3. Lifting Scheme of Wavelet Transform
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- Ginger Miller
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1 3. Lifting Scheme of Wavelet Transform 3. Introduction The Wim Sweldens 76 developed the lifting scheme for the construction of biorthogonal wavelets. The main feature of the lifting scheme is that all constructions are derived in the spatial domain. It does not require complex mathematical calculations that are required in traditional methods. Lifting scheme is simplest and efficient algorithm to calculate wavelet transforms 8. It does not depend on Fourier transforms. Lifting scheme is used to generate second-generation wavelets, which are not necessarily translation and dilation of one particular function. It was started as a method to improve a given discrete wavelet transforms to obtain specific properties. Later it became an efficient algorithm to calculate any wavelet transform as a sequence of simple lifting steps. Digital signals are usually a sequence of integer numbers, while wavelet transforms result in floating point numbers. For an efficient reversible implementation, it is of great importance to have a transform algorithm that converts integers to integers. Fortunately, a lifting step can be modified to operate on integers, while preserving the reversibility. Thus, the lifting scheme became a method to implement reversible integer wavelet transforms. Constructing wavelets using lifting scheme consists of three steps: The first step is split phase that split data into odd and even sets. The second step is predict step, in which odd set is predicted from even set. Predict phase ensures polynomial cancellation in high pass. The third step is update phase that will update even set using wavelet coefficient to calculate scaling function. Update stage ensures preservation of moments in low pass. 3. Reasons for the choice of Lifting scheme We have used lifting scheme of wavelet transform for the digital speech compression because lifting scheme is having following advantages over conventional wavelet transform technique. 8
2 It allows a faster implementation of the wavelet transform. It requires half number of computations as compare to traditional convolution based discrete wavelet transform. This is very attractive for real time low power applications. The lifting scheme allows a fully in-place calculation of the wavelet transform. In other words, no auxiliary memory is needed and the original signal can be replaced with its wavelet transform. Lifting scheme allows us to implement reversible integer wavelet transforms. In conventional scheme it involves floating point operations, which introduces rounding errors due to floating point arithmetic. While in case of lifting scheme perfect reconstruction is possible for loss-less compression. It is easier to store and process integer numbers compared to floating point numbers. Easier to understand and implement. It can be used for irregular sampling 3.3 Lifting scheme based Haar wavelet transform Alfred Haar introduced the first wavelet systems 3 in the year 90. Wavelet systems of the Haar have been generalied to higher order dimension and rank. Two types of coefficients are obtained from the wavelet transform. Scaling coefficients are obtained by averaging two adjacent samples. These scaling coefficients represent a coarse approximation of the speech. Wavelet coefficients are obtained from the subtraction of two adjacent samples. Wavelet coefficients contain the fine details of the speech signal. The Haar wavelet is famous for its simplicity and speed of computation. Computation of the scaling coefficients requires adding two samples values and dividing by two. Calculation of the wavelet coefficients requires subtracting two samples values and dividing by two. The inverse transform simply requires subtraction and addition. Using logical shifts to perform division eliminates the need for a complex divide unit. Furthermore, 83
