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Euler Number Computation

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Analogy between Euler number for graphs & binary images. Concavities ... Image 1: Number of Convexities = 74. Number of Concavities = 74. Euler Number = 0. 16 ... – PowerPoint PPT presentation

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Title: 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.
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