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. | ||
− | == | + | == Glossary of Notation == |
− | + | {| border="1px" cellpadding="3" style="text-align:left" | |
− | + | |- | |
− | + | | <math>p=(x,y,z)</math> | |
− | = | + | | the coordinate of a voxel (three-dimensional point in a volume image) |
− | + | |- | |
− | + | | <math>N_p</math> | |
− | + | | a neighbor voxel of <math>p</math> | |
− | + | |- | |
− | + | | <math>l_p</math> | |
− | + | | the segmentation label at <math>p</math> | |
− | + | |- | |
− | + | | <math>I_i(p)</math> | |
− | + | | the intensity value of <math>p</math> at <math>i^{th}</math> | |
− | + | |- | |
− | == | + | | <math>\Omega = \{p|l_p = o\}</math> |
− | + | | the set of voxels of an object <math>o</math> | |
− | == | + | |- |
− | + | | <math>\Omega_s = \{l_p = o; \exists N_p, l_{N_p} \neq o\}</math> | |
− | + | | the set of surface voxels of the object | |
− | = | + | |- |
− | + | | <math>\Omega_{in} = \Omega - \Omega_s</math> | |
− | + | | the set of interior voxels of an object | |
− | + | |- | |
− | + | | <math>\overline{p}</math> | |
− | + | | the center of mass of the object | |
− | + | |- | |
− | + | | <math>P(I)</math> | |
− | + | | Probability Density Function (PDF) of intensity values <math>I</math> | |
− | + | |- | |
− | + | | <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> | |
− | + | | Raw Moment of discrete image <math>I</math> | |
− | + | |- | |
− | + | | <math>\lambda_i</math> | |
− | + | | <math>i^{th}</math> eigenvalue of covariance matrix | |
− | + | |- | |
− | + | | <math>\overline{v_i}</math> | |
− | + | | eigenvector corresponding to <math>\lambda_i</math> | |
− | + | |} | |
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− | === | + | |
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− | == | + | == Features == |
− | + | {| border="1px" cellpadding="3" style="text-align:left" | |
− | + | |- | |
− | + | | '''Name''' | |
− | + | | '''Units''' | |
− | + | | '''Description''' | |
− | + | | '''Formula''' | |
− | + | |- | |
− | < | + | | Volume |
− | + | | voxels | |
− | < | + | | 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 | ||
+ | | voxels | ||
+ | | 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 | ||
+ | | voxels | ||
+ | | 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 | ||
+ | | voxels | ||
+ | | The length of the axes of the ND hyper-ellipsoid fit to the object [1] | ||
+ | | <math>4\sqrt{\lambda_i}</math> | ||
+ | |- | ||
+ | | Eccentricity | ||
+ | | | ||
+ | | Ratio of the distance between the foci of the best-fit hyper-ellipsoid to the length of its major axis. (2D) [1] | ||
+ | | <math>\sqrt{\frac{\lambda_1 - \lambda_0}{\lambda_1}}</math> | ||
+ | |- | ||
+ | | Elongation | ||
+ | | | ||
+ | | Ratio of the major axis length to minor axis length of the best-fit hyper-ellipsoid. (2D) [1] | ||
+ | | <math>\frac{\lambda_1}{\lambda_0}</math> | ||
+ | |- | ||
+ | | Orientation | ||
+ | | radians | ||
+ | | Angle between the major axis of the best-fit hyper-ellipsoid and origin. (2D) [1] | ||
+ | | <math>tan^{-1}\left(\frac{\overline{v_1}(1)}{\overline{v_1}(0)}\right)</math> | ||
+ | |- | ||
+ | | Bounding Box Volume | ||
+ | | voxels | ||
+ | | Number of voxels in the bounding box of the object [1] | ||
+ | | (max(X)-min(X)+1) * (max(Y)-min(Y)+1) * ... | ||
+ | |- | ||
+ | | Oriented Bounding Box Volume | ||
+ | | voxels | ||
+ | | 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] | ||
+ | | <math>\sum I(\Omega)</math> or <math>M_{000}|\{I=intensity\}</math> | ||
+ | |- | ||
+ | | Mean | ||
+ | | | ||
+ | | Average intensity of voxels in the object [2] | ||
+ | | <math>\frac{1}{|\Omega|}\sum I(\Omega)</math> | ||
+ | |- | ||
+ | | 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] | ||
+ | | <math>\sigma_I</math> | ||
+ | |- | ||
+ | | Variance | ||
+ | | | ||
+ | | Variance of intensity of voxels in the object [2] | ||
+ | | <math>\sigma_I^2</math> | ||
+ | |- | ||
+ | | Radius Variation | ||
+ | | voxels | ||
+ | | Standard deviation of distance from surface voxels to centroid | ||
+ | | stddev<math>(\|\Omega_s - \overline{p}\|)</math> | ||
+ | |- | ||
+ | | Skew | ||
+ | | | ||
+ | | Skew of the PDF [3] | ||
+ | | <math>\frac{1}{\sigma_I^3}\sum_{I=0}^{255}(I-\overline{I})^3P(I)</math> | ||
+ | |- | ||
+ | | Energy | ||
+ | | | ||
+ | | Energy of the PDF[3] | ||
+ | | <math>\sum_{I=0}^{255}[P(I)]^2</math> | ||
+ | |- | ||
+ | | Entropy | ||
+ | | | ||
+ | | Entropy of the PDF [3] | ||
+ | | <math>-\sum_{I=0}^{255}P(I)\log_2{P(I)}</math> | ||
+ | |- | ||
+ | | Surface Gradient | ||
+ | | | ||
