Sunday, March 15, 2009

Skeletonization and Medial Axis Transform

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Brief Description
Skeletonization is a process for reducing foreground regions in a binary image to a skeletal remnant that largely preserves the extent and connectivity of the original region while throwing away most of the original foreground pixels. To see how this works, imagine that the foreground regions in the input binary image are made of some uniform slow-burning material. Light fires simultaneously at all points along the boundary of this region and watch the fire move into the interior. At points where the fire traveling from two different boundaries meets itself, the fire will extinguish itself and the points at which this happens form the so called `quench line'. This line is the skeleton. Under this definition it is clear that thinning produces a sort of skeleton.
Another way to think about the skeleton is as the loci of centers of bi-tangent circles that fit entirely within the foreground region being considered. Figure 1 illustrates this for a rectangular shape.

Figure 1 Skeleton of a rectangle defined in terms of bi-tangent circles.

The terms medial axis transform (MAT) and skeletonization are often used interchangeably but we will distinguish between them slightly. The skeleton is simply a binary image showing the simple skeleton. The MAT on the other hand is a graylevel image where each point on the skeleton has an intensity which represents its distance to a boundary in the original object.

How It Works
The skeleton/MAT can be produced in two main ways. The first is to use some kind of morphological thinning that successively erodes away pixels from the boundary (while preserving the end points of line segments) until no more thinning is possible, at which point what is left approximates the skeleton. The alternative method is to first calculate the distance transform of the image. The skeleton then lies along the singularities (i.e. creases or curvature discontinuities) in the distance transform. This latter approach is more suited to calculating the MAT since the MAT is the same as the distance transform but with all points off the skeleton suppressed to zero.
Note: The MAT is often described as being the `locus of local maxima' on the distance transform. This is not really true in any normal sense of the phrase `local maximum'. If the distance transform is displayed as a 3-D surface plot with the third dimension representing the grayvalue, the MAT can be imagined as the ridges on the 3-D surface.
Guidelines for Use
Just as there are many different types of distance transform there are many types of skeletonization algorithm, all of which produce slightly different results. However, the general effects are all similar, as are the uses to which the skeletons are put.
The skeleton is useful because it provides a simple and compact representation of a shape that preserves many of the topological and size characteristics of the original shape. Thus, for instance, we can get a rough idea of the length of a shape by considering just the end points of the skeleton and finding the maximally separated pair of end points on the skeleton. Similarly, we can distinguish many qualitatively different shapes from one another on the basis of how many `triple points' there are, i.e. points where at least three branches of the skeleton meet.
In addition, to this, the MAT (not the pure skeleton) has the property that it can be used to exactly reconstruct the original shape if necessary.
As with thinning, slight irregularities in a boundary will lead to spurious spurs in the final image which may interfere with recognition processes based on the topological properties of the skeleton. Despurring or pruning can be carried out to remove spurs of less than a certain length but this is not always effective since small perturbations in the boundary of an image can lead to large spurs in the skeleton.
Note that some implementations of skeletonization algorithms produce skeletons that are not guaranteed to be continuous, even if the shape they are derived from is. This is due to the fact that the algorithms must of necessity run on a discrete grid. The MAT is actually the locus of slope discontinuities in the distance transform.

Distance Transform

Brief Description
The distance transform is an operator normally only applied to binary images. The result of the transform is a graylevel image that looks similar to the input image, except that the graylevel intensities of points inside foreground regions are changed to show the distance to the closest boundary from each point.
One way to think about the distance transform is to first imagine that foreground regions in the input binary image are made of some uniform slow burning inflammable material. Then consider simultaneously starting a fire at all points on the boundary of a foreground region and letting the fire burn its way into the interior. If we then label each point in the interior with the amount of time that the fire took to first reach that point, then we have effectively computed the distance transform of that region. Figure 1 shows a distance transform for a simple rectangular shape.

Figure 1 The distance transform of a simple shape. Note that we are using the `chessboard' distance metric.

There is a dual to the distance transform described above which produces the distance transform for the background region rather than the foreground region. It can be considered as a process of inverting the original image and then applying the standard transform as above.
How It Works
There are several different sorts of distance transform, depending upon which distance metric is being used to determine the distance between pixels. The example shown in Figure 1 uses the `chessboard' distance metric but both the Euclidean and `city block' metrics can be used as well.
Even once the metric has been chosen, there are many ways of computing the distance transform of a binary image. One intuitive but extremely inefficient way of doing it is to perform multiple successive erosions with a suitable structuring element until all foreground regions of the image have been eroded away. If each pixel is labeled with the number of erosions that had to be performed before it disappeared, then this is just the distance transform. The actual structuring element that should be used depends upon which distance metric has been chosen. A 3×3 square element gives the `chessboard' distance transform, a cross shaped element gives the `city block' distance transform, and a disk shaped element gives the Euclidean distance transform. Of course it is not actually possible to generate a good disk shaped element on a discrete grid on a small scale, but there are algorithms that vary the structuring element on each erosion so as to approximate a circular element.
The distance transform can be calculated much more efficiently using clever algorithms in only two passes (e.g. Rosenfeld and Pfaltz 1968). This algorithm, which is based on recursive morphology, will not be described here.

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