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Single-level inverse discrete 2-D wavelet transform

performs a single-level two-dimensional wavelet reconstruction based on the
approximation matrix `x`

= idwt2(`cA`

,`cH`

,`cV`

,`cD`

,`wname`

)`cA`

and details matrices
`cH`

, `cV`

, and
`cD`

(horizontal, vertical, and diagonal, respectively)
using the wavelet specified by `wname`

. For additional
information, see `dwt2`

.

Let `sa = size(`

, and let `cA`

) =
size(`cH`

) = size(`cV`

) =
size(`cD`

)`lf`

equal the length of the reconstruction filters associated with
`wname`

. If the DWT extension mode is set to
periodization, the size of `x`

, `sx`

is
equal to `2*sa`

. For other extension modes, ```
sx =
2*sa-lf+2
```

. For additional information, see `dwtmode`

.

returns the size-`x`

= idwt2(___,`s`

)`s`

central portion of the reconstruction
using any of the previous syntaxes.

returns the single-level reconstructed approximation coefficients matrix
`x`

= idwt2(`cA`

,[],[],[],___)`x`

based on the approximation coefficients matrix
`cA`

.

returns the single-level reconstructed approximation coefficients matrix
`x`

= idwt2([],`cH`

,[],[],___)`x`

based on horizontal detail coefficients matrix
`cH`

.

returns the single-level reconstructed approximation coefficients matrix
`x`

= idwt2([],[],`cV`

,[],___)`x`

based on vertical detail coefficients matrix
`cV`

.

The 2-D wavelet reconstruction algorithm for images is similar to the one-dimensional
case. The two-dimensional wavelet and scaling functions are obtained by taking the
tensor products of the one-dimensional wavelet and scaling functions. This kind of
two-dimensional inverse DWT leads to a reconstruction of approximation coefficients at
level *j* from four components: the approximation at level
*j*+1, and the details in three orientations (horizontal, vertical,
and diagonal). The following chart describes the basic reconstruction steps for
images.

where

— Upsample columns: insert zeros at odd-indexed columns

— Upsample rows: insert zeros at odd-indexed rows

— Convolve with filter

*X*the rows of the entry— Convolve with filter

*X*the columns of the entry

[1] Daubechies, Ingrid.
*Ten Lectures on Wavelets*. CBMS-NSF Regional Conference Series
in Applied Mathematics 61. Philadelphia, Pa: Society for Industrial and Applied
Mathematics, 1992.

[2] Mallat, S.G. “A Theory for
Multiresolution Signal Decomposition: The Wavelet Representation.” *IEEE
Transactions on Pattern Analysis and Machine Intelligence* 11, no. 7
(July 1989): 674–93. https://doi.org/10.1109/34.192463.

[3] Meyer, Y. *Wavelets
and Operators*. Translated by D. H. Salinger. Cambridge, UK: Cambridge
University Press, 1995.