L Measure functions

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About L-Measure

L-Measure is a research tool for obtaining quantitative morphological measurements from neuronal reconstructions. There are 43 L-Measure functions including area measurements of soma, length measurements of processes, angular twists and turns of processes, calculation of the number of terminals and bifurcations, and many more.

L-Measure website

Quick Table of L Measures and Additional Functions(*)

Features	Formula or Equation	Description
Dimensions
Width X	w	Measured along the x-axis of a neuron.
Height Y	h	Measured along the y-axis of a neuron.
Depth Z	d	Measured along the z-axis of a neuron.
Soma Features
Soma Position (*)		XYZ coordinate of the soma centroid.
Soma Radii (*)		Average radius of the soma.
Soma Surface Area (*)	Number of surface voxels of the soma	Surface area of the soma.
Soma Volume (*)	Number of voxels of the soma	Volume of the soma.
Skewness X, Y, Z (*)		Center of the soma deviation from the center of the cell (see Dimensions).
Euclidean Skewness (*)	Failed to parse (Cannot write to or create math temp directory): \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}	Distance from the center of the soma to the center of the cell.
Counts
Stems	n	Number of stems attached to the soma.
Branching Stems (*)	n	Number of stems that have bifurcations.
Branch Pt	n	Number of bifurcations of a neuron.
Segments (or Branch)	n	Number of branches of a neuron. (A branch is between two critical points or between a critical point and the tip.)
Tips	n	Number of terminal tips of a neuron.
Compartment features
Type	t = 1, 2, 3, or 4	Compartments are assigned four different types: 1=soma, 2=axon, 3=dendrites, and 4=apical dendrites (are dendrites that emerge from the apex of a pyramidal cell).
Diameter	D	Diameter of each compartment of the neuron.
Diameter Power	$D 1.5$	Diameter of each compartment of the neuron raised to the power of 1.5.
Length	l	Measured length of each compartment.
Surface Area	$2\pi\,\!rh$ (area of cylinder w/o top and bottom)	Surface area of each compartment.
Section Area	$\pi\,\!r^2$	Cross-sectional area of each compartment.
Volume	$\pi\,\!r^2h$	Volume of the compartments.
Burke Taper	$\dfrac{D_a - D_b}{L}$	The change in diameter over pathlength between two critical points (the current and previous one).
Hillman Taper	$\dfrac{D_a - D_b}{D_a}$	The ratio of the change in diameter to the initial diameter of two critical points. The initial diameter is usually larger.
Branch (Segment) features
Branch Order	n = 0,1,2,3...	Order of branch with respect to the soma starting at 0 for branches attached to soma.
Euclidean Distance	$\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$ Straight line distance	Straight line distance between the two endpoints of a branch.
Path Length	A + B + C + D + E + ...	Summation of the individual compartment lengths that form a branch.
Contraction	$EucDistance\over Pathlength$	Ratio between the Euclidean distance of a branch and its path length.
Fragmentation	n	Total number of compartments that form a branch between two bifurcation points, or between a bifurcation point and a terminal tip.
Helix	${( \overrightarrow{A} \times \overrightarrow{B} ) \times \overrightarrow{C}} \over 3(\|\overrightarrow{A}\| \cdot \|\overrightarrow{B}\| \cdot \|\overrightarrow{C}\|)$	Helicity of the branches of the neuronal tree. Needs to be at least 3 compartments long to compute the helicity.
Fractal_Dim	$\log EucDistance \over \log PathDistance$ or $(n-1) \sum e_ip_i - \sum e_i \sum p_i \over {(n-1) \sum e_i^2 - (\sum e_i)^2}$ (e_i is EucDistance and p_i is PathDistance)	Fractal dimension metric of the branches in the dendrite trees.
Bifurcation features
Daughter Ratio	$D a / D b$ Da = larger daughter diameter Db = smaller daughter diameter	Ratio between the diameter of the bigger daughter and the smaller daughter of the current bifurcation. The diameters are measured at the initial part of the branches right when they exit the bifurcation.
Parent Daughter Ratio	$D a / D p, D b / D p$	Ratio between the diameter of a daughter and its father for each critical point. The diameters are measured immediately before and after the bifurcation point.
Partition Asymmetry	${\|n1 - n2\|} \over {n1+n2-2}$	Compares the symmetry of the processes in terms of the number of tips on the left daughter to the number of tips on the right daughter of a bifurcation.
Rall Power	$D_p^n = D_a^n + D_b^n$	Best value that fits the equation with extreme 0-5, resolution 1/1000 over the range.
Pk (Rall's Ratio)	${D_a^n+D_b^n} \over {D_p^n}$	n = [0, 5]
Pk_classic (Rall's Ratio)	${D_a^{1.5}+D_b^{1.5}} \over {D_p^{1.5}}$	Rall power is set to 1.5
Pk_2 (Rall's Ratio)	${D_a^2+D_b^2} \over {D_p^2}$	Rall power is set to 2

