Wavelet Toolbox Getting Started Preface About the Authors Notes by Yves Meyer Notes by Ingrid Daubechies Acknowledgements What Is the Wavelet Toolbox? Using This Guide Caution For More Background Installing the Wavelet Toolbox System Recommendations Platform-Specific Details Typographical Conventions Related Products List Wavelets: A New Tool for Signal Analysis Wavelet Applications Scale aspects Time aspects Wavelet Decomposition as a Whole Fourier Analysis Short-Time Fourier Analysis Wavelet Analysis What Can Wavelet Analysis Do? What Is Wavelet Analysis? Number of Dimensions The Continuous Wavelet Transform Scaling Shifting Five Easy Steps to a Continuous Wavelet Transform Scale and Frequency The Scale of Nature What's Continuous About the Continuous Wavelet Transform? The Discrete Wavelet Transform One-Stage Filtering: Approximations and Details Multiple-Level Decomposition Wavelet Reconstruction Reconstruction Filters Reconstructing Approximations and Details Relationship of Filters to Wavelet Shapes Multistep Decomposition and Reconstruction Wavelet Packet Analysis History of Wavelets An Introduction to the Wavelet Families Haar Daubechies Biorthogonal Coiflets Symlets Morlet Mexican Hat Meyer Other Real Wavelets Complex Wavelets Using the Wavelet Toolbox Using Wavelets One-Dimensional Continuous Wavelet Analysis Continuous Analysis Using the Command Line Continuous Analysis Using the Graphical Interface Importing and Exporting Information from the Graphical Interface One-Dimensional Complex Continuous Wavelet Analysis Complex Continuous Analysis Using the Command Line Complex Continuous Analysis Using the Graphical Interface Importing and Exporting Information from the Graphical Interface One-Dimensional Discrete Wavelet Analysis One-Dimensional Analysis Using the Command Line One-Dimensional Analysis Using the Graphical Interface Importing and Exporting Information from the Graphical Interface Two-Dimensional Discrete Wavelet Analysis Two-Dimensional Analysis Using the Command Line Two-Dimensional Analysis Using the Graphical Interface Importing and Exporting Information from the Graphical Interface Wavelets: Working with Images Understanding Images in MATLAB Indexed Images Wavelet Decomposition of Indexed Images Other Images Image Conversion One-Dimensional Discrete Stationary Wavelet Analysis One-Dimensional Analysis Using the Command Line One-Dimensional Analysis for De-noising Using the Graphical Interface Importing and Exporting Information from the Graphical Interface Two-Dimensional Discrete Stationary Wavelet Analysis Two-Dimensional Analysis Using the Command Line Two-Dimensional Analysis for De-noising Using the Graphical Interface Importing and Exporting Information from the Graphical Interface One-Dimensional Wavelet Regression Estimation One-Dimensional Estimation Using the GUI for Equally Spaced Observations (Fixed Design) One-Dimensional Estimation Using the GUI for Randomly Spaced Observations (Stochastic Design) Importing and Exporting Information from the Graphical Interface One-Dimensional Wavelet Density Estimation One-Dimensional Estimation Using the Graphical Interface Importing and Exporting Information from the Graphical Interface One-Dimensional Variance Adaptive Thresholding of Wavelet Coefficients One-Dimensional Local Thresholding for De-noising Using the Graphical Interface Importing and Exporting Information from the Graphical Interface One-Dimensional Selection of Wavelet Coefficients Using the Graphical Interface Two-Dimensional Selection of Wavelet Coefficients Using the Graphical Interface One-Dimensional Extension One-Dimensional Extension Using the Command Line One-Dimensional Extension Using the Graphical Interface Importing and Exporting Information from the Graphical Interface Two-Dimensional Extension Two-Dimensional Extension Using the Command Line Two-Dimensional Extension Using the Graphical Interface Importing and Exporting Information from the Graphical Interface Wavelet Applications Detecting Discontinuities and Breakdown Points I Discussion Detecting Discontinuities and Breakdown Points II Discussion Detecting Long-Term Evolution Discussion Detecting Self-Similarity Wavelet Coefficients and Self-Similarity Discussion Identifying Pure Frequencies Discussion Suppressing Signals Discussion De-Noising Signals Discussion De-noising Images Discussion Compressing Images Discussion Fast Multiplication of Large Matrices Wavelets in Action: Examples and Case Studies Illustrated Examples Advice to the Reader Example 1: A Sum of Sines Example 2: A Frequency Breakdown Example 3: Uniform White Noise Example 4: Colored AR(3) Noise Example 5: Polynomial + White Noise Example 6: A Step Signal Example 7: Two Proximal Discontinuities Example 8: A Second-Derivative Discontinuity Example 9: A Ramp + White Noise Example 10: A Ramp + Colored Noise Example 11: A Sine + White Noise Example 12: A Triangle + A Sine Example 13: A Triangle + A Sine + Noise Example 14: A Real Electricity Consumption Signal Case Study: An Electrical Signal Data and the External Information Analysis of the Midday Period