Wavelet Toolbox 4 Analyze and synthesize signals and images using wavelet techniques Wavelet Toolbox extends the MATLAB® technical computing environment with graphical tools and command-line functions for developing wavelet-based algorithms for the analysis, synthesis, denoising, and compression of signals and images. Wavelet analysis provides more precise information about signal data than other signal analysis techniques, such as Fourier. Wavelet Toolbox supports the interactive exploration of wavelet properties and applications. It is useful for speech and audio processing, image and video processing, bio- medical imaging, and one-dimensional (1-D) and two-dimensional (2-D) applications in communications and geophysics. Wavelet Toolbox authors are Michel Misiti, École Centrale de Lyon; Georges Oppenheim, Université de Marne-La-Vallée; Jean-Michel Poggi, Université René Descartes, Paris 5 Université; and Yves Misiti, Université Paris-Sud. Applying Wavelet Methods Wavelet methods provide powerful tools for analyzing, encoding, compressing, recon- structing, and modeling signals and images. They are useful in capturing, identifying, and analyzing local, multiscale, and nonsta- tionary processes, enabling you to explore aspects of data that other analysis techniques miss, such as trends, breakdown points, discontinuities in higher derivatives, and self-similarity. Wavelet Toolbox supports a full suite of wavelet analysis and synthesis operations. You can use it to: • Enhance edge detection in image processing Achieve high rates of signal or image • compression with virtually no loss of significant data Restore noisy signals and degraded images Discover trends in noisy or faulty data • • Key features ■ Standard wavelet families, including Daubechies wavelet filters, complex Morlet and Gaussian, real reverse biorthogonal, and discrete Meyer ■ Wavelet and signal processing utilities, including a function to convert scale to frequency ■ Methods for adding wavelet families ■ Lifting methods for constructing wavelets ■ Customizable presentation and visualization of data ■ Interactive tools for continuous and discrete wavelet analysis ■ Wavelet packets, implemented as MATLAB objects ■ One-dimensional multisignal analysis, compression, and denoising ■ Multiscale principal component analysis ■ Multivariate denoising Fractal signal decomposed using the Continuous Wavelet Transform, with a scalogram showing the self-similarity of the signal at various scales. The bottom axes display the coefficient line and local maxima lines, respectively, for exploring continuous wavelet coefficients. Accelerating the pace of engineering and science 

Wavelet decomposition using wavelet packet analysis. • • • Study the fractal properties of signals and images Extract information-rich features for use in classification and pattern recognition applications Perform multivariate denoising of signals with multiscale principal component analysis Analyzing Signals and Images The Wavelet Toolbox graphical user interface (GUI) provides a comprehensive set of tools for analyzing 1-D and 2-D signals, includ- ing tools for wavelet analysis, wavelet packet analysis, denoising, and compression. For 1-D signals, you can use the GUI tools to: • Perform discrete wavelet analysis of signals Perform continuous wavelet analysis of real • signals using complex wavelets Denoise signals Estimate wavelet-based density Perform wavelet reconstruction schemes based on various wavelet coefficient selec- tion strategies • • • • • • Randomly generate fractional Brownian motion Perform 1-D signal extension and trunca- tion using periodic, symmetric, smooth, and zeropadding methods Perform 1-D signal clustering and clas- sification using wavelet analyses (with Statistics Toolbox, available separately) For 2-D signals, you can use the GUI tools to: Perform discrete wavelet analysis of images • • Fuse two images Perform translation-invariant denoising • of images, using the stationary wavelet transform Reconstruct wavelet schemes based on various wavelet coefficient selection strategies Required Products MATLAB • Wavelet denoising, with instant visualization of the results. Threshold settings can be applied using the denoising and compression tools in the Wavelet Toolbox graphical user interface (GUI). Image from the U.S. Federal Bureau of Investigation finger- print database. The automatic thresholding feature of Wavelet Toolbox produces a compressed image with about 72% zeros and 98% of the original signal. Related Products Image Processing Toolbox. Perform image processing, analysis, and algorithm development Signal Processing Toolbox. Perform signal processing, analysis, and algorithm development Statistics Toolbox. Apply statistical algo- rithms and probability models For more information on related products visit www.mathworks.com/products/wavelet Platform and System Requirements For platform and system requirements, visit www.mathworks.com/products/wavelet ■ Resources visit www.mathworks.com technical support www.mathworks.com/support online user community www.mathworks.com/matlabcentral Demos www.mathworks.com/demos training services www.mathworks.com/training thirD-party proDucts anD services www.mathworks.com/connections WorlDWiDe contacts www.mathworks.com/contact e-mail info@mathworks.com Accelerating the pace of engineering and science © 2007 MATLAB, Simulink, Stateflow, Handle Graphics, Real-Time Workshop, and xPC TargetBox are registered trademarks and SimBiology, SimEvents, and SimHydraulics are trademarks of The MathWorks, Inc. Other product or brand names are trademarks or registered trademarks of their respective holders. 8797v04 03/07