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.

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 decomposition positive integer

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

filters — Decomposition (analysis) and reconstruction (synthesis) filters structure

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

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.

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.

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.

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 coefficients cell 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.

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