# ImMinkowski

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

## Contents

- 1 Presentation
- 2 Quick start
- 3 Function List
- 3.1 Perimeter in 2D images
- 3.2 Area in 2D images
- 3.3 Euler-Poincare characteristic in 2D images
- 3.4 Volume in 3D images
- 3.5 Surface area in 3D images
- 3.6 Mean breadth (integral of mean curvature) in 3D images
- 3.7 Euler-Poincare characteristic in 3D images
- 3.8 Euler-Poincare characteristic and length in 1D images

- 4 References

## Presentation

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); imPerimeter(lbl) ans = 184.8668 154.9495 185.1921 267.1690 187.3183 179.5038 182.7406 180.8445 155.5049 155.5049 % 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 = 2.7291e+004

## 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

## References

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: http://www.ias-iss.org/ojs/IAS/article/view/811

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