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 = 7.3612e-12
wt — Wavelet transformWavelet 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 decompositionLevel of the wavelet decomposition, specified as a positive integer.
filters — Decomposition (analysis) and reconstruction (synthesis) filtersDecomposition (analysis) and reconstruction (synthesis) filters, specified as a structure with these fields:
Fdf — First-stage analysis filtersFirst-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 > 1Analysis 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 filtersFirst-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 > 1Reconstruction 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 coefficientsWavelet 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 imageSynthesized image, returned as a matrix.
Data Types: double
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