Function `imfindcircles`

uses a Circular
Hough Transform (CHT) based algorithm for finding circles in images.
This approach is used because of its robustness in the presence of
noise, occlusion and varying illumination.

The CHT is not a rigorously specified algorithm, rather there
are a number of different approaches that can be taken in its implementation.
However, by and large, there are three essential steps which are
common to all.

Accumulator Array Computation.

Foreground pixels of high gradient are designated as being
candidate pixels and are allowed to cast ‘votes' in
the accumulator array. In a classical CHT implementation, the candidate
pixels vote in pattern around them that forms a full circle of a fixed
radius. Figure 1a shows an example of a candidate pixel lying on an
actual circle (solid circle) and the classical CHT voting pattern
(dashed circles) for the candidate pixel.

Figure 1: classical CHT voting pattern

Center Estimation

The votes of candidate pixels belonging to an image circle
tend to accumulate at the accumulator array bin corresponding to the
circle's center. Therefore, the circle centers are estimated
by detecting the peaks in the accumulator array. Figure 1b shows an
example of the candidate pixels (solid dots) lying on an actual circle
(solid circle), and their voting patterns (dashed circles) which coincide
at the center of the actual circle.

Radius Estimation

If the same accumulator array is used for more than one radius
value, as is commonly done in CHT algorithms, radii of the detected
circles have to be estimated as a separate step.

Function `imfindcircles`

provides two algorithms
for finding circles in images: Phase-Coding (default) and Two-Stage.
Both share some common computational steps, but each has its own unique
aspects as well.

The common computational features shared by both algorithms
are as follow:

Use of 2-D Accumulator Array:

The classical Hough Transform requires a 3-D array for storing
votes for multiple radii, which results in large storage requirements
and long processing times. Both the Phase-Coding and Two-Stage methods
solve this problem by using a single 2-D accumulator array for all
the radii. Although this approach requires an additional step of
radius estimation, the overall computational load is typically lower,
especially when working over large radius range. This is a widely
adopted practice in modern CHT implementations.

Use of Edge Pixels

Overall memory requirements and speed is strongly governed
by the number of candidate pixels. To limit their number, the gradient
magnitude of the input image is threshold so that only pixels of
high gradient are included in tallying votes.

Use of Edge Orientation Information:

Another way to optimize performance is to restrict the number
of bins available to candidate pixels. This is accomplished by utilizing
locally available edge information to only permit voting in a limited
interval along direction of the gradient (Figure 2).

Figure 2: Voting mode: multiple radii, along direction of the
gradient

r_{min} | Minimum search radius |

r_{max} | Maximum search radius |

r_{actual} | Radius of the circle that the candidate pixel belongs to |

c_{min} | Center of the circle of radius r_{min} |

c_{max} | Center of the circle of radius r_{max} |

c_{actual} | Center of the circle of radius r_{actual} |

The two CHT methods employed by function `imfindcircles`

fundamentally
differ in the manner by which the circle radii are computed.

Two-Stage

Radii are explicitly estimated utilizing the estimated circle
centers along with image information. The technique is based on computing
radial histograms; see references 1 & 2 for a detailed explanation.

Phase-Coding

The key idea in Phase Coding is the use of complex values in
the accumulator array with the radius information encoded in the phase
of the array entries. The votes cast by the edge pixels contain information
not only about the possible center locations but also about the radius
of the circle associated with the center location. Unlike the Two-Stage
method where radius has to be estimated explicitly using radial histograms,
in Phase Coding the radius can be estimated by simply decoding the
phase information from the estimated center location in the accumulator
array. (see reference 3).