Azimuth (*)	ϴ (in degrees)	Azimuth of the daughter segment at a bifurcation point on the x-y plane.
Elevation (*)	ϴ	Elevation of the daughter segment at a bifurcation point extending into the z plane from the azimuth.
Bif_ampl_local	ϴ	The local angle formed right at the vertex of the branches; initial angle formed as the two daughter branches exit the bifurcation.
Bif_ampl_remote	ϴ	From the bifurcation, it is the angle to the tips of the two daughter branches. Angle between two branches; angle between the ends of two growing segments (in degrees) at the current bifurcation; angle between the two daughter bifurcations and/or terminal ends (whichever comes first) at the current bifurcation. (Daughter bifurcation means the next bifurcation encountered after the current bifurcation.)
Bif_tilt_local	ϴ	The angles between the end of the parent branch and the initial part of the daughter branches at the bifurcation; angle between the last compartment of the parent branch and the first compartment of each daughter branch at the bifurcation. There are two daughter branches for each bifurcation, which gives two angles.
Bif_tilt_remote	ϴ	The angles between the previous node of the current bifurcating father and the daughter nodes.
Bif_torque_local	ϴ	Angle between the current plane of bifurcation and the previous plane of bifurcation; the planes are defined by the initial part of the daughter branches as they exit the point of bifurcation.
Bif_torque_remote	ϴ	Angle between the current plane of bifurcation and the previous plane of bifurcation; the plane of bifurcation is defined by a bifurcation and its two daughter bifurcations or terminal ends (whichever comes first).
Terminal bifurcation features
Leaf path length	n	Path length from the tip to the soma point.
Terminal degree	n	Total number of tips each segment will terminate into.
Terminal Compartments	0 (if not terminal), 1 (if terminal)	Number of compartments that comprise the terminal branch.
Terminal Parent Diameter	D	Diameter of last bifurcation before the terminal tips.
Diameter Threshold	D	Diameter of first compartment after the terminal bifurcation leading to a terminal tip; initial diameter of the branch exiting the terminal bifurcation.
Hillman Threshold	$0.5 P + 0.25 D 1 + 0.25 D 2$ P: parent diameter, D1 = 1st daughter diameter, D2 = 2nd daughter diameter	Computes the weighted average diameter between 50% of father and 25% of daughter diameters of the terminal bifurcation.
Other features
Distance to Device	D	Euclidean distance of the soma point to the surface of the device.
Azimuth to Device	ϴ	The azimuth of the euclidean distance of the soma point to the surface of the device.
Elevation to Device	ϴ	The elevation of the euclidean distance of the soma point to the surface of the device.
Overall Cell Tips to Device Angle	ϴ	The angle between the overall tip direction and the closest point to the device with the soma point as the vertex.
Tip to Soma Euclidean Distance	D	Straight line distance from the terminal tip to the soma point.
Branch Pt to Soma Euclidean Distance	D	Straight line distance from the branch point to the soma point.
Magnitude of Tips (*)	M	The overall direction of the tips combined with the soma point as the origin.
Azimuth of Tips (*)	ϴ	The overall direction of the tips combined with the soma point as the origin.
Elevation of Tips (*)	ϴ	The overall direction of the tips combined with the soma point as the origin.
Convex Hull Magnitude (*)	M	The overall direction of the surface points with the soma point as the origin.
Convex Hull Azimuth (*)	ϴ	The overall direction of the surface points with the soma point as the origin.
Convex Hull Elevation (*)	ϴ	The overall direction of the surface points with the soma point as the origin.
Convex Hull Surface Area (*)	a	The surface area of the convex hull.
Convex Hull Volume (*)	v	The volume of the convex hull.
Ellipsoid Major Length (*)	a	The longest length of the 3D ellipse.
Ellipsoid Minor Length (*)	b	The medium length of the 3D ellipse.
Ellipsoid Normal Length (*)	c	The shortest length of the 3D ellipse.
Ellipsoid Major Azimuth (*)	ϴ	The direction of the major axis of the 3D ellipse.
Ellipsoid Major Elevation (*)	ϴ	The direction of the major axis of the 3D ellipse.
Ellipsoid Eccentricity (*)	Failed to parse (Cannot write to or create math temp directory): \sqrt{1-b^2/a^2}	A measure of how much the ellipsoid deviates from being circular. (Eccentricity of a circle is zero.)

