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Chapter 5 : Image Transforms

(from Anil. K. Jain)

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Wen-Nung Lie

2-D orthogonal and unitary

transforms

Orthogonal series expansion for anNNimageu(m, n)

v(k, l)s are the transform coefficients,

represents the transformed image

is a set of orthonormal functions,

representing the image transform

),(),(),(1

0

1

0

,nmanmulkv

N

m

N

n

lk

=

=

= 1,0 Nlk

),(),(),(1

0

1

0

*

,nmalkvnmu

N

k

N

l

lk

=

=

= 1,0 Nnm

)},({ lkvV

)},({,

nmalk

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Wen-Nung Lie

Orthonormality and

completeness

must satisfies)},({ , nma lk

=

=

=

1

0

1

0

*

,,),(),(),(:

N

m

N

n

lklkllkknmanmalityorthonorma

=

=

=1

0

1

0

*

,,),(),(),(:

N

k

N

l

lklknnmmnmanmasscompletene

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Wen-Nung Lie

Matrix representation of image

transform

is the matrix inner product

Image U can be described as a linear combination

ofN2 matrix , k, l = 0,...,N-1

are called the basis images

v(k, l) can be considered as the projection ofu(m, n) onthe (k, l)-th basis image

=

==

1

0

1

0

*

,),(

N

k

N

l

lklkv AU

>=< *,

,),(lk

lkv AU

=

=

>=