Summarily, the proposed interactive vessel extraction method can be decomposed to two steps. The first step is to get the rough minimal path C which pass the vicinity of the true centreline of the expected vessel as demonstrated in Fig.3.12a.
Next step is to build the tubular neighbourhood U of the just computed minimal path C by dilation operator and perform the region-constrained minimal path model to obtain both the centreline and boundaries of the expected vessel. The neighbourhood U is shown in Fig. 3.12b and the final vessel extraction result is demonstrated in Fig. 3.12c.
Note that the region-constrained minimal path model can be considered as the refined procedure of the minimal path obtained by using the dynamic Riemannian metric. Since the inhomogeneous vesselness distribution along the desired vessel, the extracted minimal path using the feature consistency penalized dynamic Rie-mannian metric may result in centreline bias as shown in Fig. 3.13. In this figure, a shows two minimal paths: one is the path extracted using the dynamic metric
Figure 3.14: Comparative Retinal vessel extraction results by the Benmansour-Cohen model and the proposed model. Blue crosses and black stars indicate the initial source points and the end points, respectively. Column 1 show the vessel extraction results by the Benmansour-Cohen model. Column 2show the vessel extraction results by the proposed dynamic anisotropic Riemannian metric. Column 3 show the refined results
by the proposed region-constrained minimal path model.
indicated by red dash line and another is the path obtained using the region-constrained minimal path mode (cyan solid line). In Figs.3.13b and3.13c we give the details of the centreline bias.
In Fig. 3.14, more comparative retinal vessel extraction results for the classical Benmansour-Cohen model (Benmansour and Cohen, 2011) and the proposed two-step method is given. The left column of Fig. 3.14 shows the vessel extraction results from the Benmansour-Cohen model where the green curves denote the centrelines and yellow contours denote the boundaries of the retinal vessels. In the middle column, we show the minimal paths extracted by the dynamic Riemannian metric where the paths are indicated by red dash lines. The right column present
the final results from the region-constrained minimal path model where the cyan lines indicate the retinal vessel centrelines and red contours denotes the boundaries of the vessels. From this figure, we can see that the proposed two-step method can obtain the desired results which completely avoid the short branches combination problem.
The proposed minimal path model is evaluated on the Test set of the DRIVE dataset (Staal et al.,2004), which includes 20 retinal images. We choose total 110 vessels which start from the optic disk of the retinal images or those cross another vessel. If the extracted minimal path exactly follows the desire vessel, we consider this is a positive extraction (PE), otherwise a negative extraction (NE). For the proposed two-step vessel extraction method, the number of PE = 102 out of total 110 vessels. For Benmansour-Cohen model, this number is 57. Additionally, we compute the following measures:
• NFT: the number of vessels that the proposed method positively extracts and the B-C model fails.
• NTF: the number of vessels that the Benmansour-Cohen model positively extracts and the proposed fails.
• NFF: the number of vessels that both models fail to extract.
• NTT: the number of vessels that both models positively extract.
In Table 3.1, we show these four measures mentioned above. It can be seen that the proposed two-step model obtains better results than the Benmansour-Cohen model (102 against 57).
3.5.4 Conclusion
We present a two-step method for interactive retinal vessel extraction including both the centreline and boundary. In the first step, we invoke a feature consistency penalized dynamic metric to find the rough centreline of the targeted vessel. Then a region-constrained minimal path model is applied to get the extraction results including both accurate centreline and boundary of the vessel.