# Hough Circle Transform

Hough Circle Transform (plugin)
Author llamero
Maintainer llamero
Source on GitHub
Initial release February 4th, 2017
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.

Hough circle transform is specific to circular objects. Left Panel: This panel shown the input data for the Hough circle transform. The data includes (clockwise from top left) a circle (radius 37 pixels), a square (length 37 pixels), an ellipse (minor axis 37 pixels), and a sectored circle (radius 37 pixels). Right Panel: This panel shows the output of a 24 step Hough circle transform. As you can see, the circle and the sectored circle converge to local maxima, while the square and ellipse do not, show the specificity of the transform for circular objects.

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.

Data processing steps to perform a Hough circle transform on an XYZ stack. Panel 1: Create an average intensity projection of the XYZ stack to generate a 2D projection. Panel 2: Use the "Find Edges" tool to preserve just the perimeter of each object. Panel 3: Threshold the image from panel 2 and create a binary mask. Panel 4: Run the Hough circle transform.

### 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.

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.

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.

Effect of transform resolution on distinguishing various n-gons. Panel 1 shows a circle and three regular polygons: a 4-gon, 8-gon, and 16-gon. Panel 2 shows a Hough circle transform with four steps. Since all the shapes are radially symmetrical with 90° rotations, they all have an equal peak score at their centroids. Panel 3 shows a Hough circle transform with eight steps. Since the circle, 8-gon, and 16-gon radially symmetrical with 45° rotations, they all have an equal peak score at their centroids. These shapes have a higher score at their centroids than the 4-gon, because it lacks 45° radial symmetry. Panel 4 shows a Hough circle transform with sixteen steps. Only the circle and 16-gon are radially symmetrical at this resolution, so their centroids have equally high scores, while the 4-gon and 8-gon have significantly lower scores. Panel 5 shows a Hough circle transform with 400 steps. The centroid of the circle now has a higher score than all of the other shapes, allowing for the circle to be distinguished even from the 16-gon.