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The construction of wavelet functions from known scaling functions is called the 'completion problem'. Completion algorithms exist for univariate wavelets, including multiwavelets. For multivariate wavelets, however, completion is not always possible. We present a new algorithm (a generalization of a method of Lai) which works in many cases.

A simple method of construction of a pair of orthogonal wavelet frames in L2(ℝd) is presented. This is a generalization of one-dimensional case to higher dimension. The construction is based on the well-known Unitary Extension Principle (UEP). The presented method produces the polyphase components of the filters of the wavelet functions, and hence the filters. A pair of orthogonal wavelet frames can be constructed with an extra condition. In the construction, the polyphase matrix is used as opposed to the modulation matrix. This is less restrictive and yields a fewer wavelet functions in the system than in the previously known constructions.