Transforms

Transforms are mathematical mappings that convert objects from one domain to another, often to simplify analysis, reveal structure, or enable operations that are difficult in the original domain. They may be linear or nonlinear, continuous or discrete, and many transforms are invertible, allowing the original data to be recovered from its transform.

Geometric transforms refer to mappings of points in space that preserve or alter geometric relations. Common examples include translations, rotations, reflections, and scalings. Affine transforms combine linear transformations with translations and preserve straight lines and parallelism; projective transforms capture perspective. In computer graphics, these transforms are used to manipulate shapes, coordinates, and camera projections.

Functional or integral transforms map functions to functions on another domain, often turning convolution into multiplication or turning differential equations into algebraic equations. The Fourier transform converts a time- or space-domain signal into its frequency components; the Laplace transform is used for causal systems and differential equations; the Z-transform operates on discrete-time sequences.

Discrete transforms specialize in finite or sampled data. The discrete Fourier transform (DFT) computes frequency components of a finite sequence; the fast Fourier transform (FFT) is an efficient algorithm for the DFT. The discrete cosine transform (DCT) is widely used in image and video compression; wavelet transforms provide multiresolution representations.

Key properties include linearity and invertibility, energy preservation in certain transforms (Parseval's theorem for the Fourier transform), and the choice of basis functions, which determines how the transformed data is represented and analyzed. The use of a transform depends on the application, data characteristics, and desired interpretation.

Transforms underpin many disciplines, including signal processing, image and audio compression, solving differential equations, communications, and computer graphics. They provide a framework for changing domain representations to simplify operations, analysis, and interpretation.