3 implementing a logical shift in hardware requires much less power and space than an arithmetic logic unit ALU. Given the computational requirements, the Haar wavelet is a simple and easy to implement transform. Computational simplicity makes the Haar transform a perfect choice for an initial design implementation. Let us consider a simple example of Haar wavelet: Let us consider that we have a discrete sequence fx which is obtained by sampling continuous signal such as speech signal. Consider two neighboring samples X and Y of this sequence. These two samples show strong correlation. Haar transform will replace value of X and Y by Average and difference: a = X+Y/, d = Y-X 3. Simple inverse Haar transform can be used calculate original value of sample X and Y: X = a d/ Y = a + d/ 3. Consider the sequence λ 0,k with n samples, where 0 k n. Apply average and difference on to each pair X=λ 0,k and Y=λ 0, k+ There are n/ such pairs. Average and difference can be calculated as follow: λ -, k = λ 0, k + λ 0, k+ / Average of neighbouring components 3.3 ϒ -, k = λ 0, k+ - λ 0, k Difference of neighbouring components 3. The input signal λ 0,k is divided splitted into two signals: λ -, k with n/ averages and ϒ -, k with n/ differences We can recover the original sequence from averages and differences. Here we can think averages as a coarser resolution signal and differences as fine detail resolution signals. If original signal has some local coherence i.e. sample values are smoothly varying, then coarse representation closely 8
4 resembles like original signal and detail is vary small. If detail signal is close to ero then it can be suppressed to compress data. The coarser signal λ -, k with n/ samples can be further divided into two signals λ -, k with n/ averages and ϒ -, k with n/ differences This process can be continued for N level i.e. up to λ -N, k and ϒ- N, k, where, N=Log n. This is known as Haar transform. In this transform, we end up with N detail signal and one coarser signal. Coarser signal represent overall average of the signal. Coarser signal contains one component λ -N,0 that is the average of all the samples of original signal which is ero frequency DC component of the signal. Haar forward transform: ϒ - ϒ - ϒ -3 ϒ -N λ 0 λ - λ - λ -3 λ -N N Samples N- Samples N- Samples N-3 Samples N-N Sample Haar inverse transform: ϒ -N ϒ -N+ ϒ - λ -N λ -N+ λ -N+ λ - λ 0 Haar transform and inverse transform can also be obtained by convolving x matrix with the signal. But this process demands more computation and memory. Hence better method is to obtain Haar transform with lifting scheme as explained below: Haar wavelet transform can be computed fast using Lifting scheme. If we want in-place computation, than we may use following steps. Store a at the same location as X. Store d at the same location as Y. Calculate d first. X, Y d=y-x Y = d replace value of Y 3.5 a=x+d/ X = a replace value of X
5 Equation 3.5 is known as predict stage in which difference between adjacent samples are stored at the odd locations. Effectively equation 3.5 can be written as Y = Y-X 3.7 Equation 3.6 is known as update stage in which average value of adjacent samples are stored at even locations as given in Equation 3.8 X = X+d/ = X + Y-X/ = X+Y/ Average value of adjacent samples 3.8 In case of in-place transform old values are overwritten by new values so it does not require intermediate buffers. Approximate value a is stored at the same place as that of X and detail value d is stored at the same place as that of Y. In order to perform in-place inverse Haar transform, above process is reversed. X represents a and Y represent d. Hence original value can be recovered by Equation 3.9 and 3.0 X = X Y/ = a d/ = X+Y/ Y-X/ 3.9 X replaces average value Y = Y+X = d+x = Y-X + X 3.0 Y replaces Difference value Looking at the equations 3.7 to 3.0, it is clear that inverse wavelet transform can be obtained by just changing the sign in forward wavelet transform. The Haar wavelet is simplest orthogonal wavelet having compact support. Though implementation of Haar wavelet is very simple, it is not attractive for the speech compression because it is not smooth and does not give good compression. 3. Factoring wavelet transform into lifting steps 8 The basic idea of wavelet transform is to exploit correlation structure present in most real life signals to build sparse approximation. The correlation structure is local in both frequency and time domain. Traditional wavelet transform use wavelet filters to build time frequency localisation. In this topic, we will discuss how to derive lifting steps from the wavelet filters. 86
