Detection of a Circle in Noisy Image Data

Question

I have an image that looks like the one below:

I'm trying to find the radius (or diameter) of the circle. I have tried using circular Hough transform (via matlab's imfindcircles(bw,[rmin rmax],'ObjectPolarity','bright')) , and by fitting to a circle or an ellipse (home made function that works pretty well for less noisy data, see below).

I've also tried some image processing to get a clearer circle, for example, see below:

se = strel('disk', 2);
bw = imdilate(bw, se);
bw = bwareaopen(bw,100000); 
bw =  edge(bw); 

However, when I feed the processed image to either techniques (Hough and circle\ellipse fitting) neither of them manage to detect the circle in a decent manner.

Here's a code snippet of the circle finder I wrote (matlab) [row col]=find(bw); contour = bwtraceboundary(bw, row(1), col(1)], 'N', connectivity, num_points);

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

    % solve for parameters a, b, and c in the least-squares sense by
    % using the backslash operator
    abc = [x y ones(length(x),1)] \ -(x.^2+y.^2);
    a = abc(1); b = abc(2); c = abc(3);

    % calculate the location of the center and the radius
    xc = -a/2;
    yc = -b/2;
    radius  =  sqrt((xc^2+yc^2)-c);

Alternative approaches will be appreciated...

Answer 1

Here is my solution, it is close to @Yoda's idea, but I changed some steps.

• Mark all pixels such that there are at least 6 pixels in their 7x7 neighborhood
• Remove all blobs, but the largest
• Fill holes
• Apply edge detection
• Find circle using Hough transform

Here is the relevant Matlab code. I am using Hough transform for circles .m file in my code.

function FindCircle()
    close all;
    im = imread('C:\circle.png');
    im = im(:,:,2);

    ims = conv2(double(im), ones(7,7),'same');
    imbw = ims>6;
    figure;imshow(imbw);title('All pixels that there are at least 6 white pixels in their hood');

    props = regionprops(imbw,'Area','PixelIdxList','MajorAxisLength','MinorAxisLength');
    [~,indexOfMax] = max([props.Area]);
    approximateRadius =  props(indexOfMax).MajorAxisLength/2;

    largestBlobIndexes  = props(indexOfMax).PixelIdxList;
    bw = false(size(im));
    bw(largestBlobIndexes) = 1;
    bw = imfill(bw,'holes');
    figure;imshow(bw);title('Leaving only largest blob and filling holes');
    figure;imshow(edge(bw));title('Edge detection');

    radiuses = round ( (approximateRadius-5):0.5:(approximateRadius+5) );
    h = circle_hough(edge(bw), radiuses,'same');
    [~,maxIndex] = max(h(:));
    [i,j,k] = ind2sub(size(h), maxIndex);
    radius = radiuses(k);
    center.x = j;
    center.y = i;

    figure;imshow(edge(bw));imellipse(gca,[center.x-radius  center.y-radius 2*radius 2*radius]);
    title('Final solution (Shown on edge image)');

    figure;imshow(im);imellipse(gca,[center.x-radius  center.y-radius 2*radius 2*radius]);
    title('Final solution (Shown on initial image)');

end

Answer 2

It's fairly straightforward to do it using image processing. The following is a proof of concept in Mathematica. You'll have to translate it to MATLAB.

• First, trim the axes and keep only the image part of it. I call this variable img.

• Binarize the image and dilate it, followed by a filling transform. I also remove stray small components that are not connected to the main blob. It should give you something like the following:

  filled = Binarize@img ~Dilation~ 3 // FillingTransform // DeleteSmallComponents
  

  

• Next, find the centroid of this blob and the equivalent disk radius of the blob (openCV, MATLAB all have equivalent commands to do this)

  {center, radius} = 1 /. ComponentMeasurements[filled, {"Centroid", "EquivalentDiskRadius"}]
  

• That's it! Now plot the original image and a circle with the above center and radius to see how it fits:

  Show[img, Graphics[{Red, Circle[center, radius]}]]