Intrinsic Features of Blobs

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These features can be calculated with two input images (Data Image and Label Image).  
 
These features can be calculated with two input images (Data Image and Label Image).  
 
They are most commonly used for blob-like regions, such as cell nuclei.
 
They are most commonly used for blob-like regions, such as cell nuclei.
 +
Equations are shown for 3-dimensional space unless otherwise noted.
  
−
{| border="1px" cellpadding="1"
+
{| border="1px" cellpadding="3" style="text-align:left"
 
|-
 
|-
 
| Name
 
| Name
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| Volume
 
| Volume
 
| Number of voxels in the object [1]
 
| Number of voxels in the object [1]
 +
| <math>|\Omega|</math> or <math>M_{000}|\{I=binary\}</math>
 +
|-
 +
| Integrated Intensity
 +
| Sum of the intensities of all voxels in the object [1]
 +
| <math>\sum I(\Omega)</math> or <math>M_{000}|\{I=intensity\}</math>
 +
|-
 +
| Centroid
 +
| Center of the object [1]
 +
| <math> \left [ \begin{array}{ccc} \frac{M_{100}}{M_{000}}, & \frac{M_{010}}{M_{000}}, & \frac{M_{001}}{M_{000}} \end{array} \right ]|\{I=binary\} </math>
 +
|-
 +
| Weighted Centroid
 +
| Uses the image intensity values to calculate the center of mass of the object [1]
 +
| <math> \left [ \begin{array}{ccc} \frac{M_{100}}{M_{000}}, & \frac{M_{010}}{M_{000}}, & \frac{M_{001}}{M_{000}} \end{array} \right ]|\{I=intensity\} </math>
 +
|-
 +
| Axes Lengths
 +
| The length of the axes of the ND hyper-ellipsoid fit to the object [1]
 
| <math>|\Omega|</math>
 
| <math>|\Omega|</math>
 +
|-
 +
| Eccentricity
 +
| Ratio of the distance between the foci of the best-fit hyper-ellipsoid to the length of its major axis. (2D) [1]
 +
|-
 +
| Elongation
 +
| Ratio of the major axis length to minor axis length of the best-fit hyper-ellipsoid. (2D) [1]
 +
|-
 +
| Orientation
 +
| Angle between the major axis of the best-fit hyper-ellipsoid and origin. (2D) [1]
 +
|-
 +
| Bounding Box Volume
 +
| Number of voxels in the bounding box of the object [1]
 +
|-
 +
| Oriented Bounding Box Volume
 +
| Number of voxels in the oriented bounding box of the object. The oriented bounding box is defined as the bounding box aligned along the axes of the object. [1]
 +
|-
 +
| Sum
 +
| Same as integrated intensity [2]
 +
|-
 +
| Mean
 +
| Average intensity of voxels in the object [2]
 +
|-
 +
| Median
 +
| Middle intensity of voxels in the object [2]
 +
|-
 +
| Minimum
 +
| Minimum intensity of voxels in the object [2]
 +
|-
 +
| Maximum
 +
| Maximum intensity of voxels in the object [2]
 +
|-
 +
| Sigma
 +
| Standard deviation of intensity of voxels in the object [2]
 +
|-
 +
| Variance
 +
| Variance of intensity of voxels in the object [2]
 +
|-
 +
| Radius Variation
 +
| Standard deviation of distance from surface voxels to centroid
 +
|-
 +
| Skew
 +
| Skew of the normalized intensity histogram
 +
|-
 +
| Energy
 +
| Energy of the normalized intensity histogram
 +
|-
 +
| Entropy
 +
| Entropy of the normalized intensity histogram
 +
|-
 +
| Surface Gradient
 +
| Average of surface gradients
 +
|-
 +
| Interior Gradient
 +
| Average of interior gradients
 +
|-
 +
| Interior Intensity
 +
| Average of interior intensities
 +
|-
 +
| Surface Intensity
 +
| Average of surface intensities
 +
|-
 +
| Intensity Ratio
 +
| Ratio of surface intensity to interior intensity
 +
|-
 +
| Shared Boundary
 +
| Ratio of surface area that touches another object to total surface area
 +
|-
 +
| Surface Area
 +
| Number of voxels on surface of the object
 +
|-
 +
| Shape
 +
| Ratio of surface voxels to total voxels - compactness or thinness of object
 
|}
 
|}
−
 
−
===''Volume''===
 
−
Number of voxels in the object. [1]<br>
 
−
<math>|\Omega|</math>
 
−
 
−
===''Integrated Intensity''===
 
−
Sum of the intensities of all voxels in the object. (Dirk)
 
−
===''Centroid (unweighted and weighted)''===
 
−
The unweighted centroid calculates the center of the object.
 
