IPSDK  4_1_0_2
IPSDK : Image Processing Software Development Kit
Classes
GreyInertia
Shape Analysis and Measurement » Intensity » Inertia

Measure allowing to compute the second order moment ponderated by the gray level of each pixel for shape. More...

Classes

class  ipsdk::imaproc::shape::analysis::GreyInertiaMsr
 Measurement object for measure GreyInertia. More...
 
class  ipsdk::imaproc::shape::analysis::GreyInertiaMsrInfo
 Information object for measure GreyInertia. More...
 
class  ipsdk::imaproc::shape::analysis::GreyInertiaMsrResults
 Measurement results object for measure GreyInertia. More...
 

Detailed Description

Measure allowing to compute the second order moment ponderated by the gray level of each pixel for shape.

The GreyInertia measurement computes the second order central moment for each shape, ponderated by the voxels intensities, thanks to the inertia matrix $ M $.

\[ M = \begin{bmatrix} m_{xx} & m_{xy} & m_{xz} \\ m_{yx} & m_{yy} & m_{yz} \\ m_{zx} & m_{zy} & m_{zz} \end{bmatrix} \]

Whose the components are computed as follows :

\[ \begin{array}{r c l} m_{xx} & = & \sum_{\lbrace x, y, z \rbrace \in Shape}{ \left(x - \bar{x} \right)^2 I(x, y, z)} \\ m_{xy} = m_{yx} & = & \sum_{\lbrace x, y, z \rbrace \in Shape}{ \left(x - \bar{x} \right) \left(y - \bar{y} \right) I(x, y, z)} \\ m_{xz} = m_{zx} & = & \sum_{\lbrace x, y, z \rbrace \in Shape}{ \left(x - \bar{x} \right) \left(z - \bar{z} \right) I(x, y, z)} \\ m_{yy} & = & \sum_{\lbrace x, y, z \rbrace \in Shape}{ \left(y - \bar{y} \right)^2 I(x, y, z)}\\ m_{yz} = m_{zy} & = & \sum_{\lbrace x, y, z \rbrace \in Shape}{ \left(y - \bar{y} \right) \left(z - \bar{z} \right) I(x, y, z)} \\ m_{zz} = m_{zz} & = & \sum_{\lbrace x, y, z \rbrace \in Shape}{ \left(z - \bar{z} \right) \left(z - \bar{z} \right) I(x, y, z)} \end{array} \]

Where $ I(x, y, z) $ is the image intensity at the position $ (x, y, z) $, $ \bar{x} $ , $ \bar{y} $ and $ \bar{z} $ are the grey barycenter coordinates (see GreyBarycenter) and $ Shape $ is the voxel collection of the current shape.

From these composents, three descriptors are extracted in 2d : the minimum and maximum eigen values of $ M $ ( $ \lambda_{Min} $ and $ \lambda_{Max} $ ) and the shape orientation $ \theta $ .

These features are calculated as follows :

\[ \begin{array}{r c l} \delta & = & \sqrt{(m_{xx} - m_{yy}) (m_{xx} - m_{yy}) + 4 m_{xy} m_{xy}} \\ \lambda_{Max} & = & \frac{1}{2} \left( m_{xx} + m_{yy} + \delta \right) \\ \lambda_{Min} & = & \frac{1}{2} \left( m_{xx} + m_{yy} - \delta \right) \\ \theta & = & \tan^{-1} \left( \frac{\lambda_{Max} - m_{xx}}{m_{xy}} \right) \end{array} \]

$ \lambda_{Min} $ and $ \lambda_{Max} $ are proportional to the squared length of the eigenvector axes, represented by green arrows in the figure below.

greyInertia2dMsr_illustration.png

In 3d case, other features are available. They are calculated as follows :

\[ \begin{array}{r c l} q & = & \frac{m_{xx} + m_{yy} + m_{zz}}{3} \\ p & = & \sqrt{\frac{(m_{xx} - q)^2 + (m_{yy} - q)^2 + (m_{zz} - q)^2 + 2 (m_{xy}^2 + m_{xz}^2 + m_{yz}^2)}{6}} \\ \lambda_{Max} & = & q + 2 p \cos(\phi + 2 \pi / 3) \\ \lambda_{Inter} & = & q + 2 p \cos(\phi - 2 \pi / 3) \\ \lambda_{Min} & = & q + 2 p \cos(\phi) \end{array} \]

Where :

\[ \phi = \begin{cases} \pi/3& \text{if } r \leq -1 \\ 0 & \text{if } r \geq 1\\ \frac{\cos^{-1}(r)}{3} &\text{otherwise} \end{cases} \]

And

\[ r = \frac{(m_{xx} - q)(m_{yy} - q)(m_{zz} - q) + 2 m_{xy} m_{xz}m_{yz} - (m_{yy} - q) m_{xz}^2 - (m_{zz} - q) m_{xy}^2 - (m_{xx} - q) m_{yz}^2}{2 p^3} \]

Moreover, the 2d rotation angle $\theta$ is replaced by the 3d rotation angles $\alpha$, $\beta$ and $\chi$, calculated with the coefficients of $M$ :

\[ \begin{array}{r c l} \alpha & = & \tan^{-1} \left( \frac{m_{yz}}{m_{zz}} \right) \\ \beta & = & - \sin \left( m_{xz} \right) \\ \chi & = & \tan^{-1} \left( \frac{m_{xy}}{m_{xx}} \right) \end{array} \]

Here is an example of the grey inertia calculated from a 2d input image :

greyInertia2dMsr.png
See also
https://en.wikipedia.org/wiki/Image_moment#Central_moments
Author
R. Abbal
Date
2015/07/30
Copyright
2015 Reactiv'IP, all right reserved.

