In this article, we introduce the concept of fractional wavelet transform. Using a two-channel unbalanced lifting structure it is possible to decompose a given discrete-time signal x[n] sampled with period T into two sub-signals x1[n] and x2[n] whose average sampling periods are pT and qT, respectively. Fractions p and q are rational numbers satisfying the condition: 1/p + 1/q = 1. The low-band sub-signal x 1[n] comes from [0, π/p] band and the high-band wavelet signal x 2[n] comes from (π/p, π] band of the original signal x[n]. Filters used in the liftingstructure are designed using the Lagrange interpolation formula. It is straightforward to extend the proposed fractional wavelet transform to two or higher dimensions in a separable or non separable manner. © 2011 Copyright Society of Photo-Optical Instrumentation Engineers (SPIE)

Habiboǧlu, Y.H.

Kose, K.

Çetin, A.E.

Bilkent University Institutional Repository

Fractional wavelet transform using an unbalanced lifting structure

In this article, we introduce the concept of fractional wavelet transform. Using a two-channel unbalanced lifting
structure it is possible to decompose a given discrete-time signal x[n] sampled with period T into two sub-signals
x1[n] and x2[n] whose average sampling periods are pT and qT , respectively. Fractions p and q are rational numbers satisfying the condition: 1/p + 1/q = 1. The low-band sub-signal x1[n] comes from [0, π/p] band and the high-band wavelet signal x2[n] comes from (π/p, π] band of the original signal x[n]. Filters used in the lifting
structure are designed using the Lagrange interpolation formula. It is straightforward to extend the proposed
fractional wavelet transform to two or higher dimensions in a separable or non separable manner