6 3.. Poly-phase representation Let us consider the sequence of samples of signal fk. Z transform of this sequence can be given as, f k = f k 3. k Let us consider finite impulse response FIR filter h having filter coefficients h={h k,....., h k }. Z transform of this filter is Laurent polynomial with degree k -k given by, k = k k h = h k 3. k = k Filtering of signal fk by filter h can be easily described in transform by the Equation 3.3, y=hx 3.3 Sub-sampling of the signal fk is corresponding to keeping only the even samples i.e. f e =fk. Z transform of such sub-sampled signal can be given as f = k e f k 3. k f=f0 0 + f - + f - + f3-3 + f f-=f0 0 - f - + f - - f3-3 + f From equation 3.5 and 3.6, f+f- = f0 0 + f - + f - + f f Similarly, f f + f k e = = f k 3.8 k k o = f f = f k k From equation 3.8 and 3.9, it is clear that signal f can be decomposed into f e and f o by Equation 3.0, f= f e + - f o
7 88 Now let us consider that signal f decomposed into two parts using high pass filter g and low pass filter h, then it can be represented as: f g h hp lp = 3. Sub-sampling step corresponds to, f h f h lp lp lp LP e + = + = = 3. f g f g hp hp lp HP o + = + = = 3.3 The Equation 3. and 3.3 can be written in matrix form, = = f f g g h h lp lp HP LP o e 3. In this case we first calculate all the coefficients and then throw away half of the work done. It will be more efficient if we perform sampling before filtering, mean that we compute only even part of lp and hp. lp e =[hf] e = h e f e + - h o f o 3.5 Similarly, hp e =[gf] e =g e f e + - g o f o 3.6 Let us denote output of sub-sampler and low pass filter as λ and output of sub-sampler and high pass filter as ϒ. Then above two equations can be represented as, = f f P o e γ λ 3.7 Where, P is poly-phase matrix given by = g g h h P e e e e 3.8
8 In order to achieve perfect reconstruction, filter h and g must be complementary filters that will result unity determinant of poly-phase matrix. Polyphase matrix corresponding to lay wavelet transform will be P = 0 0 This poly-phase matrix will split input samples into odd and even set. 3.. Primal Lifting Update and Dual Lifting Predict Using primal lifting, new filter h new that is complementary to g is created from filters h and g. h new = h + s g 3.9 Using dual lifting, new filter g new that is complementary to h is created from filters h and g. g new = g + t h 3.30 Following poly-phase representation can represent Equations 3.9 and P new = h new e h new o = s h e h o 3.3 g e g e 0 g e g o P new = h e h o = 0 h e h o 3.3 g new e g new e t g e g o 3.. Decomposition into lifting steps Any poly-phase matrix representing wavelet transform can be factored in finite product of upper and lower triangular matrix and diagonal normaliation matrix 3. P= K 0 0 K i= m s i 0 0 t i Normaliation primal lifting Dual lifting
9 To find out s i and t i, poly-phase matrix can be written as, P = h = h e h new e h o o s 3.3 g g e g new e g o o 0 Thus, h o = s h e + h o new 3.35 g o = s g e + g o new 3.36 with h o new < h o g o new < g o h h new e h o e h o 0 P = g = g new e g o 3.37 e g o t Thus, h e = h e new + t h o 3.38 g e = g e new + s g o 3.39 with h e new < h e g e new < g e Let us discuss example of Cohen Daubechies Feauveau bi-orthogonal wavelet filters. h = { -0.5,0.5,0.75,0.5,0.5} g = {0.5, -0.5, 0.5} h = g = + 3. Corresponding polyphase matrix for this wavelet is, P = 3. + Using Equations 3.38 and 3.39, above poly-phase matrix can be written as, = h new e + t = g new e + t 90
10 We can calculate quotient t and the remainder h e new by division. t =, h new e =, g new e =0 P = Further it can be decomposed as: 0 P = / 0 Let us consider input sequence of samples of signal fk. Then, above decomposition can be represented by following lifting steps: Forward wavelet transform: Inverse wavelet transform: Split: λ k fk 3.5 ϒ k fk+ 3.6 Dual lifting Predict: ϒ k ϒ k - ½ λ k + λ k+ 3.7 Primal lifting Update: λ k λ k +/ϒ k- + ϒ k 3.8 Inverse Primal lifting Undo Update: λ k λ k -/ϒ k- + ϒ k 3.9 Inverse Dual lifting Undo Predict: ϒ k ϒ k + ½ λ k + λ k Merge: fk λ k 3.5 fk+ ϒ k 3.5 Quantitative explanation of these steps is given in following section Lifting scheme based discrete wavelet transform Discrete wavelet transform performs multi-stage signal decomposition. Discrete wavelet transform using filter bank was discussed in section..5. In this section, fast and efficient way of finding discrete wavelet transform using lifting scheme is discussed. It de-correlates the signal at different resolution level. Basic polynomial interpolation is used to find high frequency values 9
