L Measure functions

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== About L Measures ==
+
== About L-Measure ==
  
−
L Measures is a research tool for making quantitative morphological measurements from neuronal reconstructions.  Features include 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 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.   
  
  
 
[http://cng.gmu.edu:8080/Lm/help/index.htm L-Measure website]
 
[http://cng.gmu.edu:8080/Lm/help/index.htm L-Measure website]
  
−
== Quick Table of L Measures Functions ==
+
== Quick Table of L Measures and Additional Functions(*) ==
 
{| class="wikitable"
 
{| class="wikitable"
 
|-
 
|-
Line 14: Line 14:
 
! Description
 
! Description
 
|-
 
|-
−
| Soma_Surface
+
| '''Dimensions'''
−
| Single soma: <math>4\pi\,\!r^2</math> (surface area of sphere)
+
|  
−
Multiple soma: <math>2\pi\,\!rh</math> (external area of cylinder)
+
|
 +
|-
 +
| Width X
 +
| ''w''
 +
| [[File:neuronwidth.png|150px]]
 +
| Measured along the x-axis of a neuron.
 +
|-
 +
| Height Y
 +
| ''h''
 +
| [[File:neuronheight.png|150px]]
 +
| Measured along the y-axis of a neuron.
 +
|-
 +
| Depth Z
 +
| ''d''
 +
| [[File:neurondepth.png|150px]]
 +
| Measured along the z-axis of a neuron.
 +
|-
 +
| '''Soma Features'''
 +
|
 +
|
 +
|-
 +
| Soma Position (*)
 +
|
 +
| [[File:Somasurface.png|150px]]
 +
| XYZ coordinate of the soma centroid.
 +
|-
 +
| Soma Radii (*)
 +
|
 +
| [[File:Somasurface.png|150px]]
 +
| Average radius of the soma.
 +
|-
 +
| Soma Surface Area (*)
 +
| Number of surface voxels of the soma
 
| [[File:Somasurface.png|150px]]
 
| [[File:Somasurface.png|150px]]
 
| Surface area of the soma.
 
| Surface area of the soma.
 
|-
 
|-
−
| N_stems
+
| Soma Volume (*)
 +
| Number of voxels of the soma
 +
| [[File:Somasurface.png|150px]]
 +
| Volume of the soma.
 +
|-
 +
| Skewness X, Y, Z (*)
 +
|
 +
| [[File:Xskew.png|150px]] [[File:Yskew.png|150px]] [[File:Zskew.png|150px]]
 +
| Center of the soma deviation from the center of the cell (see Dimensions).
 +
|-
 +
| Euclidean Skewness (*)
 +
| <math>\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}</math>
 +
| [[File:euclideanSkewness.png|150px]]
 +
| Distance from the center of the soma to the center of the cell.
 +
|-
 +
| '''Counts'''
 +
|
 +
|
 +
|-
 +
| Stems
 
| ''n''
 
| ''n''
 
| [[File:Nstems.png|150px]]
 
| [[File:Nstems.png|150px]]
 
| Number of stems attached to the soma.
 
| Number of stems attached to the soma.
 
|-
 
|-
−
| N_bifs
+
| Branching Stems (*)
 +
| ''n''
 +
| [[File:Nbranchstems.png|150px]]
 +
| Number of stems that have bifurcations.
 +
|-
 +
| Branch Pt
 
| ''n''
 
| ''n''
 
| [[File:Nbif.png|150px]]
 
| [[File:Nbif.png|150px]]
 
| Number of bifurcations of a neuron.
 
| Number of bifurcations of a neuron.
 
|-
 
|-
−
| N_branch
+
| Segments (or Branch)
 
| ''n''
 
| ''n''
 
| [[File:Nbranch.png|150px]]
 
| [[File:Nbranch.png|150px]]
 
| Number of branches of a neuron. (A branch is between two critical points or between a critical point and the tip.)
 