Habiboglu, Y. Hakan

Kose, Kivanc

Cetin, A. Enis

ZENODO

Fractional Wavelet Transform Using an Unbalanced LiftingStructureY. Hakan Habibog˘lu, Kivanc Kose and A. Enis C¸etinDepartment of Electrical and Electronics EngineeringBilkent UniversityBilkent, 06800, Ankara, TurkeyABSTRACTIn this article, we introduce the concept of fractional wavelet transform. Using a two-channel unbalanced liftingstructure it is possible to decompose a given discrete-time signal x[n] sampled with period T into two sub-signalsx1[n] and x2[n] whose average sampling periods are pT and qT , respectively. Fractions p and q are rationalnumbers satisfying the condition: 1/p+ 1/q = 1. The low-band sub-signal x1[n] comes from [0, π/p] band andthe high-band wavelet signal x2[n] comes from (π/p, π] band of the original signal x[n]. Filters used in the liftingstructure are designed using the Lagrange interpolation formula. It is straightforward to extend the proposedfractional wavelet transform to two or higher dimensions in a separable or non separable manner.Keywords: Lifting, wavelet transform, multirate signal processing1. INTRODUCTIONLifting structures provide computationally efficient implementation of the wavelet transform1–7 . They foundapplications in image and video coding and signal and image analysis applications.8–10 Modified versions of thebalanced wavelet structures were also proposed for fractional scalability of the images, and videos11 . However,lifting structures providing scalable subsignals have not been developed.In this article, new wavelet lifting structures producing unbalanced decomposition of the input signal areintroduced. In standard lifting structures the input signal is first decomposed into even and odd indexed samplesusing the lazy wavelet transform as shown in Fig. 1. In the proposed unbalanced lifting structure, a structuresimilar to the lazy filterbank forwards every pth sample of the original signal to the upper-branch and remainingp− 1 samples out of p samples go to the lower branch. Discrete-time update and prediction filters interconnectthe upper and lower branches, respectively. Discrete-time filters operating in the lower sampling rates aredesigned using the Lagrange interpolation formula which is also used in many filterbank designs including ourfilterbanks, Smith-Barnwell and Daubechies filterbanks. In this way, two sub-signals with different samplingrates are obtained from the original signal. When the input signal has a sampling rate of T the upper- andlower- branches of the unbalanced lifting filterbank have sampling rates of pT and Tp/(p− 1), respectively. Instandard balanced lifting the sampling periods of upper and lower branches are the same: 2T .The unbalanced lifting decomposition can be easily generalized to other sampling strategies in which theupper-branch has a sampling rate of pT and the lower-branch has a sampling rate of qT with the property that1p+1q= 1 (1)Perfect reconstruction can be easily achieved by changing the signs of the filters in the reconstruction part ofthe ordinary balanced lifting structures.In Section 2, an example filterbank design with p = 3 : 1 and q = 3 : 2 is presented. In Section 3, anotherfilterbank example with p = 3 : 2 and q = 3 : 1 is presented. In Section 4, two-dimensional (2D) filterbank designexamples are presented.Further author information:Y. Hakan Hakan Habibog˘lu: E-mail: yhakan@ee.bilkent.edu.tr, Telephone: +90 312 2901525Kivanc Kose: E-mail: kkivanc@ee.bilkent.edu.tr, Telephone: +90 312 2901164A. Enis C¸etin: E-mail:cetin@bilkent.edu.tr, Telephone: +90 312 29014772. UNBALANCED LAZY FILTERBANK AND LIFTING STRUCTURESAn unbalanced lazy filterbank for p = 3 : 1 and q = 3 : 2 is shown in Figure 2. In the upper-branch a regulardownsampling block by a factor of three is used. In the lower-branch the downsampling block for q = 3 : 2 isused. In Figure 2, the signal x[n] is fed to the lazy filterbank and outputs of downsampling blocks are shown.Every 3rd sample of the original signal goes to the upper-branch and remaining samples appear in the lowerbranch.We describe the update and the prediction filter design for the unbalanced lifting structure in the followingtwo subsections.2.1 Update FilterWe first review with the regular balanced lifting filterbank to illustrate the framework of the unbalanced lifting.Consider the lazy filterbank shown in Figure 1.-2 -1 0 1 2 3 4 5 6 7012345678abcdefx[n]2:12:1+UxUN-1 0 1 2 302468ace-1 0 1 2 302468b dfz-1-1 0 1 2 302468ace^^^. .. . ... .. . .. . .. . ... ... ..Figure 1. Lazy filterbank based regular lifting structure.The signal x[n] is divided into two sub-signals containing even and odd indexed samples of the original signal.The upper branch should be aliasing free as much as possible. The update filter tries to cancel aliasing in Figure1. In many two-channel perfect reconstruction filter banks the Lagrange filter hl[n] ={14 ,12 ,14}is used forsmoothing the signal4 . For example, in Figure 1, the even indexed sample c is linearly estimated using the oddindexed samples b and d by the update filter, and it is replaced by the value ( b+d2 + c)/2 orxu[1] = cˆ =b4+c2+d4(2)after updating. All the other samples of the upper-branch sub-signal xu[n] are updated in a similar manner usingthe linear Lagrange filter.The above update filter is based on linear interpolation. Other higher order Lagrange filters can be also usedto construct the update filter6, 12, 13 . For example, the Lagrange filter hl[n] ={−132 , 0,932 ,12 ,932 , 0,−132}based onhigher order polynomial interpolation is widely used in filter bank design1–7, 14–17 . In Figure 1, this means thatthe sample c is estimated by the update filter c′ = 932 (b+d)− 132 (f + x[−1]) and the corresponding upper branchsignal is given by cˆ = c+c′2 .