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Wavelet Transform
5.7 DISCRETE WAVELET TRANSFORM IN MATLAB ®
In MATLAB, a complete toolbox is dedicated to the WT and its applications in signal and image processing. Even though we are not to cover the detailed description
of this toolbox, we encourage the reader to explore these capabilities using the visual
user interface provided for most of these capabilities. The command “wavedemo”
provides a demonstration of these capabilities, and “wavemenu” activates the user
interface of the wavelet toolbox. While wavedemo helps the reader understand the
different DWT commands and options provided by MATLAB, wavemenu provides
the means to visually conduct almost all aforementioned applications of the DWT.
These applications include denoising, compression, and filtering. MATLAB also
provides a set of commands for direct calculation of the DWT coefficients as well
as the IDWT. These commands include dwt, idwt, dwt2, and idwt2. In using
each of these commands, one needs to identify the type of the mother wavelet and
the level of decomposition (or reconstruction).
The readers are further guided to systematically explore these capabilities in the
Problems section.
5.8 SUMMARY
In this chapter, the concept of continuous and discrete WTs was presented. We
started with the need for such a transformation and then described the mathematical formulation as well as the applications of WT such as denoising, filtering, and
compression.
PROBLEMS
5.1 A biomedical image that was corrupted by additive noise is given in the file
“p_5_1.mat.” This image shows a fluoroscopic image of arteries.*
a. Read the image using “load” command.
b. Calculate the 2-D DWT of the image using “dwt2” command (assume
only one level of decomposition). Use Daubechies 2 (“db2” in MATLAB)
as the mother wavelet.
c. Apply hard and soft thresholding for denoising of the image assuming ξ = 0.01,
ξ = 0.05, and ξ = 0.1.
d.
Use “idwt2” command to reconstruct the filtered image for all three
values of ξ for both hard and soft thresholding, compare the results, and
identify the setting with the best results.
5.2 An EEG signal is to be denoised using DWT. † The signal is given in
“p_5_2.mat.”
a. Read the EEG signal using “load” command and plot the signal.
* Courtesy of Andre D’Avila, MD, Heart Institute (InCor), University of Sao Paulo, Medical School, Sao
Paulo, Brazil.
† Courtesy of Dr. Henri Begleiter, Neurodynamics Laboratory, State University of New York Health
Center at Brooklyn, Brooklyn, NY.
Wavelet Transform
5.7 DISCRETE WAVELET TRANSFORM IN MATLAB ®
In MATLAB, a complete toolbox is dedicated to the WT and its applications in signal and image processing. Even though we are not to cover the detailed description
of this toolbox, we encourage the reader to explore these capabilities using the visual
user interface provided for most of these capabilities. The command “wavedemo”
provides a demonstration of these capabilities, and “wavemenu” activates the user
interface of the wavelet toolbox. While wavedemo helps the reader understand the
different DWT commands and options provided by MATLAB, wavemenu provides
the means to visually conduct almost all aforementioned applications of the DWT.
These applications include denoising, compression, and filtering. MATLAB also
provides a set of commands for direct calculation of the DWT coefficients as well
as the IDWT. These commands include dwt, idwt, dwt2, and idwt2. In using
each of these commands, one needs to identify the type of the mother wavelet and
the level of decomposition (or reconstruction).
The readers are further guided to systematically explore these capabilities in the
Problems section.
5.8 SUMMARY
In this chapter, the concept of continuous and discrete WTs was presented. We
started with the need for such a transformation and then described the mathematical formulation as well as the applications of WT such as denoising, filtering, and
compression.
PROBLEMS
5.1 A biomedical image that was corrupted by additive noise is given in the file
“p_5_1.mat.” This image shows a fluoroscopic image of arteries.*
a. Read the image using “load” command.
b. Calculate the 2-D DWT of the image using “dwt2” command (assume
only one level of decomposition). Use Daubechies 2 (“db2” in MATLAB)
as the mother wavelet.
c. Apply hard and soft thresholding for denoising of the image assuming ξ = 0.01,
ξ = 0.05, and ξ = 0.1.
d.
Use “idwt2” command to reconstruct the filtered image for all three
values of ξ for both hard and soft thresholding, compare the results, and
identify the setting with the best results.
5.2 An EEG signal is to be denoised using DWT. † The signal is given in
“p_5_2.mat.”
a. Read the EEG signal using “load” command and plot the signal.
* Courtesy of Andre D’Avila, MD, Heart Institute (InCor), University of Sao Paulo, Medical School, Sao
Paulo, Brazil.
† Courtesy of Dr. Henri Begleiter, Neurodynamics Laboratory, State University of New York Health
Center at Brooklyn, Brooklyn, NY.