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Multilevel 2-D wavelet decomposition

`[C,S] = wavedec2(X,N,`

* 'wname'*)

[C,S] = wavedec2(X,N,Lo_D,Hi_D)

`wavedec2`

is a two-dimensional wavelet analysis
function.

`[C,S] = wavedec2(X,N,`

returns
the wavelet decomposition of the matrix * 'wname'*)

`X`

at level `N`

,
using the wavelet named in character vector `'wname'`

`wfilters`

for more information).Outputs are the decomposition vector `C`

and
the corresponding bookkeeping matrix `S`

.

`N`

must be a strictly positive integer (see `wmaxlev`

for more information).

Instead of giving the wavelet name, you can give the filters.

For `[C,S] = wavedec2(X,N,Lo_D,Hi_D)`

, `Lo_D`

is
the decomposition low-pass filter and `Hi_D`

is the
decomposition high-pass filter.

Vector `C`

is organized as a vector with A(N),
H(N), V(N), D(N), H(N-1), V(N-1), D(N-1), ..., H(1), V(1), D(1), where `A`

, `H`

, `V`

,
and `D`

are each a row vector. Each vector is the
vector column-wise storage of a matrix.

`A`

contains the approximation coefficients`H`

contains the horizontal detail coefficients`V`

contains the vertical detail coefficients`D`

contains the diagonal detail coefficients

Matrix `S`

is such that

`S(1,:)`

= size of approximation coefficients(`N`

).`S(i,:)`

= size of detail coefficients(`N-i+2`

) for`i`

= 2, ...`N+1`

and`S(N+2,:) = size(X)`

.

This example shows the structure of `wavedec2`

output matrices.

Load original image from the `woman.mat`

file, which contains variables named `X`

and `map`

.

`load woman;`

Get current discrete wavelet transform extension mode.

origMode = dwtmode('status','nodisplay');

Change to periodic boundary handling.

`dwtmode('per');`

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! WARNING: Change DWT Extension Mode ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ***************************************** ** DWT Extension Mode: Periodization ** *****************************************

Perform decomposition at level 2 of `X`

using db1.

`[c,s] = wavedec2(X,2,'db1');`

Get the decomposition structure organization.

sizex = size(X)

```
sizex =
256 256
```

sizec = size(c)

```
sizec =
1 65536
```

Reset discrete wavelet transform extension mode to its original mode.

dwtmode(origMode);

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! WARNING: Change DWT Extension Mode ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ******************************************************* ** DWT Extension Mode: Symmetrization (half-point) ** *******************************************************

Extract and display images of wavelet decomposition level details. The resulting images are similar to the visualizations in the *At level 2, with haar --> woman* indexed image example in the Wavelet 2-D interactive tool. Use `waveletAnalyzer`

to launch this tool.

load woman; [c,s]=wavedec2(X,2,'haar');

[H1,V1,D1] = detcoef2('all',c,s,1); A1 = appcoef2(c,s,'haar',1); V1img = wcodemat(V1,255,'mat',1); H1img = wcodemat(H1,255,'mat',1); D1img = wcodemat(D1,255,'mat',1); A1img = wcodemat(A1,255,'mat',1);

[H2,V2,D2] = detcoef2('all',c,s,2); A2 = appcoef2(c,s,'haar',2); V2img = wcodemat(V2,255,'mat',1); H2img = wcodemat(H2,255,'mat',1); D2img = wcodemat(D2,255,'mat',1); A2img = wcodemat(A2,255,'mat',1);

subplot(2,2,1); imagesc(A1img); colormap pink(255); title('Approximation Coef. of Level 1'); subplot(2,2,2); imagesc(H1img); title('Horizontal detail Coef. of Level 1'); subplot(2,2,3); imagesc(V1img); title('Vertical detail Coef. of Level 1'); subplot(2,2,4); imagesc(D1img); title('Diagonal detail Coef. of Level 1');

figure; subplot(2,2,1); imagesc(A2img); colormap pink(255); title('Approximation Coef. of Level 2'); subplot(2,2,2) imagesc(H2img); title('Horizontal detail Coef. of Level 2'); subplot(2,2,3) imagesc(V2img); title('Vertical detail Coef. of Level 2'); subplot(2,2,4) imagesc(D2img); title('Diagonal detail Coef. of Level 2');

When X represents an indexed image, X, as well as the output
arrays cA,cH,cV, and 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. The size of vector C and the size of matrix `S`

depend
on the type of analyzed image.

For a truecolor image, the decomposition vector `C`

and
the corresponding bookkeeping matrix `S`

can be represented
as follows.

For more information on image formats, see the `image`

and `imfinfo`

reference
pages.

For images, an algorithm similar to the one-dimensional case is possible for two-dimensional wavelets and scaling functions obtained from one-dimensional ones 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 following chart describes the basic decomposition step for images:

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

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.)

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