# How to determine eulers number

• November 4th 2009, 01:03 AM
taurus
How to determine eulers number
I have this question related to computer vision/computer science:

Determine the Euler Number for the following characters: 0, 4, 8, A, B, C, D, E

How do I do that?
• November 4th 2009, 02:41 AM
CaptainBlack
Quote:

Originally Posted by taurus
I have this question related to computer vision/computer science:

Determine the Euler Number for the following characters: 0, 4, 8, A, B, C, D, E

How do I do that?

0 - one body and one hole, so Euler number is 1-1=0

:

8 - one body two holes, so Euler number is 1-2=-1

:

C - one body zero holes, so Euler number is 1-0=1

:

CB
• November 4th 2009, 02:47 AM
taurus
Ah so for: 0, 4, 8, A, B, C, D, E

It should be: 00-10-1101

Is that all there is to it?
• November 4th 2009, 05:33 AM
CaptainBlack
Quote:

Originally Posted by taurus
Ah so for: 0, 4, 8, A, B, C, D, E

It should be: 00-10-1101

Is that all there is to it?

I don't understand that.

CB
• November 4th 2009, 05:55 AM
HallsofIvy
Quote:

Originally Posted by taurus
I have this question related to computer vision/computer science:

Determine the Euler Number for the following characters: 0, 4, 8, A, B, C, D, E

How do I do that?

Looking at the definition of the Euler number leaps to mind! The number "e", the base of the natural logarithms, is sometimes referred to as "Euler's number" and, in addition, the limit, as n goes to infinity, of
$\sum_{i=1}^n \frac{1}{n}- ln(n)$ is also referred to as "Euler's number".

But here, you mean what is more often called the "Euler characteristic". For a plane figure that is "the number of faces minus the number of edges plus the number of vertices": F- E+ V.

The digit "0" is topologically equivalent to a rectangle which has 1 face, 4 edges and 4 vertices: F- E+ V= 1- 4+ 4= 1.

The digit "1" has 0 faces, 1 edge and 2 vertices: F- E+ V= 0-1+ 2= 1.

The digit "8" is topologically equivalent to two rectangles sharing a common edge. It has 2 faces , 7 edges, and 6 vertices: F- E+ V= 2- 7+ 6= 1.

The letter "A" has one face, 5 edges, and 5 vertices: F- E+ V= 1- 5+ 5= 1.

The letter "B" is topologically equivalent to "8" and has the same Euler characteristic: 1.

The letter "C" is topologically equivalent to "1" and has the same Euler characteristic: 1.

The letter "D" is topologically equivalent to "0" and has the same Euler characteristic: 1.

The letter "E" has 0 faces, 5 edges, and 6 vertices: F- E+ V= 0- 5+ 6= 1.

In fact, the Euler characteristic of any such polygonal figures is "1"!

That fact that this does not match Captain Black's answers makes me nervous but I am going to stand by my answer. Captain Black's reference to "bodies" and "holes" makes me think he is interpreting these figures as solids in 3 dimensions, rather than figures in the plane.
• November 4th 2009, 06:14 AM
CaptainBlack
Quote:

Originally Posted by HallsofIvy
Looking at the definition of the Euler number leaps to mind! The number "e", the base of the natural logarithms, is sometimes referred to as "Euler's number" and, in addition, the limit, as n goes to infinity, of
$\sum_{i=1}^n \frac{1}{n}- ln(n)$ is also referred to as "Euler's number".

But here, you mean what is more often called the "Euler characteristic". For a plane figure that is "the number of faces minus the number of edges plus the number of vertices": F- E+ V.

The digit "0" is topologically equivalent to a rectangle which has 1 face, 4 edges and 4 vertices: F- E+ V= 1- 4+ 4= 1.

The digit "1" has 0 faces, 1 edge and 2 vertices: F- E+ V= 0-1+ 2= 1.

The digit "8" is topologically equivalent to two rectangles sharing a common edge. It has 2 faces , 7 edges, and 6 vertices: F- E+ V= 2- 7+ 6= 1.

The letter "A" has one face, 5 edges, and 5 vertices: F- E+ V= 1- 5+ 5= 1.

The letter "B" is topologically equivalent to "8" and has the same Euler characteristic: 1.

The letter "C" is topologically equivalent to "1" and has the same Euler characteristic: 1.

The letter "D" is topologically equivalent to "0" and has the same Euler characteristic: 1.

The letter "E" has 0 faces, 5 edges, and 6 vertices: F- E+ V= 0- 5+ 6= 1.

In fact, the Euler characteristic of any such polygonal figures is "1"!

That fact that this does not match Captain Black's answers makes me nervous but I am going to stand by my answer. Captain Black's reference to "bodies" and "holes" makes me think he is interpreting these figures as solids in 3 dimensions, rather than figures in the plane.

I believe we are talking about Euler number for OCR working on binary data which reduces to the number of distinct lumps minus the numeber of holes.

CB