dwt2

Single-level 2-D discrete wavelet transform

collapse all in page

Syntax

[cA,cH,cV,cD] = dwt2(X,wname)
[cA,cH,cV,cD] = dwt2(X,LoD,HiD)
[cA,cH,cV,cD] = dwt2(___,"mode",extmode)

Description

dwt2 computes the single-level 2-D wavelet decomposition. Compare dwt2 with wavedec2 which may be more useful for your application. The decomposition is done with respect to either a particular wavelet (see wfilters for more information) or particular wavelet decomposition filters.

[cA,cH,cV,cD] = dwt2(X,wname) computes the single-level 2-D discrete wavelet transform (DWT) of the input data X using the wname wavelet. dwt2 returns the approximation coefficients matrix cA and detail coefficients matrices cH, cV, and cD (horizontal, vertical, and diagonal, respectively).

example

[cA,cH,cV,cD] = dwt2(X,LoD,HiD) computes the single-level 2-D DWT using the wavelet decomposition lowpass filter LoD and highpass filter HiD. The decomposition filters must have the same length and an even number of samples.

example

[cA,cH,cV,cD] = dwt2(___,"mode",extmode) computes the single-level 2-D DWT with the extension mode extmode. Include this argument after all other arguments.

Note

For gpuArray inputs, the supported modes are "symh" ("sym") and "per". All "mode" options except "per" are converted to "symh". See the example Single-Level 2-D Discrete Wavelet Transform on a GPU.

Examples

collapse all

Open Live Script

Load and display an image.

load tartan
imagesc(X)
colormap(map)

Figure contains an axes object. The axes object contains an object of type image.

Obtain the single-level 2-D discrete wavelet transform of the image using the order 4 symlet and periodic extension.

[cA,cH,cV,cD] = dwt2(X,"sym4","mode","per");

Display the diagonal detail coefficients and the approximation coefficients.

imagesc(cD)
title("Diagonal Detail Coefficients")

Figure contains an axes object. The axes object with title Diagonal Detail Coefficients contains an object of type image.

imagesc(cA)
title("Approximation Coefficients")

Figure contains an axes object. The axes object with title Approximation Coefficients contains an object of type image.

Open Live Script

Load and display an image.

load xbox
imagesc(xbox)
colormap gray

Figure contains an axes object. The axes object contains an object of type image.

Generate the lowpass and highpass decomposition filters for the Haar wavelet.

[LoD,HiD] = wfilters("haar","d");

Use the filters to perform a single-level 2-D wavelet decomposition. Use half-point symmetric extension. Display the approximation and detail coefficients.

[cA,cH,cV,cD] = dwt2(xbox,LoD,HiD,"mode","symh");
tiledlayout(2,2)
nexttile
imagesc(cA)
colormap gray
title("Approximation")
nexttile
imagesc(cH)
colormap gray
title("Horizontal")
nexttile
imagesc(cV)
colormap gray
title("Vertical")
nexttile
imagesc(cD)
colormap gray
title("Diagonal")

Figure contains 4 axes objects. Axes object 1 with title Approximation contains an object of type image. Axes object 2 with title Horizontal contains an object of type image. Axes object 3 with title Vertical contains an object of type image. Axes object 4 with title Diagonal contains an object of type image.

This example uses:

Open Live Script

Refer to GPU Computing Requirements (Parallel Computing Toolbox) to see what GPUs are supported.

Load an image. Put the image on the GPU using gpuArray. Save the current extension mode.

load mask
imgg = gpuArray(X);
origMode = dwtmode("status","nodisp");

Use dwtmode to change the extension mode to zero-padding. Obtain the single-level 2-D DWT of the image on the GPU using the db2 wavelet.

dwtmode("zpd","nodisp")
[cA,cH,cV,cD] = dwt2(imgg,"db2");

The current extension mode zpd is not supported for gpuArray input. Therefore, the DWT is instead performed using the "sym" extension mode. Confirm this by taking the DWT of imgg with the extension mode set to "sym" and compare with the previous result.

[cAsym,cHsym,cVsym,cDsym] = dwt2(imgg,"db2","mode","sym");
[max(abs(cA(:)-cAsym(:))) max(abs(cH(:)-cHsym(:))) ...
    max(abs(cV(:)-cVsym(:))) max(abs(cD(:)-cDsym(:)))]
ans =

     0     0     0     0

An unsupported extension mode specified as an input argument is converted to "sym". Confirm that taking the DWT of imgg with "mode" set to an unsupported mode also defaults to the "sym" extension mode.

