Euler number of digital images

Redirect to:

• Euler and Lefschetz numbers

Suppose a binary image - of any dimension - is represented by its cell decomposition (see Cell decomposition of images). Then the Euler number, or Euler characteristic, of the image is defined as the alternating sum of the numbers of cells of each dimension.

For example, in 2D it's

                        # of vertices - # of edges + # of pixels

In 3D it's

                        # of vertices - # of edges + # of pixels - # of voxels

Turns out the Euler number is independent of the chosen cell decomposition. It depends only on the topology of the image.

This idea can be traced back to the original Euler formula: for any convex polyhedron,

# of vertices - # of edges + # of faces = 2

Also, the Euler number of a circle (disk) is always 1. Same for ellipse, square, blob etc - regardless of size and shape. The Euler number of the sphere (or any balloon) is 0.

The formula applies to cell complexes too.

The Euler number is well known and even popular. The popularity comes from the fact that this is a topological characteristics of the image (preserved under deformations) and the simplicity of its computation. The Euler number can serve as a quick test to verify whether two images are NOT the same topologically.

The meaning of the Euler number itself is however obscure apart from the classification of surfaces and the Euler-Poincaré formula:

Euler number of a 2D binary image
                        = # of vertices - # of edges + # of pixels
                        = # of objects - # of holes

Euler number of a 3D binary image
                        = # of vertices - # of edges + # of pixels - # of voxels
                        = # of objects - # of tunnels + # of voids

Meanwhile, the meanings of "# of objects", "# of tunnels", "# of voids" in the formula are easy to understand (well, may be not tunnels). These numbers are not only truly meaningful but clearly very important for image analysis (see for example Cell counting). These 3 (or more) numbers are topological characteristics of the image as well. They are called Betti numbers. The Euler number can't serve as a substitute for the Betti numbers while you can recover the Euler number from the Betti numbers. They aren't as easy to compute though.

More in Euler characteristic in topology.