11 wavelet or ϒ coefficients. It is also used to construct scaling functions in order to find out low frequency values λ coefficients Forward wavelet transform 76 Consider the sequence of samples λ 0,k Where, k=0 to n- at some approximation level say level 0. This sequence can be transformed into two other sets at level - *. First of all, this sequence is divided into set of odd and even samples. Such splitting is sometimes known as lay wavelet transform. Doing this it will not help us to represent signal compactly. This sequence is transformed into coarser signal λ -,k Where value of k= 0 to n/- and detail signal ϒ -,k Where k=0 to n/- by predict and update stage of lifting scheme of wavelet transform. Lifting scheme consist of three steps: Split Lay wavelet transform Predict Dual lifting Update Primal lifting Split Lay wavelet transform: This stage splits entire set of signal into two frames. One frame consists of even index samples such as λ 0,0, λ 0,.λ 0,k. We will call this frame as λ -,k. Other frame consists of odd samples such as λ 0,, λ 0,3,.λ 0,k+. We will call this frame as ϒ -,k. Each group consists of one half samples of the original signal. Splitting the signal into two parts is called lay wavelets, because we have not performed any mathematical operations. If we remove any of the frames, then signal information will be lost. Even samples + λ - λ 0 Split Sequence Predict Odd value Update Stage Odd samples _- ϒ - Fig 3. Lifting steps for forward wavelet transform New sequence can be given as: λ * -,k = λ 0,k where k Z. * Minus sign indicates that new data set is smaller compare to the original data set 9
12 The sequence ϒ -,k can be given as: ϒ -,k = λ 0,k +, Where k Z. ϒ -,k sequence gives us idea about how much information was lost or in other words we can say that it gives information about how we can recover original sequence λ 0,k from transformed sequence λ -, k. The difference between λ 0,k and λ -, k can be encoded as wavelet coefficient. Splitting of data sequences can be done with different methods. We can cut data sequence into left part and right part directly, but in that case, values are hardly correlated. Predicting right half of signal from left half and vice-versa is a tough job. Hence, better method is to interlace the two sets by odd and even frames. Predict Dual lifting: The even and odd samples are interleaved. If the signal is having locally correlated structure, then even and odd samples are highly correlated. In that case, it is very easy to predict odd samples from even samples. For example, in case of triangular and saw-tooth waves, odd sample is average of two neighbouring even samples, hence majority of the wavelet coefficients are ero. In order to achieve maximum data compression, prediction must be powerful. In order to build powerful predictor, we must know nature of the signal. The prediction does not necessarily have to be linear. Cubic or other higher order predictor can be used. Interpolating subdivision is used to find out predictor. We would like to represent data more compactly in order to reduce storage requirement as well as to reduce transmission rate. If we can use powerful prediction mechanism, we can represent data more compactly. Consider that ϒ -,k does not contain any information Odd values perfectly predicted from even values then we can simply replace λ 0,k by smaller set λ -,k. This situation is practically impossible for real time data. So we have to find out some correlation present in the data and build prediction mechanism based on correlation structure. If we can find out prediction operator independent of data so that ϒ -,k = λ -, k
13 Sequence of samples: λ 0,k Split: Even samples k λ 0,k Linear prediction of Odd samples: k ½ λ -,k + λ -, k+ λ 0,k+ Difference calculation detail coefficients: k ϒ -,k = λ 0,k+ ½λ -,k + λ -, k+ k Fig 3. Example of Lifting scheme for Wavelet transform 9
