boxdim_listepoints (FRACLAB Functions)

FRACLAB Functions

boxdim_binaire

Box dimension, computed with the box method, of the white points in a black and white image or, more generally, of non-zero-points in an N-dimensional array.

Syntax
[boxdim,Nboxes,handlefig]=boxdim_binaire(BWimg, Size, Ratio, Axes, Waitbar ,reg, lrstr, optvarargin)

Inputs

BWimg
N-dimensional array. Usually, it is a matrix containing 0 and 1. If BWImg contains values different from 0 and 1, all non-zero values are trated as ones.

Size, Ratio :
Ratio defines the lengths of the sides of a "first box". If BWimg is an image, it is a 1x2 array. (1xN array if BWimg is an N dimensional array). The successive boxes will be deduced from this first box by homotheties with ration Size(i). More precisely, the box at at the i - th iteration will have a length along the k - th axis equal to Ratio(k)*Size(i)

Axes :

If BWImg is an array, Axes is a 1x2 array : [Xmin,Xmax]. Xmin is the abscissa corresponding to Signal(1) and Xmax the abscissa corresponding to Signal(end).
If BWImg is a matrix(black and white image), Axes is a 2 x 2 matrix containing the coordinates of its corners : [Xmin Xmax ; Ymin Ymax].
If BWImg is an N-dimensional array, Axes is an Nx2 matrix.

For a graphical explanation of Size, Ratio and Axes, see Example 1

Waitbar :
1 if you want a waitbar to appear during the calculation, else 0

reg :
The way you choose the bounds. See the help on fl_regression for more information.

lrstr,optvarargin :
Regression parameters. They define the regression type. Type " help monolr" for more information

Outputs

boxdim :
The estimated box dimension

Nboxes :
Nboxes is a vector. Nboxes(i) is the number of non-empty boxes at the i - th iteration. A non-empty box is a box that contains at least one point of the graph of the function.

handlefig :
If reg~=0, a window appears to help you choosing the linearity bounds. handlefig is the handle of this figure.

bounds :
Bounds of linearity of log2(Nboxes) in function of log2(Size). Interesting if reg==1.

Optional arguments

All arguments except BWImg are optional. For example, the syntax : boxdim_classique(PointsList) is correct. If you don't want to precise an argument, you can also type []. The missing arguments take their default values:

Size=[1 1/2 1/4 ... 1/2048 1/4096] Ratio =[1 1 ...] Axes=[0 1;0 1 ; ...] Waitbar = 0 reg=0 lrstr='ls'

These default values are not always relevant, especially Size, so when you don't know how to define Size, try to use reg ~= 0 in order to precise manually the correct box sizes.

Examples

Example 1

This drawing may help you to understand the roles of Size and Ratio.

It corresponds to the following intructions :

load('image_Weierstrass.mat'); imagesc(C);colormap('gray');%display C Size=[1/2,1/4]; Ratio=[1,4]; Axes=[0,1;0,4]; % To avoid the extrema points to be counted in separated boxes epsi=10^(-3); Axes=Axes+[epsi,-epsi;epsi,-epsi]; % Calculation [boxdim,Nboxes,handlefig]=boxdim_binaire(C, Size, Ratio,Axes)

These instructions will return Nboxes=[4,10]. It is consistent with the drawing. The computed dimension isn't meaningfull, because of the low number of iterations. The goal of this example is just to show how the boxes are defined.

Example 2

Example: Computes the box dimension of a self-similar image. Its theorical dimension is log(7)/log(3)=1.77


% load the image

load('fleche.mat');

% Plot the image

figure;imagesc(fleche);colormap('gray');pause;

% Compute its box dimension

reg=1;

Waitbar=1;

[boxdim,Ntailles,handlefig,bounds]=boxdim_binaire(fleche,[],[],[],Waitbar,reg);

boxdim

bounds


% You should find bounds = -9 -3. It means that

% the progression is linear when log2(size) is in [-9 -3]. You may keep these

% bounds, take more points and try another regression type.

reg=0;

Size=2.^[-9 : 0.5 : 0-3];

boxdim=boxdim_binaire(fleche,Size,[],[],Waitbar,reg,'pls',30);

boxdim