−
The weighted centroid uses the image intensity values to calculate the intensity center of the object. (Dirk)
 
−
===''Axes Lengths''===
 
−
The length of the axes of the ND hyper-ellipsoid fit to the object. (Dirk)
 
−
===''Eccentricity''===
 
−
Ratio of the distance between the foci of the best-fit hyper-ellipsoid to the length of its major axis. (2D) (Dirk)
 
−
===''Elongation''===
 
−
Ratio of the major axis length to minor axis length of the best-fit hyper-ellipsoid. (2D) (Dirk)
 
−
===''Orientation''===
 
−
Angle between the major axis of the best-fit hyper-ellipsoid and origin. (2D) (Dirk)
 
−
===''Bounding Box Volume''===
 
−
Number of voxels in the bounding box of the object. (Dirk)
 
−
===''Oriented Bounding Box Volume''===
 
−
Number of voxels in the oriented bounding box of the object.
 
−
The oriented bounding box is defined as the bounding box aligned along the axes of the object. (Dirk)
 
−
===''Sum''===
 
−
Same as integrated intensity (ITK statistics)
 
−
===''Mean''===
 
−
Average intensity of voxels in the object (ITK statistics)
 
−
===''Median''===
 
−
Middle intensity of voxels in the object (ITK statistics)
 
−
===''Minimum''===
 
−
Minimum intensity of voxels in the object (ITK statistics)
 
−
===''Maximum''===
 
−
Maximum intensity of voxels in the object (ITK statistics)
 
−
===''Sigma''===
 
−
Standard deviation of intensity of voxels in the object (ITK statistics)
 
−
===''Variance''===
 
−
Variance of intensity of voxels in the object (ITK statistics)
 
−
===''Radius Variation''===
 
−
Standard deviation of distance from surface voxels to centroid (Isaac)
 
−
===''Skew''===
 
−
Skew of the normalized intensity histogram (Isaac) (See Supplement B2)
 
−
===''Energy''===
 
−
Energy of the normalized intensity histogram (Isaac) (See Supplement B2)
 
−
===''Entropy''===
 
−
Entropy of the normalized intensity histogram (Isaac) (See Supplement B2)
 
−
===''Surface Gradient''===
 
−
Average of surface gradients (Isaac)
 
−
===''Interior Gradient''===
 
−
Average of interior gradients (Isaac)
 
−
===''Interior Intensity''===
 
−
Average of interior intensities (Isaac)
 
−
===''Surface Intensity''===
 
−
Average of surface intensities (Isaac)
 
−
===''Intensity Ratio''===
 
−
Ratio of surface intensity to interior intensity (Isaac)
 
−
===''Shared Boundary''===
 
−
Ratio of surface area that touches another object to total surface area (Isaac)
 
−
===''Surface Area''===
 
−
Number of voxels on surface of the object (Isaac)
 
−
===''Shape''===
 
−
Ratio of surface voxels to total voxels - compactness or thinness of object (Isaac) (See Supplement B2)
 
  
 
==Glossary of Notation==
 
==Glossary of Notation==
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<FONT SIZE="+1"><math>P</math></FONT> - the normalized histogram of the intensities<br />
 
<FONT SIZE="+1"><math>P</math></FONT> - the normalized histogram of the intensities<br />
 
<FONT SIZE="+1"><math>P(I)</math></FONT> - normalized histogram of intensity values <math>I</math><br />
 
<FONT SIZE="+1"><math>P(I)</math></FONT> - normalized histogram of intensity values <math>I</math><br />
 +
<FONT SIZE="+1"><math>M_{p,q,r} = \sum_{z=0}^{Z-1}\sum_{y=0}^{Y-1}\sum_{x=0}^{X-1}x^p y^q z^r I(x,y,z)</math></FONT> - Raw Moment of discrete image <math>I</math><br>
 +
<FONT SIZE="+1"><math>\lambda_i</math></FONT> - <math>i^{th}</math> eigenvalue of covariance matrix<br>
  
