8/9/2019 Theory - Virtual Lab in Image Processing
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VIRTUAL LAB in IMAGE PROCESSING
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Distance and Connectivity: Theory
Connectivity of pixels:
Pixel connectivity describes a relation between two or more
pixels. For two pixels to be connected they have to fulfill certain
conditions on th
pixel value and spatial adjacency.
The pixel brightness condition: Their pixel values must be from
the same set of values C. For a grayscale image, C might be any
range opossible gra y-level values 0 to 255, e.g.C= {12,13,...40},
for a binary image we simply have C={1}.
In order to understand the spatial adjacency condition, we first
explain the notion of neighborhood. For a pixel pwith the
coordinates (x, y),thset of neighborhood pixels Nd (p) can be
defined as follows:
4- neighbors
For a pixel p (x,y) shown in green circle, N4(p) is the set
of
4-neigbors which share a face (or edge of the pixel) with p.
They
are four in number and are shown by orange circles. Two
pixels
p a nd q are 4-connectedif q is in N4(p) and both p, q C
8 neighbors:
For a pixel p (x, y) shown in green circle, N8(p) is the set
of
neighbours which share a face or a vertex/corner. There are
8
such neighbours and they are shown by orange circles. Two
pixels p and q a re 8-connectedif q is in N8(p) and both p,q
C
m neighbors (mixed neighbors):
For a pixel p (x,y) shown in green circle, Nd(p) is the set
of
diagonal neigbors shown by red circles. Two pixels p and q
with
values from C are m-connected if:
I. q is in N4(p), or
II. q is in Nd(p), and the set N4(p) N4(q) has no pixelswhose
values are from C.
Properties of connectivity:
If we consider a pixel to be connected to itself (trivial
connectivity) then connectivity is reflexive. A pixel and its
neighbor are mutuall
connected and hence connectivity is also symmetric.
4-connectivity and 8-connectivity are also transitive: if pixel A
is connected to pixel B, an
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pixel B is connected to pixel C, then there exists a connected
path between pixels A and C.
Pixel connectivity is an equivalence relation as it is
reflexive, symmetric, and transitive.
Simplification of the definition of region and boundary in an
image:
Let S represent a subset of pixels in an image. Two pixels are
said to be connected in S if there exists a path between them
consisting entirely o
pixels in S. For a pixel p S, the s et of pixels pi S that are
connected to it is called connected component.
In the event there is only one connected component, the set S is
said to be a connected set and S is called a region in the given
image.
The boundary of a region R is the set of pixels in the region
that have at least one neighbor that is not in R.
Distance Metrics:
On a discrete grid, the distance between two points a: (x a, ya)
and b: (xb,yb) can be defined in a variety of ways.
Euclidean distance: In vector form, this is referred to as L2
norm.
City-block distance: This is the 4-connected distance and
sometime called Manhattan distance.
In vector form, it is referred to as L1 norm.
Chess-board distance: This is the 8-connected distance D8, also
called as chess-board distance.
In vector form, it is referred as L norm.
Using these distance metrics, for a given pixel p, one can find
pixels at equal distance from this pixel. These pixels form the
iso-distance pixe
set for the point p.
Properties of a true distance metric
Let a, b and c be three pixels with coordinates (xa, ya), (xa,
yb) and (xc, yc) respectively. A distance function D is a metric if
it has the followin
properties:
a. D(a,b) 0 and D(a,b) = 0 iff a = b
b. D(a, b) = D(a, b)
c. D(a, c) D(a, b) + D(b, c)
