Fastest available algorithm for distance transform

Solution 1:

The OpenCV library uses for its approximate cv::distanceTransform function a algorithm which passes the image from top left to bottom right and back. The algorithm is described in the paper "Distance transformations in digital images" from Gunilla Borgefors (Comput. Vision Graph. Image Process. 34 3, pp 344–371, 1986).

The algorithm calculates the distance through a combination of some basic jumps (horizontal, vertical, diagonal and the knight move). Each jump incurs costs. The following table shows the costs for the different jumps.

+------+------+------+------+------+
| 2.8  |2.1969|   2  |2.1969|  2.8 |
+------+------+------+------+------+
|2.1969|  1.4 |   1  |  1.4 |2.1969|
+------+------+------+------+------+
|   2  |   1  |   0  |   1  |   2  |
+------+------+------+------+------+
|2.1969|  1.4 |   1  |  1.4 |2.1969|
+------+------+------+------+------+
| 2.8  |2.1969|   2  |2.1969|  2.8 |
+------+------+------+------+------+

The distance from one pixel to another is the sum of the costs of the jumps necessary. The following image shows the distance from the 0-cells to each other cell. The arrows are showing the way to some cells. The colored numbers reflect the exact (euclidean) distance.

The algorithm works like this: Following mask

+------+------+------+
|   0  |   1  |   2  |
+------+------+------+
|   1  |  1.4 |2.1969|
+------+------+------+
|   2  |2.1969|  2.8 |
+------+------+------+

is moved from top left of the image to bottom right. During this pass, cells lying inside the boundaries of the mask either keep their value (if it is known and smaller) or they get the value calculated by summing the mask-value and the cell-value (if it is known) from the cell below the mask-0-cell. After that, a second pass from bottom right to top left (with a vertical and horizontal flipped mask) is performed. After the second pass the distances are calculated.

Solution 2:

This paper reviews the known exact distance transform algorithms:

"2D Euclidean distance transform algorithms: A comparative survey"
https://rfabbri.github.io/stuff/fabbri-EDT-survey-ACMCSurvFeb2008.pdf

The fastest exact distance transform is from Meijster:

"A General Algorithm for Computing Distance Transforms in Linear Time."
http://fab.cba.mit.edu/classes/S62.12/docs/Meijster_distance.pdf

The design of the algorithm is particularly well suited for parallel calculation.

This is implemented in my open source library which tries to emulate Photoshop's "Layer Style:"

https://github.com/vinniefalco/LayerEffects

Solution 3:

There's tons of newer work on computing distance functions.

• Fast marching algorithms that originally came from Tsitsiklis (not Sethian like Wikipedia says). Tons of implementations are available for this.
• Fast sweeping algorithms from Zhao
• O(n) (approximate) fast marching by Yatziv

By the way, you'd really want to use these instead of the work by Rosenfeld, specifically when you want to compute distances in the presence of obstacles.