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Methods for lossy compression: Transform coding – This is the most commonly used method. Discrete Cosine Transform (DCT) – The most widely used form of lossy compression. It is a type of Fourier-related transform, and was originally developed by Nasir Ahmed, T. Natarajan and K. R. Rao in 1974. [2]
For example, several lossy image and sound compression methods employ the discrete Fourier transform: the signal is cut into short segments, each is transformed, and then the Fourier coefficients of high frequencies, which are assumed to be unnoticeable, are discarded. The decompressor computes the inverse transform based on this reduced number ...
The image is split into blocks of 8×8 pixels, and for each block, each of the Y, C B, and C R data undergoes the discrete cosine transform (DCT). A DCT is similar to a Fourier transform in the sense that it produces a kind of spatial frequency spectrum. The amplitudes of the frequency components are quantized. Human vision is much more ...
An example FFT algorithm structure, using a decomposition into half-size FFTs A discrete Fourier analysis of a sum of cosine waves at 10, 20, 30, 40, and 50 Hz. A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT).
Representing signals in frequency domain is a common approach to data compression. As graph signals can be sparse in their graph spectral domain, the graph Fourier transform can also be used for image compression. [6] [7]
An example application of the Fourier transform is determining the constituent pitches in a musical waveform.This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord.
Formally, the optical transfer function is defined as the Fourier transform of the point spread function (PSF, that is, the impulse response of the optics, the image of a point source). As a Fourier transform, the OTF is generally complex-valued; however, it is real-valued in the common case of a PSF that is symmetric about its center.
For example, JPEG compression uses a variant of the Fourier transformation (discrete cosine transform) of small square pieces of a digital image. The Fourier components of each square are rounded to lower arithmetic precision , and weak components are eliminated, so that the remaining components can be stored very compactly.