lwtcoef2

Extract 2-D LWT wavelet coefficients and orthogonal projections

Since R2021b

Syntax

y = lwtcoef2(ll,lh,hl,hh)

y = lwtcoef2(ll,lh,hl,hh,Name=Value)

Description

y = lwtcoef2(ll,lh,hl,hh) returns the level 1 reconstructed approximation coefficients that correspond to the approximation coefficients ll and the horizontal (lh), vertical (hl), and diagonal (hh) wavelet coefficients. The coefficients in ll, lh, hl, and hh are the outputs of lwt2 using default values.

y = lwtcoef2(ll,lh,hl,hh,Name=Value) specifies options using one or more name-value arguments. For example, lwtcoef2(ll,lh,hl,hh,Type="detail",OutputType="projection") returns the projection of the detail coefficients at the finest scale using the db1 wavelet.

example

Examples

collapse all

Reconstruct Image from Orthogonal Projections

Open Live Script

Load and plot a grayscale image.

load gatlin
figure
image(X)
colormap(map)
title("1964 Gatlinburg Conference on Numerical Algebra",...
    "Wilkinson, Givens, Forsythe, Householder, Henrici, and Bauer")

Figure contains an axes object. The axes object with title 1964 Gatlinburg Conference on Numerical Algebra contains an object of type image.

Create a lifting scheme associated with the bior3.7 wavelet. Use the lifting scheme to obtain the wavelet decomposition of the image to the maximum level.

lscheme = liftingScheme(Wavelet="bior3.7");
[ll,lh,hl,hh] = lwt2(X,LiftingScheme=lscheme);

Extract and display the approximation coefficients at level 2. Confirm the row and column dimensions are one-quarter the size of those of the original image.

approxCF = lwtcoef2(ll,lh,hl,hh,...
    LiftingScheme=lscheme,OutputType="coefficients",Level=2);
figure
image(approxCF)
colormap(map)

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

size(X)./size(approxCF)

ans = 1×2

     4     4

Obtain the orthogonal projection of the level 1 approximation coefficients. Also obtain the orthogonal projections of the detail coefficients at level 1. Display the projections corresponding to the LH and HL detail coefficients. Observe that the prominent features in the LH- and HL-derived images correspond to the horizontal, and vertical features, respectively, of the original image.

approx = lwtcoef2(ll,lh,hl,hh,...
    LiftingScheme=lscheme,OutputType="projection",Level=1);
dLH = lwtcoef2(ll,lh,hl,hh,...
    LiftingScheme=lscheme,OutputType="projection",Level=1,Type="LH");
dHL = lwtcoef2(ll,lh,hl,hh,...
    LiftingScheme=lscheme,OutputType="projection",Level=1,Type="HL");
dHH = lwtcoef2(ll,lh,hl,hh,...
    LiftingScheme=lscheme,OutputType="projection",Level=1,Type="HH");
subplot(1,2,1)
imagesc(dLH)
title("LH - Horizontal")
subplot(1,2,2)
imagesc(dHL)
title("HL - Vertical")

Figure contains 2 axes objects. Axes object 1 with title LH - Horizontal contains an object of type image. Axes object 2 with title HL - Vertical contains an object of type image.

Confirm the sum of the four projections equals the original image.

max(max(abs(X-(approx+dLH+dHL+dHH))))

ans = 
2.3448e-13

Input Arguments

collapse all

`ll` — Approximation coefficients
scalar | vector | matrix

Approximation coefficients at the coarsest scale, specified as a scalar, vector, or matrix. The coefficients are the output of lwt2.

Data Types: single | double
Complex Number Support: Yes

`lh` — Horizontal detail coefficients
cell array

Horizontal detail coefficients by level, specified as a LEV-by-1 cell array, where LEV is the level of the decomposition. The elements of lh are in order of decreasing resolution. The coefficients are the output of lwt2.