+ | | Average of surface gradients | ||
+ | | <math>mean(G(\Omega_s))</math> | ||
+ | |- | ||
+ | | Interior Gradient | ||
+ | | | ||
+ | | Average of interior gradients | ||
+ | | <math>mean(G(\Omega_{in}))</math> | ||
+ | |- | ||
+ | | Interior Intensity | ||
+ | | | ||
+ | | Average of interior intensities | ||
+ | | <math>mean(I(\Omega_{in}))</math> | ||
+ | |- | ||
+ | | Surface Intensity | ||
+ | | | ||
+ | | Average of surface intensities | ||
+ | | <math>mean(I(\Omega_s))</math> | ||
+ | |- | ||
+ | | Intensity Ratio | ||
+ | | | ||
+ | | Ratio of surface intensity to interior intensity | ||
+ | | <math>\frac{mean(I(\Omega_s))}{mean(I(\Omega_{in}))}</math> | ||
+ | |- | ||
+ | | Shared Boundary | ||
+ | | | ||
+ | | Ratio of object "edges" that touch another object to total number of object "edges | ||
+ | | | ||
+ | |- | ||
+ | | Surface Area | ||
+ | | voxels | ||
+ | | Number of voxels on surface of the object [4] | ||
+ | | <math>|\Omega_s|</math> | ||
+ | |- | ||
+ | | Shape | ||
+ | | | ||
+ | | Ratio of surface voxels to total voxels - compactness or thinness of object [5] | ||
+ | | <math>\frac{|\Omega_s|^3}{36\pi|\Omega|^2}</math> | ||
+ | |} | ||
− | == | + | == References == |
− | [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] Umbaugh, S. E., Y.-S. Wei, et al. (1997). "Feature extraction in image analysis. A program for facilitating data reduction in medical image classification." Engineering in Medicine and Biology Magazine, IEEE 16(4): 62-73.<br> |
+ | [4] Lohmann, G. (1998). Volumetric Image Analysis, Wiley <br> | ||
+ | [5] Theodoridis, S. and K. Koutroumbas (1999). Pattern recognition. San Diego, Academic Press. <br> | ||
+ | [6] [http://kitware.com/products/archive/kitware_quarterly0109.pdf Kitware Source Newsletter] |
Latest revision as of 18:39, 27 May 2009
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.
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 |
Ω = {p | lp = o} | the set of voxels of an object o |
the set of surface voxels of the object | |
Ωin = Ω − Ωs | the set of interior voxels of an object |
the center of mass of the object | |
P(I) | Probability Density Function (PDF) of intensity values I |
Raw Moment of discrete image I | |
λi | ith eigenvalue of covariance matrix |
eigenvector corresponding to λi |
Features
Name | Units | Description | Formula |
Volume | voxels | 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] | or M000 | {I = intensity} | |
Centroid | voxels | Center of the object [1] | |
Weighted Centroid | voxels | Uses the image intensity values to calculate the center of mass of the object [1] | |
Axes Lengths | voxels | The length of the axes of the ND hyper-ellipsoid fit to the object [1] | |
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 | radians | Angle between the major axis of the best-fit hyper-ellipsoid and origin. (2D) [1] | |
Bounding Box Volume | voxels | Number of voxels in the bounding box of the object [1] | (max(X)-min(X)+1) * (max(Y)-min(Y)+1) * ... |
Oriented Bounding Box Volume | voxels | 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] | or M000 | {I = intensity} | |
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] | σI | |
Variance | Variance of intensity of voxels in the object [2] | ||
Radius Variation | voxels | Standard deviation of distance from surface voxels to centroid | stddev |
Skew | Skew of the PDF [3] | ||
Energy | Energy of the PDF[3] | ||
Entropy | Entropy of the PDF [3] | ||
Surface Gradient | Average of surface gradients | mean(G(Ωs)) | |
Interior Gradient | Average of interior gradients | mean(G(Ωin)) | |
Interior Intensity | Average of interior intensities | mean(I(Ωin)) | |
Surface Intensity | Average of surface intensities | mean(I(Ωs)) | |
Intensity Ratio | Ratio of surface intensity to interior intensity | ||
Shared Boundary | Ratio of object "edges" that touch another object to total number of object "edges | ||
Surface Area | voxels | Number of voxels on surface of the object [4] | | Ωs | |
Shape | Ratio of surface voxels to total voxels - compactness or thinness of object [5] |
References
[1] itkLabelGeometryImageFilter
[2] itkLabelStatisticsImageFilter
[3] Umbaugh, S. E., Y.-S. Wei, et al. (1997). "Feature extraction in image analysis. A program for facilitating data reduction in medical image classification." Engineering in Medicine and Biology Magazine, IEEE 16(4): 62-73.
[4] Lohmann, G. (1998). Volumetric Image Analysis, Wiley
[5] Theodoridis, S. and K. Koutroumbas (1999). Pattern recognition. San Diego, Academic Press.
[6] Kitware Source Newsletter