| Number of branches of a neuron. (A branch is between two critical points or between a critical point and the tip.)
 
|-
 
|-
−
| N_tips
+
| Tips
 
| ''n''
 
| ''n''
 
| [[File:Ntip.png|150px]]
 
| [[File:Ntip.png|150px]]
 
| Number of terminal tips of a neuron.
 
| Number of terminal tips of a neuron.
−
|-
 
−
| '''Dimensions'''
 
−
|
 
−
|
 
−
|-
 
−
| Width
 
−
| ''w''
 
−
| [[File:neuronwidth.png|150px]]
 
−
| Measured along the x-axis of a neuron.
 
−
|-
 
−
| Height
 
−
| ''h''
 
−
| [[File:neuronheight.png|150px]]
 
−
| Measured along the y-axis of a neuron.
 
−
|-
 
−
| Depth
 
−
| ''d''
 
−
| [[File:neurondepth.png|150px]]
 
−
| Measured along the z-axis of a neuron.
 
 
|-
 
|-
 
| '''Compartment features'''
 
| '''Compartment features'''
Line 64: Line 101:
 
|-
 
|-
 
| Type
 
| Type
−
| t = 0, 1, 2, 3, or 4
+
| 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).
 
| 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).
Line 73: Line 110:
 
| Diameter of each compartment of the neuron.
 
| Diameter of each compartment of the neuron.
 
|-
 
|-
−
| Diameter_pow
+
| Diameter Power
 
| <math>D^{1.5}</math>
 
| <math>D^{1.5}</math>
 
|
 
|
Line 83: Line 120:
 
| Measured length of each compartment.
 
| Measured length of each compartment.
 
|-
 
|-
−
| Surface
+
| Surface Area
 
| <math>2\pi\,\!rh</math>  
 
| <math>2\pi\,\!rh</math>  
 
(area of cylinder w/o top and bottom)
 
(area of cylinder w/o top and bottom)
Line 99: Line 136:
 
| Volume of the compartments.
 
| Volume of the compartments.
 
|-  
 
|-  
−
| Taper_1: Burke Taper
+
| Burke Taper
 
| <math>\dfrac{D_a - D_b}{L} </math>
 
| <math>\dfrac{D_a - D_b}{L} </math>
 
| [[File:burketaper.png|150px]]
 
| [[File:burketaper.png|150px]]
 
| The change in diameter over pathlength between two critical points (the current and previous one).
 
| The change in diameter over pathlength between two critical points (the current and previous one).
 
|-  
 
|-  
−
| Taper_2: Hillman Taper
+
| Hillman Taper
 
| <math>\dfrac{D_a - D_b}{D_a} </math>
 
| <math>\dfrac{D_a - D_b}{D_a} </math>
 
| [[File:hillmantaper.png|150px]]
 
| [[File:hillmantaper.png|150px]]
Line 113: Line 150:
 
|
 
|
 
|-  
 
|-  
−
| Branch_Order
+
| Branch Order
 
| n = 0,1,2,3...
 
| n = 0,1,2,3...
 
| [[File:branchorder.png|150px]]
 
| [[File:branchorder.png|150px]]
 
| Order of branch with respect to the soma starting at 0 for branches attached to soma.
 
| Order of branch with respect to the soma starting at 0 for branches attached to soma.
 
|-
 
|-
−
| EucDistance
+
| Euclidean Distance
 
| <math>\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}</math>
 
| <math>\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}</math>
 
Straight line distance
 
Straight line distance
−
| [[File:Eucdistance.png|150px]]
+
| [[File:EuclideanDistance.png|150px]]
−
| Straight line distance between two endpoints of a trace.
+
| Straight line distance between the two endpoints of a branch.
 
|-
 
|-
−
| PathDistance
+
| Path Length
 
| A + B + C + D + E + ...
 
