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# Fast mutual information of two images or signals

07 Dec 2006 (Updated 24 Aug 2011)

Optimized function for mutual information of two images or signals

File Information
Description

Usage: I=mi(A,B), where A and B are equally sized images/signals.

Function hist2 (included) is used to determine the joint histogram of the images/signals.

Assumptions: 1) 0*log(0)=0, 2) mutual information is obtained on the intersection between the supports of partial histograms.

Example (in mi_test.m):

disp('Test: Mutual information between two images')
A=D(:,:,8);
B=D(:,:,9);
mi(A,B)

disp('Test: Mutual information between two signals')
nasdaq = price2ret(NASDAQ);
nyse = price2ret(NYSE);
mi(nasdaq,nyse)

MATLAB release MATLAB 7.7 (R2008b)
28 May 2013

Subscripted assignment dimension mismatch.

Error in hist2 (line 32)
n(i+1,:) = histc(B(A==i),x,1);

Error in mi (line 31)
n2 = hist2(A,B,L);

Does anyone have the same problem?

10 Mar 2013

I think it will be more accurate to use

A=floor((A-ma)*L/(MA-ma+eps));
B=floor((B-mb)*L/(MB-mb+eps));

A=round((A-ma)*(L-1)/(MA-ma+eps));
B=round((B-mb)*(L-1)/(MB-mb+eps));

because round makes the first and the last bins twice less than the others.

06 Jan 2013

on running the code from mi_test.m, I got error message: ??? Undefined function or method 'mi' for input arguments of
type 'uint8'.
I am working in Matlab 7.10.0(R 2010a). I don't know what is the problem. Any help will be highly appreciated.

13 Dec 2012

ok. just ignore what i said for the last message. I made some mistake.

13 Dec 2012

Hi Jose,
Thanks for sharing the code. I think these is still something wrong with the code. when i tried this a = [1,2,3;1,2,3;1,2,3], and calculate mi(a,a), I get 1.0566, so I think there is still something wrong with it. thanks.

17 Jul 2012

Hanbo, many thanks for your comments. I'd like to add this improvement and test it. Would you provide some test data?

16 Jul 2012

Thank you for sharing this. I modified based on this to calculate normalized mutual information. However, I found the results could be larger than 1 sometimes (which is wrong). I checked my modified part and everything is correct, which means the mutual information given by this code could be wrong or not accurate. I checked the code and found that it is mainly caused by different definition of bin boundary of hist (matlab defined) and hist2 (user defined). I replaced hist() with following myhist() and the result is now excellent on simulated toy testing data.

function n=myhist(A,L)
ma=min(A(:));
MA=max(A(:));
A=round((A-ma)*(L-1)/(MA-ma+eps));
x=0:L-1;
n=histc(A,x,1);
n=n';
end

04 Nov 2011

Great Job ! However, I'm just wondering why you're using hist in mi function and histc in hist2 function. I would have used histc everywhere. Is there any difference, or it is an arbitrary choice ?

25 Aug 2011

Andrew, many thanks. I'm sorry about my delay. I added the improvements you suggested. The code is much faster now.

Divyesh Varade, mutual information is a measure based on pixel statistics, so it can't be applied pixel by pixel. Applying it to small windows could be a good idea for some applications.

06 Jul 2011

Thanks for the code it was very helpful. However, I want to compare two images using MI. If I do it pixel by pixel using the given code, I mostly get a warning: divide by zero and receive a value NaN for MI. Now I can try and work with very small windows also but that would seem to reduce the quality of my results. Is there another way or by enhancing this code somehow to analyse the two images together almost pixelwise.

22 Feb 2010

oh, and somwhere there is a re-scaling bug ; http://www.flickr.com/photos/47796226@N04/4380054294/
which shows the MI between a fixed window in one image, and a sliding window in each of 6 other images.
In some images it works, but in others it doesn't.

Some images have negative intensity; i'll investigate further and post here again if i find the cause. (it happens with both my code and the original, changing the re-binning can slightly change the pattern of the unusual elements.)

22 Feb 2010

the current implementation of both MI and hist2 are quite slow;

function n=fast_hist2(A,B,L)
ma=min(A(:));
MA=max(A(:));
mb=min(B(:));
MB=max(B(:));

% For sensorimotor variables, in [-pi,pi]
% ma=-pi;
% MA=pi;
% mb=-pi;
% MB=pi;

% Scale and round to fit in {0,...,L-1}
A=round((A-ma)*(L-1)/(MA-ma+eps));
B=round((B-mb)*(L-1)/(MB-mb+eps));
n=zeros(L);
x=0:L-1;
for i=0:L-1
n(i+1,:) = histc(B(A==i),x,1);
end
end

is about 10x quicker, and gives the same result (assuming that the input is sensible, i believe you get different behavior if you pass in all nan's, or anything which would cause matlabs 'hist' function to error) ; given that the mi function calls hist on both the inputs individually first, this should not be a problem.

also there is need for a bugfix in mi, 256 bins are hard coded, irrespective of what you pass in.

the minf function also does unnecessary allocation.

Using the functions above and below rather than the attached the running time is about 22 seconds rather than 240, for my set of MI calculations. (2500 calls for MI between two 50x50 windows)

function I=fast_mi(A,B,L)
%MI Determines the mutual information of two images or signals
%
% I=mi(A,B)
%
% Assumption: 0*log(0)=0
%

% jfd, 15-11-2006
% 01-09-2009, added case of non-double images

A=double(A);
B=double(B);

na = fast_hist(A(:),L);
na = na/sum(na);

nb = fast_hist(B(:),L);
nb = nb/sum(nb);

n2 = fast_hist2(A,B,L);
n2 = n2/sum(n2(:));

I=sum(minf(n2,na'*nb));
%I=sum(I);

% -----------------------

function y=minf(pab,papb)

I=find(papb(:)>1e-12 & pab(:)>1e-12); % function support
y=pab(I).*log2(pab(I)./papb(I));

01 Sep 2009

I solved problems related with data types. Now mi() accepts different types of images and signals (not only double images or signals). However, I am testing it with Matlab 7 (R14). Please leave a comment if it doesn't work with other versions.

26 Feb 2009

Program bombs with 2007a

??? Error using ==> mtimes
Integers can only be combined with integers of the same class, or scalar
doubles.

Error in ==> hist at 73
xx = miny + binwidth*(0:x);

Error in ==> mi at 12
na = hist(A(:),256);

06 May 2008

the following codes raise errors:
"
A = uint8(zeros(4));
mi(A,A);
"

where uint8 is the usual type for images

"
A = zeros(4);
mi(A,A);
"

gives NaN.

10 Feb 2008

thank you for your well-written and clear code.

08 Jul 2007

thank you very much for the code, really appreaciate it.

20 May 2007
07 Mar 2007
08 Oct 2008

Needed modification of input arguments. Example added.

01 Sep 2009

Added support for signals (1D matrices) and for non-double images.

24 Aug 2011

Speed improvements suggested in comments by Andrew Hill.