Describe Discrete Cosine Transform (DCT) with Example

The Discrete Cosine Transform (DCT) in Image Processing helps separate the image into parts (or spectral sub-bands) of differing importance (with respect to the image’s visual quality).

The Discrete Cosine TransformDCT is similar to the Discrete Fourier Transform: it transforms a signal or image from the spatial domain to the frequency domain.

Discrete Cosine Transform

Discrete Cosine Transform (DCT) Encoding with Example


The general equation for a 1D (N data items) DCT is defined by the following equation:

Describe Discrete Cosine Transform (DCT)

and the corresponding inverse 1D DCT transform is simple F-1(u), i.e.:

where

The general equation for a 2D (N by M image) DCT is defined by the following equation:

and the corresponding inverse 2D DCT transform is simple F-1(u,v), i.e.:

where

The Basic Operation of the Discrete Cosine Transform (DCT) is as follows:


  • The input image is N by M;
  • f(i,j) is the intensity of the pixel in row i and column j;
  • F(u,v) is the DCT coefficient in row k1 and column k2 of the DCT matrix.
  • For most images, much of the signal energy lies at low frequencies; these appear in the upper left corner of the DCT.
  • Compression is achieved since the lower right values represent higher frequencies, and are often small – small enough to be neglected with little visible distortion.
  • The DCT input is an 8 by 8 array of integers. This array contains each pixel’s gray scale level;
  • 8 bit pixels have levels from 0 to 255.
  • Therefore an 8 point DCT would be:where

 

  • Question: What is F[0,0]?answer: They define DC and AC components.
  • The output array of DCT coefficients contains integers; these can range from -1024 to 1023.
  • It is computationally easier to implement and more efficient to regard the DCT as a set of basis functions which given a known input array size (8 x 8) can be precomputed and stored. This involves simply computing values for a convolution mask (8 x8 window) that get applied (sum values x pixel the window overlap with image apply window across all rows/columns of image). The values as simply calculated from the DCT formula. The 64 (8 x 8) DCT basis functions are illustrated in Fig.

 

Discrete Cosine Transform (DCT) Basis Functions


  • Why DCT not FFT? DCT is similar to the Fast Fourier Transform (FFT), but can approximate lines well with fewer coefficients (FigĀ 7.10)

 

DCT/FFT Comparison


  • Computing the 2D DCT
    • Factoring reduces problem to a series of 1D DCTs (FigĀ 7.11):
      • apply 1D DCT (Vertically) to Columns
      • apply 1D DCT (Horizontally) to resultant Vertical DCT above.
      • or alternatively Horizontal to Vertical.

      The equations are given by:

  • Most software implementations use fixed point arithmetic. Some fast implementations approximate coefficients so all multiplies are shifts and adds.
  • World record is 11 multiplies and 29 adds. (C. Loeffler, A. Ligtenberg and G. Moschytz, “Practical Fast 1-D DCT Algorithms with 11 Multiplications”, Proc. Int’l. Conf. on Acoustics, Speech, and Signal Processing 1989 (ICASSP `89), pp. 988-991)

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