Analysis and Algorithmic Generation of Hepatic Vascular Systems

Table 4

The tables show the invariance analysis for two example features, edge radii and the bifurcation exponent 𝛾. In (a), we consider the one-population case of clinical PV, and (b) treats the two-population case clinical versus corrosion cast PV. The average number of trees as well as the average number of edges per tree is those considered for the respective feature and may thus vary due to geometric degeneracies. We moreover list the averaging weights 𝑤_𝑠 from (3) and (5), respectively, to determine the average invariance 𝑚 from the individual invariances 𝑚_𝑓,𝑠 for the different Strahler* orders 𝑠. The plots show the empirical cumulative distribution functions of the features, red for the clinical and green for the corrosion cast data sets.

(a) Detailed similarity analysis for clinical PV

Strahler* order 𝑠	0	1	2	3	4	5	6
𝑚edgeradii𝑟(𝒯)=0.270
No. of trees	167	166	166	164	149	62	5
avg. no. of edges	5.365	16.771	31.627	67.329	134.134	277.323	408.200
𝑤𝑠(𝒯)	0.045	0.139	0.263	0.553	0	0	0
𝑚edgeradii,𝑠(𝒯)	0.682	0.428	0.301	0.183	0.114	0.081	0.100
𝑚𝛾(𝒯)=0.895
No. of trees	161	166	164	149	62	5	0
avg.  no. of edges	3.627	11.711	23.939	53.966	123.613	193.200	0
𝑤𝑠(𝒯)	0.090	0.301	0.608	0	0	0	0
𝑚𝛾,𝑠(𝒯)	0.986	0.928	0.865	0.688	0.470	0.400	n/a

(b) Detailed similarity analysis for clinical PV/corrosion cast PV

Strahler* order 𝑠	0	1	2	3	4	5	6
𝑚edgeradii𝑟(𝒯0𝒯1)=0.174
No. of trees	167,7	166,7	166,7	164,7	149,7	62,0	5,0
avg.  no. of edges	5.365,6.286	16.771,39.000	31.627,70.429	67.329,158.714	134.134,369.571	277.323,0	408.200,0
𝑤𝑠(𝒯0,𝒯1)	0.032	0.141	0.260	0.567	0	0	0
𝑚𝑟,𝑠(𝒯0,𝒯1)	0.640	0.297	0.243	0.085	0.009	0.005	0
𝑚𝛾(𝒯)=0.684
No. of trees	161,5	166,7	164,7	149,7	62,7	5,0	0,0
avg.  no. of edges	3.627,4.200	11.711,24.000	23.939,48.000	53.966,107.000	123.613,275.857	193.200,0	0,0
𝑤𝑠(𝒯0,𝒯1)	0.060	0.312	0.627	0	0	0	0
𝑚𝛾,𝑠(𝒯0,𝒯1)	0.974	0.870	0.563	0.319	0.177	0.080	n/a