A Discrete Wavelet Transform without edge effects using wavelet extrapolation

The Discrete Wavelet Transform (DWT) is of considerable practical use in image and signal processing applications. For example, significant compression can be achieved through the use of the DWT. A fundamental problem with the DWT, however, is the treatment of finite length data sequences. Commonly used techniques such as circular convolution and symmetric extension can produce undesirable edge effects which propagate into the interior of the transformed data as the number of DWT iterations increases. In this paper, we develop a DWT applicable to Daubechies’ orthogonal wavelets which does not exhibit edge effects. The underlying idea is to extrapolate the data at the boundaries by determining the coefficients of a best fit polynomial through data points in the vicinity of the boundary. This approach can be regarded as a solution to the problem of orthogonal wavelets on an interval. However, it has the advantage that it does not involve the explicit construction of boundary wavelets. The extrapolated DWT is designed to be well conditioned and to produce a critically sampled output. The methods we describe are equally applicable to biorthogonal wavelet bases.