How to calculate the curvature of a boundaries in binary images?

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Meshooo 2014-11-26

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评论： Image Analyst 2022-12-10

采纳的回答： Image Analyst

Dear all,

I have a binary image that contains one shape as I show below:

I want to represent the curvature using color coding of the boundariessimilar to image I got from the video in this link http://en.wikipedia.org/wiki/Curvature

Does anyone knows how to do that?

Any help or suggestion will be appreciated.

Meshoo

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Answer 1

Image Analyst 2014-11-27

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https://ww2.mathworks.cn/matlabcentral/answers/164349-how-to-calculate-the-curvature-of-a-boundaries-in-binary-images#answer_160389

Use bwboundaries to get a list of the (x,y) boundary coordinates. Run around that border taking a section of the curve, say 9 elements, and fit a polynomial to it. Knowing the derivative of the curve model you used, and the coefficients polyfit() gave you, you can get the curvature at each point. Then run around the list of (x,y) edge coordinates using a marker color that corresponds to the curvature. Try to program that up. If you can't do it all , then post what you have and I'll try to help with the rest.

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Meshooo 2014-11-28

在 MATLAB Online 中打开

I was thinking to use your code in different approach.

I = imread('conv.tif');  %same input image
binaryImage = im2bw(I);
%figure, imshow(binaryImage, []);
%
boundaries = bwboundaries(binaryImage);
%
x = boundaries{1}(:, 2);
y = boundaries{1}(:, 1);
windowSize = 19;
halfWidth = floor(windowSize/2);   %half window size
curvatures = zeros(size(x));
%
%k =10
for k = halfWidth+1 : length(x) - halfWidth
  theseX = x(k-halfWidth:k+halfWidth);   % 19 element of x
  theseY = y(k-halfWidth:k+halfWidth);   % 19 elemebt of y
  % Get a fit.
  coefficients = polyfit(theseX, theseY, 2);
  % Get the curvature
  curvatures(k) = coefficients(1);
  xc(k) = x(k);
  yc(k) = y(k);  
end
% Get rid of ridiculous curvatures (straight line segments).
curvatures(abs(curvatures) > 20) = 0;

Now reconstruct the original image from the boundaries using bound2im from (DIPUM)

bound = bwboundaries(binaryImage);
bound = bound{1};
[M N] = size(binaryImage);
B = bound2im(bound, M, N, min(bound(:, 1)), min(bound(:, 2)));
B_gray = uint8(B);  % convert the image to 8 bit intensity
[L V] = size(bound);
pix_Value = []
for k = 1:L
Value = B_gray(y(k), x(k)) % Y then X values
pix_Value(k,1) = Value
end
curv10 = curvatures*1000;  % increase the intensity 1000 times
%but the values of the contour back to the image
for k = 1:L
B_gray(y(k), x(k)) = curv10(k);
end
figure, imshow(B_gray, []) % this will show the image

Now how to color code properly this intensity image?

Meshooo 2014-12-2

在 MATLAB Online 中打开

However, I tried to measure the curvature based on the distance from the centroid and I got the following result:

But I think this is different from the real curvature. Right?

My code is below:

% this software calculates the curvature of a colsed shape based on the
% distance of contour pixels from the centroid using signature
A = imread('curv_circle.tif');
A = im2bw(A);
%
binaryImage = A;
%
s  = regionprops(A, 'centroid');
centroids = cat(1, s.Centroid);
imshow(A)
hold on 
plot(centroids(:,1), centroids(:,2), 'r*')
hold off
X0 = centroids(:,1)
Y0 = centroids(:,2)
%*************************************************************************
% *********** Curvature measurement based on contour signature ***********
%*************************************************************************
%I = imread('curv_circle2.tif');
%
%binaryImage = im2bw(I);
%figure, imshow(binaryImage)
%
bSq = bwboundaries(binaryImage, 'noholes');
%[curvatures, angleSq] = signature(bSq{1}, X0, Y0);
[curvatures, angleSq] = signature(bSq{1})
figure
plot(angleSq, curvatures)
axis ([0 500 0 600]);
grid on;
boundaries = bwboundaries(binaryImage);
%
x = boundaries{1}(:, 2);
y = boundaries{1}(:, 1);
bound = bwboundaries(binaryImage);
bound = bound{1};
[M N] = size(binaryImage);
B = bound2im(bound, M, N, min(bound(:, 1)), min(bound(:, 2)));
B_gray = uint8(B);  % convert the image to 8 bit intensity
[L V] = size(bound);
pix_Value = []
for k = 1:L
Value = B_gray(y(k), x(k)) % Y then X values
pix_Value(k,1) = Value
end
%%adjust the size of curvatures
[C R] = size(pix_Value);
curvatures = imresize(curvatures, [C 1])
%but the values of the contour back to the image
for k = 1:L
B_gray(y(k), x(k)) = curvatures(k);
end
figure, imshow(B_gray, []) % this will show the image
%
SE = strel('square', 5);
%
Dil = imdilate(B_gray, SE);
%
RGB = label2rgb(Dil, 'cool');
figure, imshow(RGB)

