cosinetransform

Cosine transform is a family of integral and discrete transforms that decompose a real-valued function or sequence into a sum of cosine basis functions. It uses cosine kernels instead of complex exponentials, reflecting even symmetry and yielding real-valued spectra. In the continuous case on a finite interval [0, L], the Fourier cosine transform is defined by F_c(ξ) = sqrt(2/L) ∫_0^L f(t) cos(ξ t) dt, with ξ = nπ/L for integers n, and the inverse relation f(t) = sqrt(2/L) ∑ F_c(ξ_n) cos(ξ_n t). This pair is orthogonal and invertible under suitable conditions, and is often interpreted as the Fourier transform of an even extension of f.

The discrete cosine transform (DCT) is the finite, sampled counterpart used for sequences of length N. The most common variant, DCT-II, is given by X_k = ∑_{n=0}^{N−1} x_n cos[π (n + 1/2) k / N], for k = 0,...,N−1, with appropriate normalization to make it orthonormal. There exist several other variants (DCT-I to DCT-VI) corresponding to different boundary conditions at the ends of the sequence. The DCT is real-valued, separable, and invertible, making it well suited for many practical tasks.

Properties and algorithms: cosine transforms are linear, orthogonal (with proper normalization), and provide energy compaction for smooth signals. Fast algorithms (FDCT) reduce computation time similarly to FFT for Fourier transforms.

Applications: the cosine transform is widely used in data compression and signal processing, most notably the Discrete Cosine Transform in image compression standards such as JPEG, as well as in audio and video coding, feature extraction, and numerical solutions of partial differential equations.