14 In practice, it might not be possible to exactly predict ϒ -,k from λ -,k. However, λ -, k is likely to be closed to ϒ -,k. Thus, we want to replace ϒ -,k with difference between itself and its predicted value. If prediction is reasonable then difference will contain much less information. We can denote this with Equation 3.5 ϒ -,k = λ 0,k+ - λ -, k 3.5 If signal is correlated, the majority of wavelet coefficients are small. For example, in case of linear signals such as saw-tooth and triangular waveform odd sample is average of two even samples, where prediction function can be given as: λ -, k = ½λ -,k + λ -, k ϒ -,k = λ 0,k+ ½λ -,k + λ -, k Predicted value In terms of frequency content, wavelet coefficients capture high frequency present in the signal. We are happy, if this wavelet coefficients are small or ero. This represents enough information to move from λ -,k to λ 0,k. The sequence λ -, k is further divided into two set λ -,k and ϒ -,k.. Where, ϒ -,k is the difference between λ -,k+ and predicted values from λ -,k. To get more compact representation, we can iterate this scheme to higher levels such as λ -3, k, λ -, k.and so on. After n steps, original data will be replaced by {λ -n, k, ϒ -n,k, ϒ -n+,k, ϒ -n+,k.ϒ -,k } Update Primal lifting: The coarser signal must have same average value that of original signal. To do this, we require lifting the λ -, k with help of wavelet coefficients ϒ -,k. After lifting process, mean value of original signal and transformed signal remains same. We require constructing update operator U for this lifting process. λ -, k = λ -, k + U ϒ -,k 3.57 In order to maintain same properties among all the λ coefficients throughout all the levels, we require to find out some scaling function. One way to achieve 95
15 this scaling function is to set all {λ 0, k, k = 0 to n} to ero except λ 0, 0 which is set to. Then run interpolating subdivision to infinity. The resulting function is scaling function that will help us to create real wavelet, which will heritage properties of original signal. In update phase, scaling function is calculated from previous value of wavelet coefficients ϒ -,k. The equation 3.58 represents update phase. λ -, k = λ 0, k + ¼ λ 0, k- + λ 0, k+ or 3.58 λ -, k = λ -, k + ¼ϒ -,k- + ϒ -,k 3.59 The calculation of update phase is in-place to reduce memory requirements. Even locations are over-written with the averages and the odd ones with the details. In this case, we have used linear interpolation, so we call it a linear wavelet transform. Let us find out the relation of this wavelet transform with conventional filter bank wavelet transform. If we substitute the equation 3.56 in the equation 3.59 then we get equation for even samples. Even samples: λ -, k = λ -, k + ¼ϒ -,k- + ϒ -,k = λ 0, k + ¼ [λ 0,k- ½λ-,k- + λ -, k + λ 0,k+ ½λ -,k + λ -,k+] = λ 0, k + ¼ [λ 0,k- ½λ 0,k- + λ 0, k + λ 0,k+ ½λ 0,k + λ 0,k+] = - 0.5λ 0,k λ 0,k λ 0, k λ 0,k λ 0,k These even samples represent output of the low pass filter. It can be seen from the Equation 3.60 that it is convolution of input sample stream with the filter coefficients. -0.5, 0.5, 0.75, 0.5, Summation of these coefficients is unity, implies that it will not affect overall average of the signal. 96
16 We can write equation for odd samples as: ϒ -,k = λ 0,k+ ½λ -,k + λ -, k+ = λ 0,k+ ½λ 0,k + λ 0, k+ 3.6 The Equation 3.6 represents convolution of input samples with high pass filter coefficients -0.5,, In order to achieve multilevel decomposition, λ -, k is further decomposed into coarser and detail parts using split, predict and update stage and we get λ -, k and ϒ -,k. This process can be repeated N number of time. Where, N=log n and n is frame sie. Fig. 3.3 represents multilevel decomposition using lifting scheme. Thus if we start with frame sie of 5 samples, at the first level it will be decomposed into 56 coarser coefficients and 56 detail coefficients. Again 56 coarser coefficients are further decomposed into 8 coarser coefficients and 8 detail coarser coefficients and so on. _+ λ -, k _+ λ -, k Split Predict Update λ 0, k Split Predict Update - _ ϒ -,k - _ ϒ -,k Fig 3.3 Multilevel wavelet transform with lifting scheme Total n samples of signal λ 0, k k= 0 to n- Even n/ samples λ -, k = λ 0, k + /ϒ -,k- +ϒ -,k Odd n/ samples ϒ -,k = λ 0, k+ + ½ λ -, k +λ -,k+, Even n/ samples λ -, k = λ -, k + /ϒ -,k- +ϒ -,k Odd n/ samples ϒ -,k = λ -, k+ + ½ λ -, k +λ -,k+, Fig 3. Lifting Process Chart 97
