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# CALCULUS I MATH 141

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This 49 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 141 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/229545/math-141-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

The Hitchhiker s Guide to the Dual Tree Complex Wavelet Transform iDON T PANICi October 267 2007 Outline The Hilbert Transform De nition The Fourier Transform Definition Invertion Fourier Approximation The Wavelet Transform Definition Invertion Wavelet Approximation The Dual Tree Complex Wavelet Transform Definition The Hilbert Transform De nition Hilbert Transform Given a real valued function of a real variable7 f R gt R de ne Hf R a R by Hm 11ml Md H 7 Irrgt6 5 The Fourier Transform De nition Fourier Transform Given a complex valued function of real variable f R gt C de ne Ff R gtbe me 0 eeW dx ltfltxgt W5 Invertion 0f the Fourier Transform Theorem F07 f good enough flt96gt 00 WWW d5 00 We e2m gt W5 d5 Hilbert Transform Via Fourier Transform f 96 gt Hfygt Hilbert Transform Via Fourier Transform f 96 gt Hfygt fflt gt fH Hilbert Transform Via Fourier Transform f 96 gt Hfygt f f f isign5ff5 Hilbert Transform Via Fourier Transform f 96 gt Hfygt f f arm T isignlt gtfflt gt Hilbert Transform Via Fourier Transform arm T isignlt gtfflt gt Discretization 0f the Fourier Transform Theorem F07 f good enough oo fflt gt 2mng OO 27rm gt 27riit 27ringt 27rin foltngt 27rmn nEZ neZ Fourier Approximation N Use partial sums Z ffn62mm to approximate f niN Fourier Approximation N Use partial sums Z ffn62mm to approximate f niN f and 3 term approximation Fourier Approximation N Use partial sums Z ffn62mm to approximate f niN f and 5 term approximation Fourier Approximation N Use partial sums Z ffn62mm to approximate f niN f and 11 term approximation Fourier Approximation N Use partial sums Z ffn62mm to approximate f niN f and 33 term approximation Fourier Approximation N Use partial sums Z ffn62mm to approximate f niN x y y 1 5 Gibbs phenomenom 05 0 A05 1 45 I 2 I 5 i as U 05 I 1 5 2 f and 33 term approximation Fourier Approximation N Use partial sums Z ffn62mm to approximate f niN f and 65 term approximation The Wavelet Transform De nition Wavelets Functions w R gt R satisfying the admissibility condition lfwlt gt2 1 Denote the shifts and dilations of the wavelet by 1mm zWT b a b e n The Wavelet Transform De nition Wavelets Functions w R gt R satisfying the admissibility condition lfwlt gt2 1 Denote the shifts and dilations of the wavelet by 1mm zWT b a b e n De nition Wavelet Transform Given a real valued function of real variable f R gt R7 de ne W f R2 gt R by Wfa7bgt f fltwgtwabltwgt doc lt1 pm Invertion 0f the Wavelet Transform Theorem F07 f good enough flt96gt Wflta7bgtwmbltacgt cw Discretization 0f the Wavelet Transform Theorem F07 f good enough fa 00 00 Wfabzpabdbj g Z ZW Z myanWrmmng mEZ nEZ Wavelet Approximation Use partial sums 2534 ZJEN Vf2m7 n27quot ibg mmgm to approximate f Industrial Mathemmi Institute I Lum a Sun m ulmu Wavelet Approximation Use partial sums 2534 ZJEN Vf2m7 n27quot wg mmgm to approximate f 2 15 if I L 7 I15 I I I I I 2 15 1 435 El I15 1 15 2 f and 1 term approximation Industrial M ih 39 l ii Institute ruinnu I Sun mam Wavelet Approximation Use partial sums 2534 ZJEN Vf2m7 n27quot wg mmgm to approximate f Is 7 1 5 us 7 u m 1 39 4 45 77 45 71 VHS El US I IS I I I I I 2 15 1 435 El I15 1 15 2 f and 2 term approximation Industrial Mathenlmi Institute ruinnu I Sun mam Wavelet Approximation Use partial sums 2534 ZJEN Vf2m7 n27quot wg mmgm to approximate f Is 7 1 5 us 7 u m 1 39 4 45 77 45 71 VHS El US I IS El 5 5 1 15 2 I I I I I 2 15 1 435 I 05 1 15 2 f and 4 term approximation Industrial Mathenlmi Institute Min1quot I Sun mam Wavelet Approximation Use partial sums 2534 ZJEN Vf2m7 n27quot wg mmgm to approximate f I I I I I 2 15 1 435 El I15 1 15 2 f and 8 term approximation Industrial M ih 39 l ii Institute ruinnu I Sun II Wavelet Approximation Use partial sums 2534 ZJEN Vf2m7 n27quot wg mmgm to approximate f Is 7 1 5 us 7 u m 1 39 4 45 77 45 71 VHS El US I IS I I