Overview

Both the local trigonometric and the conjugate quadrature filter algorithms extend to multidimensional signals. We consider three methods of extension. The first two consist of tensor products of one-dimensional basis elements, combined so that the d-dimensional basis function is really the product of d one-dimensional basis functions: b( x) = b( x ₁, ..., x _d) = b ₁( x ₁) b _d( x _d). Such tensor product basis elements are called separable because we can factor them across sums and integrals, to obtain a sequence of d one-dimensional problems by treating each variable separately.

Consider the two-dimensional case for simplicity. Let E = { e _k : k ? I} and F = { f _k : k ? J} be bases for L ²( R), where I and J are index sets for the basis elements. Then we can produce two types of separable bases for L ²( R ²):

• E ? F ? { e _n ? f _m : ( n, m) ? I J}, the separable tensor product of the bases E and F;

• { e _n ? f _m : ( n, m) ? B}, a basis of separable elementary tensors from a basis subset B which is not necessarily all of I