- Remote sensing
- Image trmission and storage for business application
- Medical imaging
- Astronomy

- The DFT and unitary DFT matrices are symmetric.
- The extensions of the DFT and unitary DFT of a sequence and their inverse trforms are periodic with period N.
- The DFT or unitary DFT of a real sequence is conjugate symmetric about N/@

Forward trform

The sequence of x(n) is given by x(n) = { x0,x1,x2,… xN-1}.

X(k) = (n=0 to N-1) _ x(n) exp(-j 2* pi* nk/N) ; k= 0,1,2,…N-1

Reverse trforms

X(n) = (1/N) (k=0 to N-1) _ x(k) exp(-j 2* pi* nk/N) ; n= 0,1,2,…N-1

- Separability
- Trlation
- Periodicity and Conjugate property
- Rotation
- Distributivity and scaling
- Average value
- Convolution and Correlation
- Laplacian

The negative of an image with gray levels in the range [0, L-1] is obtained by using the negative trformation, which is given by the expression.

s = L-1-r

Where s is output pixel.

r is input pixel.

- Slant trform is real and orthogonal.
- Slant trform is a fast trform
- Slant trform has very good energy compaction for images
- The basic vectors of Slant matrix are not sequency ordered.

Hadamard trform matrices Hn are NXN matrices where N=2^n , n= 1,2,3,… is defined as Hn= Hn-1 * H1 = H1* Hn-1

= 1/ _ 2 Hn-1 Hn-1

H2 = 1 1

1 –1

- To reduce band width
- To reduce redundancy
- To extract feature.

The number of bits required to store a digital image is

b=M X N X k

When M=N, this equation becomes

b=N^2k

- Hadamard trform contains any one value.
- No multiplications are required in the trform calculations.
- The no: of additions or subtractions required can be reduced from N^2 to Nlog2N
- Very good energy compaction for highly correlated images.

Trpose of matrix = Inverse of a matrix. Orthoganality.

@Periodicity

WN^(K+N)= WN^K

@Symmetry

WN^(K+N/2)= -WN^K

- Determinant and the Eigen values of a unitary matrix have unity magnitude
- The entropy of a random vector is preserved under a unitary Trformation
- Since the entropy is a measure of average information, this me information is preserved under a unitary trformation.

f(x,y )= f(0,0) f(0,1)………………f(0,N-1)

f(1,0) f(1,1)………………f(1,N-1)

.

.

.

f(M-1) f(M-1,1)…………f(M-1,N-1)

- Real, symmetric and orthogonal.
- Not the imaginary part of the unitary DFT.
- Fast trform.

- Real and orthogonal
- Very fast trform
- Basis vectors are sequentially ordered
- Has fair energy compaction for image
- Useful in feature extraction,image coding and image analysis problem

The digital image is an array of real or complex numbers that is represented by a finite no of bits.

- Symmetric
- Periodic extensions
- Sampled Fourier trform
- Conjugate symmetry.

- Real & orthogonal.
- Fast trform.
- Has excellent energy compaction for highly correlated data

For a function f (x, y), the gradient f at co-ordinate (x, y) is defined as the

vector_f = _f/_x

_f/_y

_f = mag (_f) = {[(_f/_x) 2 +(_f/_y) 2 ]} ½

The NXN cosine trform c(k) is called the discrete cosine trform and is defined as

C(k) = 1/_N , k=0, 0 _ n _ N-1 = _ (2/N) cos (pi (2n+1)/2N 1_ k _ N-1, 0_ n _ N-1

The categories of Image Enhancement are

- Spatial domain
- Frequency domain Spatial domain: It refers to the image plane, itself and it is based on direct manipulation of pixels of an image.

Frequency domain techniques are based on modifying the Fourier trform of an image.

N = 4 = 2n

=> n = 2

- Haar trform is real and orthogonal.
- Haar trform is a very fast trform
- Haar trform has very poor energy compaction for images
- The basic vectors of Haar matrix sequency ordered.