17 3.5. Inverse wavelet transform Inverse wavelet transform is exactly reverse process of forward wavelet transform. In lifting scheme, it is very easy to find out inverse wavelet transform because it can be obtained by just changing sign. Inverse wavelet transform consists of following steps: Undo Update Inverse Primal lifting Undo Predict Inverse Dual lifting Merge Undo Update Inverse Primal Lifting: Original even samples are recovered by simply subtracting the update information. The Equation 3.6 represents undo update, which is obtained by changing sign in Equation λ -, k = λ 0, k - ¼ λ 0, k- + λ 0, k+ 3.6 λ - - Undo Update Undo Predict Merge λ 0 ϒ - + Fig 3.5 Inverse Wavelet Transform with Lifting scheme Undo Predict Inverse Dual Lifting: Odd samples can be recovered by adding prediction information to loss of information that can be given by Equation 3.63 by changing sign in equation 3.56 λ 0,k+ = ϒ -, k + ½ λ -,k + λ -, k After recovering odd and even samples, final job is to merge them together to obtain original signal. To merge the signal, even samples are interpolated by inserting eros in between the samples and then odd samples are placed in place of eros. 98
18 The two-stage inverse wavelet transform is shown in the Fig λ - + Undo Update Undo Predict Merge ϒ - - λ - + Undo Update Undo Predict Merge λ 0 ϒ - - Fig 3.6 Multilevel Inverse Wavelet transform with Lifting scheme Daubechies wavelet transform 3 The Daubechies wavelet transform use Daubechies wavelet that was invented by the mathematician Ingrid Daubechies. In wavelet analysis scaling function known as father wavelet. The father wavelet is discontinuous but it has compact support. Shannon father function is smooth but it is not compactly supported. The properties of Daubechies wavelet are compromise between smoothness and compact support. Daubechies wavelet is not expressed by single mathematical equation but it is a set of filter coefficients. As discussed in the chapter, Daubechies wavelet filter coefficients satisfies three conditions i compact support condition ii Orthogonality condition and iii Regularity condition Daubechies D transform has four scaling and four wavelet coefficients: Daubechies D scaling coefficients: + h 0 = =0.83 h = 3 3 = h = 3 =0. h 3 = 3 =
19 Daubechies D wavelet coefficients: g 0 = h 3 = 3 3 =-0.9 g = -h = - 3 = g = h = = g 3 = -h 0 = - = The filter coefficients values mentioned above are approximate values. The actual coefficients values used in the software are given in the Appendix D. In classical wavelet transform technique, above filter coefficients are convolved with input speech signal. Each step of wavelet transform applies the scaling filter coefficients and wavelet filter coefficients to the data input. If input data speech samples has length N then it will results into N/ smooth approximate values and N/ detail values. N/ smooth values are further processed. In case of pyramidal algorithm, matrix of filter coefficients is multiplied with input speech samples. Both this methods demands more amount of computation. As filter length increases, computational complexity increases. Daubechies wavelet transform can be implemented with lifting scheme by factoriation. Daubechies wavelet filters can be written as: h = h 0 + h - + h - + h g = g 0 + g - + g - + g The equations 3.6 and 3.65 can be represented compactly as: k = 3 k k h = h k and = 3 k g = g k 3.66 k= 0 k = 0 Poly-phase representation of equation 3.6 and 3.65 can be written as: h = h e + - h o 3.67 g = g e + - g o 3.68 k k Where, = k h e = hk = g e = g k = 0 k = 0 k k 00
20 k = k k h o = hk + = g o = k= 0 k= 0 g k + k After poly-phase representation, corresponding poly-phase matrix can be given by, h h P = = 0 + h - -h 3 e h o - h 3.69 g h + h 3 - h e g o - h 0 Factorisation of above matrix can be given by, P = Approximate and detail coefficients can be obtained by multiplying analysis poly-phase matrix with set of even and odd samples as represented by the following equation. λ = P γ f e f o 3.7 Where analysis poly-phase matrix P - can be given as: P - = According to poly-phase matrix represented by equation 3.7, we can implement in-place forward wavelet transform by following steps: Split: λ -,k fk Even samples and ϒ -,k fk+ Odd samples Update : λ -,k λ -,k + 3ϒ -,k Predict: ϒ -,k ϒ -,k - 3 λ-, k 3 λ-, k- Update : λ -,k λ -,k - ϒ -,k+ 0
