Documentation

## 2-D Continuous Wavelet Transform App

The 2-D continuous wavelet transform (CWT) app enables you to analyze your image data and export the results of that analysis to the MATLAB® workspace. The app provides all the functionality of the command line functions cwtft2 and cwtftinfo2. Access the 2-D CWT app in the apps gallery by selecting Wavelet Design & Analysis in the Signal Processing and Communications section or entering

cwtfttool2

at the MATLAB command prompt.

### 2-D Continuous Wavelet Transform

The 2-D continuous wavelet transform is a representation of 2-D data (image data) in 4 variables: dilation, rotation, and position. Dilation and rotation are real-valued scalars and position is a 2-D vector with real-valued elements. Let x denote a two-element vector of real-numbers. If

$f\left(x\right)\in {L}^{2}\left({ℝ}^{2}\right)$

is square-integrable on the plane, the 2-D CWT is defined as

${\text{WT}}_{f}\left(a,b,\theta \right)={\int }_{{ℝ}^{2}}f\left(x\right)\frac{1}{a}\overline{\psi }\left({r}_{-\theta }\left(\frac{x-b}{a}\right)\right)\text{\hspace{0.17em}}dx\text{ }a\in {ℝ}^{+},\text{ }\text{\hspace{0.17em}}x,b\in {ℝ}^{2}$

where the bar denotes the complex conjugate and rθ is the 2-D rotation matrix

${r}_{\theta }=\left(\begin{array}{cc}\mathrm{cos}\left(\theta \right)& -\mathrm{sin}\left(\theta \right)\\ \mathrm{sin}\left(\theta \right)& \mathrm{cos}\left(\theta \right)\end{array}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\theta \in \left[0,2\pi \right)$

The 2-D CWT is a space-scale representation of an image. You can view the inverse of the scale and the rotation angle taken together as a spatial-frequency variable, which gives the 2-D CWT an interpretation as a space-frequency representation. For all admissible 2-D wavelets, the 2-D CWT acts as a local filter for an image in scale and position. If the wavelet is isotropic, there is no dependence on angle in the analysis. The Mexican hat wavelet is an example of an isotropic wavelet. Isotropic wavelets are suitable for pointwise analysis of images. If the wavelet is anisotropic, there is a dependence on angle in the analysis, and the 2-D CWT acts a local filter for an image in scale, position, and angle. The Cauchy wavelet is an example of an anisotropic wavelet. In the Fourier domain, this means that the spatial frequency support of the wavelet is a convex cone with the apex at the origin. Anisotropic wavelets are suitable for detecting directional features in an image. See Two-Dimensional CWT of Noisy Pattern for an illustration of the difference between isotropic and anisotropic wavelets.

### 2-D CWT App Example

This example shows how to analyze an image using the 2-D CWT app.

Load the triangle image in the MATLAB workspace.

Launch the 2-D CWT app by selecting Wavelet Design & Analysis in the Signal Processing and Communications section of the apps gallery. From the 2-D section, select Continuous Wavelet Transform 2-D. Alternatively, enter

cwtfttool2

at the MATLAB command prompt.

Select File –> Import Data to import the imdata variable.

From the Wavelet drop down menu, select the cauchy wavelet.

For the Angles and Scales, select the Manual option.

Click Define to specify a vector of angles. Select Manual from the Type drop-down list and specify a vector of angles from 0 to 7*pi/8 radians in increments of pi/8 radians, 0:pi/8:(7*pi)/8. Click Apply to apply your choice of angles.

Click Define to specify a vector of scales from 0.5 to 4 in increments of 0.5. Select Linear from the Type drop-down list. Set First Scale equal to 0.5, Gap between two scales equal to 0.5, and Number of Scales equal to 8. Equivalently, you can select Manual from the Type drop-down list and specify the vector of scales as 0.5:0.5:4. Click Apply to apply your choice of scales.

Click Analyze to obtain the 2-D CWT.

Set the Index of Scale to be 1 and click More on Angles. Click Movie to step through the manually-defined angles for the 2-D CWT coefficients at scale 0.5.

Select File –> Export Data –> Export CWTFT Struct to Workspace to export the analysis to the MATLAB workspace. You can find an explanation of the structure fields in the function reference for cwtft2.

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