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 Sine wave Waveletdb10 Wavelet db10

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 (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 periodic function has 4 important attributes Amplitude– max value it has in any periodAmplitude– max value it has in any period Period---2PPeriod---2P Frequency f=1/P(inverse)– cycles per second, HzFrequency f=1/P(inverse)– cycles per second, Hz Phase—Cos is a Sin function with a phasePhase—Cos is a Sin function with a phaseFunctions

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

6.  continuous wavelet transform (CWT) of a function f(t) a mother wavelet mother wavelet may be real or complex with the following properties mother wavelet may be real or complex with the following properties 1.the total area under the curve=0,1.the total area under the curve=0, 2. the total area of is finite2. the total area of is finite 3. Admissible condition3. Admissible condition oscillate above and below the t-axisoscillate above and below the t-axis energy of the function is finite  function is localizeenergy of the function is finite  function is localize Infinite number of functions satisfies above conditions– some of them used for wavelet transform Infinite number of functions satisfies above conditions– some of them used for wavelet transform example example Morlet waveletMorlet wavelet Mexican hat waveletMexican hat waveletCWT

7.  once a wavelet has been chosen, the CWT of a square integrable function f(t) is defined as * denotes complex conjugate * 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 1  stretch the wavelet and 0<a<1 shrink it

8. WaveletsWavelets Fourier Transform CWT = C( scale, position)= Scaling wave means simply Stretching (or Shrinking) it (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 b jk

10.  Haar Transform Digitized sound, image are discrete.  we need discrete wavelet Digitized sound, image are discrete.  we need discrete wavelet where c k and d j,k are coefficients to be calculated where c k and d j,k are coefficients to be calculated example:- consider the array of 8 values (1,2,3,4,5,6,7,8) example:- consider the array of 8 values (1,2,3,4,5,6,7,8) 4 average values  4 difference ( detail coefficients) 4 average values  4 difference ( detail coefficients) calculate average, and difference for 4 averages calculate average, and difference for 4 averages continue this way continue this way Method is called PYRAMID DECOMPOSITION 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 1. What is the purpose of y's 2. Can we get back x'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 1. Lossless( Orthogonal)-- Transformed Signal has the same length 2. Invertible (bi-orthogonal)-- length and angle may change-- no information lost 2. Invertible (bi-orthogonal)-- length and angle may change-- no information lost 3. Lossy ( Not invertible)-- 3. Lossy ( Not invertible)--  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 1. What is the purpose of y's 2. Can we get back x'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 1. Lossless( Orthogonal)-- Transformed Signal has the same length 2. Invertible (bi-orthogonal)-- length and angle may change-- no information lost 2. Invertible (bi-orthogonal)-- length and angle may change-- no information lost 3. Lossy ( Not invertible)-- 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]  SCALE  Key idea for wavelets is the concept of " SCALE "  Multiresolution  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 decoderinterchangeable format= compressed data + all tables need for decoder abbreviated format= compressed data+ not tables ( few tables)abbreviated format= compressed data+ not tables ( few tables) abbreviated format =just tables + no compressed dataabbreviated 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 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 2. wavelet coeff are quantized, -- depends on bit rate 3. use arithmetic encoder for wavelet coefficients 3. use arithmetic encoder for wavelet coefficients 4. construct bit stream– do certain region, no order 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. H V D A

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  orthogonal orthogonal symmetric  "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 x sums y0 and y2 difference y1 and y3 sums z0=0 sums z1=0.4 difference z2=4.4 difference z3=0

36. Filters and Filter Banks Filters and Filter Banks  Filter  Filter is a linear time-invariant operator  xy h  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....  h* x  y(n) = Σh(k) x(n-k) = convolution h* x in the time domain  Filter Bank  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