DWTn

DWTn, short for discrete wavelet transform in n dimensions, is a family of multi-dimensional signal representations that generalizes the conventional one-dimensional discrete wavelet transform to data sets defined in any number of dimensions. Like the standard DWT, DWTn decomposes a signal into approximation and detail components at successive scales, enabling multiresolution analysis and efficient data sparsification. In practice, DWTn is implemented via separable filter banks along each dimension or via non-separable, cross-dimensional wavelets for more complex correlations.

In separable implementations, a 1D wavelet filter pair is applied along each axis in sequence, producing a set of subbands that capture horizontal, vertical, and diagonal features for 2D data, and analogous directions for higher dimensions. The result is a tensor of coefficients corresponding to different scales and orientations. Non-separable variants use specially designed multi-dimensional wavelets to capture features that cross dimensions more effectively.

Applications include image and video compression, denoising, medical imaging, geophysical data analysis, and scientific visualization, where handling three- or higher-dimensional data efficiently is important. DWTn enables sparse representations and fast transforms via optimized filter banks, enabling real-time or large-scale processing in some systems. Typical variants include orthogonal and biorthogonal DWTn designs, depending on the desired reconstruction properties and symmetry.

Key considerations when using DWTn include choice of mother wavelet, boundary handling for finite-sized data, and the trade-off between separable simplicity and non-separable expressiveness. Computational complexity grows with the number of dimensions, and artifacts can arise near edges unless appropriate padding strategies are used.