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 Transform - 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 with Example
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 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 (FigÂ 7.10)
- Computing the 2D DCT
Factoring reduces problem to a series of 1D DCTs (FigÂ 7.11):
- The equations are given by:
- apply 1D DCT (Vertically) to Columnsapply 1D DCT (Horizontally) to resultant Vertical DCT above.or alternatively Horizontal to Vertical.
- 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)