| A + B + C + D + E + ...
−
| [[File:pathdistance.png|150px]]
+
| [[File:PathLength.png|150px]]
−
| Summation of the individual compartment distances that form a trace.
+
| Summation of the individual compartment lengths that form a branch.
−
|-
+
−
| Branch_pathlength
+
−
| AB + BC + CD + DE + ...
+
−
| [[File:branchpathlength1.png|150px]]
+
−
| Summation of the lengths of all the compartments comprising a branch.
+
 
|-  
 
|-  
 
| Contraction
 
| Contraction
Line 152: Line 184:
 
| <math>\log EucDistance \over \log PathDistance</math> or
 
| <math>\log EucDistance \over \log PathDistance</math> or
 
<math>(n-1) \sum e_ip_i - \sum e_i \sum p_i \over {(n-1) \sum e_i^2 - (\sum e_i)^2}</math>
 
<math>(n-1) \sum e_ip_i - \sum e_i \sum p_i \over {(n-1) \sum e_i^2 - (\sum e_i)^2}</math>
 +
 +
(e<sub>i</sub> is EucDistance and p<sub>i</sub> is PathDistance)
 
|
 
|
 
| Fractal dimension metric of the branches in the dendrite trees.
 
| Fractal dimension metric of the branches in the dendrite trees.
Line 159: Line 193:
 
|  
 
|  
 
|-  
 
|-  
−
| Daughter_Ratio
+
| Daughter Ratio
 
| <math>D_a/D_b</math>
 
| <math>D_a/D_b</math>
 
Da = larger daughter diameter <br />
 
Da = larger daughter diameter <br />
Line 166: Line 200:
 
| 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.
 
| 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
+
| Parent Daughter Ratio
 
| <math>D_a/D_p ,D_b/D_p</math>
 
| <math>D_a/D_p ,D_b/D_p</math>
 
| [[File:parentdaughterratio.png|150px]]
 
| [[File:parentdaughterratio.png|150px]]
 
| 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.
 
| 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
+
| Partition Asymmetry
 
| <math>{|n1 - n2|} \over {n1+n2-2}</math>
 
| <math>{|n1 - n2|} \over {n1+n2-2}</math>
 
|
 
|
 
| 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.
 
| 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
+
| Rall Power
 
| <math>D_p^n = D_a^n + D_b^n</math>
 
| <math>D_p^n = D_a^n + D_b^n</math>
 
| [[File:parentdaughterratio.png|150px]]
 
| [[File:parentdaughterratio.png|150px]]
 
| Best value that fits the equation with extreme 0-5, resolution 1/1000 over the range.
 
| Best value that fits the equation with extreme 0-5, resolution 1/1000 over the range.
 
|-  
 
|-  
−
| Pk
+
| Pk (Rall's Ratio)
 
| <math>{D_a^n+D_b^n} \over {D_p^n}</math>
 
| <math>{D_a^n+D_b^n} \over {D_p^n}</math>
 
| [[File:parentdaughterratio.png|150px]]
 
| [[File:parentdaughterratio.png|150px]]
 
| n = [0, 5]
 
| n = [0, 5]
 
|-  
 
|-  
−
| Pk_classic
+
| Pk_classic (Rall's Ratio)
 
| <math>{D_a^{1.5}+D_b^{1.5}} \over {D_p^{1.5}}</math>
 
| <math>{D_a^{1.5}+D_b^{1.5}} \over {D_p^{1.5}}</math>
 
| [[File:parentdaughterratio.png|150px]]
 
| [[File:parentdaughterratio.png|150px]]
 
| Rall power is set to 1.5
 
| Rall power is set to 1.5
 
|-  
 
|-  
−
| Pk_2
+
| Pk_2 (Rall's Ratio)
 
| <math>{D_a^2+D_b^2} \over {D_p^2}</math>
 
| <math>{D_a^2+D_b^2} \over {D_p^2}</math>
 
| [[File:parentdaughterratio.png|150px]]
 
