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I am looking for a good introduction to wavelets and wavelet transforms.

that covers the following:

Basics

  • Vector Spaces – Properties– Dot Product – Basis – Dimension, Orthogonality and Orthonormality
  • Relationship Between Vectors and Signals – Signal Spaces
  • Concept of Convergence
  • Hilbert Spaces for Energy Signals
  • Fourier Theory: Fourier series expansion, Fourier transform, Short time Fourier transform, Time-frequency analysis.

Multi-resolution analysis

  • Definition of Multi Resolution Analysis (MRA)
  • Haar Basis
  • Construction of General Orthonormal MRA
  • Wavelet Basis for MRA
  • Continuous Time MRA Interpretation for the DTWT
  • Discrete Time MRA
  • Basis Functions for the DTWT
  • PRQMF Filter Bank

Continuous wavelet transforms

  • Wavelet Transform – Definition and Properties – Concept of Scale and its Relation with Frequency
  • Continuous Wavelet Transform (CWT)
  • Scaling Function and Wavelet Functions (Daubechies-Coiflet, Mexican Hat, Sinc, Gaussian, Bi Orthogonal)
  • Tiling of Time – Scale Plane for CWT

Discrete wavelet transform

  • Filter Bank and Sub Band Coding Principles
  • Wavelet Filters
  • Inverse DWT Computation by Filter Banks
  • Basic Properties of Filter Coefficients; Choice of Wavelet Function Coefficients
  • Derivations of Daubechies Wavelets
  • Mallat's Algorithm for DWT
  • Multi Band Wavelet Transforms Lifting Scheme
  • Wavelet Transform Using Polyphase Matrix Factorization
  • Geometrical Foundations of Lifting Scheme
  • Lifting Scheme in Z –Domain.

Applications

  • Wavelet methods for signal processing
  • Image Procession: Compression Techniques: EZW–SPHIT Coding; Denoising Techniques: Noise Estimation – Shrinkage Rules – Shrinkage Functions – Edge Detection and Object Isolation, Image Fusion, and Object Detection.

Please suggest the steps,resources and materials to do the same. And the time frame to master in this.

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    $\begingroup$ Your detailed list of "covered material" has relatively little to do with wavelets per se. The first third of your list is covered in any linear algebra textbook/course; the middle third seems to be about signal processing; and the last third is applied Fourier theory. I would say the list you gave are more like prerequisites for starting to study wavelets. $\endgroup$ – Willie Wong Jan 30 '14 at 15:59
  • $\begingroup$ yes this is the first intro part i'll add the rest of the part now thanks @WillieWong $\endgroup$ – DeeRam Jan 30 '14 at 16:16
  • $\begingroup$ My suggestion is to stay away from Mallat. I found that book to be horribly written with very awkward and unnecessarily complicated notation. $\endgroup$ – AnonSubmitter85 Jan 31 '14 at 2:20
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In that case, I recommend "A Friendly Guide to Wavelets" by Gerald Kaiser. It includes a clear introduction to linear algebra, some fundamentals of Fourier Analysis and Windowed Fourier Transform, and a presentation of wavelets for those who had not heard of that.

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How about learning the subject from Wavelets and Filter Banks, by Gilbert Strang and Truong Nguyen. The legendary MIT Professor has a great knack of explaining stuff.

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