Single-level discrete 2-D wavelet transform

The 2-D wavelet decomposition algorithm for images is similar to the one-dimensional
case. The two-dimensional wavelet and scaling functions are obtained by taking the
tensor products of the one-dimensional wavelet and scaling functions. 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 following chart describes the basic decomposition steps for
images.

— 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 columns of the entry

The decomposition is initialized by setting the approximation
coefficients equal to the image *s*: $$C{A}_{0}=s$$.

To deal with signal-end effects introduced by a convolution-based algorithm, the
1-D and 2-D DWT use a global variable managed by `dwtmode`

. This variable defines
the kind of signal extension mode used. The possible options include zero-padding
and symmetric extension, which is the default mode.

[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.