What is Wavelet? ( Wavelet Analysis)

1. What is Wavelet? ( Wavelet Analysis) • Wavelets are functions that satisfy certain mathematical requirements and are used to represent data or other functions • Idea is not new--- Joseph Fourier--- 1800's • Wavelet-- the scale we use to see data plays an important role • FT non local -- very poor job on sharp spikes Waveletdb10 Sine wave

2. History of wavelets • 1807 Joseph Fourier- theory of frequency analysis-- any 2pi functions f(x) is the sum of its Fourier Series • 1909 Alfred Haar-- PhD thesis-- defined Haar basis function---- it is compact support( vanish outside finite interval) • 1930 Paul Levy-Physicist investigated Brownian motion ( random signal) and concluded Haar basis is better than FT • 1930's Littlewood Paley, Stein ==> calculated the energy of the function 1960 Guido Weiss, Ronald Coifman-- studied simplest element of functions space called atom • 1980 Grossman (physicist) Moorlet( Engineer)-- broadly defined wavelet in terms of quantum mechanics • 1985 Stephen Mallat--defined wavelet for his Digital Signal Processing work for his Ph.D. • Y Meyer constructed first non trivial wavelet • 1988 Ingrid Daubechies-- used Mallat work constructed set of wavelets • The name emerged from the literature of geophysics, by a route through France. The word onde led to ondelette. Translation wave led to wavelet

3. Fourier Series and Energy

4. Functions • Functions (Science and Engg) often use time as their parameter • g(t)-> represent time domain • since typical function oscillate – think it as wave– so G(f) where f= frequency of the wave, the function represented in the frequency domain • A function g(t) is periodic, there exits a nonzero constant P s.t. g(t+P)=g(t) for all t, where P is called period • periodic function has 4 important attributes • Amplitude– max value it has in any period • Period---2P • Frequency f=1/P(inverse)– cycles per second, Hz • Phase—Cos is a Sin function with a phase

5. Fourier, Haar • Amplitude, time  amplitude , frequency • 1965 Cooley and Tukey – Fast Fourier Transform • Haar

6. CWT • continuous wavelet transform (CWT) of a function f(t) a mother wavelet • mother wavelet may be real or complex with the following properties • 1.the total area under the curve=0, • 2. the total area of is finite • 3. Admissible condition • oscillate above and below the t-axis • energy of the function is finite function is localize • Infinite number of functions satisfies above conditions– some of them used for wavelet transform • example • Morlet wavelet • Mexican hat wavelet

7. once a wavelet has been chosen , the CWT of a square integrable function f(t) is defined as * denotes complex conjugate For any a, Thus b is a translation parameter Setting b=0, Here a is a scaling parameter a>1 stretch the wavelet and 0<a<1 shrink it

8. Wavelets Fourier Transform CWT = C( scale, position)= Scaling wave means simply Stretching (or Shrinking) it Shifting f (t) f(t-k)

9. Wavelets Continue • Wavelets are basis functions in continuous time • A basis is a set of linearly independent function that can be used to produce a function f(t) • f(t) = combination of basis function = • is constructed from a single mother wave w(t) -- normally it is a small wave-- it start at 0 and ends at t=N • Shrunken ( scaled) • shifted • A typical wavelet compressed j times and shifted k times is • Property:- Remarkable property is orthogonality i.e. their inner-products are zero • This leads to a simple formula for bjk

10. Haar Transform • Digitized sound, image are discrete.  we need discrete wavelet • where ck and dj,k are coefficients to be calculated • example:- consider the array of 8 values (1,2,3,4,5,6,7,8) • 4 average values 4 difference ( detail coefficients) • calculate average, and difference for 4 averages • continue this way • Method is called PYRAMID DECOMPOSITION • Haar transform depends on coeff ½, ½ and ½, - ½ • if we replace 2 by √2 then it is called coarse detail and fine detail

11. Transforms • Transform of a signal is a new representation of that signal • Example:- signal x0,x1,x2,x3 define y0,y1,y2,y3 • Questions • 1. What is the purpose of y's • 2. Can we get back x's • Answer for 2: The Transform is invertible-- perfect reconstruction • Divide Transform in to 3 groups • 1. Lossless( Orthogonal)-- Transformed Signal has the same length • 2. Invertible (bi-orthogonal)-- length and angle may change-- no information lost • 3. Lossy ( Not invertible)--

12. Answer to Q1: Purpose • IT SEES LARGE vs SMALL • X0=1.2, X1= 1.0, x2=-1.0, x3=-1.2 • Y=[2.2 0 -2.2 0] • Key idea for wavelets is the concept of " SCALE" • We can take sum and difference again==> recursion => Multiresolution • Main idea of Wavelet analysis– analyze a function at different scales– mother wavelet use to construct wavelet in different scale and translate each relative to the function being analyzed • Z=[ 0 0 4.4 0 ] • Reconstruct =====>compression 4:1

