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MA304: Topics in Applied Mathematics

Unit 6: Wavelet Methods   This final unit is concerned with the study of multi-scale data analysis.  This unit will introduce you to the notions of multiresolution analysis (MRA) and wavelet transform (WT) as well as the associated wavelet decomposition and reconstruction algorithms.  The WT of a finite Fourier series is discussed to introduce the relationship between scale and frequency, in particular with a group of frequencies, called a frequency band.  We will also derive the inversion formula for recovering the function from its WT.  The MRA architecture is demonstrated by using B-spline functions.  Construction of wavelets by appealing to the MRA is achieved by matrix extension.  To reduce the computational complexity of the wavelet decomposition and reconstruction algorithms, you will also study lifting schemes.  To extend to the wavelet transform of functions of two variables, we use tensor-products of the wavelets with the corresponding scaling functions of the MRA.  This unit ends with embedding a digital image in the wavelet-domain for image manipulation, such as progressive transmission, image edge extraction, and image enhancement.

Unit6 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Give the relationship between scale and frequency for a given wavelet filter. - Perform matrix extension to compute wavelet filters. - Compute multi-scale data representation by applying the wavelet decomposition algorithm for the Haar wavelet. - Identify the order of vanishing moments of a given wavelet.

6.1 Time-Scale Analysis   - Reading: Time-Scale Analysis The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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6.1.1 Wavelet Transform   6.1.2 Frequency versus Scale   6.1.3 Partition into Frequency Bands   6.2 Multiresolution Analysis (MRA)   - Reading: Multiresolution Analysis (MRA) The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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6.2.1 Function Refinement   6.2.2 B-spline Examples   6.2.3 The MRA Architecture   6.3 Wavelet Construction   - Reading: Wavelet Construction The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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6.3.1 Matrix Extension   6.3.2 Quadrature Mirror Filter   6.3.3 Orthogonal and Bi-Orthogonal Wavelets   6.4 Wavelet Algorithms   - Reading: MIT: Professor Gilbert Strang’s “Lecture Notes: Handouts 1–16” Link: MIT: Professor Gilbert Strang’s “Lecture Notes: Handouts 1–16” (PDF)

 Instructions:  Please click on the link above, and select the PDF
links for the slides and handouts for 1–16.  Study these lecture
notes and handouts to learn about computational schemes of wavelet
decomposition and reconstruction, filter banks, and the lifting
scheme.    
 Studying these lecture slides and reflecting on the material should
take approximately 4 hours to complete.  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.

6.4.1 Wavelet Decomposition and Reconstruction   - Reading: Wavelet Decomposition and Reconstruction The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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6.4.2 Filter Banks   - Reading: Filter Banks The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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6.4.3 The Lifting Scheme   - Reading: The Lifting Scheme The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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6.5 Application to Image Coding   - Reading: Application to Image Coding The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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6.5.1 Mapping Digital Images to the Wavelet Domain   6.5.2 Progressive Image Transmission   6.5.3 Lossless JPEG-2000 Compression