How should we evaluate the similarity of two graphs?

I estimated the ellipse, and thought about it in my own way for comparison with the actual ellipse.

If you plot the graph in the Cartesian coordinate system through the attached file, you can see that the difference is not large.

In the figure below, the blue line is the estimated ellipse, and the red line is the actual ellipse.

To check the difference between the two graphs, I changed the polar coordinates, and calculated the difference in the lengths of the two graphs assuming that thetha is the same in polar coordinates. Likewise, blue is the estimated ellipse, and red is the polar coordinates of the actual ellipse.

I simply calculated the difference between the two data's r value and then plotted it, but the difference is too large than I thought, which is confusing.

The MSE is also much larger than I thought, but in fact, it is unlikely that the MSE of the two data will be too large.

It seems to be a data sorting problem, but I would like to ask how to solve it.

Also, the difference between each angle is up to 3mm, so please help me to see that.

load ellipse_original.mat
load ellipse_estimation.mat

m = ellipse_original(:,1);
n = ellipse_original(:,2);

x = ellipse_estimation(:,1);
y = ellipse_estimation(:,2);

plot(x,y,'.b')
xlim([-150 150])
ylim([-150 150])
pbaspect([1 1 1])
grid on; grid minor
hold on
plot(m,n,'.r')
hold off

%% 
[th r] = cart2pol(m,n);
[xth yr] = cart2pol(x,y);

plot(th,r,'.b')
hold on
plot(xth,yr,'.r')
hold off

%%

plot(th,(r-yr),'.')

Answer 1

[Matt J]

I don't know why you think the MSE is "large", but perhaps the percent area difference would be more suitable to you:

load ellipse_original.mat
load ellipse_estimation.mat

ellipse0=polyshape( ellipse_original );

Warning: Polyshape has duplicate vertices, intersections, or other inconsistencies that may produce inaccurate or unexpected results. Input data has been modified to create a well-defined polyshape.

ellipse1=polyshape( ellipse_estimation );

Warning: Polyshape has duplicate vertices, intersections, or other inconsistencies that may produce inaccurate or unexpected results. Input data has been modified to create a well-defined polyshape.

plot([ellipse0,ellipse1])

percentError = abs( area(intersect(ellipse0,ellipse1))/area(ellipse0)*100-100)

percentError = 0.9974

Answer 2

[Chunru]

load ellipse_original
load ellipse_estimation
whos

  Name                       Size            Bytes  Class     Attributes

  ans                        1x47               94  char                
  cmdout                     1x33               66  char                
  ellipse_estimation      1000x2             16000  double              
  ellipse_original        1000x2             16000  double              

plot(ellipse_original(:,1), ellipse_original(:,2), 'r', ellipse_estimation(:, 1), ellipse_estimation(:,2), 'b');

% conver the geometric shape into image

im1 = shp2im(ellipse_original);

im2 = shp2im(ellipse_estimation);

% Now compare two images
black1 = im1 == 0;
black2 = im2 == 0;
common = black1 & black2;
similarity = sum(common(:))/sum(black1(:))

similarity = 0.9976

function im = shp2im(x)
% Convert shape into binary image
figure
fill(x(:,1), x(:,2), 'k'); axis equal
F = getframe(gcf);
F = im2gray(F.cdata);
im = F>0;        % binary: 0 for black & 1 for white
imagesc(im); colormap([0 0 0; 1 1 1]); axis equal
end