Measure allowing to compute the second order moment ponderated by the gray level of each pixel for shape

Measure synthesis :

Measure Type Measure Unit Type Parameter Type Result Type Shape Requirements
Intensity.png
Intensity
none.png
None
none.png
None
Custom.png
Custom
RowIntersections.png
Row Intersections
See Shape measurement for additional information on these pictograms

Measure Type :

This is an intensity measure

This measure can be used in 2d and 3d case

Measure Unit Type:

Measure GreyInertia is not associated to any unit [ipsdk::shape::analysis::eMsrUnitFormat::eMUF_NoUnit]

Measure Parameter Type :

Measure GreyInertia has no parameters

Measure Result Type :

Measure GreyInertia is associated to GreyInertiaMsrResults results

Measure Shape Requirements :

Measure GreyInertia requires row intersections from shape data

Measure Dependencies :

Measure GreyInertia depends on following measures :

Measure Mode Measure Name Measure Type Measure Parameters
eMVP_2d3d GreyBarycenter GreyBarycenter
Note
See Shape Analysis 2d for more information on general shape 2d analysis and measurement usage.
See Shape Analysis 3d for more information on general shape 3d analysis and measurement usage.

Example of Python code :

Generic example in 2d case :

import PyIPSDK
import PyIPSDK.IPSDKIPLShapeAnalysis as shapeanalysis
# Create the infoset
inMeasureInfoSet2d = PyIPSDK.createMeasureInfoSet2d()
PyIPSDK.createMeasureInfo(inMeasureInfoSet2d, "GreyInertiaMsr")
#Perform the analysis
outMeasureSet = shapeanalysis.labelAnalysis2d(inGreyImg, inLabelImg2d, inMeasureInfoSet2d)
# save results to csv format
PyIPSDK.saveCsvMeasureFile(os.path.join(tmpPath, "shape_analysis_results.csv"), outMeasureSet)
# retrieve measure results
outMsr = outMeasureSet.getMeasure("GreyInertiaMsr")
# retrieve measure values
outMsrValues = outMsr.getMeasureResult().getColl(0)
print("First label inertia mxx component measurement equal " + str(outMsrValues[1].mxx()))

Generic example in 3d case :

import PyIPSDK
import PyIPSDK.IPSDKIPLShapeAnalysis as shapeanalysis
# Create the infoset
inMeasureInfoSet3d = PyIPSDK.createMeasureInfoSet3d()
PyIPSDK.createMeasureInfo(inMeasureInfoSet3d, "GreyInertiaMsr")
#Perform the analysis
outMeasureSet = shapeanalysis.labelAnalysis3d(inGreyImg, inLabelImg, inMeasureInfoSet3d)
# save results to csv format
PyIPSDK.saveCsvMeasureFile(os.path.join(tmpPath, "shape_analysis_results.csv"), outMeasureSet)
# retrieve measure results
outMsr = outMeasureSet.getMeasure("GreyInertiaMsr")
# retrieve measure values
outMsrValues = outMsr.getMeasureResult().getColl(0)
print("First label inertia mxx component measurement equal " + str(outMsrValues[1].mxx()))

Example of C++ code :

Example informations

Associated library

IPSDKIPLShapeAnalysis

Code Example

// opening grey level input image
ImagePtr pInGreyImg2d = loadTiffImageFile(inputGreyImgPath);
// read entity shape 2d collection used for processing
Shape2dCollPtr pShape2dColl = boost::make_shared<Shape2dColl>();
readFromXmlFile(inputShape2dCollPath, *pShape2dColl);
// define a measure info set
MeasureInfoSetPtr pMeasureInfoSet = MeasureInfoSet::create2dInstance();
createMeasureInfo(pMeasureInfoSet, "GreyInertiaMsr");
// compute measure on shape 2d collection
MeasureSetPtr pOutMeasureSet = shapeAnalysis2d(pInGreyImg2d, pShape2dColl, pMeasureInfoSet);
// retrieve associated results
const MeasureConstPtr& pGreyInertiaOutMsr = pOutMeasureSet->getMeasure("GreyInertiaMsr");
const GreyInertiaMsrResults& outResults = static_cast<const GreyInertiaMsrResults&>(pGreyInertiaOutMsr->getMeasureResult());
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