Wavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)

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1 Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2π-periodic square-integrable function is generated by a superposition of integral dilations of the basic function e jx. The time information is lost; no knowledge of when a particular event took place. Look for wavelets---small waves---to generate the basis. Use a single function ψ (mother wavelet) with fast decay (compact support) to generate all the basis elements. Need to shift as well as scale the mother wavelet. Continuous and discrete wavelets Wavelet requirements Zero mean, oscillatory (wave) ψ (x) dx = Ψ(ω) 2 Admissibility condition Fast decay (let) compact support Regularity condition ω dω < 2 Stationary signal 2 Hz + Hz + 2Hz Example signals 3 6 Occur at all times Stationary Signal Signals with frequency content unchanged in time All frequency components exist at all times Stationary Non-stationary Signal Frequency changes in time One example: the Chirp Signal.-.4: 2 Hz : Hz +.7-.: 2 Hz Non- Stationary Time Frequency (Hz) Do not appear at all times Time Frequency (Hz) 4

2 Chirp signal Resolution of time & frquency Frequency: 2 Hz to 2 Hz Frequency: 2 Hz to 2 Hz Different in Time Domain Better time resolution; Poor frequency resolution Frequency Time Frequency (Hz) -.5 Same in Frequency Domain At what time the frequency components occur? FT can not tell! Time Frequency (Hz) Better frequency resolution; Poor time resolution Time 6 Decomposing non-stationary signals Decomposing non-stationary signals f L f L Signal:.-.4: 2 Hz.4-.7: Hz.7-.: 2 Hz Signal:.-.4: 2 Hz.4-.7: Hz.7-.: 2Hz f H Wavelet: db4 Level: 6 7 f H Wavelet: db4 Level: 6 8 2

3 Discrete wavelets: binary dilation and dyadic translation Use basis functions ψ j,k (x) = 2 j/2 ψ (2 j x-k), j,k є Z compression by 2 j (scale = 2 -j ) translation by k2 -j Example of compression and translation scaling by 2 j/2 to preserve L 2 norm A function ψ is called an orthogonal wavelet if the above family of functions is an orthonormal basis for L 2 (R), <ψ j,k, ψ l,m > = δ j,l δ k,m j,k,l,m є Z δ j,k = if j = l and k = m, and otherwise. the family of functions spans L 2 (R). Orthogonal and nested subspaces Let W j be the span of {ψ j,k, k є Z} W j W k for j k (case of orthogonal wavelets) L 2 (R) = W j = W - W W more details j є Z denotes orthogonal sum Let V j = W j-2 W j- V j+ = V j W j V = V - W - = V -2 W -2 W - = Consider the following conditions. {} V - V V (nested subspaces) 2. closure ( V j ) = L 2 (R) j є Z j є Z 3. V j = {} 4. φ j,k (x) = 2 j/2 φ (2 j x-k), j,k є Z 5. {φ j,k (x) k є Z} generates an orthonormal basis for V j. 9 Scaling function A function φ is called a scaling function (also called father wavelet) if it generates a nested sequence of subspaces that satisfies conditions on the previous slide. φ generates V j ψ generates W j Example of space generated by Haar scaling function. Decomposition I(x) = c 2 2 (x) + c 2 2 (x) + c (x) + c (x) V V - W - V -2 W -2 W - 2 3

4 Decomposition Decomposition c N c N- c N-2 d N- d N-2 d N-M c N-M Decompose each basis vector of V j into basis vectors of V j- and W j-. This results in a coefficient for each basis vector of V j- and W j-. Could also take inner products directly with the basis vectors. 3 Since V j is the orthogonal sum of V j- and W j-, there are decomposition sequences {p i }, {q i } such that φ j,k (x) = [ p k,m φ j-,m (x) + q k,m ψ j-,m (x) ] m This is called the decomposition relation 4 Decomposition f j (x) Reconstruction c N c N- c N-2 c N-M d N- d N-2 d N-M c j,k φ j,k (x) c j,k+ φ j,k+ (x) c j,k+2 φ j,k+2 (x) p k,m p k+,m p k+2,m φ j-,m (x) c j-,m = p l,m c j,l l ψ j-,m (x) d j-,m = q l,m c j,l l 5 We have a coefficient for each basis vector of V j- and W j-. Decompose each basis vector of V j- into basis vectors of V j. Repeat for W j-. Coefficient of each basis of V j as a function of coefficients of bases of V j- and W j- This results in a coefficient for each basis vector of V j. 6 4

