Kingsbury Q-shift 1-D dual-tree complex wavelet transform

`[`

returns the 1-D dual-tree complex wavelet transform (DTCWT) of `A`

,`D`

] = dualtree(`X`

)`X`

.
The output `A`

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

is
an *L*-by-1 cell array of complex-valued wavelet coefficients, where
*L* is the level of the transform.

The input `X`

must have at least two samples. The DTCWT is obtained
by default down to level
`floor(log`

, where
_{2}*N*)*N* is the length of `X`

if `X`

is
a vector and the row dimension of `X`

if `X`

is a
matrix. If *N* is odd, `X`

is extended by one sample by
reflecting the last element of `X`

.

By default, `dualtree`

uses the near-symmetric biorthogonal 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.

`[___] = dualtree(`

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

,`Name,Value`

)`'Level',10`

specifies a decomposition down to level 10.

[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.

`dualtree2`

| `dualtree3`

| `idualtree`

| `qbiorthfilt`

| `qorthwavf`