Dimensionality of Image transforms
The
 computations for  can also be reduced by restricting the choice of
 to fast transforms.This implies that  has a structure that allows factorization of the type,
 =
where
  ,......  are matrices with just a few non zero entries say
 where
 .
Therefore a multiplication of the type :
 =
 is accomplished in
 operations.
For several transforms like Fourier, Sine, Cosine, Hadamard etc,
 ,
and operations reduce to the order of
 or
 (for  images).
Depending on the transform, an operation is defined as 1 multiplication and 1 addition or, 1 addition or subtraction as in Hadamard Transform .
Kronecker products:
If and  are  and  matrices we define Kronecker product as:
    
Consider the transform,
 =
  
or,
 
|
(4.1.7) |
If  and  denote  and  row vectors of  and  then (7) becomes ,
 =
  =
 where  is the  block of 
If  and  are row ordered into vectors  and  respectively, then  =   = (  )  .
The number of operations required for implementing equation(4.1.7) reduces from  to  .
| 
