# waverec3

Multilevel 3-D discrete wavelet transform reconstruction

## Description

reconstructs or extracts at level `c`

= waverec3(`wdec`

,`type`

,`n`

)`n`

the multilevel components
specified by `type`

. If `type`

begins with
`'c'`

or `'C'`

, `waverec3`

extracts the specified components. Otherwise, `waverec3`

reconstructs the components.

`x = waverec3(wdec,'a',0)`

and ```
x =
waverec3(wdec,'ca',0)
```

are equivalent to ```
x =
waverec3(wdec)
```

, where `'a'`

specifies the lowpass
component. `x`

is the reconstruction of the coefficients in
`wdec`

at level 0.

`c = waverec3(wdec,type)`

is equivalent to ```
c =
waverec3(wdec,type,wdec.level)
```

.

## Examples

### Perfect Reconstruction with 3-D Discrete Wavelet Transform

Construct a 3-D matrix, obtain the wavelet transform down to level 2 using the `'db2'`

wavelet, and reconstruct the matrix to verify perfect reconstruction.

Create 3-D matrix.

M = magic(8); X = repmat(M,[1 1 8]);

Obtain the 3-D discrete wavelet transform of the matrix and reconstruct the input based on the 3-D approximation and detail coefficients.

```
wd = wavedec3(X,2,'db2');
XR = waverec3(wd);
```

Verify perfect reconstruction using the wavelet decomposition down to level 2.

err1 = max(abs(X(:)-XR(:)))

err1 = 8.6050e-11

Verify that the data matrix is the sum of the approximation and the details from levels 2 and 1. Reconstruct the sum of components different from the lowpass component and check that `X`

= `A`

+ `D`

.

A = waverec3(wd,'LLL'); D = waverec3(wd,'d'); err2 = max(abs(X(:)-A(:)-D(:)))

err2 = 8.6054e-11

### Compare waverec3 and idwt3

Compare level-1 reconstructions based on the filtering operations `'LLH'`

using `idwt3`

and `waverec3`

.

M = magic(8); X = repmat(M,[1 1 8]); wd = wavedec3(X,2,'db2','mode','per'); dwtOut = dwt3(X,'db2'); Xr = idwt3(dwtOut,'LLH'); Xrec = waverec3(wd,'LLH',1); norm(Xr(:)-Xrec(:))

ans = 2.2773e-14

## Input Arguments

`wdec`

— Wavelet decomposition

structure

Wavelet decomposition, specified as a structure. The structure is the
output of `wavedec3`

and has the
following fields:

`sizeINI`

— Size

vector

Size of the 3-D array, specified as a 1-by-3 vector.

`level`

— Level of the decomposition

integer

Level of the decomposition, specified as an integer.

`mode`

— Name of the wavelet transform extension mode

character vector

Name of the wavelet transform extension mode, specified as a character vector.

`filters`

— Wavelet filters

structure

Wavelet filters used for the decomposition, specified as a structure with the following fields:

`LoD`

— lowpass decomposition filter`HiD`

— highpass decomposition filter`LoR`

— lowpass decomposition filter`HiR`

— highpass decomposition filter

`dec`

— Decomposition coefficients

cell array

Decomposition coefficients, specified as an
*N*-by-1 cell array, where
*N* equals 7× `wdec.level`

+1.

`dec{1}`

contains the lowpass component
(approximation) at the level of the decomposition. The
approximation is equivalent to the filtering operations
`'LLL'`

.

`dec{k+2},...,dec{k+8}`

with ```
k =
0,7,14,...,7*(wdec.level-1)
```

contain the 3-D
wavelet coefficients for the multiresolution starting with the
coarsest level when `k=0`

.

For example, if `wdec.level=3`

,
`dec{2},...,dec{8}`

contain the wavelet
coefficients for level 3 (`k=0`

),
`dec{9},...,dec{15}`

contain the wavelet
coefficients for level 2 (`k=7`

), and
`dec{16},...,dec{22}`

contain the wavelet
coefficients for level 1
(`k=7*(wdec.level-1)`

).

At each level, the wavelet coefficients in
`dec{k+2},...,dec{k+8}`

are in the
following order:
`'HLL'`

,`'LHL'`

,`'HHL'`

,`'LLH'`

,`'HLH'`

,`'LHH'`

,`'HHH'`

.

The sequence of letters gives the order in which the separable
filtering operations are applied from left to right. For
example, `'LHH'`

means that the lowpass
(scaling) filter with downsampling is applied to the rows of
`x`

, followed by the highpass (wavelet)
filter with downsampling applied to the columns of
`x`

. Finally, the highpass filter with
downsampling is applied to the 3rd dimension of
`x`

.

`sizes`

— Successive sizes

matrix

Successive sizes of the decomposition components, specified as
a `wdec.level`

+1-by-2 matrix.

`type`

— Type of reconstruction or extraction

character vector | string scalar

Type of reconstruction or extraction, specified as a character vector or
string scalar. For reconstruction, valid values of `type`

are:

A group of three characters

`'xyz'`

, one per direction, with`'x'`

,`'y'`

and`'z'`

selected in the set {`'a'`

,`'d'`

,`'l'`

,`'h'`

} or in the corresponding uppercase set {`'A'`

,`'D'`

,`'L'`

,`'H'`

}, where`'A'`

(or`'L'`

) is a lowpass filter and`'D'`

(or`'H'`

) is a highpass filter.The char

`'d'`

(or`'h'`

or`'D'`

or`'H'`

) gives the sum of all the components different from the lowpass component.The char

`'a'`

(or`'l'`

or`'A'`

or`'L'`

) gives the lowpass component (the approximation at level`n`

).

To extract coefficients, the valid values for
`type`

are the same but prefixed by
`'c'`

or `'C'`

.

`n`

— Decomposition level

`wdec.level`

(default) | integer

Decomposition level, specified as an integer.

## Output Arguments

`x`

— Reconstruction

3-D array

Reconstruction, returned as a 3-D array of size
`sz(1)`

-by-`sz(2)`

-by-`sz(3)`

,
where `sz = wpdec.sizeINI`

.

`c`

— Extracted coefficients

3-D array

Extracted coefficients, returned as a 3-D array.

## Version History

**Introduced in R2010a**

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