Hough Circle Transform
Hough Circle Transform (plugin) | |
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Author | llamero |
Maintainer | llamero |
Source | on GitHub |
Initial release | February 4^{th}, 2017 |
Latest version | February 7^{th}, 2017 (v0.5.1) |
Development status | stable, active |
Category | Analysis, Feature Extraction |
Introduction
A Hough circle transform is an image transform that allows for circular objects to be extracted from an image, even if the circle is incomplete. The transform is also selective for circles, and will generally ignore elongated ellipses. The transform effectively searches for objects with a high degree of radial symmetry, with each degree of symmetry receiving one "vote" in the search space. By searching a 3D Hough search space, the transform can measure the centroid and radius of each circlular object in an image.
The method works by transforming an image around in a circle. Each time a transformed pixel with an intensity greater than zero lands on a Cartesian coordinate, that coordinate gets one vote. As the image continues to be transformed in a circle of a given radius, if a circle in the image has the same radius, then votes will accumulate at the centroid of this circle. Therefore, by finding the maxima in the transform (points with the highest number of votes) you can find the centroid of circles within the image. A Hough circle transform can also be used to find circles of an unknown radius by searching a 3D transform space, where the the third dimension is the range of radii to be tested.
Image Processing Workflow
The Hough circle transform finds circles based on the rotational symmetry of the perimeter. Therefore, the data needs to be converted to this format for the transform to work.
Step 0: Convert XYZ(T) data to XY(T) data
If the data is a 3D stack, then collapse the data to 2D space using a maximum, sum, or average intensity projection. The plugin can handle multiple frames (time-points) in a stack, but it can only search in 2D space.
Step 1: Find Edges
If the circular objects are solid rather than hollow, take the derivative of the image by running: Process › Find Edges. This will preserve just the perimeter of each object.
Step 2: Threshold
The algorithm does not weight the transform based on the intensity of the pixels, as this would result in bright, non-circular objects getting a very high score. Therefore, any pixel with an intensity > 0 is given one vote per transform. This means that any pixel you do not want to be part of the transform needs to be set to zero, which is best done by thresholding the image and creating a mask by running: Image › Adjust › Threshold. After choosing the right threshold for the data press "Apply" to create a mask with an inverting LUT (0 is white, 255 is black). The LUT can be changed back to a normal gray-scale by going to Image › Lookup Tables › Grays.
Step 3: Run the Hough Transform
Now you are ready to run the Hough transform (see below for detailed information on the various options). If the number of circles is unknown or varies from frame to frame, then the best option is to set the search to one circle and perform the transform with the output set to show the results table, and the centroids marked on the original image. This will give you the highest Hough score in the whole image, and will allow you to confirm that the circle found is correct and what its score was. If the circle is incorrect, adjust the search parameters to narrow the Hough search space.
If the circle found is correct, then gradually reduce the threshold until all the circles in the image are found. This will give you the upper threshold bound. Continue to decrease the threshold until an errant circle is detected, this will give you the lower threshold bound. Set the threshold between the upper and lower bounds, and then run the transform on the full data set.
Running the Hough circle transform plugin
The plugin runs on the current active image, and can also process stacks, but it cannot handle hyperstacks. The plugin is also recordable for macro implementation, and multi-threaded to fast searching on the 3D Hough space.
A key component to this plugin is that it always searches for the highest scoring circle first, then the second highest, and so on. When a circle is found, all neighboring circles are removed to prevent the same circle from being found repeatedly. This parameter can be adjusted to allow for increasing degrees of overlap between neighboring circles.
Search Parameters
The Hough circle plugin is designed to be adaptable to a variety of segmentation tasks, and as such, there are seven search parameters that can be adjusted to tune the search space.
Minimum/maximum search radius:
The minimum and maximum search radii are the lower and upper cutoff for the radii you expect to find in the image. Ideally, you want to make the Hough search space as specific as possible, so be sure to set these values to specifically the range of radii you expect to find in your data.
Radius search increment:
This determines the radius step size to use when creating the 3D Hough space from the minimum radius to the maximum radius. This allows a trade-off between speed and resolution, where larger steps will give a linear increase in speed, but also decrease the precision of the measured radii.
Maximum number of circles to be found:
This option sets the upper limit for the number of circles that can be found in the search. If there are fewer than the specified number of circles in the image with a score above the threshold (see below), then the algorithm will only return the number of circles that were above the threshold.
Hough score threshold:
This option sets the minimum cutoff for the Hough score (i.e. number of votes) that a circle can have to count as a valid object. The search starts with the highest scoring circle, and then the second highest, and so on, so if there are a greater number of circles with a score above the threshold than the limit set in "Maximum number of circles to be found", only the highest scoring circles will be returned.
NOTE: When the transform resolution is changed (see below) the scores will change to. This is because the resolution sets the number of rounds of voting. Therefore, as the number of voting rounds increases, the score per object will increase as well.
Hough transform resolution
This option sets the number of steps in each circle transform. To reduce unnecessary computation, if the resolution is set arbitrarily high (such as the default value of 1000), the algorithm will automatically find the nearest number of unique transforms possible (i.e. unique integer x,y coordinates) for the maximum radius, and will use this value as the actual resolution in the transform series. Reducing the resolution below the maximum value can greatly speed up the algorithm, but it will also decrease the transform's specificity and sensitivity.
Clear neighbors radius ratio:
The 3D Hough search space approaches a local maxima. Therefore, when a circle is found, the space around the local maxima needs to be removed to prevent the circle from being found a second time. The radius of the Hough search space that is cleared is defined as a ratio of the radius of the circle found, meaning that large circles clear out more Hough search space than small circles.
By default, this ratio is set to be one, meaning that a circle of the same size and location as the one found is cleared from the entire search space. This has the effect of eliminating all potential centroids within one radius of the found centroid. This effectively excludes overlapping circles of a similar radius from the search. To allow overlapping circles, this ratio can be reduced. A ratio of "0" will result in the same circle being found repeatedly. This means that perfectly concentric circles cannot be found in one run of the plugin, and rather need to be found iteratively by removing the found circles from the image and re-running the plugin.
Reduce transform resolution:
Since all images will be pixels in a Cartesian coordinate system, transforms can only be to discrete integer coordinates (i.e. x=5,y=10). Therefore, with an infinitely fine resolution, then when the next step of a transform is rounded to the nearest integer coordinate, it is possible that these are the same coordinates as the previous transform step. This means that when the transform is performed, it will perform identical transformations for both these steps.
To speed up a Hough transform, you can find and remove all of the redundant transform steps before performing the transform, keeping only the set of unique transform steps. The plugin performs this check by removing all redundant transform steps for the maximum search radius, and then setting this as the new resolution for the subsequent radii. This means the smaller radii will perform redundant transforms, but this is essential to ensure that each radius gets and equal number of voting rounds (or else large circles will always score higher than small circles).
While speeding up the algorithm, the trade-off is that the transform steps are distributed anisotropically for the maximum radius (all subsequent radii will be isotropic). Unchecking this box will result in the transform perform all of the specified transform steps, which can be be very computationally intensive for high resolution values.