[cA,cH,cV,cD] = dwt2(imgg,"db2","mode","spd");
[max(abs(cA(:)-cAsym(:))) max(abs(cH(:)-cHsym(:))) ...
    max(abs(cV(:)-cVsym(:))) max(abs(cD(:)-cDsym(:)))]
ans =

     0     0     0     0

Change the current extension mode to periodic. Obtain the single-level DWT of the image on the GPU using the db2 wavelet.

dwtmode("per","nodisp")
[cA,cH,cV,cD] = dwt2(imgg,"db2");

Confirm the current extension mode "per" is supported for gpuArray input.

[cAper,cHper,cVper,cDper] = dwt2(imgg,"db2","mode","per");
[max(abs(cA(:)-cAper(:))) max(abs(cH(:)-cHper(:))) ...
    max(abs(cV(:)-cVper(:))) max(abs(cD(:)-cDper(:)))]
ans =

     0     0     0     0

Restore the extension mode to the original setting.

dwtmode(origMode,"nodisp")

Input Arguments

collapse all

Input data, specified as a numeric or logical array. X can be an m-by-n array representing an indexed image or an m-by-n-by-3 array representing a truecolor image. For more information on truecolor images, see Working with Image Types in MATLAB.

Data Types: double | single | uint8

Analyzing wavelet used to compute the 2-D DWT, specified as a character vector or string scalar. The analyzing wavelet is from one of the following wavelet families: Best-localized Daubechies, Beylkin, Coiflets, Daubechies, Fejér-Korovkin, Haar, Han linear-phase moments, Morris minimum-bandwidth, Symlets, Vaidyanathan, Discrete Meyer, Biorthogonal, and Reverse Biorthogonal. See wfilters for the wavelets available in each family.

Data Types: char | string

Wavelet decomposition lowpass filter, specified as an even-length real-valued vector. LoD must be of the same length as HiD.

Data Types: double | single

Wavelet decomposition highpass filter, specified as an even-length real-valued vector. HiD must be of the same length as LoD.

Data Types: double | single

Extension mode used when performing the wavelet decomposition, specified as:

extmode

DWT Extension Mode

"zpd"

Zero padding

"sp0"

Smooth extension of order 0

"spd" (or "sp1")

Smooth extension of order 1

"sym" or "symh"

Symmetric extension (half point): boundary value symmetric replication

"symw"

Symmetric extension (whole point): boundary value symmetric replication

"asym" or "asymh"

Antisymmetric extension (half point): boundary value antisymmetric replication

"asymw"

Antisymmetric extension (whole point): boundary value antisymmetric replication

"ppd", "per"

Periodized extension

If the signal length is odd and mode is "per", an extra sample equal to the last value is added to the right and the extension is performed in "ppd" mode. If the signal length is even, "per" reduces to "ppd". This rule also applies to images.

The global variable managed by dwtmode specifies the default extension mode.

Output Arguments

collapse all

Approximation coefficients, returned as an array whose size depends on X. Let sx = size(X) and lf = the length of the decomposition filters.

  • If the DWT extension mode is set to periodization, then this output is of size ceil(sx/2).

  • For the other extension modes, this output is of size floor((sx+lf-1)/2).

Data Types: double

Horizontal detail coefficients, returned as an array whose size depends on X. Let sx = size(X) and lf = the length of the decomposition filters.

  • If the DWT extension mode is set to periodization, then this output is of size ceil(sx/2).

  • For the other extension modes, this output is of size floor((sx+lf-1)/2).

Data Types: double

Vertical detail coefficients, returned as an array whose size depends on X. Let sx = size(X) and lf = the length of the decomposition filters.

  • If the DWT extension mode is set to periodization, then this output is of size ceil(sx/2).

  • For the other extension modes, this output is of size floor((sx+lf-1)/2).

Data Types: double

Diagonal detail coefficients, returned as an array whose size depends on X. Let sx = size(X) and lf = the length of the decomposition filters.

  • If the DWT extension mode is set to periodization, then this output is of size ceil(sx/2).

  • For the other extension modes, this output is of size floor((sx+lf-1)/2).

Data Types: double

Algorithms

The 2-D wavelet decomposition 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 DWT leads to a decomposition of approximation coefficients at level j in four components: the approximation at level j + 1, and the details in three orientations (horizontal, vertical, and diagonal). The following chart describes the basic decomposition steps for images.

where

  • — Downsample columns: keep the even-indexed columns

  • — Downsample rows: keep the even-indexed rows

  • — Convolve with filter X the rows of the entry

  • — Convolve with filter X the columns of the entry

The decomposition is initialized by setting the approximation coefficients equal to the image s: cA0 = s.

Note

To deal with signal-end effects introduced by a convolution-based algorithm, the 1-D and 2-D DWT use a global variable managed by dwtmode. This variable defines the kind of signal extension mode used. The possible options include zero-padding and symmetric extension, which is the default mode.

References

[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.

Extended Capabilities

expand all

Version History

Introduced before R2006a

See Also

| | | | | | |