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Discrete stationary wavelet transform 2-D

`SWC = swt2(X,N,'`

* wname*')

[A,H,V,D] = swt2(X,N,'

`wname`

SWC = swt2(X,N,Lo_D,Hi_D)

[A,H,V,D] = swt2(X,N,Lo_D,Hi_D)

`swt2`

performs a multilevel
2-D stationary wavelet decomposition using either an orthogonal or
a biorthogonal wavelet. Specify the wavelet using its name(* 'wname'*,
see

`wfilters`

for more information)
or its decomposition filters.`SWC = swt2(X,N,'`

or
* wname*')

```
[A,H,V,D]
= swt2(X,N,'
````wname`

')

compute the
stationary wavelet decomposition of the real-valued 2-D or 3-D matrix `X`

at
level `N`

, using `'wname'`

If `X`

is a 3-D matrix, the third dimension of `X`

must
equal 3.

`N`

must be a strictly positive integer (see `wmaxlev`

for more information), and 2^{N }must
divide `size(X,1)`

and `size(X,2)`

.

The dimension of `X`

and level `N`

determine the
dimensions of the outputs.

If

`X`

is a 2-D matrix and`N`

is greater than 1, the outputs`[A,H,V,D]`

are 3-D arrays, which contain the coefficients:For

`1`

≤`i`

≤`N`

, the output matrix`A(:,:,i)`

contains the coefficients of approximation of level`i`

.The output matrices

`H(:,:,i)`

,`V(:,:,i)`

and`D(:,:,i)`

contain the coefficients of details of level`i`

(horizontal, vertical, and diagonal):SWC = [H(:,:,1:N) ; V(:,:,1:N) ; D(:,:,1:N) ; A(:,:,N)]

If

`X`

is a 2-D matrix and`N`

is equal to 1, the outputs`[A,H,V,D]`

are 2-D arrays where`A`

contains the approximation coefficients, and`H`

,`V`

, and`D`

contain the horizontal, vertical, and diagonal detail coefficients, respectively.If

`X`

is a 3-D matrix of dimension`m`

-by-`n`

-by-3, and`N`

is greater than 1, the outputs`[A,H,V,D]`

are 4-D arrays of dimension`m`

-by-`n`

-by-3-by-`N`

, which contain the coefficients:For

`1`

≤`i`

≤`N`

and`j = 1, 2, 3`

, the output matrix`A(:,:,j,i)`

contains the coefficients of approximation of level`i`

.The output matrices

`H(:,:,j,i)`

,`V(:,:,j,i)`

and`D(:,:,j,i)`

contain the coefficients of details of level`i`

(horizontal, vertical, and diagonal):SWC = [H(:,:,1:3,1:N) ; V(:,:,1:3,1:N) ; D(:,:,1:3,1:N) ; A(:,:,1:3,N)]

If

`X`

is a 3-D matrix of dimension`m`

-by-`n`

-by-3, and`N`

is equal to 1, the outputs`[A,H,V,D]`

are 4-D arrays of dimension`m`

-by-`n`

-by-1-by-3, which contain the coefficients:For

`j = 1, 2, 3`

, the output matrix`A(:,:,1,j)`

contains the approximation coefficients.The output matrices

`H(:,:,1,j)`

,`V(:,:,1,j)`

and`D(:,:,1,j)`

contain the horizontal, vertical, and diagonal detail coefficients, respectively.SWC = [H(:,:,1,1:3) ; V(:,:,1,1:3) ; D(:,:,1,1:3) ; A(:,:,1,1:3)]

`swt2`

uses double-precision arithmetic internally and returns double-precision coefficient matrices.`swt2`

warns if there is a loss of precision when converting to double.To distinguish a single-level decomposition of a truecolor image from a multilevel decomposition of an indexed image, the approximation and detail coefficient arrays of truecolor images are 4-D arrays. See the Release Notes for details. Also see examples Stationary Wavelet Transform of an Image and Inverse Stationary Wavelet Transform of an Image.

If an

`K`

-level decomposition is performed, the dimensions of the`A`

,`H`

,`V`

, and`D`

coefficient arrays are`m`

-by-`n`

-by-3-by-`K`

.If a single-level decomposition is performed, the dimensions of the

`A`

,`H`

,`V`

, and`D`

coefficient arrays are`m`

-by-`n`

-by-1-by-3. Since MATLAB^{®}removes singleton last dimensions by default, the third dimension of the coefficient arrays is singleton.

`SWC = swt2(X,N,Lo_D,Hi_D)`

or ```
[A,H,V,D]
= swt2(X,N,Lo_D,Hi_D)
```

, computes the stationary wavelet
decomposition as in the previous syntax, given these filters as input:

`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.

`swt2`

is defined using periodic extension.
The size of the approximation and details coefficients computed at each level equals the
size of the input data.

When X represents an indexed image, X is an `m`

-by-`n`

matrix. If the level of decomposition `N`

is greater than 1, the output
arrays SWC or cA, cH, cV, and cD are
`m`

-by-`n`

-by-`N`

arrays. If the level of
decomposition `N`

is equal to 1, the output arrays SWC or cA, cH, cV, and cD
are `m`

-by-`n`

arrays.

When X represents a truecolor image, it becomes an
`m`

-by-`n`

-by-3 array. This array 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. If the level of decomposition
`N`

is greater than 1, the output arrays SWC or cA, cH, cV, and cD are
`m`

-by-`n`

-by-3-by-`N`

. If the level of
decomposition `N`

is equal to 1, the output arrays SWC or cA, cH, cV, and cD
are `m`

-by-`n`

-by-1-by-3.

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 SWT 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:

Nason, G.P.; B.W. Silverman (1995), “The stationary wavelet
transform and some statistical applications,” *Lecture
Notes in Statistics*, 103, pp. 281–299.

Coifman, R.R.; Donoho, D.L. (1995), “Translation invariant
de-noising,” *Lecture Notes in Statistics*,
103, pp. 125–150.

Pesquet, J.C.; H. Krim, H. Carfatan (1996), “Time-invariant
orthonormal wavelet representations,” *IEEE Trans.
Sign. Proc.*, vol. 44, 8, pp. 1964–1970.

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