Keypoints Definition with a Path Score

The score PS of a path γ = (Γ, r) is obtained by averaging the vesselness map V_nesscomputed from the optimally oriented flux filter proposed byLaw and Chung

(2008), see (3.3) for details. One has

The threshold parameter Th>0 and selectorδ(·,·) are used to eliminate irrelevant parts of the path

Our algorithm combines three main ingredients:

• The classical keypoint searching scheme (Benmansour and Cohen, 2009);

• State-of-the-art anisotropic Fast Marching method (Mirebeau, 2014a) with integrated geodesic curve length computation;

• Vessel tree extraction based on original and new keypoints selection criteria as well as the stopping criteria.

In practice, our keypoints detection method is embedded within the inner loop of the fast marching method, which it augments with several robust criteria for keypoints detection, adaptation of a set of path score thresholds, and termination.

Keypoints Selection Method

The approximated geodesic distance U to the currently extracted tree structure is estimated using the fast marching algorithm update scheme: initialization and steps 2-12of the loop in Algorithm 3.

Following the dynamic programming principle, image pixels are tagged as either Trial or Accepted. The Trial point ˆxmin currently minimizing U is tagged as Accepted (i.e. frozen), and the value U(ˆy) at neighbouring points ˆy is suitably updated. In addition, line10, we estimate the geodesic curve lengthU(ˆy) and Eu-clidean curve length L(ˆy), and potentially tag ˆy asTrial. The original keypoints selection method (KPSM) (Benmansour and Cohen, 2009) adds the currently ac-tive point (the latest Accepted point) ˆxmin to the set K of keypoints as soon as L(ˆxmin) ≥ λ, where λ is the user chosen Euclidean curve length threshold. The reason of choosing such a point as keypoint is that among all the points with the

(a) (b) (c)

Figure 3.3: Steps of keypoints searching scheme. (a) The first keypoint is found.

(b) Two keypoints are found. (c) Seven keypoints are found. Cyan contours are the boundaries of the tubular structure and red lines are the centrelines. (d)-(f ) are the

optimal distance maps Uopt for corresponding to the images of (a)-(c).

same minimal action map value, a point ˆqglobally maximizing the Euclidean curve length map L will be located in the centreline of the tubular structure (Benman-sour and Cohen, 2009; Chen et al.,2014; Kaul et al., 2012; Li et al., 2009).

Keypoints Searching Scheme Based on A Path Score

A point ˆx = (x, r) is marked as a new keypoint if its Euclidean curve length satisfies L(ˆx)2[λ,3λ), and if additionally it obeys

minn

PS(x,Thi−1),V_ness(x)o

>Thi, (3.27)

where V_ness is the vesselness map. The key idea behind this selection criterion is that, among all points for which the action map U is within a given bound, the point which maximizes the Euclidean curve length map L should be inside the tubular structure. Indeed the large ratio L(ˆx)/U(ˆx) reflects the fact that the geodesicCxˆ, joining ˆxto a previous keypoint, has a small actionRAR(C(·),C⁰(·)) in average. Thus, by construction of the metric, this geodesic must lie on the vessel centrelines and be aligned with the vessel orientations.

Here we consider the minimal action map U for a tubular structure tree with the initial point at the root of this tree. For all the points ˆy in the level set C ={yˆ| U(ˆy) = c}wherec > 0 is a constant, there may be many local maximums ofL, some of which are the intersections of the level setCand the tubular structure branch centrelines. Thus we select the maximal local maximum which is inside a tubular structure judging by the equation (3.27). In our keypoints searching method, one or several path score thresholds T h1 ≥T h2 ≥ · · · ≥T hk are given as input; generally k 2 {1,2}, and by convention T h0 = +1. If the currently active point ˆxmin has an excessive estimated curve length L(ˆxmin) ≥ 3λ, see line 13, then the next threshold T hi+1 is activated in order to discover finer, less visible vessel structures; unless i = k in which case the method ends. If the currently active point ˆx_min has an appropriate estimated curve length λ  L(ˆx_min) < 3λ, then it is considered as a candidate new keypoint (i.e. a potential node of the vessel tree structure), see line 23. We extract the geodesic C^xˆ linking ˆx to the already extracted tree structure, and evaluate its relevance using a path score.

The selection test line 25 requires the path score to exceed the current selection threshold T hi. The selectorδ(T,·) appearing in (3.25) is applied with T :=T hi−1

so as to push the keypoint selection towards finer and less visible structures, when i > 1, and leave the vicinity of the tree extracted with the previous threshold T hi−1. Using a hierarchy of successive thresholds, an anisotropic metric, allows in the end to reliably handle a much smaller curve length threshold λ than the inspirational KPSM (Benmansour and Cohen, 2009), without leaking outside of the vessel or tubular structure, but staying right in its centreline. In Fig. 3.3, we show the keypoint searching results using the proposed method. In this figure, the green dot is the user-input initial source point, red dots indicate the searched keypoints with two path score thresholds. Cyan contours are the boundaries of the tubular structures. In Figs. 3.3a to 3.3c, we show one, two and seven keypoints with the corresponding centrelines and contours. In Fig.3.3d to3.3f, we show the optimal minimal action map Uopt of U with respect to Figs. 3.3a to 3.3c, where Uopt is defined as

The keypoint selection process has to be stopped after all vessel branches have been explored, but before spurious artifacts appear in the reconstructed vessel tree, which requires an adequate stopping criterion. The proposed algorithm ter-minates when all provided path score thresholds (Thi)^k_i=1 have been used, and the value of Euclidean curve length is L(ˆxmin) > 3λ, where ˆxmin is the latest ac-cepted point in the Fast Marching propagation. In Fig.3.4, we show the complete keypoints searching result. Green point is the initial source point. Red points

Figure 3.4: Keypoints searching result with two path score thresholds. Green dot indi-cates the initial source point, red dots are the keypoints searched using large path score threshold and blue dots are the keypoints obtained using small path score threshold.

are the keypoints with respect to path score threshold T h1. Blue points are the keypoints associated to the path score threshold T h2. From Fig. 3.4, we can see that no leakage occurs, even on weak branches which are reliably extracted by our algorithm.

Semi-Automatic Parameter Setting

The proposed algorithm requires a few parameters: a curve length threshold λ, and a collection of thresholds (T h_i)^k_i=1, one initial source point. The curve length threshold λ should be slightly more than twice the largest radius of the vessels to be detected. We use one or two path score thresholds, depending on the test case, which are automatically selected as quantiles of the vesselness map V_ness distribution on image pixels. Finally, the initial source point used in the Fast Marching algorithm can be user provided or automatically selected as the point which maximizes the value of the vesselness map V_ness (3.3).

(a) (b)

(c) (d)

Figure 3.5: Comparison between our algorithm and the classical KPSM. (a) and (b) are the results of KPSM with curve length threshold 26 and 60, respectively. (c) and

(d) are the results of our algorithm with 26 and 60, respectively.