-2 -1 0 1 2 3 4 5 6 7012345678abcdefx[n]3:13:2U+xU+P-xL-1 0 1 202468ad^^. .. . ... ... ..-1 0 1 2 3 4012345678ad. ... ..-1 0 1 2 3 4 5 6 7012345678bcef. ... ..Figure 2. Unbalanced lifting scheme with p = 3 : 1 and q = 3 : 2 downsampling ratios.In Figure 2 an unbalanced lifting structure is shown with downsampling ratios 3 : 1 and 3 : 2 in the upperand lower branches, respectively. Similar to the regular balanced filterbank case, the upper branch sample d canbe estimated using the neighboring lower branch samples b, c, e, and f and an estimate of the sample d is givenas follows:d′ =b+ 2c+ 2e+ f6, (3)which is the output of the update filter linking the lower branch to the upper branch. Since samples c and e arecloser to the sample d compared to f and b more weight is given to the samples c and e. The sample d and theoutput of the filter is linearly combined and the updated sample is obtained as followsdˆ =2d′3+d3(4)Since the downsampling is by a factor of three in the upper-branch original signal must be filtered with a low-passfilter with a cut-off frequency of π/3. We could have used dˆ = d′2 +d2 but this does not produce as good frequencyresponse as (4). Therefore, the upper branch output sample is given bydˆ =2b18+4c18+6d18+4e18+2f18(5)The corresponding filter coefficients are given as followshl[n] = { 218,418,618,418,218} (6)The frequency response of this filter is plotted in Figure 3. As it can be observed from this figure the cut-offfrequency of this filter is π/3.Other samples of the upper branch of the filterbank are smoothed in a similar manner. In Equation (5),each sample of the upper branch sub-signal is updated using four neighboring samples of the lower branch. It ispossible to smooth the samples of the upper branch further by using more samples from the lower branch without0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91Normalized Frequency (×π rad/sample)MagnitudeMagnitude ResponseFigure 3. Frequency response of the filter in Equation (6).effecting the perfect reconstruction capability of the filterbank. In this case, Lagrange interpolation formula orother interpolation methods can be used to determine the update filter coefficients. In general,xU [n] =19x[3n− 2] + 29x[3n− 1] + 39x[3n] +29x[3n+ 1] +19x[3n+ 2] (7)where x[n] is the original input signal to the filterbank.2.2 Prediction FilterSamples of the lower branch are estimated from the upper branch and difference is transmitted to the receiver.Lower branch samples can be predicted from the upper branch using the updated samples aˆ, dˆ, gˆ, ... (see Fig.2).The prediction filter can be as simple as the identity operator selecting the nearest upper branch sample as anestimate of the lower branch sample:xL[0] = b− aˆ,xL[1] = c− dˆ,xL[2] = e− dˆ,xL[3] = f − gˆ,. .. .. .(8)Other samples of xL[n] can be determined in a similar manner. Although the above prediction strategy isvery simple and computationally efficient, the above predictor is not a good estimator. we can use Lagrangeinterpolation and obtain:xL[0] = b− (2aˆ+ dˆ)/3xL[1] = c− (2dˆ+ aˆ)/3xL[2] = e − (2dˆ+ gˆ)/3xL[3] = f − (2gˆ + dˆ)/3. .. .. .(9)In general,xL(n) ={x[ 3n+22 ]−2xU [n2 ]+xU [n2 +1]3 , n is evenx[ 3n+12 ]−2xU [n+12 ]+xU [n−12 ]3 , n is odd(10)where x[n] is the original input signal to the filterbank.To determine xL[0] the sample b is estimated using the two nearest upper branch samples aˆ and dˆ. Let P (t)be the Lagrange interpolator based on the samples aˆ and dˆ:P (t) = aˆ o(t) + dˆ 1(t) (11)where i(t) are the Lagrange basis polynomials. The function P (t) is constructed using the input data pairs(aˆ, 0) and (dˆ, 3T ). Since the sample b occurs at t = T , o(T ) = 2/3 and 1(T ) = 1/3. Therefore the Lagrangeinterpolation gives more weight to the sample aˆ because it is nearer to the sample b compared to dˆ to determinexL[0] in (9). Similarly, more weight is given to the sample dˆ because it is nearer to the sample c compared to aˆto determine xL[1] etc. As it can be seen from the above equations the predictor is a time varying filter. Thepredictor can even be an adaptive LMS-type filter as described in references4, 14, 18 . The adaptive predictiontries to remove as much information as possible from the lower-branch using xL[n] samples. In this case thecomputational complexity is higher than the predictors in (8) and (9). Other samples of xU [n] can be also usedby the predictor filter. In this case higher order Lagrange interpolation formula needs to be used4, 13 .It is trivial to design the reconstruction filterbank as in regular lifting structures. At the reconstruction stagesigns of filters U and P are changed and sub-signal samples are realigned to obtain the original signal x[n].3. UNBALANCED FILTERBANK WITH p = 3 : 2 AND q = 3 : 1x[n]3:23:1U+xU+P-xL-2 -1 0 1 2 3 4 5 6 7 8012345678abcdef g. ... ..-1 0 1 2 3 4 5 6 7 8012345678abdeg. ... ..