# how to find contour inflection points?

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Alessandro on 30 May 2014
Commented: Alessandro on 31 May 2014
Hi, i have a bw image describing a shape contour. How find the coordinates of inflection points? And then, how plot the fingerprint representarion? Thanks to all!!!

Bjorn Gustavsson on 30 May 2014
Edited: Bjorn Gustavsson on 30 May 2014
Inflection points ought to be rather simple to calculate (function below should get the job done), but the actual contour-lines returned from contour et al. might have a rather large number of inflection-points as you can verify by looking at the example.
HTH,
function r_inflections = contourinflectionpoints(C,fK)
% CONTOURINFLECTIONPOINTS - find inflection-points on contourlines
% CONTOURINFLECTIONPOINTS identifies inflection points on
% contourlines by identifying line-segments where the curvature
% changes direction. Each contourline is interpreted as a number
% of connected straight-line-segments and an inflection-point is
% identified as the mid-point of a line-segment where the
% neihbouring points fall on opposite sides of the extension of
% the line-segment.
%
% Calling:
% r_inflections = contourinflectionpoints(C,fK)
% Input:
% C - contourline matrix as returned by contourc or the
% contour-family of functions on the format:
% C = [level1 x1 x2 x3 ... level2 x2 x2 x3 ...;
% pairs1 y1 y2 y3 ... pairs2 y2 y2 y3 ...]
% fK - filter-factors used to smoothen the contour-lines (simple
% experimentation reveal that contourlines can have a large
% number of spurious inflection-points, smoothing of the
% contour-lines reduces this problem, user tuning required).
% Output:
% r_inflections - [n x 2] double array with identified
% inflection-points.
%
% Example:
% M = peaks(127);
% C = contour(M,8);
% r_i = contourinflectionpoints(C);
% fK = [1/2,1,1/2]/2;
% r_s = contourinflectionpoints(C,fK);
% hold on
% plot(r_i(:,1),r_i(:,2),'b.')
% plot(r_s(:,1),r_s(:,2),'ro','linewidth',3)
%
% No argument checks or error-controls, if you want that you have
% to pay me good money.
% Copyright © B. Gustavsson 20140530, GPL version 3 or later applies
%%Extract the individual contour-lines
iC = 1;
idxC = 1;
while iC < size(C,2)
nP = C(2,iC); % number of points in current contour
xC{idxC} = C(1,iC+(1:nP)); % x coordinates of current contour
yC{idxC} = C(2,iC+(1:nP)); % y coordinates of current contour
if nargin > 1 && ~isempty(fK)
xC{idxC} = filtfilt(fK/sum(fK),1,xC{idxC});
yC{idxC} = filtfilt(fK/sum(fK),1,yC{idxC});
end
iC = iC+nP+1; % Start-point of next contour
idxC = idxC + 1; % next contourline index
% plot(xC{idxC-1},yC{idxC-1},'b.-')
% pause
end
iInflections = 1;
for iC = 1:length(xC),
for iP = 2:(length(xC{iC})-2)
% First take 2 points on the current contour :
r0 = [xC{iC}(iP);yC{iC}(iP)];
r1 = [xC{iC}(iP+1);yC{iC}(iP+1)];
% let them define a stratight line on the form:
% dot(r,e_n) - 1 = 0
e_n = [cos(pi/2) sin(pi/2);-sin(pi/2) cos(pi/2)]*(r1-r0)/norm(r1-r0);
l = dot(r0,e_n);
% the points on the contour on either side of the line-segment:
r_p = [xC{iC}(iP+2);yC{iC}(iP+2)];
r_m = [xC{iC}(iP-1);yC{iC}(iP-1)];
% lengths along e_n to points r_p and r_m
l_p = dot(r_p,e_n);
l_m = dot(r_m,e_n);
% if they are on either side of the line-segment then we have
% an inflection point
if (l_p - l)*(l_m - l) <0
% take the mid-point of that line-segment as the
% inflection-point, change this according to taste...
r_inflections(iInflections,:) = (r0+r1)'/2;
iInflections = iInflections + 1;
end
end
end

Image Analyst on 30 May 2014
Image Analyst on 30 May 2014

Alessandro on 30 May 2014
Edited: Image Analyst on 30 May 2014
Hi, thanks for responses! I don't know (and still understand) the 'countour'/'contourc' Matlab function.... I have not regular shapes (triangles, squares, etc...), in my situation i have just a bw image (obtained from such edge detection filters and others..) and i just need to locate the inflection points of a shape in orther to make a descriptor for it. Can i use contour functions also in this situation?... i didn't understand what contour function does with just a bw image.
To be more clear, i need something like figure 3) and 4) of this page http://vgg.fiit.stuba.sk/wordpress/2013-04/css-%E2%80%93-curvature-scale-space-in-opencv/. Alessandro on 31 May 2014
Ok... i got it!!... but i still don't understand the mathematical meaning og e_n and l. Thanks a lot!!

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