Modified Haussdorf Fractal Dimension

% 150507: Reza Farrahi Moghaddam (imriss@ieee.org) (May 7th, 2015)
% Modified Haussdorf Fractal Dimension
%
% Main features:
% 1. Valuing both White and Black pixels,
% 2. Probabilistically discarding boxes of less value.
%
% SYNTAX: [D, D_old, I, D_ref] = hausDim_Modified_R(I, method_flag)
% I: Input image (White pixels are assumed as information),
% method_flag: 'edg' to use edge of I, 'ske' to use skeleton of I, 'pre' to preprocess I, 'full' to preprocess and then use the edge and skeleton.
% D: Modified Haussdorf Fractal Dimension of I after preprocessing (if any),
% D_old: Haussdorf Fractal Dimension of I after preprocessing (if any),
% D_ref: Haussdorf Fractal Dimension of the unprocessed I,
% I: Processed output of I.
%
% Based on: http://www.mathworks.com/matlabcentral/fileexchange/30329-hausdorff--box-counting--fractal-dimension
%
% Requires:
% 1. Hausdorff (Box-Counting) Fractal Dimension (http://www.mathworks.com/matlabcentral/fileexchange/30329-hausdorff--box-counting--fractal-dimension)
% 2. inpaint_nans (http://www.mathworks.com/matlabcentral/fileexchange/4551-inpaint-nans)
% 3. Universal Color to Gray Conversion (http://www.mathworks.com/matlabcentral/fileexchange/27578-universal-color-to-gray-conversion)
%
% Examples:
% I = mat2gray(double(imread('http://4.bp.blogspot.com/-aHCfmDvyzFU/Un_U-Neo_GI/AAAAAAAAGpQ/DWzjztkh4HM/s1600/sierpinski.png')));
% [D, D_old, I, D_ref] = hausDim_Modified_R(I, 'edg'); % D = 1.5843, D_ref = 1.5999
%
% I = mat2gray(double(imread('http://upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Sierpinski_carpet.png/480px-Sierpinski_carpet.png')));
% [D, D_old, I, D_ref] = hausDim_Modified_R(I, 'edg'); % D = 1.6794, D_ref = 1.8811
%
% I = mat2gray(double(imread('http://www.math.upenn.edu/~pstorm/images/round_Sierpinski_carpet.png')));
% [D, D_old, I, D_ref] = hausDim_Modified_R(I, 'edg'); % D = 1.6263, D_ref = 1.7129
%