2-D wavelet decomposition

For images, an algorithm similar to the one-dimensional case is possible for
two-dimensional wavelets and scaling functions obtained from one-dimensional vectors by
tensor product. This kind of two-dimensional DWT leads to a decomposition of
approximation coefficients at level *j* in four components: the
approximation at level *j*+1 and the details in three orientations
(horizontal, vertical, and diagonal).

The chart describes the basic decomposition step for images:

where

— Downsample columns: keep the even-indexed columns.

— Downsample rows: keep the even-indexed rows.

— Convolve with filter

*X*the rows of the entry.— Convolve with filter

*X*the rows of the entry.

and

**Initialization**:
*cA*_{0} = *s*.

So, for *J* = 2, the two-dimensional wavelet tree has the form

[1] Daubechies, I. *Ten
Lectures on Wavelets*, CBMS-NSF Regional Conference Series in Applied
Mathematics. Philadelphia, PA: SIAM Ed, 1992.

[2] Mallat, S. G. “A Theory
for Multiresolution Signal Decomposition: The Wavelet Representation,”
*IEEE Transactions on Pattern Analysis and Machine Intelligence*.
Vol. 11, Issue 7, July 1989, pp. 674–693.

[3] Meyer, Y. *Wavelets
and Operators*. Translated by D. H. Salinger. Cambridge, UK: Cambridge
University Press, 1995.