Euler Number Computation

1
Euler Number Computation

• By
• Kishore Kulkarni
• (Under the Guidance of Dr. Longin Jan Latecki)

2
Outline

• Connected Components Holes
• Euler Number its significance
• For Binary Images
• For Graphs
• Analogy between Euler number for graphs binary
  images
• Concavities and Convexities
• Relation between Euler Number, Concavities
  Convexities
• Euler Number computation for test images.
• Justification of correctness of algorithm used
  for computing Euler Number

3
Connected Components Holes

• Connected Components 4, 6 and 8 Connectivity.
• Holes

4
Euler Number

• The Euler Number (also called as the
  connectivity factor) is defined as the difference
  between connected components and holes.
• Formally, Euler Number is given by ncomp
• E ncomp - ? nihole where
• i 1
• ncompgt number of foreground connected components
• nihole gt number of holes for ith connected
  component

5
Euler Number
E 5 11 -6
E 11 4 7
6
Significance of Euler Number

• Euler Number is a topological parameter, which
  describes the structure of an object, regardless
  of its specific geometric shape.
• Euler number (for discrete case) is a local
  measure and is obtained as the sum of measures in
  the local neighbourhoods.
• Although it is a local measure , it can be used
  to obtain global properties.
• For objects with no holes, it can be used to
  calculate the number of connected components.

7
Euler Number for Graphs

• If V, E, and F represent the set of vertices,
  edges and faces respectively then for any graph
  we have
• V F E R
• where R is the Euler Number for graphs
• V 7, F 3(or 4), E 9
• Euler Number ,
• R 7 3(or 4) 9 1(or 2) depending
  upon whether or not we treat the area
  outside the graph as a face.

8
Equivalence of Euler Number for graphs and Binary
images

• Any binary image can be represented as a graphs
  where
• Every black pixel (1) is represents the vertex of
  the graph
• Two neighbouring black pixels (vertices of the
  graphs) are joined to form an edge
• The Euler Number for the binary image and its
  corresponding graph is the same.

9
Euler Number Contd

• Consider the following image.
• Euler Number E 2 0 2.
• For the corresponding graph,
• we have
• V 28, E 30, F 4
• R 28 30 4 2

10
Convexities Concavities

• A region R is said to be convex iff, for every
  x1, x2 in R, x1, x2 belongs to R.
• A convex hull for a region R is defined as the
  convex closure of region R.
• Convexity (or the convex number) N2(X, a) is
  defined as the number of first entries in the
  object in a given direction a.

11
Convexity concavity.

• Concavity (or the concave number) N2-(X, a) is
  defined as the number of exits in a given
  direction a.
• In the following figure,
• N2(X, 90) 3
• N2-(X, 90) 2

12
Convexities and Concavities

• For a binary image with 4-connectivity the
  convexities and concavities can be represented by
  the following 22 masks.
• Convexities Three 0s and one 1.
• Concavities Three 1s and one 0.

13
Euler Number in terms of convexity and concavity

• The relationship between Euler Number,
  Convexities and concavities is given by
• N2(X) N2(X) - N2-(X)
• So now we have the following relation
• E c h v e f N2(X) - N2-(X)
• The upstream convexity -------- (1)
• The upstream concavity ----------(2)

14
Algorithm for computing Euler Number (for
4-connectivity)

• For every pixel (x, y) of the image,
• Do
• if p(x,y) 1 and p(x-1, y) p(x , y-1)
  p(x-1 , y-1) 0
• convexities
• if p(x,y) 0 p(x1, y) p(x , y1) p(x1
  , y1) 1
• concavities
• end
• Euler number convexities - concavities

15
Euler Number for test images

• Image 1
• Number of Convexities
• 74
• Number of Concavities
• 74
• Euler Number 0

16
Euler Number for test images

• Image 2
• Number of Convexities
• 86
• Number of Concavities
• 85
• Euler Number 1

17
Euler Number for test images

• Image 3
• Number of Convexities
• 135
• Number of Concavities
• ?
• Euler Number ?

18
Euler Number for test images

• Image 4
• Number of Convexities
• 74
• Number of Concavities
• 73
• Euler Number 1

19
Euler Number for test images

• Image 5
• Number of Convexities
• 27
• Number of Concavities
• 16
• Euler Number 11
• Number of Objects

20
Euler Number for test images

• Image 6
• Number of Convexities
• 51
• Number of Concavities
• 61
• Euler Number -10
• Number of Holes
• 1-(-10) 11

21
Why does this approach always returns Euler
Number ?

• To prove
• Euler Number for an image Number of upstream
  convexities - Number of upstream concavities
• Proof
• First we prove that Euler Number for the whole
  image is the sum of Euler numbers for every
  component. Then we prove the above equality for
  one component to complete the proof.

22
Proof - Contd.

• Euler number for component is given by
• Ei 1 nihole
• where nihole is the number of holes in component
  i
• Consider ncomp ncomp
• ? Ei ? (1 - nihole) i 1 i 1
• ncomp
• ncomp - ? nihole
• i 1
• Eimage

23
Proof - Contd.

• The upstream convexity -------- (1)
• The upstream concavity ----------(2)
• The first pixel with value 1 in the image
  should be of type (1). Hence this convexity
  refers to the connected component. Now any pixel
  with the mask can
• be a Hole
• be a Background
• If (1) then we are done..

24
Proof - Contd.

• If it is a background pixel then we have an
  extra concavity count. Hence we need to prove
  that there exists an equivalent convexity to
  balanced the same. Now if it a background pixel,
  then the two edges in the 22 mask (referring to
  graphical representation of a binary image)
  should not form a cycle with any of the
  previously visited black pixels (otherwise we get
  a hole). If this is the case there should be a
  pixel with the mask . Hence we can prove
  that number of concavities exceeding number of
  holes and convexities exceeding 1 are equal by
  similar argument.
• Thus they cancel out. Hence the result.