Module 7:Data Representation Lecture 35: Wavelets. The Lecture Contains: Wavelets. Discrete Wavelet Transform (DWT) Haar wavelets: Example

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1 The Lecture Contains: Wavelets Discrete Wavelet Transform (DWT) Haar wavelets: Example Haar wavelets: Theory Matrix form Haar wavelet matrices Dimensionality reduction using Haar wavelets file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_1.htm[6/14/2012 3:54:25 PM]

2 Wavelets Fourier transform analyzes frequency resolutions, but not time Wavelets analyze a function in both time and frequency domains Good time resolution and poor frequency resolution at high frequencies Good frequency resolution and poor time resolution at low frequencies Wavelets are useful for short duration signals of high frequency and long duration signals of short frequency Wavelets are generated from a mother wavelet function Zero mean (oscillatory, i.e., wave nature): Unit length: Basis functions are generated by scaling (s) and shifting ( l ) the mother wavelet file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_2.htm[6/14/2012 3:54:25 PM]

3 Discrete Wavelet Transform (DWT) DWT generates a set of basis function or vectors Two functions: Wavelet function Scaling function Space spanned by basis vectors at level can be spanned by two sets of basis vectors and at level and are wavelet and scaling functions respectively DWT generates basis vectors for wavelet and scaling functions at different levels file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_3.htm[6/14/2012 3:54:25 PM]

4 Haar wavelets: Example Compute sum and difference between each consecutive pairs of entries (and scale) Repeat the steps for the sum coefficients Length is preserved: Invertible: is losslessly obtained from file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_4.htm[6/14/2012 3:54:25 PM]

5 Haar wavelets: Theory Wavelet ( ) and scaling ( ) functions: Shifting (i) and scaling ( j ): Binary dilation (scaling) and dyadic translation (shifting) Coefficients corresponding to are averages or sum coefficients Coefficients corresponding to are differences or detail coefficients file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_5.htm[6/14/2012 3:54:25 PM]

6 Matrix form etc. form the basis vectors Only is needed since others are expressed in terms of When size of vector is, j levels of basis vectors are needed There are basis vectors corresponding to detail coefficients and 1 basis vector corresponding to sum coefficient at level Transformation matrix H has these basis vectors as columns Transformed vector for data (row) vector file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_6.htm[6/14/2012 3:54:26 PM]

7 Matrix form etc. form the basis vectors Only is needed since others are expressed in terms of When size of vector is, j levels of basis vectors are needed There are basis vectors corresponding to detail coefficients and 1 basis vector corresponding to sum coefficient at level Transformation matrix H has these basis vectors as columns Transformed vector for data (row) vector Each step can be defined as multiplication by a matrix Composition of these matrices gives the final transformation matrix H is orthonormal Hence, inverse operation is easy file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_7.htm[6/14/2012 3:54:26 PM]

8 Haar wavelet matrices file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_8.htm[6/14/2012 3:54:26 PM]

9 Haar wavelet matrices file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_9.htm[6/14/2012 3:54:26 PM]

10 Haar wavelet matrices file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_10.htm[6/14/2012 3:54:26 PM]

11 Haar wavelet matrices Inversely, file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_11.htm[6/14/2012 3:54:26 PM]

12 Dimensionality reduction using Haar wavelets Retain sum and lower level detail coefficients For example, retain 1 sum (at level 0) and 3 detail (1 at level 0 and 2 at level 1) coefficients Contractive file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_12.htm[6/14/2012 3:54:26 PM]

13 Dimensionality reduction using Haar wavelets Retain sum and lower level detail coefficients For example, retain 1 sum (at level 0) and 3 detail (1 at level 0 and 2 at level 1) coefficients Contractive Alternatively, for a database of objects, retain those coefficients whose variances are highest Another option is to zero out coefficients whose absolute values are below a threshold What happens when dimensionality is not a power of 2? file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_13.htm[6/14/2012 3:54:27 PM]

14 Dimensionality reduction using Haar wavelets Retain sum and lower level detail coefficients For example, retain 1 sum (at level 0) and 3 detail (1 at level 0 and 2 at level 1) coefficients Contractive Alternatively, for a database of objects, retain those coefficients whose variances are highest Another option is to zero out coefficients whose absolute values are below a threshold What happens when dimensionality is not a power of 2? Pad zeros at end: latter half becomes less important Pad equal amount of zeros in each half: should be done recursively file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_14.htm[6/14/2012 3:54:27 PM]

15 Dimensionality reduction using Haar wavelets Retain sum and lower level detail coefficients For example, retain 1 sum (at level 0) and 3 detail (1 at level 0 and 2 at level 1) coefficients Contractive Alternatively, for a database of objects, retain those coefficients whose variances are highest Another option is to zero out coefficients whose absolute values are below a threshold What happens when dimensionality is not a power of 2? Pad zeros at end: latter half becomes less important Pad equal amount of zeros in each half: should be done recursively Interestingly, shuffing the dimensions produce different wavelet coefficients How to shuffle to satisfy some critera? file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_15.htm[6/14/2012 3:54:27 PM]