20. Real electricity consumption • peak in the center, followed by two drops, shallow drop, and then a considerably weaker peak • d1 d2 shows the noise • d3– presents high value in the beginning and at the end of the main peak, thus allowing us to locate the corresponding peak • d4 shows 3 successive peak– this fits the shape of the curve remarkably • a1,a2 strong resemblance • a3 reasonable---- a4 lost lots of information

25. JPEG (Joint Photographic Experts Group) • 1. Color images ( RGB) change into luminance, chrominance, color space • 2. color images are down sampled by creating low resolution pixels – not luminance part– horizontally and vertically, ( 2:1 or 2:1, 1:1)– 1/3 +(2/3)*(1/4)= ½ size of original size • 3. group 8x8 pixels called data sets– if not multiple of 8– bottom row and right col are duplicated • 4. apply DCT for each data set– 64 coefficients • 5. each of 64 frequency components in a data unit is divided by a separate number called quantization coefficients (QC) and then rounded into integer • 6. QC encode using RLE, Huffman encoding, Arithmetic Encoding ( QM coder) • 7. Add Headers, parameters, and output the result • interchangeable format= compressed data + all tables need for decoder • abbreviated format= compressed data+ not tables ( few tables) • abbreviated format =just tables + no compressed data • DECODER DO THE REVERSE OF THE ABOVE STEPS

26. JPEG 2000 or JPEG Y2k • divide into 3 colors • each color is partitioned into rectangular, non-overlapping regions called tiles– that are compressed individually • A tile is compressed into 4 main steps • 1. compute wavelet transform – sub band of wavelets– integer, fp,---L+1 levels, L is the parameter determined by the encoder • 2. wavelet coeff are quantized, -- depends on bit rate • 3. use arithmetic encoder for wavelet coefficients • 4. construct bit stream– do certain region, no order • Bit streams are organized into layers, each layer contains higher resolution image information • thus decoding layer by layer is a natural way to achieve progressive image transformation and decompression

29. A H D V

31. Lowpass Filter = Moving Average • y(n)= x(n)/2 + x(n-1)/2 here h(0)=1/2 and h(1)=1/2 • Fits standard form for k=0,1 • x= unit impulse • x=( ...0 0 0 0 1 0 0 0...) then y=( ...0 0 1/2 1/2 0 0..) • average filter= 1/2 (identity) + 1/2 (delay) • Every linear operator acting on a single vector x can be rep by y=Hx • main diagonal come from identity--subdiagonal come from delay • we have finite ( two ) coefficients--> FIR finite impulse response • low pass==> scaling function • It smooth out bumps in the signal(high freq component

32. Highpass Filter Moving Difference • y(n)= 1/2[x(n)-x(n-1)] • h(0)=1/2 • h(1)=-1/2 • y=H1x • Filter Bank === Lowpass and Highpass • they separate the signal into frequency bank • Problem:-- Signal length doubled, • both are same size as signal ==> gives double size of the original signal • Solution:-- Down Sampling

33. Down Sampling • We can keep half of Ho and H1 and still recover x • Save only even-numbered components ( delete odd numbered elements) -- denoted by (↓2)-- decimation • (↓2)y = (... y(-4) y(-2)y(0)y(2).......) • Filtering + Down sampling ==> Analysis Bank ( brings half size signal) • Inverse of this process==> Synthesis bank • i,e, Up sampling + Filtering • Add even numbered components zeros ( It will bring full size) denoted by (↑2) • y = (↓2 y)= (↑2)(↓2 y)

34. Scaling function and Wavelets • corresponding to low pass--> there is scaling function • corresponding to high pass--> there is wavelet function • dilation equation--> scaling function • In terms of original low pass filters • we have • for h(0) and h(1) = 1/2 we have • the graph compressed by 2 gives and shifted by 1/2 gives • By similar way the wavelet equation

35. Wavelet Packet • Walsh-Hadamard transform-- complete binary tree --> wavelet packet • "Hadamard matrix"==> all entries are 1 and -1 and all rows are orthogonal-- divide two time by sqrt(2)==> orthogonal & symmetric • Compare with wavelet-- computations sums z0=0 sums y0 and y2 difference z2=4.4 x sums z1=0.4 difference y1 and y3 difference z3=0

36. Filters and Filter Banks • Filter is a linear time-invariant operator • It acts on input vector x --- Out put vector y is the convolution of x with a fixed vector h • h--> contains filter coefficients-- our filters are digital not analog-- h(n) are discrete time t= nT, • T is sampling period assume it is 1 here • x(n) and y(n) comes all the time t= 0, +_ 1.... • y(n) = Σh(k) x(n-k) = convolution h* x in the time domain • Filter Bank= Set of all filters • Convolution by hand--- arrange it as ordinary multiplication -- but don't carry digits from one column to another • x= 3 2 4 h= 1 5 2 • x * h = 3 17 20 24 8