# cwtft2

2-D continuous wavelet transform

## Description

## Examples

### 2-D CWT with Morlet Wavelet

Load and display the star image.

```
img = imread("star.jpg");
image(img)
```

Obtain the 2-D CWT of the star image using the default function values. Visualize the magnitudes of the coefficients at the finest scale.

cwtout = cwtft2(img); sca = 1; imagesc(abs(cwtout.cfs(:,:,1,1,sca)))

### Compare Isotropic and Anisotropic Wavelets

This example shows how an isotropic wavelet does not discern the orientation of features while an anisotropic wavelet does. In the example, you use the Marr isotropic wavelet and the directional (anisotropic) Cauchy wavelet.

Load and view the hexagon image.

```
img = imread("hexagon.jpg");
imagesc(img)
```

Obtain the 2-D CWT of the image using both the Marr and Cauchy wavelets. Specify a scale equal to 1. Specify a vector of angles going from 0 to $$15\pi /8$$ radians in $$\pi /8$$ increments.

cwtScales = 1; cwtAngles = 0:pi/8:2*pi-pi/8; cwtCauchy = cwtft2(img,wavelet="cauchy",scales=cwtScales, ... angles=cwtAngles); cwtMarr = cwtft2(img,wavelet="marr",scales=cwtScales, ... angles=cwtAngles);

There are 16 angles. Visualize the scale-one 2-D CWT coefficient magnitudes at any two consecutive angles. Confirm that using the Marr isotropic wavelet does not discern the orientation of features, but the Cauchy wavelet does.

angz = {"0", "pi/8", "pi/4", "3pi/8", "pi/2", "5pi/8", "3pi/4", ... "7pi/8","pi", "9pi/8", "5pi/4", "11pi/8", "3pi/2", ... "13pi/8" "7pi/4", "15pi/8"}; indexAngle1 = 7; indexAngle2 = 8; tiledlayout(2,2) for k=[indexAngle1 indexAngle2] nexttile imagesc(abs(cwtMarr.cfs(:,:,1,1,k))) title(["Marr Wavelet at " angz(k) "radians"]) nexttile imagesc(abs(cwtCauchy.cfs(:,:,1,1,k))) title(["Cauchy Wavelet at " angz(k) "radians"]) end

Visualize the 2-D CWT coefficient magnitudes obtained using the Marr isotropic wavelet at any two angles. Confirm the wavelet does not discern the orientation of features.

indexAngle1 = 2; indexAngle2 = 7; tiledlayout(1,2) for k=[indexAngle1 indexAngle2] nexttile imagesc(abs(cwtMarr.cfs(:,:,1,1,k))) title(["Marr Wavelet at " angz(k) "radians"]) end

## Input Arguments

`X`

— Input data

array

Input data, specified as a numeric 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.

**Data Types: **`double`

| `single`

| `uint8`

| `uint16`

### 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: **`wavelet="paul",scales=2.^(0:5)`

specifies the Paul
wavelet and a vector of scales.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`"wavelet","paul","scales",2.^(0:5)`

specifies the Paul wavelet and a
vector of scales.

`angles`

— Angles

`0`

(default) | scalar | vector

Angles in radians used in the 2-D CWT, specified as a scalar or a vector.

**Example: **`angles=[0 pi/2 pi]`

`norm`

— Normalization

`"L2"`

(default) | `"L1"`

| `"L0"`

Normalization to use in the 2-D CWT, specified as one of these:

`"L2"`

— The Fourier transform of the analyzing wavelet at a given scale is multiplied by the corresponding scale.`"L2"`

is the default normalization.`"L1"`

— The Fourier transform of the analyzing wavelet is multiplied by 1 at all scales.`"L0"`

— The Fourier transform of the analyzing wavelet at a given scale is multiplied by the square of the corresponding scale.

**Example: **`norm="L1"`

`scales`

— Scales

`2.^(0:5)`

(default) | real-valued scalar | real-valued vector

Scales to use in the 2-D CWT, specified as a real-valued scalar or vector. Scales must be greater than or equal to 1.

**Example: **`scales=[1 2 3 5 8]`

**Data Types: **`double`

| `single`

`wavelet`

— Analyzing wavelet

`"morlet"`

(default) | character vector | string scalar | structure | cell array

Analyzing wavelet, specified as a character vector, a string scalar, a structure, or a cell
array. `cwtftinfo2`

provides a
comprehensive list of supported wavelets and associated
parameters.

If you specify `wavelet`

as a structure, the structure must contain two fields:

`name`

— Character vector or string scalar corresponding to a supported wavelet.`param`

— Cell array containing optional parameters, which depend on the wavelet. If you do not wish to specify optional parameters, use an empty cell array in the field.

If you specify `wavelet`

as a cell array, `wav`

, the cell
array must contain two elements:

`wav{1}`

— Character vector or string scalar corresponding to a supported wavelet.`wav{2}`

— Cell array with the parameters of the wavelet.

**Example: **`wavelet={"morlet",{6,1,1}}`

specifies the Morlet wavelet as a cell
array.

**Example: **`wavelet=struct("name","paul","param",{{2}})`

specifies the Paul
wavelet as a structure array.

## Output Arguments

`cwtstruct`

— 2-D CWT

structure

The 2-D CWT, returned as a structure with the following fields:

`wav`

— Analyzing wavelet and parameters

structure

Analyzing wavelet and parameters, returned as a structure with the following fields:

`wname`

— Wavelet name`param`

— Wavelet parameters

`wav_norm`

— Normalization constants

matrix

Normalization constants, returned as an *M*-by-*N* matrix,
where *M* is the number of scales and
*N* is the number of angles.

`cfs`

— CWT coefficients

array

CWT coefficients, returned as an N-D array.

The row and column dimensions of the array equal the row and column dimensions of the input data.

The third page of the array is equal to 1 or 3, depending on whether the input data is a grayscale or truecolor image, respectively.

The fourth page of the array is equal to the number of scales.

The fifth page of the array is equal to the number of angles.

`scales`

— Scales

vector

Scales for the 2-D CWT, returned as a row vector.

`angles`

— Angles

vector

Angles for the 2-D CWT, returned as a row vector.

`meanSIG`

— Mean

scalar

Mean of the input data, returned as a scalar.

## Version History

**Introduced in R2013b**

### R2023b: Support for data type `uint16`

You can use the `cwtft2`

function to obtain the 2-D CWT of a
16-bit image.

### R2023b: Plotting is not recommended

The plotting syntax for the `cwtft2`

function continues to
work, but is no longer recommended. Use the Wavelet Image
Analyzer app instead.

## See Also

### Apps

### Functions

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