Chapter 15: Wavelets (i) Fourier spectrum provides all the frequencies present in a signal but does not tell where they are present. (ii) Fourier transform requires that the entire signal to be transformed be readily available. Windowed Fourier transform suffers from Small range – poor frequency resolution Large range – poor localization

Wave Wavelet Wavelet: wave that is only nonzero in a small region Types of wavelets:

Haar: Morlet: Mexican hat: DOG, LOG

○ Operations on wavelet: (a) Dilation: i) Squashing ii) Expanding (b) Translation: i) Shift to the right ii) Shift to the left (c) Magnitude change: i) Amplification ii) Minification

Any function can be expressed as a sum of wavelets of the form

Inverse wavelet transform: 2 new variables: scale translation Wavelets: Mother wavelet: Inverse wavelet transform:

Discrete wavelet transform: Approximation coefficients (cA) : scaling functions Detail coefficients (cD) : wavelet functions Inverse discrete wavelet transform:

Scales and positions are often based on a power of 2 Example: Haar wavelet Scaling functions

Wavelet functions

Haar basis functions: Haar spaces: Properties: i) , ii) iii)

Fast Wavelet Transform (FWT) Discrete signal to be transformed into wavelet coefficients cA: approximate coef. cD: detailed coef.

Inverse discrete wavelet transform:

2-D:

Wavelet coefficients: (s, d). ○ Wavelet transforms work by taking low pass filtering (e.g., average) and high pass filtering (e.g., difference) of input values Example: Input data: a, b Average: s = (a + b) / 2 (low pass filtering) Difference: d = a – s (high pass filtering) Wavelet coefficients: (s, d). ○ Inverse wavelet transforms work by taking addition and subtraction of wavelet coefficients Example: Input (s, d) Addition: s + d = s + (a – s) = a, Subtraction: s – d = s – (a – s) = 2s – a = 2 (a + b) / 2 – a = b Inverse wavelet coefficients: (a, b).

。Example: Input data 14, 22 Average: s = (14＋22)/2 = 18, Difference: d = 14-18= -4 Wavelet transform result: (18, -4). To recover the input numbers: s ＋ d = 18+(-4) = 14, s － d = 18-(-4) = 22 Inverse wavelet transform result: (14, 22). 。Example: (multiple data) Input vector: v = [71 67 24 26 36 32 14 18] Average vector: s1 = [(71+67)/2 (24+26)/2 (36+32)/2 14+18)/2] = [69 25 34 16] Difference vector: d1 = [71-69 24-25 36-34 14-16] = [2 -1 2 -2]

Wavelet transform at 1 scale: v1 = [ s1 d1 ] = [ 69 25 34 16 2 -1 2 -2 ] Average vector: s2 = [ (69+25)/2 (34+16)/2 ] = [ 47 25] Difference vector: d1 = [ 69-47 34-25 ] = [ 22 9 ] Wavelet transform at 2 scale: v2 = [ s2 d2 ] = [ 47 25 22 9 2 -1 2 -2 ] Wavelet transform at 3 scale: v3 = [ s3 d3 ] = [ 36 11 22 9 2 -1 2 -2 ]

Recover the original input signal From v3 = [36 11 22 9 2 -1 2 -2] [36+11 36-11 22 9 2 -1 2 -2] = [47 25 22 9 2 -1 2 -2] [47+22 47-22 25+9 25-9 2 -1 2 -2] = [69 25 34 16 2 -1 2 -2 ] Inverse wavelet transform results: [ 69+2 69-2 25-1 25+2 34+2 34-2 16-2 16+2] = [ 71 67 24 26 36 32 14 18 ]