Note: Branches are modeled as a series of compartments. Critical points refer to bifurcation points and terminal tips.

Hierarchy of Feature Computation

Only features computed in the Trace Editor are shown in the diagram and table below.

* indicates important feature?
ID	Features	Level	Type	Description	Derived from ID	L-Measure?
0	Radius	Node		length
1	X,Y,Z	Node		point
2	Average Radius	Compartment		length	0
3	Average Diameter	Compartment		length	2	Y
4	Diameter Power	Compartment		length	3	Y
5	Section Area	Compartment		size	2	Y
6	Surface Area	Compartment		size	2,8	Y
7	Volume	Compartment		size	5,8	Y
8	Length	Compartment		length	1	Y
9	Stem Distance	Segment	Soma	length	1
10	Branch Order *	Segment	Branch	label		Y
11	Average Diameter	Segment	Branch	length	3
12	Average Diameter Power	Segment	Branch	length	4
13	Section Area	Segment	Branch	size	5
14	Surface Area	Segment	Branch	size	6
15	Volume	Segment	Branch	size	7
16	Euclidean Length	Segment	Branch	length	1	Y
17	Path Length	Segment	Branch	length	8	Y
18	Contraction	Segment	Branch	ratio	16,17	Y
19	Fragmentation	Segment	Branch	count		Y
20	Burke Taper	Segment	Branch	ratio	3,17	Y
21	Hillman Taper	Segment	Branch	ratio	3	Y
22	Terminal Degree	Segment	Branch	count		Y
23	Daughter Ratio	Branch Point	Branch	ratio	3	Y
24	Parent Daughter Ratio	Branch Point	Branch	ratio	3	Y
25	Rall Power	Branch Point	Branch	ratio	3	Y
26	Rall's Ratio	Branch Point	Branch	ratio	3	Y
27	Rall's Ratio (ralls = 1.5)	Branch Point	Branch	ratio	3	Y
28	Rall's Ratio (ralls = 2)	Branch Point	Branch	ratio	3	Y
29	Partition Asymmetry	Branch Point	Branch	ratio		Y
30	Azimuth	Branch Point	Branch	angle	1
31	Elevation	Branch Point	Branch	angle	1
32	Local Bifurcation Amplitude	Branch Point	Branch	angle	1	Y
33	Local Bifurcation Tilt Smaller	Branch Point	Branch	angle	1	Y
34	Local Bifurcation Tilt Larger	Branch Point	Branch	angle	1
35	Local Bifurcation Torque Smaller	Branch Point	Branch	angle	1	Y
36	Local Bifurcation Torque Larger	Branch Point	Branch	angle	1
37	Remote Bifurcation Amplitude	Branch Point	Branch	angle	1	Y
38	Remote Bifurcation Tilt Smaller	Branch Point	Branch	angle	1	Y
39	Remote Bifurcation Tilt Larger	Branch Point	Branch	angle	1
40	Remote Bifurcation Torque Smaller	Branch Point	Branch	angle	1	Y
41	Remote Bifurcation Torque Larger	Branch Point	Branch	angle	1
42	Width	Cell		length		Y
43	Height	Cell		length		Y
44	Depth	Cell		length		Y
45	Stems *	Cell	Soma	count		Y
46	Branch Pt *	Cell	Branch	count		Y