I I I 2 15 1 435 El I15 1 15 2 f and 16 term approximation Industrial Mathenlmi Institute ruinnu I Sun mam Wavelet Approximation 1024 x 1 024 1048576 pixels Wavelet Approximation 1024 gtlt 1 024 1 048 576 pixels 1 Wavelet coef cient Wavelet Approximation 1024 gtlt 1 024 1 048 576 pixels 1 4 5 Wavelet coef cients Wavelet Approximation 1024 gtlt 1 024 1 048 576 pixels 1 4 16 21 Wavelet coef cients Wavelet Approximation 1024 gtlt 1 024 1 048 576 pixels 1 4 16 64 85 Wavelet coef cients my lalhrmmvrslnmlnm Wavelet Approximation 1024 gtlt 1 024 1 048 576 pixels 1 4 16 64 256 341 Wavelet coef cients my lalhrmmvrslnmlnm Wavelet Approximation 1024 gtlt 1 024 1 048 576 pixels 1 4 16 64 256 1024 1365 Wavelet coef cients mhlm lalhrmmvrslnmlnm Wavelet Approximation 1 024 gtlt 1 024 1048 576 pixels 1 4 16 64 256 1024 4096 5461 Wavelet coef cients IMI mhlm lalhrmmvrslnmlnm Problems With Real Wavelets Poor Directional Selectivity The standard tensor product construction of multi variate wavelets produces a checkerboard pattern that is simultaneously oriented along several directions This lack of directional selectivity complicates processing of geometric image features like ridges and edges N Kingsbmy Complex Wavelets for Shift Invariant Analysis and Filtering of Signals77 Attempts to solve this problem in the last 10 years gt Brushlets Meyer amp Coifmatn7 1997 Attempts to solve this problem in the last 10 years gt Brushlets Meyer amp Coifmatn7 1997 gt Ridgelets Catnd s7 1998 Attempts to solve this problem in the last 10 years gt Brushlets Meyer amp Coifmatn7 1997 gt Ridgelets Catnd s7 1998 gt Curvelets Cand s amp Donoho7 1999 Attempts to solve this problem in the last 10 years gt Brushlets Meyer amp Coifmatn7 1997 gt Ridgelets Catnd s7 1998 gt Curvelets Cand s amp Donoho7 1999 gt Wedgelets Donoho7 1999 Attempts to solve this problem in the last 10 years Brushlets Meyer amp Coifmatn7 1997 Ridgelets Catnd s7 1998 Curvelets Cand s amp Donoho7 1999 Wedgelets Donoho7 1999 Beamlets Donoho amp H1107 2001 VVVVV Attempts to solve this problem in the last 10 years Brushlets Meyer amp Coifmatn7 1997 Ridgelets Catnd s7 1998 Curvelets Cand s amp D01r10h07 1999 Wedgelets Donoho7 1999 Beamlets Donoho amp H1107 2001 Sur ets Betratlniuk7 2004 VVVVVV Attempts to solve this problem in the last 10 years Brushlets Meyer amp Coifmatn7 1997 Ridgelets Catnd s7 1998 Curvelets Cand s amp D01r10h07 1999 Wedgelets Donoho7 1999 Beamlets Donoho amp H1107 2001 Sur ets Betratlniuk7 2004 Shearlets G Kutyniok7 2005 VVVVVVV Attempts to solve this problem in the last 10 years Brushlets Meyer amp Coifmatn7 1997 Ridgelets Catnd s7 1998 Curvelets Cand s amp D01r10h07 1999 Wedgelets Donoho7 1999 Beamlets Donoho amp H1107 2001 Sur ets Betratlniuk7 2004 Shearlets G Kutyniok7 2005 Needlets P Petrushev7 2005 VVVVVVVV Attempts to solve this problem in the last 10 years Brushlets Meyer amp Coifmatn7 1997 Ridgelets Catlnde s7 1998 Curvelets Cande s amp D01r10h07 1999 Wedgelets Donoho7 1999 Beamlets Donoho amp H1107 2001 Sur ets Betratlniuk7 2004 Shearlets G Kutyniok7 2005 Needlets P Petrushev7 2005 Your name herelets VVVVVVVVV The Dual Tree Complex Wavelet Transform N Kingsbuly Complex Wavelets for Shift Invariant Analysis and Filtering of Signals77 The key Hilbert Transform pairs Use complex valued functions 11 R gt C satisfying Mac tH1 ltl Where both it and Hit are real valued That s a neat idea If both in and Hw are wavelets7 we perform two different wavelet transforrns7 one with 1 one with Hw For each choice a7 1 E R combine the corresponding real valued wavelet coef cients waib and f7Hwaib to form a single complex valued coef cient ltf7 glab ltf7 wmb Hwabgt39 That s a neat idea If both w and Hw are wavelets7 we perform two different wavelet transforms7 one with 1 one with Hw For each choice a7 1 E R combine the corresponding real valued wavelet coef cients waib and f7Hwaib to form a single complex valued coef cient ltf7 me ltf7 wmb Hwabgt39 WARNING The Dual Tree Complex Wavelet Transform is not a transform per se It is a smart way to combine the information we obtain from two transforms

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