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The imMinkwoski module contains various functions for measuring or estimating geometric quantities from 2D or 3D images.


For 2D images, parameters are the area, the perimeter and the (2D) Euler Number. For 3D images, parameters are the volume, the surface area (called surface), the mean breadth (also known as integral of mean curvature), and the (3D) Euler Number. For the sake of completeness, parameters for 1D images are also included: length and number (1D Euler Number).

Several functions are provided for each parameter, depending on how the parameter is considered:

  • im<Param>: global measure of the parameter in the whole image. If the

structure touches the border of the image, it is considered as a structure border. Such functions should fit most needs.

  • im<Param>Estimate: estimation of the parameter by considering image

is a representative window of a larger structure. Interesection of the structure with iamge border are not taken into account for measurements.

  • im<Param>Density: Same as im<Param>Estimate, but the result is

normalised by the area or volume of the observed window.

Most functions work both for binary and label images. It is possible to specify options (connectivity for Euler Number, number of directions for perimeter or surface area), as well as image resolution in each direction.

Quick start

     % compute perimeter of several coins 
     lbl = bwlabel(imread('coins.png') > 100);
     ans = 
     % Surface area measured in 3D binary image (result in pixel^2)
     img = analyze75read(analyze75info('brainMRI.hdr'));
     bin = imclose(img>0, ones([5 5 3]));
     S = imSurface(bin, [1 1 2.5])      % specify resolution
     ans = 

Function List

Perimeter in 2D images

   imPerimeter         - Perimeter of a 2D image using Crofton formula
   imPerimeterDensity  - Perimeter density of a 2D binary structure, using Crofton formula
   imPerimeterEstimate - Perimeter estimate of a 2D binary structure

Area in 2D images

   imArea              - Compute area of binary 2D image 
   imAreaDensity       - Compute area density in a 2D image
   imAreaEstimate      - Estimate area of binary 2D structure with edge correction

Euler-Poincare characteristic in 2D images

   imEuler2d           - Euler number of a binary 2D image
   imEuler2dDensity    - Euler density in a 2D image
   imEuler2dEstimate   - Estimate Euler number in a 2D image

Volume in 3D images

   imVolume            - Volume measure of a 3D binary structure
   imVolumeDensity     - Compute volume density of a 3D image
   imVolumeEstimate    - Estimate volume of a 3D binary structure with edge correction

Surface area in 3D images

   imSurface           - Surface area of a 3D binary structure
   imSurfaceDensity    - Surface area density of a 3D binary structure
   imSurfaceEstimate   - Estimate surface area of a binary 3D structure
   imJointSurface      - Surface area of the interface between two labels

Mean breadth (integral of mean curvature) in 3D images

   imMeanBreadth       - Mean breadth of a 3D binary or label image

Euler-Poincare characteristic in 3D images

   imEuler3d           - Euler number of a binary 3D image
   imEuler3dDensity    - Compute Euler density in a 3D image
   imEuler3dEstimate   - Estimate Euler number in a 3D image

Euler-Poincare characteristic and length in 1D images

   imEuler1d           - Compute Euler number of a binary 1D image
   imEuler1dEstimate   - Compute Euler number of a binary 1D image
   imLength            - Compute total length of a binary 1D structure
   imLengthDensity     - Estimate length density of a binary 1D structure using edge correction
   imLengthEstimate    - Estimate total length  of a binary 1D structure using edge correction


If you use this package, please be kind to cite following reference: "Computation of Minkowski measures on 2D and 3D binary images". David Legland, Kien Kieu and Marie-Francoise Devaux (2007) Image Analysis and Stereology, Vol 26(2), June 2007 web:

Following reference can also be of interest: "Statistical Analysis of Microstructures in Material Sciences" Joachim Ohser and Frank Muecklich (2000) John Wiley and Sons