47	Segments *	Cell	Branch	count		Y
48	Tips *	Cell	Branch	count		Y
49	Soma X Pos *	Cell	Soma	point	1
50	Soma Y Pos *	Cell	Soma	point	1
51	Soma Z Pos *	Cell	Soma	point	1
52	Skewness X *	Cell	Soma	length	49
53	Skewness Y *	Cell	Soma	length	50
54	Skewness Z *	Cell	Soma	length	51
55	Euclidean Skewness *	Cell	Soma	length	52,53,54
56	Soma Radii	Cell	Soma	length	1
57	Soma Surface Area *	Cell	Soma	size
58	Soma Volume *	Cell	Soma	size
59	Average Stem Distance	Cell	Soma	length	9
60	Average Diameter *	Cell	Branch	length	11
61	Average Diameter Power	Cell	Branch	length	12
62	Average Section Area	Cell	Branch	size	13
63	Average Surface Area	Cell	Branch	size	14
64	Average Volume	Cell	Branch	size	15
65	Average Euclidean Length	Cell	Branch	length	16	Y
66	Average Path Length *	Cell	Branch	length	17	Y
67	Average Contraction *	Cell	Branch	ratio	18	Y
68	Average Fragmentation	Cell	Branch	count	19	Y
69	Average Burke Taper *	Cell	Branch	ratio	20	Y
70	Average Hillman Taper	Cell	Branch	ratio	21	Y
71	Average Daughter Ratio	Cell	Branch	ratio	23	Y
72	Average Parent Daughter Ratio *	Cell	Branch	ratio	24	Y
73	Average Rall Power	Cell	Branch	ratio	25	Y
74	Average Rall's Ratio	Cell	Branch	ratio	26	Y
75	Average Rall's Ratio (ralls = 1.5)	Cell	Branch	ratio	27	Y
76	Average Rall's Ratio (ralls = 2)	Cell	Branch	ratio	28	Y
77	Average Partition Asymmetry	Cell	Branch	ratio	29	Y
78	Average Azimuth	Cell	Branch	angle	30
79	Average Elevation	Cell	Branch	angle	31
80	Average Local Bifurcation Amplitude *	Cell	Branch	angle	32	Y
81	Average Local Bifurcation Tilt	Cell	Branch	angle	33,34
82	Average Local Bifurcation Torque	Cell	Branch	angle	35,36
83	Average Remote Bifurcation Amplitude	Cell	Branch	angle	37	Y
84	Average Remote Bifurcation Tilt	Cell	Branch	angle	38,39
85	Average Remote Bifurcation Torque	Cell	Branch	angle	40,41
86	Average Leaf Level *	Cell	Branch	label	10
87	Average Leaf Path Length *	Cell	Branch	length	3
88	Average Terminal Compartments	Cell	Branch	count	19	Y
89	Average Diameter Threshold *	Cell	Branch	length	3	Y
90	Average Terminal Parent Diameter	Cell	Branch	length	3	Y
91	Average Terminal Hillman Threshold	Cell	Branch	ratio	3	Y
92	Average Tip to Soma Euclidean Distance	Cell	Branch	length	1
93	Average Branch Pt to Soma Euclidean Distance	Cell	Branch	length	1

L Measure functions

About L-Measure

Quick Table of L Measures and Additional Functions(*)

Hierarchy of Feature Computation

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