Data Types: single | double
Complex Number Support: Yes

`hl` — Vertical detail coefficients
cell array

Vertical detail coefficients by level, specified as a LEV-by-1 cell array, where LEV is the level of the decomposition. The elements of hl are in order of decreasing resolution. The coefficients are the output of lwt2.

Data Types: single | double
Complex Number Support: Yes

`hh` — Diagonal detail coefficients
cell array

Diagonal detail coefficients by level, specified as a LEV-by-1 cell array, where LEV is the level of the decomposition. The elements of hh are in order of decreasing resolution. The coefficients are the output of lwt2.

Data Types: single | double
Complex Number Support: Yes

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: y = lwtcoef2(ll,lh,hl,hh,OutputType="projection",LiftingScheme=lscheme)

`Wavelet` — Wavelet
`"db1"` (default) | character vector | string scalar

Orthogonal or biorthogonal wavelet, specified as a character vector or string scalar. See the Wavelet property of liftingScheme for the list of supported wavelets. The specified wavelet must match the value that you used to obtain the coefficients ll, lh, hl, and hh.

You cannot specify Wavelet and LiftingScheme at the same time.

Example: y = lwtcoef2(ll,lh,hl,hh,Wavelet="bior3.5") uses the bior3.5 biorthogonal wavelet.

Data Types: char | string

`LiftingScheme` — Lifting scheme
`liftingScheme` object

Lifting scheme, specified as a liftingScheme object. The specified lifting scheme must be the same lifting scheme that you used to obtain the coefficients ll, lh, hl, and hh.

You cannot specify LiftingScheme and Wavelet at the same time.

Example: y = lwtcoef2(ll,lh,hl,hh,LiftingScheme=lScheme) uses the lScheme lifting scheme.

`OutputType` — Output type
`"coefficients"` (default) | `"projection"`

Output type, specified as one of these:

"coefficients" — Extract the approximation or details coefficients
"projection" — Return the projection (reconstruction) of the approximation or details coefficients

Example: y = lwtcoef2(ll,lh,hl,hh,OutputType="projection",Type="detail") returns the projection corresponding to the detail coefficients at the finest scale.

`Type` — Type of coefficients
`"ll"` (default) | `"lh"` | `"hl"` | `"hh"`

Type of coefficients to extract or reconstruct, specified as one of these:

"ll" — Approximation coefficients
"lh" — Horizontal coefficients
"hl" — Vertical coefficients
"hh" — Diagonal coefficients

Example: y = lwtcoef2(ll,lh,hl,hh,Type="hh") extracts the diagonal coefficients at the finest scale.

`Level` — Level
`1` (default) | positive integer

Level of coefficients to extract or reconstruct, specified as a positive integer less than or equal to length(hh).

Example: y = lwtcoef2(ll,lh,hl,hh,LiftingScheme=lsc,Level=3) uses the lifting scheme lsc to extract the approximation coefficients at level 3.

Data Types: double

`Extension` — Extension mode
`"periodic"` (default) | `"zeropad"` | `"symmetric"`

Extension mode to use to extract or reconstruct the coefficients, specified as one of these:

"periodic" — Periodized extension
"zeropad" — Zero extension
"symmetric" — Symmetric extension

This argument specifies how to extend the signal at the boundaries. The extension mode must match the value you used to generate ll, lh, hl, and hh.

Example: y = lwtcoef2(ll,lh,hl,hh,Extension="zeropad") specifies zero extension.

`Int2Int` — Handling integer-valued data
`false` or `0` (default) | `true` or `1`

Handling integer-valued data, specified as one of these:

1 (true) — Preserve integer-valued data
0 (false) — Do not preserve integer-valued data

Int2Int must match the value you used to generate ll, lh, hl, and hh.

Example: y = lwtcoef2(ll,lh,hl,hh,Int2Int=true) preserves integer-valued data.

Output Arguments

collapse all

`y` — Extracted coefficients or projection
matrix

Extracted coefficients or projection, returns as a matrix. y has the same dimensionality as the input used by the lwt2 function to generate the approximation and details coefficients.