| [[File:parentdaughterratio.png|150px]]
Line 200: Line 234:
 
|  
 
|  
 
|-  
 
|-  
−
| Bif_ampl_local
+
| Azimuth (*)
 
| ϴ (in degrees)
 
| ϴ (in degrees)
 +
| [[File:azimuth.png|150px]]
 +
| Azimuth of the daughter segment at a bifurcation point on the x-y plane.
 +
|-
 +
| Elevation (*)
 +
| ϴ
 +
| [[File:elevation.png|150px]]
 +
| Elevation of the daughter segment at a bifurcation point extending into the z plane from the azimuth.
 +
|-
 +
| Bif_ampl_local
 +
| ϴ
 
| [[File:bifamplocal.png|150px]]
 
| [[File:bifamplocal.png|150px]]
 
| The local angle formed right at the vertex of the branches; initial angle formed as the two daughter branches exit the bifurcation.  
 
| The local angle formed right at the vertex of the branches; initial angle formed as the two daughter branches exit the bifurcation.  
Line 213: Line 257:
 
| ϴ
 
| ϴ
 
| [[File:biftiltlocal.png|150px]]
 
| [[File:biftiltlocal.png|150px]]
−
| The smaller of the two 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.
+
| 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.
 
*There are two daughter branches for each bifurcation, which gives two angles.
 
|-  
 
|-  
Line 219: Line 263:
 
| ϴ
 
| ϴ
 
| [[File:biftiltremote.png|150px]]
 
| [[File:biftiltremote.png|150px]]
−
| The smaller of the two angles between the first daughter compartment of the current bifurcating father and the previous father compartment of the previous father bifurcation; angle between the last part of the previous branch and the initial part of the next branches (skipping the branch in-between).
+
| The angles between the previous node of the current bifurcating father and the daughter nodes.
 
|-  
 
|-  
 
| Bif_torque_local
 
| Bif_torque_local
Line 235: Line 279:
 
|  
 
|  
 
|-  
 
|-  
−
| Terminal_degree
+
| Leaf path length
 +
| ''n''
 +
| [[File:leafpathlength.png|150px]]
 +
| Path length from the tip to the soma point.
 +
|-
 +
| Terminal degree
 
| ''n''
 
| ''n''
 
| [[File:terminaldegree.png|150px]]
 
| [[File:terminaldegree.png|150px]]
−
| Total number of tips each compartment will terminate into.
+
| Total number of tips each segment will terminate into.
 
|-  
 
|-  
−
| TerminalSegment
+
| Terminal Compartments
 
| 0 (if not terminal), 1 (if terminal)
 
| 0 (if not terminal), 1 (if terminal)
 
| [[File:terminalsegment.png|150px]]
 
| [[File:terminalsegment.png|150px]]
−
| Assigns "1" for all the compartments in the terminal branch.
+
| Number of compartments that comprise the terminal branch.
 
|-  
 
|-  
−
| Last_parent_diam
+
| Terminal Parent Diameter
 
| ''D''
 
| ''D''
 
| [[File:lastparentdiameter.png|150px]]
 
| [[File:lastparentdiameter.png|150px]]
 
| Diameter of last bifurcation before the terminal tips.
 
| Diameter of last bifurcation before the terminal tips.
 
|-  
 
|-  
−
| Diam_threshold
+
| Diameter Threshold
 
| ''D''
 
| ''D''
 
| [[File:diameterthreshold.png|150px]]
 
| [[File:diameterthreshold.png|150px]]
 
| Diameter of first compartment after the terminal bifurcation leading to a terminal tip; initial diameter of the branch exiting the terminal bifurcation.
 
| Diameter of first compartment after the terminal bifurcation leading to a terminal tip; initial diameter of the branch exiting the terminal bifurcation.
 
