Intrinsic Features of Blobs

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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)
$N p$	a neighbor voxel of $p$
$l p$	the segmentation label at $p$
$I i (p)$	the intensity value of $p$ at $i t h$
$Ω = {p \| l p = o}$	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
$Ω i n = Ω - Ω s$	the set of interior voxels of an object
$\overline{p}$	the center of mass of the object
$P (I)$	Probability Density Function (PDF) 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 $I$
$λ i$	$i t h$ eigenvalue of covariance matrix
$\overline{v_i}$	eigenvector corresponding to $λ i$

Features

Name	Units	Description	Formula
Volume	voxels	Number of voxels in the object [1]	$\| Ω \|$ or $M 000 \| {I = b i n a r y}$
Integrated Intensity		Sum of the intensities of all voxels in the object [1]	$\sum I(\Omega)$ or $M 000 \| {I = i n t e n s i t y}$
Centroid	voxels	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	voxels	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	voxels	The length of the axes of the ND hyper-ellipsoid fit to the object [1]	$4\sqrt{\lambda_i}$
Eccentricity		Ratio of the distance between the foci of the best-fit hyper-ellipsoid to the length of its major axis. (2D) [1]	$\sqrt{\frac{\lambda_1 - \lambda_0}{\lambda_1}}$
Elongation		Ratio of the major axis length to minor axis length of the best-fit hyper-ellipsoid. (2D) [1]	$\frac{\lambda_1}{\lambda_0}$
Orientation	radians	Angle between the major axis of the best-fit hyper-ellipsoid and origin. (2D) [1]	$tan^{-1}\left(\frac{\overline{v_1}(1)}{\overline{v_1}(0)}\right)$
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]	$\sum I(\Omega)$ or $M 000 \| {I = i n t e n s i t y}$
Mean		Average intensity of voxels in the object [2]	$\frac{1}{\|\Omega\|}\sum I(\Omega)$
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]	$σ I$
Variance		Variance of intensity of voxels in the object [2]	$\sigma_I^2$
Radius Variation	voxels	Standard deviation of distance from surface voxels to centroid	stddev $(\\|\Omega_s - \overline{p}\\|)$
Skew		Skew of the PDF [3]	$\frac{1}{\sigma_I^3}\sum_{I=0}^{255}(I-\overline{I})^3P(I)$
Energy		Energy of the PDF[3]	$\sum_{I=0}^{255}[P(I)]^2$
Entropy		Entropy of the PDF [3]	$-\sum_{I=0}^{255}P(I)\log_2{P(I)}$
Surface Gradient		Average of surface gradients	$m e a n (G (Ω s))$
Interior Gradient		Average of interior gradients	$m e a n (G (Ω i n))$
Interior Intensity		Average of interior intensities	$m e a n (I (Ω i n))$
Surface Intensity		Average of surface intensities	$m e a n (I (Ω s))$
Intensity Ratio		Ratio of surface intensity to interior intensity	$\frac{mean(I(\Omega_s))}{mean(I(\Omega_{in}))}$
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]	$\| Ω s \|$
Shape		Ratio of surface voxels to total voxels - compactness or thinness of object [5]	$\frac{\|\Omega_s\|^3}{36\pi\|\Omega\|^2}$

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

Intrinsic Features of Blobs

Glossary of Notation

Features

References

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