2008 • 135 Pages • 1.88 MB • English

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THE DISCRETE HAAR WAVELET TRANSFORMATION Patrick J. Van Fleet Center for Applied Mathematics University of St. Thomas St. Paul, MN USA Joint Mathematical Meetings, 7 & 9 January 2008 7 JANUARY 2008 (SESSION 1) THE DHWT JMM MINICOURSE #4 1 / 14

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NAIVE DATA APPROXIMATION THE PROBLEM ◮ Suppose you are given N values x = (x1, x2, . . . , xN) where N is even. ◮ Your task: Send an approximation s (a list of numbers) of this data via the internet to a colleague. ◮ In order to reduce transfer time, the length of your approximation must be N/2. ◮ How do you suggest we do it? 7 JANUARY 2008 (SESSION 1) THE DHWT JMM MINICOURSE #4 2 / 14

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NAIVE DATA APPROXIMATION THE PROBLEM ◮ Suppose you are given N values x = (x1, x2, . . . , xN) where N is even. ◮ Your task: Send an approximation s (a list of numbers) of this data via the internet to a colleague. ◮ In order to reduce transfer time, the length of your approximation must be N/2. ◮ How do you suggest we do it? 7 JANUARY 2008 (SESSION 1) THE DHWT JMM MINICOURSE #4 2 / 14

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NAIVE DATA APPROXIMATION THE PROBLEM ◮ Suppose you are given N values x = (x1, x2, . . . , xN) where N is even. ◮ Your task: Send an approximation s (a list of numbers) of this data via the internet to a colleague. ◮ In order to reduce transfer time, the length of your approximation must be N/2. ◮ How do you suggest we do it? 7 JANUARY 2008 (SESSION 1) THE DHWT JMM MINICOURSE #4 2 / 14

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NAIVE DATA APPROXIMATION THE PROBLEM ◮ Suppose you are given N values x = (x1, x2, . . . , xN) where N is even. ◮ Your task: Send an approximation s (a list of numbers) of this data via the internet to a colleague. ◮ In order to reduce transfer time, the length of your approximation must be N/2. ◮ How do you suggest we do it? 7 JANUARY 2008 (SESSION 1) THE DHWT JMM MINICOURSE #4 2 / 14

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NAIVE DATA APPROXIMATION THE PROBLEM ◮ One solution is to pair-wise average the numbers: x2k−1 + x2k sk = , k = 1, . . . , N/2 2 ◮ For example: x = (6, 12, 15, 15, 14, 12, 120, 116) → s = (9, 15, 13, 118) 7 JANUARY 2008 (SESSION 1) THE DHWT JMM MINICOURSE #4 2 / 14

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NAIVE DATA APPROXIMATION THE PROBLEM ◮ One solution is to pair-wise average the numbers: x2k−1 + x2k sk = , k = 1, . . . , N/2 2 ◮ For example: x = (6, 12, 15, 15, 14, 12, 120, 116) → s = (9, 15, 13, 118) 7 JANUARY 2008 (SESSION 1) THE DHWT JMM MINICOURSE #4 2 / 14

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NAIVE DATA APPROXIMATION ADDING MORE INFORMATION ◮ Suppose now you were allowed to send extra data in addition to the pair-wise averages list s. ◮ The idea is to send a second list of data d so that the original list x can be recovered from s and d. ◮ How would you do it? 7 JANUARY 2008 (SESSION 1) THE DHWT JMM MINICOURSE #4 3 / 14

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NAIVE DATA APPROXIMATION ADDING MORE INFORMATION ◮ Suppose now you were allowed to send extra data in addition to the pair-wise averages list s. ◮ The idea is to send a second list of data d so that the original list x can be recovered from s and d. ◮ How would you do it? 7 JANUARY 2008 (SESSION 1) THE DHWT JMM MINICOURSE #4 3 / 14

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NAIVE DATA APPROXIMATION ADDING MORE INFORMATION ◮ Suppose now you were allowed to send extra data in addition to the pair-wise averages list s. ◮ The idea is to send a second list of data d so that the original list x can be recovered from s and d. ◮ How would you do it? 7 JANUARY 2008 (SESSION 1) THE DHWT JMM MINICOURSE #4 3 / 14

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