 
==External Links==
 
==External Links==
−
[1] [http://www.insight-journal.org/browse/publication/301 itkLabelGeometryImageFilter]
+
[1] [http://www.insight-journal.org/browse/publication/301 itkLabelGeometryImageFilter]<br>
−
 
+
[2] [http://www.itk.org/Doxygen312/html/classitk_1_1LabelStatisticsImageFilter.html itkLabelStatisticsImageFilter]<br>
−
[http://kitware.com/products/archive/kitware_quarterly0109.pdf Kitware Source Newsletter]
+
[3] [http://kitware.com/products/archive/kitware_quarterly0109.pdf Kitware Source Newsletter]

Revision as of 17:50, 28 April 2009

Intrinsic Features for Blobs

These features can be calculated with two input images (Data Image and Label Image). They are most commonly used for blob-like regions, such as cell nuclei. Equations are shown for 3-dimensional space unless otherwise noted.

Name Description Formula
Volume Number of voxels in the object [1] | Ω | or M000 | {I = binary}
Integrated Intensity Sum of the intensities of all voxels in the object [1] \sum I(\Omega) or M000 | {I = intensity}
Centroid Center of the object [1] \left [ \begin{array}{ccc} \frac{M_{100}}{M_{000}}, & \frac{M_{010}}{M_{000}}, & \frac{M_{001}}{M_{000}} \end{array} \right ]|\{I=binary\}
Weighted Centroid Uses the image intensity values to calculate the center of mass of the object [1] \left [ \begin{array}{ccc} \frac{M_{100}}{M_{000}}, & \frac{M_{010}}{M_{000}}, & \frac{M_{001}}{M_{000}} \end{array} \right ]|\{I=intensity\}
Axes Lengths The length of the axes of the ND hyper-ellipsoid fit to the object [1] | Ω |
Eccentricity Ratio of the distance between the foci of the best-fit hyper-ellipsoid to the length of its major axis. (2D) [1]
Elongation Ratio of the major axis length to minor axis length of the best-fit hyper-ellipsoid. (2D) [1]
Orientation Angle between the major axis of the best-fit hyper-ellipsoid and origin. (2D) [1]
Bounding Box Volume Number of voxels in the bounding box of the object [1]
Oriented Bounding Box Volume Number of voxels in the oriented bounding box of the object. The oriented bounding box is defined as the bounding box aligned along the axes of the object. [1]
Sum Same as integrated intensity [2]
Mean Average intensity of voxels in the object [2]
Median Middle intensity of voxels in the object [2]
Minimum Minimum intensity of voxels in the object [2]
Maximum Maximum intensity of voxels in the object [2]
Sigma Standard deviation of intensity of voxels in the object [2]
Variance Variance of intensity of voxels in the object [2]
Radius Variation Standard deviation of distance from surface voxels to centroid
Skew Skew of the normalized intensity histogram
Energy Energy of the normalized intensity histogram
Entropy Entropy of the normalized intensity histogram
Surface Gradient Average of surface gradients
Interior Gradient Average of interior gradients
Interior Intensity Average of interior intensities
Surface Intensity Average of surface intensities
Intensity Ratio Ratio of surface intensity to interior intensity
Shared Boundary Ratio of surface area that touches another object to total surface area
Surface Area Number of voxels on surface of the object
Shape Ratio of surface voxels to total voxels - compactness or thinness of object

Glossary of Notation

p = (x,y,z) - the coordinate of a voxel (three-dimensional point in a volume image)
Np - a neighbor voxel of p
lp - the segmentation label at p
Ii(p) - the intensity value of p at ith channel
Ω = {p | lp = o} - the set of voxels of an object o
\Omega_s = \{l_p = o; \exists N_p, l_{N_p} \neq o\} - the set of surface voxels of the object
Ωin = Ω − Ωs - the set of interior voxels of an object
G - the magnitude of intensity gradient at p
\bar{p} - the center of mass of the object
P - the normalized histogram of the intensities
P(I) - normalized histogram of intensity values I
M_{p,q,r} = \sum_{z=0}^{Z-1}\sum_{y=0}^{Y-1}\sum_{x=0}^{X-1}x^p y^q z^r I(x,y,z) - Raw Moment of discrete image I
λi - ith eigenvalue of covariance matrix

External Links

[1] itkLabelGeometryImageFilter
[2] itkLabelStatisticsImageFilter
[3] Kitware Source Newsletter

Retrieved from "http://farsight-toolkit.ee.uh.edu/wiki/Intrinsic_Features_of_Blobs"
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