# Documentation

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# dwt2

Single-level discrete 2-D wavelet transform

## Syntax

```[cA,cH,cV,cD] = dwt2(X,'wname') [cA,cH,cV,cD] = dwt2(X,Lo_D,Hi_D) [cA,cH,cV,cD] = dwt2(...,'mode',MODE) ```

## Description

The `dwt2` command performs a single-level two-dimensional wavelet decomposition. Compare this function to `wavedec2`, which may be more useful for your application. The decomposition is done with respect to either a particular wavelet (`'wname'`, see `wfilters` for more information) or particular wavelet decomposition filters (`Lo_D` and `Hi_D`) you specify.

`[cA,cH,cV,cD] = dwt2(X,'wname')` computes the approximation coefficients matrix `cA` and details coefficients matrices `cH`, `cV`, and `cD` (horizontal, vertical, and diagonal, respectively), obtained by wavelet decomposition of the input matrix `X`. The `'wname'` character vector contains the wavelet name.

`[cA,cH,cV,cD] = dwt2(X,Lo_D,Hi_D)` computes the two-dimensional wavelet decomposition as above, based on wavelet decomposition filters that you specify.

• `Lo_D` is the decomposition low-pass filter.

• `Hi_D` is the decomposition high-pass filter.

`Lo_D` and `Hi_D` must be the same length.

Let `sx = size(X)` and `lf = `the length of filters; then `size(cA) = size(cH) = size(cV) = size(cD) = sa` where ```sa = ceil(sx/2)```, if the DWT extension mode is set to periodization. For the other extension modes, `sa = floor((sx+lf-1)/2)`.

For information about the different Discrete Wavelet Transform extension modes, see `dwtmode`.

`[cA,cH,cV,cD] = dwt2(...,'mode',MODE)` computes the wavelet decomposition with the extension mode `MODE` that you specify.

`MODE` is a character vector containing the desired extension mode.

An example of valid use is

```[cA,cH,cV,cD] = dwt2(x,'db1','mode','sym'); ```

## Examples

collapse all

This example shows how to obtain the 2-D DWT of an image.

Load the "woman" image and obtain the 2-D DWT using the `sym4` wavelet. Use the periodic extension mode.

```load woman wname = 'sym4'; [CA,CH,CV,CD] = dwt2(X,wname,'mode','per');```

Display the vertical detail image and the lowpass approximation.

```subplot(211) imagesc(CV) title('Vertical Detail Image') colormap gray subplot(212) imagesc(CA) title('Lowpass Approximation')```

## Tips

When X represents an indexed image, then X, as well as the output arrays cA,cH,cV,cD are `m`-by-`n` matrices. When X represents a truecolor image, it is an `m`-by-`n`-by-3 array, where each `m`-by-`n` matrix represents a red, green, or blue color plane concatenated along the third dimension.

For more information on image formats, see the `image` and `imfinfo` reference pages.

## Algorithms

For images, there exist an algorithm similar to the one-dimensional case for two-dimensional wavelets and scaling functions obtained from one- dimensional ones by tensorial 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 following chart describes the basic decomposition steps for images:

### Note

To deal with signal-end effects involved by a convolution-based algorithm, a global variable managed by `dwtmode` is used. This variable defines the kind of signal extension mode used. The possible options include zero-padding (used in the previous example) and symmetric extension, which is the default mode.

## References

Daubechies, I. (1992), Ten lectures on wavelets, CBMS-NSF conference series in applied mathematics. SIAM Ed.

Mallat, S. (1989), “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Pattern Anal. and Machine Intell., vol. 11, no. 7, pp. 674–693.

Meyer, Y. (1990), Ondelettes et opérateurs, Tome 1, Hermann Ed. (English translation: Wavelets and operators, Cambridge Univ. Press. 1993.)