Inverse dual-tree and double-density 2-D wavelet transform

Demonstrate perfect reconstruction of an image using a complex oriented dual-tree wavelet transform.

Load the image and obtain the complex oriented dual-tree wavelet transform down to level 5 using `dddtree2`

. Reconstruct the image using `idddtree2`

and demonstrate perfect reconstruction.

load woman; wt = dddtree2('cplxdt',X,5,'dtf2'); xrec = idddtree2(wt); max(max(abs(X-xrec)))

ans = 7.4181e-12

`wt`

— Wavelet transformstructure

Wavelet transform, returned as a structure from `dddtree2`

with these fields:

`type`

— Type of wavelet decomposition (filter bank)`'dwt'`

| `'ddt'`

| `'realdt'`

| `'cplxdt'`

| `'realdddt'`

| `'cplxdddt'`

Type of wavelet decomposition (filter bank), specified as one
of `'dwt'`

, `'ddt'`

, `'realdt'`

, `'cplxdt'`

, `'realdddt'`

,
or `'cplxdddt'`

. `'dwt'`

is the
critically sampled DWT. `'ddt'`

produces a double-density
wavelet transform with one scaling and two wavelet filters for both
row and column filtering. `'realdt'`

and `'cplxdt'`

produce
oriented dual-tree wavelet transforms consisting of two and four separable
wavelet transforms. `'realdddt'`

and `'cplxdddt'`

produce
double-density dual-tree wavelet transforms consisting of two and
four separable wavelet transforms.

`level`

— Level of the wavelet decompositionpositive integer

Level of the wavelet decomposition, specified as a positive integer.

`filters`

— Decomposition (analysis) and reconstruction (synthesis) filtersstructure

Decomposition (analysis) and reconstruction (synthesis) filters, specified as a structure with these fields:

`Fdf`

— First-stage analysis filtersmatrix | cell array

First-stage analysis filters, specified as an *N*-by-2
or *N*-by-3 matrix for single-tree wavelet transforms,
or a 1-by-2 cell array of two *N*-by-2 or *N*-by-3
matrices for dual-tree wavelet transforms. The matrices are *N*-by-3
for the double-density wavelet transforms. For an *N*-by-2
matrix, the first column of the matrix is the scaling (lowpass) filter
and the second column is the wavelet (highpass) filter. For an *N*-by-3
matrix, the first column of the matrix is the scaling (lowpass) filter
and the second and third columns are the wavelet (highpass) filters.
For the dual-tree transforms, each element of the cell array contains
the first-stage analysis filters for the corresponding tree.

`Df`

— Analysis filters for levels > 1matrix | cell array

Analysis filters for levels > 1, specified as an *N*-by-2
or *N*-by-3 matrix for single-tree wavelet transforms,
or a 1-by-2 cell array of two *N*-by-2 or *N*-by-3
matrices for dual-tree wavelet transforms. The matrices are *N*-by-3
for the double-density wavelet transforms. For an *N*-by-2
matrix, the first column of the matrix is the scaling (lowpass) filter
and the second column is the wavelet (highpass) filter. For an *N*-by-3
matrix, the first column of the matrix is the scaling (lowpass) filter
and the second and third columns are the wavelet (highpass) filters.
For the dual-tree transforms, each element of the cell array contains
the analysis filters for the corresponding tree.

`Frf`

— First-level reconstruction filtersmatrix | cell array

First-level reconstruction filters, specified as an *N*-by-2
or *N*-by-3 matrix for single-tree wavelet transforms,
or a 1-by-2 cell array of two *N*-by-2 or *N*-by-3
matrices for dual-tree wavelet transforms. The matrices are *N*-by-3
for the double-density wavelet transforms. For an *N*-by-2
matrix, the first column of the matrix is the scaling (lowpass) filter
and the second column is the wavelet (highpass) filter. For an *N*-by-3
matrix, the first column of the matrix is the scaling (lowpass) filter
and the second and third columns are the wavelet (highpass) filters.
For the dual-tree transforms, each element of the cell array contains
the first-stage synthesis filters for the corresponding tree.

`Rf`

— Reconstruction filters for levels > 1matrix | cell array

Reconstruction filters for levels > 1, specified as an *N*-by-2
or *N*-by-3 matrix for single-tree wavelet transforms,
or a 1-by-2 cell array of two *N*-by-2 or *N*-by-3
matrices for dual-tree wavelet transforms. The matrices are *N*-by-3
for the double-density wavelet transforms. For an *N*-by-2
matrix, the first column of the matrix is the scaling (lowpass) filter
and the second column is the wavelet (highpass) filter. For an *N*-by-3
matrix, the first column of the matrix is the scaling (lowpass) filter
and the second and third columns are the wavelet (highpass) filters.
For the dual-tree transforms, each element of the cell array contains
the first-stage analysis filters for the corresponding tree.

`cfs`

— Wavelet transform coefficientscell array of matrices

Wavelet transform coefficients, specified as a 1-by-(`level`

+1)
cell array of matrices. The size and structure of the matrix elements
of the cell array depend on the type of wavelet transform as follows:

`'dwt'`

—`cfs{j}(:,:,d)`

j = 1,2,...

`level`

is the level.d = 1,2,3 is the orientation.

`cfs{level+1}(:,:)`

are the lowpass, or scaling, coefficients.

`'ddt'`

—`cfs{j}(:,:,d)`

j = 1,2,...

`level`

is the level.d = 1,2,3,4,5,6,7,8 is the orientation.

`cfs{level+1}(:,:)`

are the lowpass, or scaling, coefficients.

`'realddt'`

—`cfs{j}(:,:,d,k)`

j = 1,2,...

`level`

is the level.d = 1,2,3 is the orientation.

k = 1,2 is the wavelet transform tree.

`cfs{level+1}(:,:)`

are the lowpass, or scaling, coefficients.

`'cplxdt'`

—`cfs{j}(:,:,d,k,m)`

j = 1,2,...

`level`

is the level.d = 1,2,3 is the orientation.

k = 1,2 is the wavelet transform tree.

m = 1,2 are the real and imaginary parts.

`cfs{level+1}(:,:)`

are the lowpass, or scaling, coefficients..

`'realdddt'`

—`cfs{j}(:,:,d,k)`

j = 1,2,...

`level`

is the level.d = 1,2,3 is the orientation.

k = 1,2 is the wavelet transform tree.

`cfs{level+1}(:,:)`

are the lowpass, or scaling, coefficients.

`'cplxdddt'`

—`cfs{j}(:,:,d,k,m)`

j = 1,2,...

`level`

is the level.d = 1,2,3 is the orientation.

k = 1,2 is the wavelet transform tree.

m = 1,2 are the real and imaginary parts.

`cfs{level+1}(:,:)`

are the lowpass, or scaling, coefficients.

`xrec`

— Synthesized 2-D imagematrix

Synthesized image, returned as a matrix.

**Data Types: **`double`

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