A Beginning with some practical situations, which call for multiresolution/ multiscale analysis – and how time-frequency analysis and wavelets arise from them. Examples: Image Compression, Wideband Correlation Processing, Magnetic Resonance Imaging, Digital Communication – Piecewise constant approximation – the Haar wavelet – Building up the concept of dyadic Multiresolution Analysis (MRA) – Relating dyadic MRA to filter banks – A review of discrete signal processing – Elements of multirate systems and two-band filter bank design for dyadic wavelets – Families of wavelets: Orthogonal and biorthogonal wavelets – Daubechies’ family of wavelets in detail – Vanishing moments and regularity – Conjugate Quadrature Filter Banks (CQF) and their design – Dyadic MRA more formally – Data compression – fingerprint compression standards, JPEG-2000 standards – The Uncertainty Principle: and its implications: the fundamental issue in this subject – the problem and the challenge that Nature imposes – The importance of the Gaussian function: the Gabor Transform and its generalization; time, frequency and scale – their interplay – The Continuous Wavelet Transform (CWT) – Condition of admissibility and its implications – Application of the CWT in wideband correlation processing.

Journey from the CWT to the DWT: Discretization in steps – Discretization of scale – generalized filter bank – Discretization of translation – generalized output sampling – Discretization of time/ space (independent variable) – sampled inputs – Going from piecewise linear to piecewise polynomial – The class of spline wavelets – a case for infinite impulse response (IIR) filter banks – Variants of the wavelet transform and its implementational structures – The wavepacket transform – Computational efficiency in realizing filter banks – Polyphase components – The lattice structure – The lifting scheme – An exploration of applications (this will be a joint effort between the instructor and the class) – Examples: Transient analysis; singularity detection; Biomedical signal processing applications; Geophysical signal analysis applications; Efficient signal design and realization: wavelet based modulation and demodulation; Applications in mathematical approximation; Applications to the solution of some differential equations; Applications in computer graphics and computer vision; Relation to the ideas of fractals and fractal phenomena.

Other Resources

Course Curriculum

Introduction Details
The Haar Wavelet Details
The Haar Multiresolution Analysis Details
Wavelets and Multirate Digital Signal Processing Details
Equivalence:Functions and Sequences Details
The Haar Filter Bank Details
Haar Filter Bank Analysis and Synthesis I Details
Relating psi, phi and the Filters Details
Iterating the filter bank from Psi, Phi Details
Z-Domain Analysis Of Multirate Filter Bank Details
Two Channel Filter Bank Details
Perfect Reconstruction:Conjugate Quadrature Details
Conjugate Quadrature Filters – Daubechies Family of MRA Details
Daubechies’ Filter Banks – Conjugate Quadrature Details
Time and Frequency Joint Perspective Details
Ideal Time Frequency Behaviour Details
The Uncertainty Principle Details
Time Bandwith Product Uncertainty Bound Details
Evaluating and Bounding squareroot t.omega Details
The Time Frequency Plane And Its Tilings Details
Short Time Fourier Transform and Wavelet Transform In General Details
Reconstruction & Admissibility Details
Admissibility in Detail Discretization of Scale Details
Logarithmic Scale Discretization Dyadic Discretization Details
The Theorem of (DYADIC) Multiresolution Analysis Details
Proof of the Theorem of (DYADIC) Multiresolution Analysis Details
Introducing Variants of The Multiresolution Analysis Concept Details
JPEG 2000 5/3 FilterBank & Spline MRA Details
Orthogonal Multiresolution Analysis With Splines Details
Building Piece wise Linear Scaling Function,Wavelet Details
The Wave Packet Transform Details
‘NOBEL’ Identities And The Haar Wavepacket Transform Details
The Lattice Structure For Orthogonal Filter Banks Details
Constructing the Lattice & its Variants Details
The ‘Lifting Structure’ And Polyphase Matrices Details
The Polyphase Approach The Modulation Approach Details
Modulation Analysis and The 3-Band Filter Bank, Applications Details
Two Applications,Data Mining,Face Recognition Details
Proof that a non-zero function can not be both time and band-limited Details
M-Band Filter Banks and Looking Ahead Details
Tutorial -Session 1 Details
Student’s Presentation Details
Tutorial on Uncertainty Product Details
Tutorial on Two band Filter Bank Details
Tutorial -Frequency Domain Analysis of Two band Filter Bank Details
Zoom in and Zoom out using Wavelet Transform Details
More Thoughts on Wavelets : Zooming In Details
Towards selecting Wavelets through vanishing moments Details
In Search of Scaling Coefficients Details
Wavelet Applications Details

This course is delivered by IIT Bombay , as part of NPTEL Free Courses sponsored by Govt of India.

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