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.

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 :

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	None	None	Custom	Row Intersections

See Shape measurement for additional information on these pictograms

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());