Kingsbury Q-shift 2-D inverse dual-tree complex wavelet transform
returns the inverse 2-D complex dual-tree transform of the final-level approximation
imrec = idualtree2(
A, and cell array of wavelet coefficients,
D are outputs of
idualtree2 uses two sets of filters:
Orthogonal Q-shift filter of length 10
Near-symmetric biorthogonal filter pair with lengths 7 (scaling synthesis filter) and 5 (wavelet synthesis filter)
This example shows how to reconstruct an approximation based on a subset of the wavelet subbands.
Load a 128-by-128 grayscale image.
load xbox imagesc(xbox) colormap gray
Obtain the dual-tree wavelet transform of the image down to level 2
lev = 2; [a,d] = dualtree2(xbox,'Level',lev);
Since there are six wavelet subbands in each level of the decomposition, create a 2-by-6 matrix of zeros.
dgains = zeros(lev,6);
To reconstruct an approximation based on the 2nd and 5th wavelet subbands, set the second and fifth rows of
dgains equal to 1. The 2nd and 5th wavelet subbands correspond to the highpass filtering of the rows and columns of the image.
dgains(:,[2 5]) = 1;
Obtain two reconstructions using the specified wavelet subbands. Include the scaling (lowpass) coefficients only in the first reconstruction.
imrec = idualtree2(a,d,'DetailGain',dgains); imrec2 = idualtree2(a,d,'DetailGain',dgains,'LowpassGain',0); figure subplot(2,1,1) imagesc(imrec) title('With Lowpass Coefficients') subplot(2,1,2) imagesc(imrec2) title('Without Lowpass Coefficients') colormap gray
A— Final-level approximation coefficients
Final-level approximation coefficients, specified as a real-valued array. The
approximation coefficients are the output of
D— Wavelet coefficients
Approximation coefficients, specified as a cell array. The wavelet coefficients are
the output of
comma-separated pairs of
the argument name and
Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
LevelOneFilter— Biorthogonal filter
Biorthogonal filter to use in the first-level synthesis, specified by one of the
values listed here. For perfect reconstruction, the first-level synthesis filters must
match the first-level analysis filters used in
'legall' — LeGall 5/3 filter
'nearsym13_19' — (13,19)-tap near-orthogonal
'nearsym5_7' — (5,7)-tap near-orthogonal filter
'antonini' — (9,7)-tap Antonini filter
FilterLength— Orthogonal Hilbert Q-shift synthesis filter pair length
Orthogonal Hilbert Q-shift synthesis filter pair length to use for levels 2 and
higher, specified as one of the listed values. For perfect reconstruction, the filter
length must match the filter length used in
DetailGain— Wavelet coefficients subband gains
Wavelet coefficients subband gains, specified as a real-valued matrix with a row
dimension of L, where L is the number of
D. There are six columns in
DetailGain for each of the six wavelet subbands. The elements of
DetailGain are real numbers in the interval [0, 1]. The
kth column elements of
DetailGain are the gains (weightings) applied to the
kth wavelet subband. By default,
DetailGain is a L-by-6 matrix of ones.
1(default) | real number
Gain to apply to final-level approximation (lowpass, scaling) coefficients, specified as a real number in the interval [0, 1].
 Antonini, M., M. Barlaud, P. Mathieu, and I. Daubechies. “Image Coding Using Wavelet Transform.” IEEE Transactions on Image Processing 1, no. 2 (April 1992): 205–20. https://doi.org/10.1109/83.136597.
 Kingsbury, Nick. “Complex Wavelets for Shift Invariant Analysis and Filtering of Signals.” Applied and Computational Harmonic Analysis 10, no. 3 (May 2001): 234–53. https://doi.org/10.1006/acha.2000.0343.
 Le Gall, D., and A. Tabatabai. “Sub-Band Coding of Digital Images Using Symmetric Short Kernel Filters and Arithmetic Coding Techniques.” In ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing, 761–64. New York, NY, USA: IEEE, 1988. https://doi.org/10.1109/ICASSP.1988.196696.