sum of the distance difference between the outline of a shape in certain direction to that of an equivalent ellipse

This measure allows to quantified the angularity of a shape using the difference between shape radius in a certain direction and that of its equivalent ellipse.

Note that equivalent ellipse associated to a shape is define such as :

ellipse center is set at shape barycenter (see Barycenter2d)
ellipse orientation is equal to orientation extracted from second order moment of shape (see Inertia Orientation 2d)
ellipse dimensions are defined such that ellipse perimeter and area match with shape perimeter (see Perimeter2d) and shape area (see Area2d).

Given a set of orientations regularly sampled around unit circle (with number of orientation defined by parameter NbOrientations), we compute an angularity value given by :

$A_i = \frac{\left |R(\theta_i) - R^{EE}(\theta_i)\right |}{R^{EE}(\theta_i)}$

where :

$\theta_i$ is the angle associated to sample number ( $\theta_i=i \times \frac{2\pi}{m}$ with set to NbOrientations)
$R(\theta_i)$ is the radius of shape at angle $\theta_i$ , which is the length of segment starting at equivalent ellipse center, oriented and directed with respect to angle $\theta_i$ and ending at intersection with shape boundary (note that we retain farest intersection in case of non convex shape which may lead to several intersection points).
$R^{EE}(\theta_i)$ is the radius of equivalent ellipse at angle $\theta_i$ .

Then the global shape angularity value is simply given by the sum of previous values over all sampled angles :

$A=\sum_{i=1}^{m}{A_i}=\sum_{i=1}^{m}{\frac{\left |R(\theta_i) - R^{EE}(\theta_i)\right |}{R^{EE}(\theta_i)}}$

Here is an example of degree of angularity 2d computation on few caracteristic shapes :

See also: Taleb Al-Rousan, Eyad Masad, Erol Tutumluer, Tongyan Pan, Evaluation of image analysis techniques for quantifying aggregate shape characteristics, Construction and Building Materials 21 (2007) 978�990.; Al-Rousan, T., Masad, T., Tutumluer, E. and Pan T. (2007). Evaluation of image analysis techniques for quantifying aggregate shape characteristics.Construction and building materials21, pp. 978-990.; Dallas Little, Joe Button, Priyantha Jayawickrama, Mansour Solaimanian, Barry Hudson, Quantify shape, angularity and surface texture of aggregates using image analysis and study their effect on performance, Texas Department of Transportation, October 2003; Tafesse Solomon, Sun Wenjuan, Fernlund Joanne, and Linbing Wang, A new image analysis technique to quantify particle angularity, Engineering Geology and Geophysics Research Group Royal Institute of Technology (KTH)

sum of the distance difference between the outline of a shape in certain direction to that of an equivalent ellipse

Measure synthesis :

Measure Type	Measure Unit Type	Parameter Type	Result Type	Shape Requirements
Geometry 2d	None	RadiusAngularityIndex2dMsrParams	Value (ipsdk::ipReal64)	Both

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, "RadiusAngularityIndex2dMsr")
    
    #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("RadiusAngularityIndex2dMsr")
    
    # retrieve measure values
    outMsrValues = outMsr.getMeasureResult().getColl(0)
    print("First label measurement equal " + str(outMsrValues[1]))

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>();
    IPSDK_REQUIRE(readFromXmlFile(inputShape2dCollPath, *pShape2dColl) == true);
    // define a measure info set
    MeasureInfoSetPtr pMeasureInfoSet = MeasureInfoSet::create2dInstance();
    createMeasureInfo(pMeasureInfoSet, "RadiusAngularityIndex2dMsr");
    
    // compute measure on shape 2d collection
    MeasureSetPtr pOutMeasureSet = shapeAnalysis2d(pInGreyImg2d, pShape2dColl, pMeasureInfoSet);
    // retrieve associated results
    const MeasureConstPtr& pRadiusAngularityIndex2dOutMsr = pOutMeasureSet->getMeasure("RadiusAngularityIndex2dMsr");
    const ipsdk::shape::analysis::ValueMeasureResult<ipsdk::ipReal64>& outResults = static_cast<const ipsdk::shape::analysis::ValueMeasureResult<ipsdk::ipReal64>&>(pRadiusAngularityIndex2dOutMsr->getMeasureResult());