21 Normalie: λ -,k 3 λ-,k and ϒ -,k 3 + ϒ-,k We can implement inverse wavelet transform by inverting the steps and changing the signs. These steps can be given as: Undo normalie: λ -,k 3 + λ-,k and ϒ -,k 3 ϒ-,k Undo Update : λ -,k λ -,k + ϒ -,k+ Undo Predict: ϒ -,k ϒ -,k + 3 λ-, k + 3 λ-, k- Undo Update : λ -,k λ -,k - 3ϒ -,k Merge: fk = λ -,k + ϒ -,k We have implemented lifting steps of the Daubechies- wavelet transform in this thesis using C language. 3.6 Prediction functions We have seen lifting scheme of wavelet transform using linear prediction in section 3.5. The real life signals are not linear signals we have to choose some complicated prediction functions. The interpolating subdivision method gives better approximation of the higher order signals. Interpolating subdivision is attractive from implementation point of view because we need a routine, which can construct an interpolating polynomial given some numbers and locations. The new sample value can be given by evaluation of this polynomial at the new refined location. In case of linear prediction, order of interpolation subdivision N is that is piecewise linear approximation. As value of order of subdivision increases, it results into smooth interpolating function and gives good prediction. To find wavelet coefficient using cubic interpolation, value of order of subdivision is four N=. 0
22 In cubic interpolation, odd samples are predicted from two neighbouring values either side i.e. total four adjacent samples are used to predict the odd samples as shown in the Fig.3.6. Hence order of subdivision is four. In this case prediction function use polynomial of order N-. Fig 3.7 Linear and Cubic Interpolation If original data resembles like polynomial of order 3 then we will get good prediction and wavelet coefficients ϒ will have smaller values. This will lead the transform coefficients to have maximum number of eros and higher compression. In speech compression, signal is processed frame by frame. Each frame is having fixed length. Interpolation scheme allows us to easily accommodate interval boundaries for finite length. We should use some boundary treatments. For the calculation of prediction for the first odd sample, there is only sample on the left. So 3 samples are used from right side for the prediction. Similarly, for calculation of prediction for last odd coefficient, there is no sample at right side hence it is predicted from left samples. All possibilities are shown in the Table 3.. Case O s on left O s on right Near left boundary 3 Middle Near right boundary 3 On right boundary 0 Table 3. Boundary treatments 03
23 As we are using four samples for the prediction, there must be some weight associated with each sample. In case of linear interpolation, weight associated with neighbouring sample was 0.5. In cubic interpolation, to find out the weight associated with each sample, solution of polynomial is required. Let us have polynomial: y = c + 3 x + cx + c3x c 3.73 We want to calculate c, c, c 3, and c. To calculate c, we would put its value to and all other to 0. We construct polynomial that best fit for cubic polynomial and start evaluating function at each point in which we are interested. = c + c + c + c c + c + c c + 3c3 + 6c + c3 0 = 8 c = 7c + c = 6c + c 3.77 Solution of these equations can be found out by using matrix. c c c 3 = = c Equation 3.73 can be written as: 3 y = 0.667x +.5x.3333x By putting values of x=.5 and.5, we get y=0.35 and , y x Now let us consider y= for x= 0 = c + c + c + c c + c + c c + 3c3 + 6c + c3 = 8 c = 7c + c = 6c + c 3.8 0
24 Solving Equations 3.79 to 3.8, we get, 3 y = 0.5x x + 9.5x 6 By putting values of x=.5 and.5, we get y= and y x Similarly in case of third and four point we will get following curves: y y x x 3 3 y = 0.5x + 3.5x 7x + y = 0.667x x x x=.5, y=-0.35 x=.5 y=0.565 x=.5, y= x=.5 y= Following table shows filter coefficients for different cases: Case Coefficients Samples Samples K-7 K-5 K-3 K- K+ K+3 K+5 K+7 on left on right Table 3. Weighting factors for neighbouring samples If we want to predict odd samples having two samples on the left and two samples on the right then equation can be written as: γ λ 0.065λ λ λ 0.065λ 3.83 j, k = j+ j, k 3 j, k j, k + j, k Integer Wavelet Transform Digital speech signal is a sequence of integer samples; hence speech compression algorithms should work with integers. Apart from speech compression, there are many applications like image compression, image processing etc. require integer wavelet transform because of nature of input 05
