Lecture11_Wavelets

Lustig eecs uc berkeley m lustig eecs uc berkeley 17

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Unformatted text preview: function M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 17 18 Haar Scaling function 8 &lt; 1 0t&lt; 1 2 1 1 t&lt;1 ( t) = 2 : 0 otherwise ( t) = ⇢ Back to Discrete • Early 80’s, theoretical work by Morlett, Grossman and Meyer (math, geophysics) • Late 80’s link to DSP by Daubechies and Mallat. 1 0t&lt;1 0 otherwise • From CWT to DWT not so trivial! • Must take care to maintain properties M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 19 20 Discrete Wavelet Transform d s ,u = a s ,u = N1 X n=0 N1 X Discrete Wavelet Transform x [ n] x [ n] s ,u [ n] d s ,u = s ,u [ n] a s ,u = n=0 d00 N1 X n=0 N1 X x [ n] s ,u [ n] x [ n] s ,u [ n] n=0 d01 d02 d00 d03 ! d01 d02 d03 ! d10 d11 d10 a10 d20 a20 ﬁnest scale d11 a11 stop here: n n M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 21 22 Example: Discrete Haar Wavelet Haar for n=2 Discrete Orthogonal Haar Wavelet Haar for n=8 1 p 2 scaling Φ20 1 p 2 scaling function Ψ00 1 p 8 1 p 2 0 ,0 0,0 1 p 2 Ψ20 Ψ01 1 p 8 1 p 2 mother wavelet Ψ10 approximation detail d00 Ψ11 ! Ψ02 1 2 1 p 2 1 2 Ψ03 a00 Equivalent to DFT2! ! t M. Lustig, EECS UC Berkeley 23 t M. Lustig, EECS UC Berkeley 24 Fast DWT with Filter Banks (more Later!) h0[n] x [ n] Fast DWT with Filter Banks h0[n] h1[n] h0[n] h1[n] h1[n] not quite... too many d0n? coefﬁcien...
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