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Discrete Cosine Transform (DCT)

corner frequency coefficient compression

Definition: Discrete Cosine Transform is a technique applied to image pixels in spatial domain in order to transform them into a frequency domain in which redundancy can be identified.

In JPEG compression, image is divided into 8×8 blocks, then the two-dimensional Discrete Cosine Transform (DCT) is applied to each of these 8×8 blocks. In JPEG decompression, the Inverse Discrete Cosine Transform (IDCT) is applied to the 8×8 DCT coefficient blocks. DCT and IDCT are defined as follows:

where f ( i, j ) and F ( u, v ) are respectively the pixel value and the DCT coefficient, and

As shown in Figure 1, in an 8×8 DCT coefficient block, F (0,0) is called DC coefficient, while other 63 DCT coefficients are called AC coefficients. The coefficients at the bottom-right corner are of high frequencies, and they are sensitive to distortions such as compression. The coefficients at the upper-left corner are of low frequencies, and they are robust to distortions. Human eyes are less sensitive to the high frequencies, while more sensitive to the low frequencies.

For still images, most of the energy is located in the low frequency area (upper-left corner). That means, there are bigger values on the upper-left corner and smaller values on the bottom-right corner. Together with human visual system, this feature can be used for image and video compression.

Compression algorithms are so designed that try to preserve the low frequency components and to depress the high frequency components.

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