BasisImages

BasisImages is a term used in image processing and linear algebra to describe a collection of images that serve as a basis for representing other images in a fixed vector space. When the basis is orthonormal or otherwise linearly independent, any admissible image can be expressed as a linear combination of the BasisImages.

Mathematically, an image is treated as a vector in R^N by stacking pixels. A BasisImages set B = {b1, ..., bk} spans a subspace. The coefficients a satisfy x ≈ sum ai bi. If B is a full basis (k = N and linearly independent), the representation is exact; otherwise it is an approximation.

BasisImages can be hand-crafted (the basis images of the Discrete Cosine Transform, Fourier, Haar wavelets) or learned from data (dictionary learning, PCA eigenimages, autoencoder-derived bases). In learned dictionaries, the components adapt to the dataset to promote sparsity.

Applications include compression, denoising, sparse representation, feature extraction, and reconstruction from incomplete data. Overcomplete dictionaries allow multiple representations and sparser coefficients.

Relation to other concepts: BasisImages are a general concept closely related to basis vectors, eigenfaces, and transforms like DCT and wavelet transforms. The term appears in both theoretical discussions and practical image representation algorithms.