5 Two-scale relation Since φ(2 j- x-k) and ψ(2 j- x-k) belong to V j, there exist reconstruction sequences {a i }, {b i } such that φ j-,k (x) = a k,m φ j,m (x) m ψ j-,k (x) = b k,m φ j,m (x) m The above relation is called the two-scale relation of the scaling function and the wavelet respectively. Reconstruction c j-,k φ j-,k (x) c j-,k+l φ j-,k+l (x) d j-,k ψ j-,k (x) a k,m a k+l,m b k,m φ j,m (x) c j,m = (a l,m c j-,l + b l,m d j-,l ) l d j-,k+l ψ j-,k+l (x) b k+l,m 7 8 Example: Haar wavelet ψ, (x) = for x <.5, - for.5 x <, otherwise. φ, (x) = for x <, otherwise. Check the orthonormality conditions Decomposition φ j,2k (x) = / 2 [φ j-,k (x) + ψ j-,k (x)] φ j,2k+ (x) = / 2 [φ j-,k (x) - ψ j-,k (x)] c j-,k (x) = / 2 [c j,2k + c j,2k+ ] d j-,k (x) = / 2 [c j,2k - c j,2k+ ] Reconstruction φ j-,k (x) = / 2 [φ j,2k (x) + φ j,2k+ (x)] ψ j-,k (x) = / 2 [φ j,2k (x) - φ j,2k+ (x)] c j,2k (x) = / 2 [c j-,k + d j-,k ] c j,2k+ (x) = / 2 [c j-,k - d j-,k ] Decomposition relation Two-scale relation c =[4,2,3,3,6,8,,7] Representation 4 φ, (x) 2 φ, (x-) 3 φ, (x-2) / 2 / 2 Decomposition 3 2φ -, (x) c - =[3 2,3 2,7 2,4 2 ] d - =[ 2,,- 2,-3 2] c -2 =[6,] d -2 =[,3] c -3 =[7/ 2] d -3 =[- 5/ 2] 2ψ -, (x) 9 2 5

6 Reconstruction Decomposition matrix c - =[3 2,3 2,7 2,4 2 ] d - =[ 2,,- 2,-3 2] / 2 / 2 / 2 -/ 2 c =[4,2,3,3,6,8,,7] / c, c, c,2 c,3 c,4 c,5 c,6 c,7 = c -, d -, c -, d -, c -,2 d -,2 c -,3 d -, Reconstruction matrix Example: Daubechies wavelet (D 4 ) - / c -, d -, c -, d -, c -,2 d -,2 c -,3 d -,3 = c, c, c,2 c,3 c,4 c,5 c,6 c,7 Scaling function φ (x) Wavelet function ψ (x) (h, h, h2, h3) = (+ 3, 3+ 3, 3-3, - 3)/

7 Decomposition matrix Reconstruction matrix h h h 2 h 3 h 3 -h 2 h -h h h h 2 h 3 h 3 -h 2 h -h h h h 2 h 3 h 3 -h 2 h -h h 2 h 3 h h c, c, c,2 c,3 c,4 c,5 c,6 c,7 = c -, d -, c -, d -, c -,2 d -,2 c -,3 d -,3 h h 3 h 2 h h -h 2 h 3 -h h 2 h h h 3 h 3 -h h -h 2 h 2 h h h 3 h 3 -h h -h 2 h 2 h h h 3 h 3 -h h -h 2 c -, d -, c -, d -, c -,2 d -,2 c -,3 d -,3 = c, c, c,2 c,3 c,4 c,5 c,6 c,7 h -h h 3 -h Wavelets in 2-d (standard decomposition) Apply wavelet transform along the rows. Decompose the signal into a set of coefficients. Apply transform along the columns on the row coefficients. Wavelets in 2-d (non-standard decomposition) Alternate between row and column transformations

8 Parseval s theorem: Properties f 2 = <f,ψ j,k > 2 f-g 2 = <f,ψ j,k > - <g,ψ j,k > 2 Also holds for the decomposition into a set of multiresolution details and a single summary. f N = g N- + g N f N-M Time complexity of decomposition Each step is O(n) The amount of work gets halved at each step Overall complexity is still O(n) Faster than DFT Time complexity of reconstruction is also O(n). Applications Images and videos Compression Smoothing and Denoising Big Data Dimensionality reduction Summarization Which would you use Fourier Wavelets (which one?) 29 3 Bibliography Wavelets and Dilation Equations: A Brief Introduction, Gilbert Strang, SIAM Review, Vol. 3, No. 4. (Dec., 989), pp

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