The Story of Wavelets Theory and Engineering Applications

The Story of Wavelets Theory and Engineering Applications Stationary discrete wavelet transform Two-dimensional wavelet transform 2D-DWT using MATLAB Implementation issues Image compressing using 2D-DWT Stationary Wavelet Transfporm (SWT) DWT is not time invariant Not Good ! What makes DWT time varying? Decimation (down sampling) DWT can be made time invariant, however, the transform must be redundant!!! Stationary wavelet transform -decimated DWTdecimated DWT -decimated DWT??? At any given level you can have two different DWT, due to choice in discarding even or odd indexed elements during subsampling

At J levels, you can have N=2J different DWTs. The particular DWT chosen can be denoted by =[1 2 N], j=1, if odd indexed elements are chosen, j=0, if even indexed elements are chosen SWT SWT is defined as the average of all -decimated DWTs For 6 level DWT64 DWTs For 10 level DWT 1024 DWTs . An efficient algorithm: cAj=a(j,n) where Hj cAj+1=a(j+1,n) Gj cDj+1=d(j+1,n)

Hj-1 2 Hj Gj-1 2 Gj Note: No subsampling is involved!!!. SWT Does it work? MATLAB DEMO Applications of SWT Denoising denoisingdenoising MATLAB demo: noisy doppler & noisy quadchirp Interval dependent thresholds

1D-DWT2D-DWTD-DWT Recall fundamental concepts of 1D-DWT Provides time-decimated DWTscale (frequency) representation of non-decimated DWTstationary signals Based on multiresolution approximation (MRA) Approximate a function at various resolutions using a scaling function, (t) Keep track of details lost using wavelet functions, (t) Reconstruct the original signal by adding approximation and detail coeff. Implemented by using a series of lowpass and highpass filters Lowpass filters are associated with the scaling function and provide approximation Highpass filters are associated with the wavelet function and provide detail lost in approximating the signal 2D-DWT-D DWT How do we generalize these concepts to 2D? 2D functions images f(x,y) I[m,n] intensity function What does it mean to take 2D-DWT of an image? How do we interpret? How can we represent an image as a function?

How do we define low frequency / high frequency in an image? How to we compute it? Why would we want to take 2D-DWT of an image anyway? Compression Denoising Feature extraction 2D-DWTD Scaling/Wavelet Functions We start by defining a two-dimensional scaling and wavelet functions: s ( x, y ) ( x) ( y ) s ( x, y ) ( x) ( y ) If (t) is orthogonal to its own translates, s ( x k , y l ) : k , l Z is also orthogonal to its own translates. Then, if fo(x,y) is the projection of f(x,y) on the space Vo generated by s(x,y): f o ( x, y )

ao (i, j ) s ( x i, y i j ao (i, j ) f ( x, y ), s ( x i, y j ) j) 2D-DWTD-DWT Just like in 1D we generated an approximation of the 2D function f(x,y). Now, how do we compute the detail lost in approximating this function? Unlike 1D case there will be three functions representing the details lost: Details lost along the horizontal direction Details lost along the vertical direction Details lost along the diagonal direction 1D Two sets of coeff.; a(k,n) & d (k,n) 2D Four sets of coefficients: a(k,n), b(k, n), c(k, n) & d(k,n)

Four Faces of 2D-DWTD-DWT One level of 2D DWT reconstruction: f 1 ( x, y ) Approximation coefficients ao (n, p) ( x n) ( x n) n p Detail coefficients along the horizontal direction bo (n, p ) ( x n) ( x n) Detail coefficients along the vertical direction Detail coefficients along the diagonal direction

co (n, p ) ( x n) ( x n) d o (n, p ) ( x n) ( x n) Implementation of 2D-DWTD-DWT COLUMNS ROWS ROWS COLUMNS ~ H 2 1 INPUT IMAGE

COLUMNS ROWS ~ G 2 1 COLUMNS INPUT IMAGE ~ H 1 2 LL Ak 1 ~ G

1 2 ( h) D LH k 1 ~ H 1 2 HL Dk(v)1 1 2 (d ) D HH k 1 ~ G LLL LLH

LL LH LHL LHH HL HH HL LH HH LLH LHL LHH LL HL LH

HH Up and Down Up and Down 2 1 Downsample columns along the rows: For each row, keep the even indexed columns, discard the odd indexed columns 1 2 Downsample columns along the rows: For each column, keep the even indexed rows, discard the odd indexed rows 2 1 Upsample columns along the rows: For each row, insert zeros at between every other sample (column) 1 2 Upsample rows along the columns: For each column, insert zeros at between every other sample (row)

Implementing 2D-DWTD-DWT Decomposition COLUMN j ROW i Reconstruction LL Ak 1 1 2 H 2 1 Dk( h)1 1 2 G

HL D (v ) k 1 1 2 H LH HH Dk( d)1 ORIGINAL IMAGE 2 1 1 2 G H G

2D-DWT-D DWT ON MATLAB Load Image (must be .mat file) Choose wavelet type Hit Analyze Choose display options

Recently Viewed Presentations

  • 2798345 - Amazon S3

    2798345 - Amazon S3

    do you have a song to sing? sing it now! do you have a gift to give? gift it now! do you have a prayer to pray? pray it now! do you have a sermon to preach? preach it now!...
  • Slide 1

    Slide 1

    Title: Slide 1 Author: Anthony Smith Last modified by: Elektra Media Ltd Created Date: 10/31/2007 2:00:44 PM Document presentation format: On-screen Show
  • Self-neglect An intervention study Professor Marian Brandon Dr

    Self-neglect An intervention study Professor Marian Brandon Dr

    Self-neglect An intervention study Professor Marian Brandon Dr Penny Sorensen Sue Bailey Professor Sara Connolly Dr Ann Anka Structure of the presentation Background to the research Methodology Findings (self-neglect) Research Questions What is the experience of the clients where this...
  • Emotional Health and Bullying

    Emotional Health and Bullying

    Take out a piece of paper and draw a shield. Create 6 sections on the shield and number them 1-6. Answer the following questions in the space. 1. Write 3 things you like about yourself or are good at. 2....
  • Quick & Healthy Meals

    Quick & Healthy Meals

    Bt is a natural bacteria in the soil that is a poison for moth and butterfly larva (caterpillars). Newest GE Approved: Salmon First GE animal in the food supply Growth hormone from a Chinook salmon & gene from an ocean...
  • Ocean Process Analysis Laboratory ,Institute for the Study of ...

    Ocean Process Analysis Laboratory ,Institute for the Study of ...

    Fig 4. 1992-2014 Time series of the altimeter derived cross-track current Vg (negative is SW flow anomaly) for passes 100, 024, 202 and 065 (from top to bottom) at the shelf-edge (across100-1000m isobaths). See altimeter track locations in Fig 1....
  • title.open ( ); revolution {execute};

    title.open ( ); revolution {execute};

    GridPP Collaboration Meeting 5 Nov 2001 Tony Doyle
  • Duaa alIftitah (with Arabic text and English, Urdu

    Duaa alIftitah (with Arabic text and English, Urdu

    Merits of Dua'aal-Iftitah. Dua'a al-Iftitah was taught by the Twelfth Imam Mehdi (A.S) to recite every night during the holy month Ramadhan. The Du'a is excellent for moulding man's attitude towards his Creator, as it discusses many aspects of the...