MSP1 References Jain (a text book?; IP per se; available) Castleman (a real text book ; image analysis; less available) Lim (unavailable?)

MSP1

References

• Jain (a text book?; IP per se; available)

• Castleman (a real text book ; image analysis; less available)

• Lim (unavailable?)

MSP2

Image Transforms – Why?

• Simplicity

• Applications

• Image compression (JPEG

• Image enhancement (e.g., filtering)

• Image analysis (e.g., feature extraction)

MSP3

Image Transforms

• Preliminary definitions Orthogonal matrix

Unitary matrix

IAAAA

AA

TT

T

or

1

IAAAA

AA

TT

T

**

*1

or

MSP4

Preliminary Definitions (cont’)

• Real orthogonal matrix is unitary

• Unitary matrix need not be orthogonal

• Columns (rows) of an NxN unitary matrix are orthogonal and form a complete set of basis vectors in an N-dimensional vector space

MSP5

Preliminary Definitions (cont’)

• Examples (Jain, 1989)

1

1

2

1

2

2

11

11

2

1321 j

jA

j

jAA

orthogonal & unitary not unitary unitary

MSP6

Image Transforms (cont’)

• ... are a class of unitary matrices used to facilitate image representation

• Representation using a discrete set of basis images (similar to orthogonal series expansion of a continuous function)

MSP7

Image Transforms (cont’)

• For a 1D sequence , a unitary transformation is written as

where (unitary). This gives

10 ),( Nnnu

TAA *1

10 ),(),()( 1

0

NknunkakvN

n

Auv

10 ),,()()( 1

0

*

NnnkakvnuN

k

vAu T*

MSP8

Basic Vectors of 8x8 Orthogonal TransformsJain, 1989Jain, 1989

MSP9

2D Orthogonal & Unitary Transformations

• A general orthogonal series expansion for an NxN image u(m,n) is a pair of transformations

where is called an image transform, the elements v(k,l) are called the transform coefficients and is the transformed image.

),(, nma lk

1

0

1

0, 1,0 ),,(),(),(

N

m

N

nlk Nlknmanmulkv

1

0

1

0

*, 1,0 ),,(),(),(

N

k

N

llk Nnmnmalkvnmu

),( lkvV

MSP10

2D Orthogonal & Unitary Transformations (cont’)

is a set of complete orthonormal discrete basis functions satisfying ),(, nma lk

)','(),(),( :lityOrthonorma1

0

1

0

*',', llkknmanma

N

m

N

nlklk

)','()','(),( :ssCompletene1

0

1

0

*,, nnmmnmanma

N

k

N

llklk

MSP11

2D Orthogonal & Unitary Transformations (cont’)

• The orthonormality property assures that any truncated series expansion of the form

will minimise the sum-square-error

for v(k,l) as above, and the completeness property guarantees that this error will be zero for P=Q=N.

NQNPnmalkvnmuP

k

Q

llkQP

1

0

1

0

*,, , ),,(),(),(

21

0

1

0,

2 ),(),(

N

m

N

nQPe nmunmu

MSP12

Basis Images

• Define the matrices , where is the kth column of , and the matrix inner product of two NxN matrices F and G as

• Then Equations 2 & 1 provide series representation for the image as

Tlklk***

, aaA *ka

*TA

.),(),(,1

0

1

0

*

N

m

N

n

nmgnmfGF

*,

*,

1

0

1

0

,),(

),(

lk

lk

N

k

N

l

lkv

lkv

AU

AU

MSP13

Basic Images of the 8x8 2D TransformsJain, 1989Jain, 1989

MSP14

The Continuous 1D Fourier Transform

• The Fourier transform pair

dsesFtf

dtetfsF

stj

stj

2

2

)()(

)()(