从第二个中心时刻计算对象统计数据

Question

我正在为GNU Octave编写一个MATLAB RegionProps函数版本.我已经完成了大部分工作,但我仍然在努力实现一些部分.我之前曾询问过一个地区的第二个中心时刻.

这在理论上是有帮助的,但我实际上在实施这些建议时遇到了麻烦.我得到的结果与MATLAB(或者常识)的结果大不相同,真的不明白为什么.

考虑这个测试图像:

我们可以看到它与X轴成45度倾斜,次轴和长轴分别为30和100.

通过MATLAB的RegionProps功能运行它确认了这一点:

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

与此同时,我甚至没有让轴正确.我正在尝试使用维基百科中的这些公式.

到目前为止我的代码是:

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);

这是我的代码试图使用这些.我在每个步骤中都包含了对值的注释:

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

我不确定我做错了什么.我似乎正确地遵循这些公式,但我的结果是无稽之谈.我没有在我的时刻功能中发现任何明显的错误,但老实说,我对时刻的理解并不是最开始的.

谁能看到我误入歧途的地方？非常感谢你.

Answer 1

编辑:

根据维基百科:

eignevalues [...] 与特征向量轴的平方长度成正比.

解释如下:

axisLength = 4 * sqrt(eigenValue)

下面显示的是我的代码版本(我将时刻函数向量化):

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

...以及测试它的代码:

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]);