1. Circle Recognition Using The ‘folding’ method
Guy Ben-Yosef
July 2005

2. A Folding of an image
We can part the image into two sub image identical on its size,
An AND operation between these two sub image is folding of an image around the given axis.
Regard non-zero as 1
reduce the amount of information leaving only the information relevant for identify the symmetry.
As a circle being an all around symmetric shape, we can use folding around its diameter as much as we want without losing valuable information for detecting it.

3. The Hough Transform Search and Score method
Voting for all possible structure
Using The gradient normal for voting a circle
Advantages: Accurate, missing parts and distortion shapes
Disadvantages: Depend no the gradient, noisy edge image can lead to false detection

4. The Use of Folding for circle detection
The double fold over X axis and Y axis
Such double fold would be unique for an origin of a circle

5. A Voting process on the double fold image
The voting process on the double fold image.
Examining each distance votes
How many votes for a circle declaration?
Using the condition
Noise can be confused with small radiuses
Using the T threshold:

6. Complexity
The Fold operation takes O(N^2)
Total complexity:

7. Performance
Synthetic Images

8. Performance (2) Noisy Images Additive noise:
Noisy Images Salt and Paper noise:

9. Performance (3)
Real image:

10. Performance – My Magnum Opus
Computer, Computer Where is the Ball?

11. Performance - My Magnum Opus
Here is the ball!!!

12. The main advantages Accurate origin, Accurate radius.
The ability to deal and recognize fat edges contour .
Low chance for false circles.
Folding makes the algorithm robust to noise on any kind.

13. The main disadvantages
Time complexity
The inability to detect distortion and missing parts circles.
Detection of small circles
Can not detect two different circle with the same origin

14. The Fold and Vote Vs. Hough
Complexity: HT (Hough transform) takes O(N3) while Fold&Vote algorithm takes (N4).
Dealing with noise image: The folding method gives for the Fold&Vote algorithm a small benefit upon HT.
Dealing with noisy edge-detection: The HT is heavily dependant on a good clear edge detection, both in edges and gradients.
Fold&Vote algorithm is capable of handing noisy edge detection, as long as the circle edge stays as a whole.
Robustness to shape distortions and to occlusions/missing parts of an object: The HT is capable of finding such circles while the Fold&Vote algorithm is not.
False detection: HT algorithm can fall into phantom circles , while the strict and simple voting process on The Fold&Vote algorithm minimize false detections

15. Conclusions and Further work
The problem of circle detection on an image is a tough problem.
The algorithm presented has its disadvantages, and the major of it , the way I see it is it’s time complexity. For example, it takes about 3-4 hours for it to detect a 260x355 image.
It seems that smaller time complexity algorithm would take its price on the performances.
I kept compare my algorithm to the Hough transform base algorithm, and to the naïve circle detection algorithm. I guess my algorithm is somewhere between.
I do think now, that progress on solving the problem ,need the use of other tools then image manipulation, such as inference, for example.
The algorithm, of course can use for detection of all kinds of symmetric shapes, after making few changes on the voting process.