Question

I m currently working on writing a version of the MATLAB RegionProps function for GNU Octave. I have most of it implemented, but I m still struggling with the implementation of a few parts. I had previously asked about the second central moments of a region.

This was helpful theoretically, but I m having trouble actually implementing the suggestions. I get results wildly different from MATLAB s (or common sense for that matter) and really don t understand why.

Consider this test image:

Slanting ellipse.

We can see it slants at 45 degrees from the X axis, with minor and major axes of 30 and 100 respectively.

Running it through MATLAB s RegionProps function confirms this:

MajorAxisLength: 101.3362
MinorAxisLength: 32.2961
Eccentricity: 0.9479
Orientation: -44.9480

Meanwhile, I don t even get the axes right. I m trying to use these formulas from Wikipedia.

My code so far is:

raw_moments.m:

function outmom = raw_moments(im,i,j)

  total = 0;
  total = int32(total);
  im = int32(im);

  [height,width] = size(im);

  for x = 1:width;
     for y = 1:height;
        amount = (x ** i) * (y ** j) * im(y,x);
        total = total + amount;
     end;
  end;

  outmom = total;

central_moments.m:

function cmom = central_moments(im,p,q);

  total = 0;
  total = double(total);
  im = int32(im);

  rawm00 = raw_moments(im,0,0);

  xbar = double(raw_moments(im,1,0)) / double(rawm00);
  ybar = double(raw_moments(im,0,1)) / double(rawm00);

  [height,width] = size(im);

  for x = 1:width;
    for y = 1:height;
      amount = ((x - xbar) ** p) *  ((y - ybar) ** q) * double(im(y,x));
      total = total + double(amount);
    end;
  end;

  cmom = double(total);

And here s my code attempting to use these. I include comments for the values I get at each step:

inim = logical(imread( 135deg100by30ell.png ));

cm00 = central_moments(inim,0,0);          % 2567

up20 = central_moments(inim,2,0) / cm00;   % 353.94
up02 = central_moments(inim,0,2) / cm00;   % 352.89
up11 = central_moments(inim,1,1) / cm00;   % 288.31

covmat = [up20, up11; up11, up02];
%[ 353.94  288.31
%  288.31  352.89 ]

eigvals = eig(covmat);          % [65.106 641.730]

minoraxislength = eigvals(1);   % 65.106
majoraxislength = eigvals(2);   % 641.730

I m not sure what I m doing wrong. I seem to be following those formulas correctly, but my results are nonsense. I haven t found any obvious errors in my moment functions, although honestly my understanding of moments isn t the greatest to begin with.

Can anyone see where I m going astray? Thank you very much.

Answer 1

EDIT:

According to Wikipedia:

the eignevalues [...] are proportional to the squared length of the eigenvector axes.

which is explained by:

axisLength = 4 * sqrt(eigenValue)

Shown below is my version of the code (I vectorized the moments functions):

my_regionprops.m

function props = my_regionprops(im)
    cm00 = central_moments(im, 0, 0);
    up20 = central_moments(im, 2, 0) / cm00;
    up02 = central_moments(im, 0, 2) / cm00;
    up11 = central_moments(im, 1, 1) / cm00;

    covMat = [up20 up11 ; up11 up02];
    [V,D] = eig( covMat );
    [D,order] = sort(diag(D),  descend );        %# sort cols high to low
    V = V(:,order);

    %# D(1) = (up20+up02)/2 + sqrt(4*up11^2 + (up20-up02)^2)/2;
    %# D(2) = (up20+up02)/2 - sqrt(4*up11^2 + (up20-up02)^2)/2;

    props = struct();
    props.MajorAxisLength = 4*sqrt(D(1));
    props.MinorAxisLength = 4*sqrt(D(2));
    props.Eccentricity = sqrt(1 - D(2)/D(1));
    %# props.Orientation = -atan(V(2,1)/V(1,1)) * (180/pi);      %# sign?
    props.Orientation = -atan(2*up11/(up20-up02))/2 * (180/pi);
end

function cmom = central_moments(im,i,j)
    rawm00 = raw_moments(im,0,0);
    centroids = [raw_moments(im,1,0)/rawm00 , raw_moments(im,0,1)/rawm00];
    cmom = sum(sum( (([1:size(im,1)]-centroids(2)) .^j * ...
                     ([1:size(im,2)]-centroids(1)).^i) .* im ));
end

function outmom = raw_moments(im,i,j)
    outmom = sum(sum( ((1:size(im,1)) .^j * (1:size(im,2)).^i) .* im ));
end

... and the code to test it:

test.m

I = imread( 135deg100by30ell.png );
I = logical(I);

>> p = regionprops(I, { Eccentricity   MajorAxisLength   MinorAxisLength   Orientation })
p = 
    MajorAxisLength: 101.34
    MinorAxisLength: 32.296
       Eccentricity: 0.94785
        Orientation: -44.948

>> props = my_regionprops(I)
props = 
    MajorAxisLength: 101.33
    MinorAxisLength: 32.275
       Eccentricity: 0.94792
        Orientation: -44.948

%# these values are by hand only ;)
subplot(121), imshow(I), imdistline(gca, [17 88],[9 82]);
subplot(122), imshow(I), imdistline(gca, [43 67],[59 37]);

screenshot

Answer 2

Are you sure about the core of your raw_moments function? You might try

amount = ((x-1) ** i) * ((y-1) ** j) * im(y,x);

This doesn t seem like enough to cause the problems you re seeing, but it might be at least a part.

raw_moments.m:

central_moments.m:

EDIT:

my_regionprops.m

test.m

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