25 data is integer samples. In addition to this the storage and encoding of integer numbers is easier compared to floating point numbers. Filter bank algorithm of wavelet transform work with floating point numbers even though input values actually are integer, carrying out filtering operations on these numbers will transform them in rational or real numbers because the filter coefficients need not be integers. To obtain an efficient implementation of the discrete wavelet transform, it is of great practical importance that the wavelet transform is represented by a set of integers. Because if we store wavelet coefficients as a floating point values it requires 3 bits per coefficients. This is not reasonable in some applications like speech compression. Hence wavelet coefficients are rounded to convert it into integer number for efficient encoding and storage. Because of this rounding process, the original signal cannot be reconstructed from its transform without an error. This is the reason for not getting loss-less speech compression in the filter bank implementation. Using lifting scheme of wavelet transform, rounding error is cancelled during the inverse transform. Hence it is possible to achieve perfect reconstruction. This is explained below: For example let us consider sequence of samples: {,, 3,, 6, 7, 8, 0, } Split: λ k ={, 3, 6, 8, } λ k+ = {,, 7, 0} Predict: Pλ k+ = {,.5, 7, 9.5} Pλ k+ = {,, 7, 9} {Rounding predicted value to integer} ϒ -,k = λ k+ - Pλ k+ = {0, 0, 0, } During inverse transform: Undo Predict: λ k+ = λ k+ +ϒ -,k = {,,7,0} Merge: {,, 3,, 6, 7, 8, 0, } Though, we have rounded wavelet coefficients, we obtained original values back because of reversible operation. Error added during forward transform is subtracted in inverse transform hence net error is ero. In this case, only predict stage is shown. This is also true for update stage. 06
26 3.8 Boundary treatment In the implementation of wavelet transform for speech compression, we have used frame length of 56 bytes. Entire speech samples are divided into number of blocks, where length of each block is 56 bytes. During application of wavelet transform, we have to consider the boundary. For example, if we want to predict last sample, we can not predict it in usual way because there is no right sample. Similarly if we want to update first sample then it is again difficult because there is no coefficient on left side. This problem is more serious in case of filter bank implementation. To avoid this problem, some kind of signal extension method is necessary. Following methods are popular for the signal extension Zero Padding This is very simplest method of signal extension. In this method, signal is extended by simply inserting string of eros at the end of the signal and at the beginning of the signal. This method is not preferable because it introduce spikes Audio clicks after the transform. Finite signal t Zero padding of finite signal t 3.8. Periodic extension In this method, we put copies of the samples of the finite signal in front of and behind the signal. After wavelet transform, we can simply discard this extra coefficients. If first and last samples are not having same value then it will produce unwanted discontinuities and hence produce audio clicks. Finite signal t Periodic extension of finite signal t 07
27 3.8.3 Symmetric extension In this method, finite signal is extended by mirroring it around its endpoints, which makes the discontinuities disappear. This is better solution compare to above two methods. Finite signal t Symmetric extension of finite signal t 3.8. Signal extension with lifting Lifting scheme allows transforming signal of finite length without extending the signal as we have seen in four different cases in the section 3.6. Where prediction of last sample is possible without using right sample No extension is needed!. Right sample was predicted by using four left samples with different weighting factors. Suppose if we do not want to use different case and if we want to calculate using classical method then we can use symmetric extension method. Depending on first and last sample being odd or even, we can consider four cases: First Even last Odd First Even Last Even First Odd Last Even First Odd Last Odd First Even Last Odd First Even Last Even First Odd Last Odd First Odd Last Even Fig 3.8 Signal extension 08
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