The discrete cosine transform (DCT) helps separate the image into parts (or spectral sub-bands) of differing importance (with respect to the image’s visual quality).

The DCT is similar to the discrete Fourier transform: it transforms a signal or image from the spatial domain to the frequency domain.

**Discrete Cosine Transform (DCT) Encoding**

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

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 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 pixelthe 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.

**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:

- Factoring reduces problem to a series of 1D DCTs (FigĀ 7.11):

- 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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