detecting the radius of an incomplete circle

Question

I need your help to detect the radius of the blue circle in images like this:

.

Reading these questions: 1, 2, 3, I figured that this code taken from @halirutan answer to question n°3 could help:

imga=Import[http://i.stack.imgur.com/moV5x.png];
circle = SelectComponents[
MorphologicalComponents[
LaplacianGaussianFilter[ColorNegate@imga, 2], 0.003`], 
"Count", # > 400 &];
Colorize[circle]

And I have this output:

And now I'm stuck, if the circle were closed it would be simple, how can I find the best circle completing it? Obviously if you have a completely different solution it is fine (for example I thought it could be possible to take only the left semicircle, but I don't know how to do it).

Please note that I have to process about 100 images like that so if possible I would need the program to run in a reasonable amount of time and with no manual adjustments.

As a bonus question: since I have to determine also a possible error on the radius, would it be possible to find the outer and the inner circles? and the eccentricity of the closed figure (if it is possible to close it)?

P.S.: since I just started working with mathematica for image processing I would also like to know a few good references or books about it. I have read a few mathematica books but none of them treated this argument in detail.

Answer 1

You're close: You can use ComponentMeasurements to calculate measurements for each component. (The same measurements that you can use in SelectComponents, too). If you can get away with the center of mass of the marked pixels, it'really quite simple:

imga = Import["http://i.stack.imgur.com/moV5x.png"];
binary = MorphologicalComponents[
  LaplacianGaussianFilter[ColorNegate@imga, 2], 0.003`]; 
components = ComponentMeasurements[
  binary, {"Count", "AreaRadiusCoverage", "Centroid", 
   "MeanCentroidDistance", "MaxCentroidDistance", 
   "MinCentroidDistance"}, #1 > 400 && #2 < 0.01 &];

The "AreaRadiusCoverage" measurement helps to select components where most pixels are near the border. This returns a list of rules, one for each component that matches the filter function:

{193 -> {4516, 0., {222.283, 211.7}, 181.482, 200.342, 167.181}}

Where 193 is the component index, 4516 is the pixel count, 0. is AreaRadiusCoverage and so on.

The found circle looks like this:

Show[HighlightImage[imga, Image[1 - Abs[binary - components[[1, 1]]]]],
 Graphics[
  {
   Yellow, Dashed,
   components /. 
     {(_ -> {count_, areaRadiusCoverage_, center_, radii__}) 
     :> (Circle[center, #] & /@ {radii})}}]]

---

If the center of mass is not good enough, you can use the "Mask" component measurement to get an image of the marked pixels in each component:

imga = Import["http://i.stack.imgur.com/moV5x.png"];
binary = MorphologicalComponents[
  LaplacianGaussianFilter[ColorNegate@imga, 2], 0.003`]; components = 
 ComponentMeasurements[
  binary, {"Count", "AreaRadiusCoverage", 
   "Mask"}, #1 > 400 && #2 < 0.01 &];

The "mask" value returned looks like this:

Image[components[[1, 2, 3]]]

Getting the coordinates of the white pixels in this image is easy:

whitePixels = PixelValuePositions[Image[components[[1, 2, 3]]], 1];

Now, what we're looking for is a circle with center (cx,cy) and radius r that minimized:

err = Total[
  Map[
      Function[whitePixel, (Norm[{cx, cy} - whitePixel] - r)^2], 
      whitePixels]];

i.e. the squared distance of the white pixels from the circle.

We can simply pass this error function to FindMinimum:

optimalCircle = FindMinimum[err, {cx, cy, r}]

To get a better estimate of the circle:

Show[HighlightImage[imga, Image[components[[1, 2, 3]]]],
 Graphics[
  {
   Yellow, Dashed,
   Circle[{cx, cy}, r] /. optimalCircle[[2]]}]]

(Finding the min/max radius should be straightfoward from here - you have the center and the white pixel coordinates.)