The document discusses key concepts in digital image processing including array vs matrix operations, linear vs nonlinear operations, and arithmetic and logical operations. Array operations are performed on a pixel-by-pixel basis while matrix operations use matrix theory. A linear operator satisfies additivity and homogeneity, processing the sum and scaling of inputs the same as individual inputs summed or scaled. Logical operations like AND, OR, and NOT are applied to binary images.

1. DIGITAL IMAGE
PROCESSING
MATHEMATIC
PRELIMINARIES
Processing of Images which are Digital in nature by
means of Digital Computer
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2. Array versus Matrix Operation
 Array operation is carried out on pixel-by-pixel
basis and matrix operation is based on matrix
theory.
 The array product of two images:
=
 The matrix product of two images:
=
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a00 a01
a10 a11
b00 b01
b10 b11
a00b00 a01b01
a10b10 a11b11
a00 a01
a10 a11
b00 b01
b10 b11
a00b00+a01b10 a00b01+a01b11
a10b00+a11b10 a10b01+a11b11

3. Linear versus Nonlinear Operation
 Let an operator H produces an output image
g(x, y) for an input image f(x, y):
H[f(x, y)] = g(x, y)
 H is said to be Linear operator if it satisfy the
property of additivity and homogeneity,
otherwise Nonlinear operator.
H[aifi(x, y)+ajfj(x, y)] = H[aifi(x, y)] + H[ajfj(x, y)]
= aiH[fi(x, y)] + ajH[fj(x, y)]
= aigi(x, y) + ajgj(x, y)
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4. Additivity and Homogeneity
 Additivity property: The output of the sum
of two inputs is same as performing the
operation on the inputs individually and
then summing the result.
H[fi(x, y)+fj(x, y)] = H[fi(x, y)] + H[fj(x, y)]
 Homogeneity property: The output of a
constant times input is same as the output of
original input multiplied by that constant.
H[af(x, y)] = aH[f(x, y)] = ag(x, y)
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5. Example of Linear Operator
 Sum operator Σ :
Σ[aifi(x, y)+ajfj(x, y)] = ai Σ[fi(x, y)] + aj Σ[fj(x, y)]
0 2
 Σ[aifi(x, y)+ajfj(x, y)] = -15
 ai Σ[fi(x, y)] + aj Σ[fj(x, y)] = 7-22 = -15
 Where Σ[fi(x, y)]= 7 , Σ[fj(x, y)] = 22
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fi(x, y) = fj(x, y) =
2 3
6 5
4 7
ai = 1 & ai = -1

6. Example of Nonlinear Operator
 Max operator Max{ } :
Max[aifi(x, y)+ajfj(x, y)] = ai Max[fi(x, y)] + aj Max[fj(x, y)]
(1) = Max
0 2
(1) Max = 3 - 7 = -4
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0 2
Max + (-1)
2 3
6 5
4 7
-6 -3
-2 -4
= -2
Max + (-1)
2 3
6 5
4 7

7. Arithmetic & Logical Operation
 The basic Arithmetic operations are:
s(x, y) = f(x, y) + g(x, y)
d(x, y) = f(x, y) - g(x, y)
p(x, y) = f(x, y) × g(x, y)
q(x, y) = f(x, y) ÷ g(x, y)
 The basic logical operations are:
C =(A) OR (B)
D =(A) AND (B)
E = NOT (A)
 Logical operations applied to binary images
only.
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8. Examples of Logical NOT
Operation
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A NOT (A)

9. Examples of Logical OR Operation
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A
B
(A)OR (B)

10. Examples of Logical AND
Operation
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(A)AND (B)
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B

11. Questions??????
1. Define linear and nonlinear operation with
examples.
2. Find (A) AND ( NOT (B))
3. Find (A) XOR (B)
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A B