A tutorial overview on the properties of the discrete cosine transform for encoded image and video processing

Discrete trigonometric transforms, such as the discrete cosine transform (DCT) and the discrete sine transform (DST), have been extensively used in signal processing for transform-based coding. The even type-II DCT, used in image and video coding, became specially popular to decorrelate the pixel data and minimize the spatial redundancy. Albeit this DCT tends to be the most often used, it integrates a broader family of transforms composed of eight DCTs and eight DSTs. However, even though most applications require little knowledge more than the actual DCT definition and its inverse, it is often widely regarded that the implementation of more complex operations on transformed data sequences (transcoding) requires a more in-depth knowledge about its precise definitions and formal mathematical properties. One of such relations is the multiplication-convolution property, often required to implement more specific and complex manipulations. Considering that such information is still spread into several documents and manuscripts, the main purpose of this article is to provide a broad set of practical and useful information in a single and self-contained source, embracing a wide range of definitions and properties related to the DCT and DST families, with a special emphasis on its application to image and video processing.

► DCT is extensively used in image and video coding and transcoding applications.
► Type-II DCT decorrelates the pixel data and minimizes the spatial redundancy.
► DCT integrates a broader family of transforms composed of eight DCTs and eight DSTs.
► Some specific and complex manipulations use the multiplication-convolution property.
► We provide the main definitions and mathematical properties related to DCT and DST.