Review Article

Fast Transforms in Image Processing: Compression, Restoration, and Resampling

Table 2

Selected fast transforms most relevant for digital image processing and their computational complexity (in 1D denotations).

Name DefinitionComputational complexity
(operations per signal sample)
“Global”: applied to entire signal of   samples“Local”: applied in mowing window of   pixels

Discrete Cosine Transform (DFT) Real number operations   Real number operations   (<WSz)

Discrete Fourier Transforms
 Canonic DFT Complex number operations   Complex number operations   <WSz
 General Shifted Scaled DFT ;  
, —shift parameters that indicate shifts , of sampling lattices in signal and spectrum domains; —sampling scale parameter ( )

Discrete Fresnel Transforms (DFrT)
 Canonic DFrT ;  Complex number operations   n/a
 Convolutional DFrT ; 

Walsh-Hadamard Transform
 Hadamard Transform
;  
Addition operations only   n/a
; ;
 Walsh Transform ;  

—binary digits of Gray code of taken in bit reversal order

Haar Transform (the simplest discrete wavelet transform) ;  
  
—index of the most significant nonzero bit of binary code of msb; —residual of
Addition operations only   n/a