2. Algorithms

This implementation is based on the classical Hough transformation on the detection of circles not lines. An edge point on a possible circle is transformed in 3-D (corresponding the center x, y and radius r of a circle) Hough space into a series of 3-D circles (asuming that the Hough parameter space is quantized with same quantum size) with different possible radii.

To successfully map between the two spaces, I divide one Hough circle (in Hough space) into four parts by drawing two lines (across the center of the circle) with + 45 and -45 degres: the upper, lower, left and right arcs. The upper and lower are symetric to X-axis, and the left and right ones to Y-axis. Thus time can be saved. For example, in the case of the upper and lower parts: we only check the calculated Y's values corresponding to the limited range of X-values, to check if the Y's are in the right range given a certain radius. Therefore, we don't worry about the scenario in which, for a certain X-value, there are two or more Y's values as in the case if the left and right parts. Because we also check X's values (in the right and left parts ) for a certain Y's value in the limited range of Y's values given the same radius. This algorithm is implemented in the function onePixelHough().

The overall pseudocode for the whole Hough transformation on detection of circles can be shown as follows (with the function names in Bold Italic style font):

Read in an image;
  To do edge detection via sobel_image()  or canny_image();
  Allocate memory according to the input parameters (quantum size...)  houghInitial();
   For every possible radius  imageHough()
      For every edget point       imageHough()
          Do Hough transformation (as described above) onePixelHough();
   Find the clusters of circles  siftCircles();
    Draw the clusters of circles drawCircles()
    Do local maxima suppression on the clusters of circles findLocalMaximaCircles();
    Draw the found circles for the objects  drawCircles()
    Print fitting errors  calculateFE().


   

3. Results

I also combined the original image, the edge image, the detected cluster circles, and the chosen circles for the coins into animated GIFs, to illustrate the effect.

3.1.  For image 0.17.pgm

Quantum size: 3
Canny detection.
Circles found:
 Found Circle at (x, y): 121 97 radius = 33
 The counting at that is: 219
 Found Circle at (x, y): 181 175 radius = 39
 The counting at that is: 212
 Found Circle at (x, y): 184 49 radius = 33
 The counting at that is: 205
 Found Circle at (x, y): 247 109 radius = 30
 The counting at that is: 146

Fitting errors:

 circle 0's fitting error = 0.743979
 circle 1's fitting error = 1.12208
 circle 2's fitting error = 1.00431
 circle 3's fitting error = 0.571544