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Kingsbury Q-shift 2-D dual-tree complex wavelet transform

`[`

returns the 2-D dual-tree complex wavelet transform (DTCWT) of `A`

,`D`

] = dualtree2(`X`

)`X`

using
Kingsbury Q-shift filters. The output `A`

is the matrix of real-valued
final-level scaling (lowpass) coefficients. The output `D`

is a
*L*-by-1 cell array of complex-valued wavelet coefficients, where
*L* is the level of the transform. For each element of
`D`

there are six wavelet subbands.

The DTCWT is obtained by default down to level
`floor(log`

, where _{2}(min([*H*
*W*])))*H* and *W*
refer to the height (row dimension) and width (column dimension) of `X`

,
respectively. If any of the row or column dimensions of `X`

are odd,
`X`

is extended along that dimension by reflecting around the last row
or column.

By default, `dualtree2`

uses the near-symmetric biorthogonal wavelet
filter pair with lengths 5 (scaling filter) and 7 (wavelet filter) for level 1 and the
orthogonal Q-shift Hilbert wavelet filter pair of length 10 for levels greater than or equal
to 2.

`[___] = dualtree2(`

specifies additional options using name-value pair arguments. For example,
`X`

,`Name,Value`

)`'LevelOneFilter','antonini'`

specifies the (9,7)-tap Antonini filter as
the biorthogonal filter to use in the first-level analysis.

[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.

[2] 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.

[3] 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.

`dualtree`

| `dualtree3`

| `idualtree2`

| `qbiorthfilt`

| `qorthwavf`