|-  
 
|-  
−
| HillmanThreshold
+
| Hillman Threshold
 
| <math>0.5P + 0.25D1 + 0.25D2</math>
 
| <math>0.5P + 0.25D1 + 0.25D2</math>
 
P: parent diameter, <br />
 
P: parent diameter, <br />
Line 263: Line 312:
 
| Computes the weighted average diameter between 50% of father and 25% of daughter diameters of the terminal bifurcation.
 
| Computes the weighted average diameter between 50% of father and 25% of daughter diameters of the terminal bifurcation.
 
|-  
 
|-  
−
| Hausdorff
+
| '''Other features'''
−
| ...
+
 
|
 
|
−
| Hausdorff dimension is a fractal dimension using the box counting method. Count the number of boxes required to cover the set.
+
|  
 +
|-
 +
| Distance to Device
 +
| ''D''
 +
| [[File:DistancetoDevice.png|150px]]
 +
| 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
 +
| [[File:TipDeviceAngle.png|150px]]
 +
| 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''
 +
| [[File:TipToSoma.png|150px]]
 +
| Straight line distance from the terminal tip to the soma point.
 +
|-
 +
| Branch Pt to Soma Euclidean Distance
 +
| ''D''
 +
| [[File:branchptToSoma.png|150px]]
 +
| Straight line distance from the branch point to the soma point.
 +
|-
 +
| Magnitude of Tips (*)
 +
| ''M''
 +
| [[File:magnitudeTip.png|150px]]
 +
| The overall direction of the tips combined with the soma point as the origin.
 +
|-
 +
| Azimuth of Tips (*)
 +
| ϴ
 +
| [[File:somaToTip.png|150px]]
 +
| The overall direction of the tips combined with the soma point as the origin.
 +
|-
 +
| Elevation of Tips (*)
 +
| ϴ
 +
| [[File:somaToTip.png|150px]]
 +
| The overall direction of the tips combined with the soma point as the origin.
 +
|-
 +
| Convex Hull Magnitude (*)
 +
| ''M''
 +
| [[File:convexHull.png|150px]]
 +
| 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''
 +
| [[File:ellipsoid.png|150px]]
 +
| 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 (*)
 +
| <math>\sqrt{1-b^2/a^2}</math>
 +
|
 +
| 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.
 
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.
 +
 +
[[File:MicrogliaOntology5.jpg|800px|center|Top-down approach]]
 +
 +
 +
[[File:OntologyLevel1.jpg|800px|center|Bottom-up approach]]
 +
 +
{|class="wikitable" style="text-align: center; width: 750px; height: 200px; margin: 1em auto 1em auto;"
 +
|+ * 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 ||
 +
|}

Latest revision as of 17:59, 5 June 2013

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 Image Description
Dimensions
Width X w Neuronwidth.png Measured along the x-axis of a neuron.
Height Y h Neuronheight.png Measured along the y-axis of a neuron.
Depth Z d Neurondepth.png Measured along the z-axis of a neuron.
Soma Features
Soma Position (*) Somasurface.png XYZ coordinate of the soma centroid.
Soma Radii (*) Somasurface.png Average radius of the soma.
Soma Surface Area (*) Number of surface voxels of the soma Somasurface.png Surface area of the soma.
Soma Volume (*) Number of voxels of the soma Somasurface.png Volume of the soma.
Skewness X, Y, Z (*) Xskew.png Yskew.png Zskew.png 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} EuclideanSkewness.png Distance from the center of the soma to the center of the cell.
Counts
Stems n Nstems.png Number of stems attached to the soma.
Branching Stems (*) n Nbranchstems.png Number of stems that have bifurcations.
Branch Pt n Nbif.png Number of bifurcations of a neuron.
Segments (or Branch) n Nbranch.png Number of branches of a neuron. (A branch is between two critical points or between a critical point and the tip.)
Tips n Ntip.png 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 Lmdiameter.png Diameter of each compartment of the neuron.
Diameter Power D1.5 Diameter of each compartment of the neuron raised to the power of 1.5.
Length l Lmlength.png Measured length of each compartment.
Surface Area 2\pi\,\!rh