Analysis of the End of the Night Period Suggestions for Further Analysis Using Wavelet Packets About Wavelet Packet Analysis One-Dimensional Wavelet Packet Analysis Compressing a Signal Using Wavelet Packets De-Noising a Signal Using Wavelet Packets Two-Dimensional Wavelet Packet Analysis Compressing an Image Using Wavelet Packets Importing and Exporting from Graphical Tools Saving Information to Disk Loading Information into the Graphical Tools Advanced Concepts Mathematical Conventions General Concepts Wavelets: A New Tool for Signal Analysis Wavelet Decomposition: A Hierarchical Organization Finer and Coarser Resolutions Wavelet Shapes Wavelets and Associated Families Wavelet Transforms: Continuous and Discrete Local and Global Analysis Synthesis: an Inverse Transform Details and Approximations The Fast Wavelet Transform (FWT) Algorithm Filters Used to Calculate the DWT and IDWT Algorithms Why Does Such an Algorithm Exist? One-Dimensional Wavelet Capabilities Two-Dimensional Wavelet Capabilities Dealing with Border Distortion Signal Extensions: Zero-Padding, Symmetrization, and Smooth Padding Discrete Stationary Wavelet Transform (SWT) e-decimated DWT How to Calculate the e-decimated DWT: SWT Inverse Discrete Stationary Wavelet Transform (ISWT) More about SWT Frequently Asked Questions Wavelet Families: Additional Discussion Daubechies Wavelets: dbN Symlet Wavelets: symN Coiflet Wavelets: coifN Biorthogonal Wavelet Pairs: biorNr.Nd Meyer Wavelet: meyr Battle-Lemarie Wavelets Mexican Hat Wavelet: mexh Morlet Wavelet: morl Other Real Wavelets Complex Wavelets Summary of Wavelet Families and Associated Properties (Part 1) Summary of Wavelet Families and Associated Properties (Part 2) Wavelet Applications: More Detail Suppressing Signals Splitting Signal Components Noise Processing De-Noising Data Compression Function Estimation: Density and Regression Available Methods for De-noising, Estimation and Compression Using GUI Tools Wavelet Packets From Wavelets to Wavelet Packets: Decomposing the Details Wavelet Packets in Action: an Introduction Building Wavelet Packets Wavelet Packet Atoms Organizing the Wavelet Packets Choosing the Optimal Decomposition Some Interesting Subtrees Wavelet Packets 2-D Decomposition Structure Wavelet Packets for Compression and De-Noising References Adding Your Own Wavelets Preparing to Add a New Wavelet Family Choose the Wavelet Family Full Name Choose the Wavelet Family Short Name Determine the Wavelet Type Define the Orders of Wavelets Within the Given Family Build a MAT-File or M-File Define the Effective Support Adding a New Wavelet Family Example 1 Example 2 After Adding a New Wavelet Family Reference Reference Functions by Category allnodes appcoef appcoef2 bestlevt besttree biorfilt biorwavf centfrq cgauwavf cmorwavf coifwavf cwt dbaux dbwavf ddencmp depo2ind detcoef detcoef2 drawtree dtree dwt dwt2 dwtmode dyaddown dyadup entrupd fbspwavf gauswavf get idwt idwt2 ind2depo instdfft intwave isnode istnode iswt iswt2 leaves mexihat meyer meyeraux morlet nodeasc nodedesc nodejoin nodepar nodesplt noleaves nstdfft ntnode ntree orthfilt plot qmf rbiowavf read readtree scal2frq set shanwavf swt swt2 symaux symwavf thselect tnodes treedpth treeord upcoef upcoef2 upwlev upwlev2 wavedec wavedec2 wavedemo wavefun waveinfo wavemenu wavemngr waverec waverec2 wbmpen wcodemat wdcbm wdcbm2 wden wdencmp wentropy wextend wfilters wkeep wmaxlev wnoise wnoisest wp2wtree wpbmpen wpcoef wpcutree wpdec wpdec2 wpdencmp wpfun wpjoin wprcoef wprec wprec2 wpsplt wpthcoef wptree wpviewcf wrcoef wrcoef2 wrev write wtbo wthcoef wthcoef2 wthresh wthrmngr Discrete Wavelet 1-D options. Discrete Stationary Wavelet 1-D options. Discrete Wavelet 2-D options. Discrete Stationary Wavelet 2-D options. Discrete Wavelet Packet 1-D options. Discrete Wavelet Packet 2-D options. wtreemgr wvarchg GUI Reference General Features Color Coding Connectedness of Plots Using the Mouse Controlling the Colormap Controlling the Number of Colors Controlling the Coloration Mode Using Menus Using the View Axes Button Using the Interval Dependent Threshold Settings Tool Continuous Wavelet Tool Features Wavelet 1-D Tool Features Tree Mode More Display Options Wavelet 2-D Tool Features Wavelet Packet Tool Features (1-D and 2-D) Node Action Functionality Wavelet Display Tool Wavelet Packet Display Tool Object-Oriented Programming Short Description of Objects in the Toolbox Simple Use of Objects Through Four Examples Example 1: plot and wpviewcf Example 2: drawtree and readtree Example 3: A Funny One Example 4: Thresholding Wavelet Packets Detailed Description of Objects in the Toolbox WTBO object NTREE object DTREE object WPTREE object Advanced Use of Objects Example 1: Building a Wavelet Tree Object (WTREE) Example 2: Building a Right Wavelet Tree Object (RWVTREE) Example 3: Building a Wavelet Tree Object (WVTREE) Example 4: Building a Wavelet Tree Object (EDWTTREE) New Features Detailed Description New Graphical User Interface Tools New Wavelets New Functions Enhancements Compatibility with Earlier Versions Printable Documentation (PDF) Product Page (Web)