-1 0 1 2 3 4 5 6 7 8012345678cf. ... ..Figure 4. Unbalanced lifting structure with p=3:2 and q=3:1.In Figure 4, the upper-branch has a downsampling ratio of 3 : 2 and the lower-breach has a downsamplingration of 3 : 1. In this case, samples of the upper-branch are downsampled in a non-uniform manner. In theupper branch the original shape of the signal has to be preserved as much as possible in practical applicationsincluding image and video processing. Therefore, the samples of the upper branch should be transformed to auniformly sampled grid.For example, in Figure 4, we need to estimate x[1.5], x[4.5], ... using the lower-breach samples and thenearest current samples of the upper-branch, which are x[1], x[4], ..., respectively.We can use linear interpolation and obtain the following updated values:xU [1] = xˆ[1.5] =b+ c2, xU [3] = xˆ[4.5] =e+ f2... (12)One can also use other lower-branch samples as well during estimation without affecting perfect reconstruction.Samples of xU [2n] can be set as x[3n] or it is also possible to update other samples of the upper-breach usingthe lower branch:xU [0] = a, xU [2] =13c+23d, xU [4] =13f +23g, ... (13)In general,xU (n) ={x[1.5n−1]+2x[1.5n]3 , n is even and n = 0x[1.5(n−1)+1]+x[1.5(n+1)−1]2 , n is odd(14)where x[n] is the original input signal to the filterbank.As usual, the prediction filter tries to remove as much information from the lower-breach as possible. Forexample, the samples c can be predicted as follows, xL[1] = c− xU [1], and, similarly xL[2] = f − xU [3], etc...Similar to the structure in Section 2, other samples of the upper-branch output xU [n] can be also used by aLagrange interpolator for prediction purposes without effecting perfect reconstruction resulting a better qualitysub-signal xL[n] as follows:xL[1] = c− (23xU [1] +13xU [2]) (15)where the predictor uses the same weights as in Equation (8) because the sample xU [1] occurs at t = 1.5T andxU [2] occurs at t = 3T , and the sample c occurs at t = 2T as shown in Fig. 4. As a result the Lagrangeinterpolator gives more weight to xU [1] compared to xU [2]. Similarly, the next sample is predicted as followsxL[2] = f − (23xU [3] +13xU [4]) (16)In general,xL[n] = x[3n− 1]− (23xU [2n− 1] + 13xU [2n]) (17)where x[n] is the original input signal to the filterbank.It is straightforward to design the reconstruction filterbank as in regular lifting structures. At the reconstruc-tion stage signs of filters U and P are changed and sub-signal samples are realigned to obtain the original signalx[n].4. EXTENSION FOR TWO-DIMENSIONAL SIGNALSA N-dimensional filter T (x1, x2...xN ) is seperable if it can be represented asT (x1, x2...xN ) = T1(x1)T2(x2)...TN(xN ) (18)Assume that the two-dimensional (2D) filter f(x, y) is seperable. Therefore it can be represented as followsT (x, y) = T1(x)T2(y) (19)For a 2D signal or an image, Equation (19) corresponds to first filtering the signal in horizontal direction byT1(x) and then filtering the output signal in vertical direction by T2(y).In Figure 5, the seperable filterbank structure for the unbalanced lifting wavelet transform with p = 3 : 2and q = 3 : 1 is given. The input image is first downsampled in the horizontal direction. Then the update andprediction filters are applied to the downsampled images. The intermediate output signals XU and XL are thendownsampled in the vertical direction and the update and prediction filters are applied again. As an example,x[n]3:23:1U++P-3:23:1U+xLL+P-xLH3:23:1U+xHL+P-xHHxUxLFigure 5. Extension of the unbalanced lifting structure with p=3:2 and q=3:1 to 2D.the image given at Figure 6 is fractional wavelet transformed using the scheme presented in Figure 5. Thetransformed image is given at Figure 7. As another example, the image given at Figure 8 is wavelet transformedin Figure 2 using the fractional wavelet transform with p = 3 : 1 and q = 3 : 2 which is a 2D extended version ofthe filterbank described in Section 2.It is also possible to develop non-seperable fractional filterbanks.CONCLUSIONIn this article, fractional wavelet transform (FWT) methods are developed using two-channel unbalanced liftingstructures. Sub-signals have different average sampling rates.The discrete time signal x[n] sampled with period T can be decomposed into two sub-signals with averagesampling periods of pT and qT , respectively. Example lifting structures for p = 3 : 1 and q = 3 : 2 and p = 3 : 2and q = 3 : 1 cases are presented.The FWT method can be easily extended to two or higher dimensions. Image transformation examples arepresented.ACKNOWLEDGMENTSThe research leading to there results has received funding from the European Community’s Seventh FrameworkProgramme (FP7-ENV-2009-1) under grant agreement no FP7-ENV-244088 “FIRESENSE - Fire Detection andManagement through a Multi-Sensor Network for the Protection of Cultural Heritage Areas from the Risk ofFire and Extreme Weather”. 11, 14–17, 19, 20Figure 6. Example image.Figure 7. 2D unbalanced lifting wavelet transformed image with p=3:2 and q=3:1. Highband subimages are amplified bya factor of three.REFERENCES[1] Mallat, S., [A Wavelet Tour of Signal Processing, Second Edition (Wavelet Analysis & Its Applications) ],Academic Press, 2 ed. (Sept. 1999).[2] Sweldens, W., “The lifting scheme: A custom-design construction of biorthogonal wavelets,” Applied andComputational Harmonic Analysis 3(2), 186 – 200 (1996).[3] Daubechies, I. and Sweldens, W., “Factoring wavelet transforms into lifting steps,” Journal of FourierAnalysis and Applications 4, 247–269 (1998). 10.1007/BF02476026.[4] Kim, C. W., Ansari, R., and Cetin, A. E., “A class of linear-phase regular biorthogonal wavelets,” inFigure 8. 2D unbalanced lifting wavelet transformed image with p=3:1 and q=3:2. Highband subimages are amplified bya factor of three.[Acoustics, Speech, and Signal Processing, 1992. ICASSP-92., 1992 IEEE International Conference on ], 4,673 –676 vol.4 (Mar. 1992).[5] Hampson, F. J. and Pesquet, J. 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English