(area of cylinder w/o top and bottom)

Lmsurface.png Surface area of each compartment.
Section Area \pi\,\!r^2 Lmsectionarea.png Cross-sectional area of each compartment.
Volume \pi\,\!r^2h Lmvolume.png Volume of the compartments.
Burke Taper \dfrac{D_a - D_b}{L} Burketaper.png The change in diameter over pathlength between two critical points (the current and previous one).
Hillman Taper \dfrac{D_a - D_b}{D_a} Hillmantaper.png 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... Branchorder.png 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

EuclideanDistance.png Straight line distance between the two endpoints of a branch.
Path Length A + B + C + D + E + ... PathLength.png Summation of the individual compartment lengths that form a branch.
Contraction EucDistance\over Pathlength Contraction.png Ratio between the Euclidean distance of a branch and its path length.
Fragmentation n Fragmentation.png 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}

(ei is EucDistance and pi is PathDistance)

Fractal dimension metric of the branches in the dendrite trees.
Bifurcation features
Daughter Ratio Da / Db

Da = larger daughter diameter
Db = smaller daughter diameter

Daughterratio.png 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 Da / Dp,Db / Dp Parentdaughterratio.png 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 Parentdaughterratio.png 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} Parentdaughterratio.png n = [0, 5]
Pk_classic (Rall's Ratio) {D_a^{1.5}+D_b^{1.5}} \over {D_p^{1.5}} Parentdaughterratio.png Rall power is set to 1.5
Pk_2 (Rall's Ratio) {D_a^2+D_b^2} \over {D_p^2} Parentdaughterratio.png Rall power is set to 2
Azimuth (*) ϴ (in degrees) Azimuth.png Azimuth of the daughter segment at a bifurcation point on the x-y plane.
Elevation (*) ϴ Elevation.png Elevation of the daughter segment at a bifurcation point extending into the z plane from the azimuth.
Bif_ampl_local ϴ Bifamplocal.png 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 Bifampremote.png 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 Biftiltlocal.png 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 Biftiltremote.png 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 Leafpathlength.png Path length from the tip to the soma point.
Terminal degree n Terminaldegree.png Total number of tips each segment will terminate into.
Terminal Compartments 0 (if not terminal), 1 (if terminal) Terminalsegment.png Number of compartments that comprise the terminal branch.
Terminal Parent Diameter D Lastparentdiameter.png Diameter of last bifurcation before the terminal tips.
Diameter Threshold D Diameterthreshold.png Diameter of first compartment after the terminal bifurcation leading to a terminal tip; initial diameter of the branch exiting the terminal bifurcation.
Hillman Threshold 0.5P + 0.25D1 + 0.25D2

P: parent diameter,
D1 = 1st daughter diameter,
D2 = 2nd daughter diameter

Parentdaughterratio.png Computes the weighted average diameter between 50% of father and 25% of daughter diameters of the terminal bifurcation.
Other features
Distance to Device D DistancetoDevice.png 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 TipDeviceAngle.png 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 TipToSoma.png Straight line distance from the terminal tip to the soma point.
Branch Pt to Soma Euclidean Distance D BranchptToSoma.png Straight line distance from the branch point to the soma point.
Magnitude of Tips (*) M MagnitudeTip.png The overall direction of the tips combined with the soma point as the origin.
Azimuth of Tips (*) ϴ SomaToTip.png The overall direction of the tips combined with the soma point as the origin.
Elevation of Tips (*) ϴ SomaToTip.png The overall direction of the tips combined with the soma point as the origin.
Convex Hull Magnitude (*) M ConvexHull.png 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 Ellipsoid.png 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.

Top-down approach


Bottom-up approach
* 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
Retrieved from "http://farsight-toolkit.ee.uh.edu/wiki/L_Measure_functions"
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