Wavelet Transform Analysis of Images Using WaveletAnalyzer Toolbox in Matlab

The wavelet transform is the decomposition of signals using wavelet techniques, that is, functions. Each of these functions are shifted and a scaled copy of a function, the mother wavelet. <!--more--> Wavelet transform is the definition of the decomposition method class. Wavelet Analyzer app is an interactive tool for using wavelets to visualize and analyze signals and images. This app can be used for the wavelet transform analysis, de-noising images and signals, e.t.c.

This article covers wavelet analysis and synthesis of images using filter banks. It also discusses how to perform wavelet transform analysis of images using the Wavelet Analyzer toolbox. It will also cover how to use the various wavelet functions such as dwt2 and idwt2.

Prerequisites #

To follow along with this tutorial, you will need:

• MATLAB installed.
• Proper understanding of MATLAB basics.

Wavelet transform of images #

Images are 2-D data i.e. f(x,y). To transform them, we need 2-D wavelets, i.e. $\varphi$(x,y). Below are the two examples of 2-D wavelets.

This approach needs a lot of computational power. Due to that, there is another approach that needs less computational power.

The other approach #

We can successfully supply the 1-D transform to the rows and columns of images as separable 2-D transform. Instead of taking 2-D wavelets, we use 1-D wavelets and perform row-wise and column-wise operations to get the final 2-D transform.

In most applications where wavelets are used for image processing, this approach is preferred because of the low computational complexity of separable transform.

Filter bank theory #

The wavelet decomposition of an image based on the multi-resolution theory can be done using digital FIR filters. Here, wavelet coefficients are calculated with a fast O(N) algorithm using a multi-rate filter bank proposed by S.Mallet.

The scheme of the filter-based approach and column-wise operation is shown below:

In the first step, we have an image of the dimension MxM. A row-wise operation is performed on this image. It is done in the low-pass wavelet decomposition filter(Lo_D) and the high pass wavelet decomposition filter(Hi-D).

Then the down-sampling of two is done for the output of both filters.

The next step is performing a column-wise operation to the down-sampled output. The down-sampling of two is also done for the output of the column-wise operation. The output of the down-sampling gives the wavelet coefficients.

These coefficients are approximated coefficient, horizontal detailed, vertical detailed and detailed diagonal coefficients.

After obtaining these coefficients, there is a reconstruction scheme to achieve the original image. The scheme is shown below:

The simplified operation at the first level of decomposition is shown below:

At the first decomposition level, all the coefficients are of the dimension M/2xM/2. At the second level decomposition, only the approximated coefficients($cA_1$) is decomposed into the four matrices as shown below:

The matrices at this point are of the size M/4xM/4. You can do a 3rd level decomposition and so on. Below is an image sampled at different decomposition levels.

How to use Matlab wavelet analyzer toolbox #

You execute the waveletAnalyzer command in the command window to open this toolbox. It opens a new window shown below:

In this toolbox, as we can see, we can analyze various signals. These signals include; 1-D, 2-D, 3-D, e.t.c. Since we want to analyze images (2-D), we select the wavelet 2-D.

Note that using this toolbox, we can perform the various wavelet operations such as wavelet transform, wavelet packet transform, curvelet wavelet transform etc.

After selecting wavelet 2-D, a new window opens up again, like the one below:

To load your data into the toolbox, click on the file at the top of the window. Next, click on the Load tab then select Image as shown below:

The command load-image allows you to select your input image from any folder. When you import your image, it should appear in the original image section as seen here:

You can select the wavelet type you want to use and the level of decomposition by clicking on the dropdown arrow. After defining these, click on the analyzer to get the output. In our case, we have used the haar wavelet and the decomposition level as three. The output is as as shown below:

Now let us see how to use the dwt2() and idwt2() function in Matlab to perform a similar operation.

Using dwt2() and idwt() functions #

The syntax for these two functions is as follows:

[cA, cH, cV, cD]=dwt2(x, 'wname', 'mode', extmode)   %Single level decomposition
Y=idwt(cA, cH, cV, cD, 'wname', 'mode', extmode)    %single level reconstruction

Where:

• cA: Approximated coefficients.
• cH: Horizontal detailed coefficients.
• cV: Vertical detailed coefficients.
• cD: Detailed Diagonal coefficients.
• X: Input image.
• Y: Reconstructed image.
• WName: This is the type of wavelet you are using.
• Mode: Signal extension mode. This extension mode can be periodic, symmetric e.t.c.

Dwt2() function is used for the single-level decomposition, while the idwt2() is used for single-level reconstruction. The dwt2() gives all the coefficients as the output.

The coefficients are used for the image reconstruction. It means they are arguments of the idwt2() function.

Program to show the use of dwt2() and idwt2() function #

The first step is to import the input image into Matlab. This will be achieved using the following lines:

[filename, pathname] = uigetfile('*.*', 'Select your input grayscale image');
filewithpath = strcat(pathname, filename);
img = imread(filewithpath);

The uigetfile is a function that allows the user to select the input image from any folder. This folder does not have to be in the current directory. This input image is then read using the imread() function.

Since the input image is not in the current directory, the imread() function uses the filename and the path as the arguments to get the image.

Our image img is grayscale. It means it is made up of the uint8 data type. We convert this datatype from uint8 to double using the double() function. It is converted as shown below:

img1 = double(img);

We are converting the image to double type because the functions dwt2() function only accepts double datatype as the arguments. Let's perform the decomposition using the dwt2() function:

[cA, cH, cV, cD] = dwt2(img1, 'haar', 'mode', 'sym'); %first level decomposition

We use the haar wavelet, and the extmode is symmetric sym. For visualization we plot these coefficients in subplots as shown below:

subplot(2,2,1)
imshow(uint8(cA));
title('Approximated coefficients');

subplot(2,2,2)
imshow(uint8(cH), []);
title('Horizontal detailed coefficients');

subplot(2,2,3)
imshow(uint8(cV), []);
title('Vertical detailed coefficients');

subplot(2,2,4)
imshow(uint8(cD), []);
title('Diagonal detailed coefficients');

These coefficients are of the double type. We have to convert them to the original type uint8 and show the output using the imshow() function. When we run the program at this point, we will have the plot of the coefficients as shown below:

To get the transformed image, we reconstruct the coefficients. For reconstruction, we use the idwt2() function as shown here:

imgr = idwt2(cA, cH, cV, cD, 'haar', 'mode', 'sym'); %first level decomposition

We then plot the output using the imshow() function:

figure(2)
imshow(uint8(imgr))
title('Reconstructed Image');

When we now run this program, we get the reconstructed image like the one below:

Conclusion #

As illustrated, the waveletAnalyzer toolbox is simple to use. Its interface is very clear. Also, this toolbox is not only used for wavelet transform but also de-noising, e.t.c.

Therefore, this toolbox is efficient even for users who are unfamiliar with Matlab. Also, you don't have to write any code while using it. Since their output is accurate, the wavelet functions dwt2() and idwt2() are efficient.

Happy coding!

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Peer Review Contributions by: Monica Masae

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