Meshooo 2014-12-2

在 MATLAB Online 中打开

I also tried to correct your code by adding small part:

I = imread('curv_circle.tif');
binaryImage = im2bw(I);
%figure, imshow(binaryImage, []);
%
boundaries = bwboundaries(binaryImage);
%
x = boundaries{1}(:, 2);
y = boundaries{1}(:, 1);
windowSize = 10;
halfWidth = floor(windowSize/2);   %half window size
curvatures = zeros(size(x));
%
%k =10
for k = halfWidth+1 : length(x) - halfWidth
  theseX = x(k-halfWidth:k+halfWidth);   % 19 element of x
  theseY = y(k-halfWidth:k+halfWidth);   % 19 elemebt of y
  % Get a fit.
  coefficients = polyfit(theseX, theseY, 2);
  % Get the curvature
  curvatures(k) = coefficients(1);
  xc(k) = x(k);
  yc(k) = y(k);  
end
clc
% Get rid of ridiculous curvatures (straight line segments).
curvatures(abs(curvatures) > 20) = 0;
%curvatures = flip(curvatures)
%
plot(x, curvatures, 'b-');
hold on 
plot(y, curvatures, 'r-');
min_Cur = min(curvatures)
%
max_Cur = max(curvatures)
[L W] = size(curvatures);
%
CurCur = curvatures;
for i = 1 :L
    for j = 1 : W
        if CurCur (i,j) < 0  %negative values
           CurCur (i,j) =  CurCur (i,j) +(abs(min_Cur))
        elseif CurCur (i,j) > 0  %positive values
            CurCur (i,j) =  CurCur (i,j) + 100
        end
    end
end
curvatures = CurCur;

Make up a colormap

minC = min(curvatures)
maxC = max(curvatures)
range = ceil(maxC - minC)
myColorMap = jet(range);
% Display the image again.
imshow(binaryImage);
hold on;
%
for k = halfWidth+1 : length(x) - halfWidth
  % Get the index in the color map.
  thisIndex = round(size(myColorMap, 1) * (curvatures(k) - minC) / range)
  
    fprintf('For point #%d, the colormap index is %d\n', k, thisIndex);
  if thisIndex <= 0
    thisIndex = 1;
  end
  if isnan(thisIndex)
    thisIndex = 1;
  end
  % Extract out the RGB triplet for this particular row in the color map.
  thisColor = myColorMap(thisIndex, :);
  plot(x(k), y(k), '.', 'MarkerSize', 25, 'Color', thisColor);
end

the result is:

Francesco Pignatelli 2022-12-9

@Image Analyst I have followed this post and I saw your comments and you mentioned that "Knowing the derivative of the curve model you used, and the coefficients polyfit() gave you, you can get the curvature at each point.". My question is, how is it possible to get the derivatives?

At the current stage, I am at:

boundaries = bwboundaries(binaryImage);

x = boundaries{1}(:, 2);

y = boundaries{1}(:, 1);

So I have x and y now.

Image Analyst 2022-12-10

curvature_of_curve.m

@Francesco Pignatelli see attached demo that does what I said. Adapt as needed. If you need more help, start your own discussion thread and attach your (x,y) data.

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Answer 2

Preetham Manjunatha 2021-8-2

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编辑：Preetham Manjunatha 2021-8-4

Here this function measures the curvature and other shape properties. In addition, it displays the curvature.

Here is the T-section with sharp corners:

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Answer 3

linda cummings cummingslinda14@gmail.com 2016-9-9

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https://ww2.mathworks.cn/matlabcentral/answers/164349-how-to-calculate-the-curvature-of-a-boundaries-in-binary-images#answer_234271

Very nice. Have you worked in gray image the this code ? Have you tried?

Linda Cummings

cummingslinda14 at gmail.com

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Answer 4

sana3 sal 2018-6-18

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Hello Mr @Image Analyst, I am wondering how to read the curvature? or how to analyse it?

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Image Analyst 2020-4-9

Did you see the code he gave for computing curvature in this link to his comment.

cyril grandjean 2020-4-10

Hello I read the code well but I was not sure that I understood correctly. So the coefficient provided in this code corresponds to the overall curvature of my form or a trend is that right?

thank you in advance

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How to calculate the curvature of a boundaries in binary images?

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