9.4 The Graph Construction
9.4.2 Construction of a Graph Line
Having found a starting node, the goal is to construct a smooth graph line which follows the center of the neuronal branch. This is done in an iterative fashion by selecting “successor” graph nodes (Figure 9.4) in the direction of the associated axes. As soon as a symmetry point is visited by the line construction algorithm, it is marked, such that multiple processing of the same points is avoided. If a symmetry point is marked but it is not chosen to be a “successor” graph node, it registers as its ”parents” the two nearest connected graph nodes between which
9.4 The Graph Construction 127
it lies. This symmetry point becomes then the “child” of its parents. A “good”
successor graph node fulfills the following conditions5:
Q0 AQ
0
AP
0
P0x
x rP Q
00
rP Q+AQ0 00
Figure 9.4: The Choice of the Successor Graph Node: Choose the best suc-cessor node (cf. Conditions C1 to C4), such that the last axial direction is closely followed. Bold arrows denote axial directions associated to symmetry points, bold lines mark the branch boundary and transversal branch sections, thin dashed lines are possible connections to symmetry points in the successor transversal sectioning planes.
The Conditions for the Choice of Graph Nodes:
The conditions for a good successor Q0 of the current graph nodeP0 are:
C1. Reliability of Axial Direction: The successorQ0 should have a reliable axial direction (low variance of direction), which is equivalent with a large norm AQ0 of its associated axial direction vectorAQ0.
C2. Consistency of axial directions: The angleαbetween the axial directionAQ0 of the successorQ0 and the axial directionAP0 of the current nodeP0 should be small (Figure 9.5.a).
C3. No abrupt direction changes: The angle ˆβ between the axial direction AP0 of the current node P0 and the next search direction (vector addition of rP0Q0 +AQ0 should be small (Figure 9.5.b);
C4. Prefered neighborhood size: Depending on the size and smoothness of the neuron, smaller or larger distances between graph nodes are more appropri-ate. Typically, for neurons with smooth branches, graph nodes should have
5Notation: As before, Pi, Qi ∈ IR3 represent points in 3D space, i.e. they are three dimensional coordinate vectors. For simplification of the notation, the arrows on top of the symbols are omitted. Arrows are only used for symbols denoting a direction in 3D space (for example axial directions, or displacement vectors).
128 Graph Construction
large distances between each other and therefore successors which are far-ther away should be prefered. For neurons with bended branches a graph with dense nodes is needed, in order to follow the more abundant changes in direction.
Figure 9.5: Criteria for the Choice of Successor Nodes:
a) Axial direction consistency (cf.
Condition C2); b) Smooth di-rectional changes (cf. Condi-tion C3).
AP α
x x
AP Q
0 0
0
AQ 0
P0
0
rP Q AP
AQ
x x
Q0
P0
0 0 0
β
0 0
AQ P Q 0
r +
a) b)
The following method is proposed for finding the next graph node.
Algorithm 9.6 (The Choice of Successor Graph Nodes)
1. Construct a set of equally spaced (at one voxel from each other) planes which are perpendicular to the axial direction of the current graph node P0 and which are laterally limited by the edges of the branch currently under consideration. All planes lie axially within a local neighborhood whose size Vδ δAP0 is proportional to the length of the current axial direction (i.e.
in regions of low axial direction variance, wider neighborhoods are consid-ered, cf. Equation 9.3). The proportionality constant δ is set by the user and should roughly be equal to the overall mean diameter of the neuronal branches (measured in voxels) in the image.
2. To comply with the four conditions stated above, the next graph node Q0 is chosen such that i) it is not yet marked (see Step 4) and ii) the validation function T is maximal w.r.t. Qi, where T is given by:
Tn,m,l,αˆ0(P0, Qi) = ( ˆαA
P0, AQi ≤αˆ0)· (9.6) min
sgn
cos ˆαA
P0, AQi
,sgn
cos ˆαA
P0,(rP0Qi+AQi)
· cos ˆαA
P0, AQiAQimΩn,l,αˆ0(AP0,(rP0Qi +AQi), rP0Qi),
Ωn,l,αˆ0(C, D, r) = (9.7)
αˆC, D ≤αˆ0·sgn
cos ˆαC, D cosnαˆC, Drl Q0 = arg max
Qi Tn,m,l,αˆ0(P0, Qi), where (9.8)
9.4 The Graph Construction 129
P0
Q0
Figure 9.6: Plane Clustering: Symmetry points (empty circles) lying in the transversal planes between two graph nodes P0 and Q0 (filled circles) are marked as “children” of the two nodes.
Qi are all symmetry points in the set of transversal planes,rP0Qi =Qi−P0 is the displacement vector of the pointQi to the current graph nodeP0, AQi, andAP0 are their associated axes, and αˆC, D is the angle between two vectors C and D. In Equation 9.6 the first term allows deviations of axes associated to potential “successor” graph nodes from the initial axial direction to lie only inside a cone (α0 ∈(0,π2)), whereas the second term prohibits backward connections. The cosine in the third term implements Condition C2, since it rewards smaller angles, and the length of the axial direction raised to the power ofm (m≥1)implements Condition C1 by amplifying the influence of longer axes. In Equation 9.7 the third term amplifies through n (n ≥1) the influence of smaller angles between the current node’s axial directionAP0 and the overall change of directionrP0Qi+AQi, implementing thus Condition C3.
The last term amplifies or lowers (for l ≥1 resp. for l ≤ −1) the influence of distance between two graph nodes and therefore implements Condition C4.
3. The successor node Q0 from Eq. 9.8 and its axial direction AQ0 become the current graph node and the current search direction in the next iteration.
4. Mark all symmetry points which belong to the transversal planes lying between Q0 and P0 and register Q−→0P0 as their “parent” segment. This avoids the multiple processing of a branch segment and the construction of spurious parallel graph segments (see Figure 9.6).
5. Repeat Steps 1 to 4 until either:
(a) All axial directions associated with the potential successor graph nodes lie outside the cone specified by the angle αˆ0 relatively to the current axial direction, or
(b) all potential successor graph nodes Qi have already been marked, or (c) no potential successor graph node is found in the current neighborhood.
Here ˆα0 = 2π5 is used, such that the cone’s angle is less than π2, and n = 3, which confers more importance to the deviation angle ˆβ between the current axial
130 Graph Construction
direction and the overall direction change, than to the new axial length (which has exponent m = 2). Finally, l = −1 makes the validation function to prefer nearer located points. The parameters’ influence on the behavior of the validation function is summarized in Table 9.1 below.
Table 9.1: The Parameters’ Influence on the Behavior of the Validation Function.
α0 = (0,π2) m= [1,∞) n= [1,∞) l ∈(−∞,−1]∪[1,∞) l ∈(−∞,−1] l∈(1,∞]
Increasing allowed devia-tions between the two node axes.
Increasing importance of the new axial direction.
Decreasing allowed devia-tions between the current axial direction and the new search direc-tion.
l = −∞
-maximum preference of nearer located successors, which is de-creasing with lower absolute values of l.
Increasing preference of farther located successors.
The stopping Condition 5a is fulfilled at varicosities, sharp bends, and branch-ing points and avoids the generation of false graph segments, due to tracbranch-ing in wrong axial directions (see Section 9.4.3 for the special treatment in these regions).
The stopping Condition 5b is typically fulfilled when an other graph segment is met. This case is discussed in detail in Section 9.4.4. Condition 5c appears due to missing symmetry points in edge gaps (at varicosities or regions of low image contrast) and leads to a premature end of the graph segment. A solution to this problem is given in Section 9.4.5.