Data Types: single | double

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2021b

expand all

R2021b: `lwtcoef2` input syntax has changed

The lwtcoef2 input syntax has changed. Use name-value arguments instead.

Functionality Result Use Instead Compatibility Considerations

Functionality	Result	Use Instead	Compatibility Considerations
`Y = lwtcoef2(TYPE,XDEC,LS,LEVEL,LEVEXT)`	Errors	`Y = lwtcoef2(CA,CH,CV,CD,Name=Value)` with the lifting decomposition `CA`, `CH`, `CV`, and `CD` in place of `XDEC`, and the following name-value arguments: Replace `LS` with `LiftingScheme`, where `LiftingScheme` is a `liftingScheme` object. Replace `LEVEXT` with `Level`. Replace `TYPE` with the `Type` and `OutputType` name-value arguments. `LEVEL` is no longer needed.	According to the value of `TYPE`, set the `Type` and `OutputType` name-value arguments as listed: `'a'` — `Type="approximation"` and `OutputType="projection"` `'ca'` — `Type="approximation"` and `OutputType="coefficients"` `'d'` — `Type="detail"` and `OutputType="projection"` `'cd'` — `Type="detail"` and `OutputType="coefficients"`

Y =
                                    lwtcoef2(TYPE,XDEC,LS,LEVEL,LEVEXT)

Errors

Y = lwtcoef2(CA,CH,CV,CD,Name=Value) with the lifting decomposition CA, CH, CV, and CD in place of XDEC, and the following name-value arguments:

Replace LS with LiftingScheme, where LiftingScheme is a liftingScheme object.
Replace LEVEXT with Level.
Replace TYPE with the Type and OutputType name-value arguments.
LEVEL is no longer needed.

According to the value of TYPE, set the Type and OutputType name-value arguments as listed:

'a' — Type="approximation" and OutputType="projection"
'ca' — Type="approximation" and OutputType="coefficients"
'd' — Type="detail" and OutputType="projection"
'cd' — Type="detail" and OutputType="coefficients"

lwtcoef2

Syntax

Description

Examples

Reconstruct Image from Orthogonal Projections

Input Arguments

ll — Approximation coefficients scalar | vector | matrix

lh — Horizontal detail coefficients cell array

hl — Vertical detail coefficients cell array

hh — Diagonal detail coefficients cell array

Name-Value Arguments

Wavelet — Wavelet "db1" (default) | character vector | string scalar

LiftingScheme — Lifting scheme liftingScheme object

OutputType — Output type "coefficients" (default) | "projection"

Type — Type of coefficients "ll" (default) | "lh" | "hl" | "hh"

Level — Level 1 (default) | positive integer

Extension — Extension mode "periodic" (default) | "zeropad" | "symmetric"

Int2Int — Handling integer-valued data false or 0 (default) | true or 1

Output Arguments

y — Extracted coefficients or projection matrix

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

Version History

R2021b: lwtcoef2 input syntax has changed

See Also

`ll` — Approximation coefficients
scalar | vector | matrix

`lh` — Horizontal detail coefficients
cell array

`hl` — Vertical detail coefficients
cell array

`hh` — Diagonal detail coefficients
cell array

`Wavelet` — Wavelet
`"db1"` (default) | character vector | string scalar

`LiftingScheme` — Lifting scheme
`liftingScheme` object

`OutputType` — Output type
`"coefficients"` (default) | `"projection"`

`Type` — Type of coefficients
`"ll"` (default) | `"lh"` | `"hl"` | `"hh"`

`Level` — Level
`1` (default) | positive integer

`Extension` — Extension mode
`"periodic"` (default) | `"zeropad"` | `"symmetric"`

`Int2Int` — Handling integer-valued data
`false` or `0` (default) | `true` or `1`

`y` — Extracted coefficients or projection
matrix

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

R2021b: `lwtcoef2` input syntax has changed