A 2-d orientation-adaptive prediction filter in lifting structures for image coding,”

A class of linear-phase regular biorthogonal wavelets,” in Figure 8. 2D unbalanced lifting wavelet transformed image with p=3:1 and q=3:2. Highband subimages are amplified by a factor of three.

A nonlinear diffusion-based three-band filter bank,” Signal Processing Letters,

A nonlinear subband decomposition with perfect reconstruction,”

A Wavelet Tour of Signal Processing, Second Edition (Wavelet Analysis & Its Applications)

Adaptive polyphase subband decomposition structures for image compression,”

Adaptive update lifting with a decision rule based on derivative filters,” Signal Processing Letters,

Compressive sensing using the entropy functional,” arXiv:1101.5079v2 [cs.IT]

Exact reconstruction techniques for tree-structured subband coders,” Acoustics, Speech and Signal Processing,

Factoring wavelet transforms into lifting steps,”

Geometrical grouplets,”

Image compression with geometrical wavelets,” in [Image Processing,

Lagrange m-th band filters and the construction of smooth m-band wavelets,” in [TimeFrequency and Time-Scale Analysis,

Lms based adaptive prediction for scalable video coding,”

Modified m-band synthesis filter bank for fractional scalability of images,” Signal Processing Letters,

Nonlinear wavelet transforms for image coding via lifting,” Image Processing,

Programme (FP7-ENV-2009-1) under grant agreement no FP7-ENV-244088 “FIRESENSE - Fire Detection and Management through a Multi-Sensor Network for the

Structure and design of two-channel filter banks derived from a triplet of halfband filters,” Circuits and Systems II: Analog and Digital Signal Processing,

The lifting scheme: A custom-design construction of biorthogonal wavelets,”

Three-dimensional lifting schemes for motion compensated video compression,”

Wavelet construction using lagrange halfband filters,” Circuits and Systems,

Fractional Wavelet Transform Using an Unbalanced Lifting Structure

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

http://repository.bilkent.edu.tr/bitstream/11693/28344/1/Fractional%20wavelet%20transform%20using%20an%20unbalanced%20lifting%20structure.pdf

Fractional wavelet transform using an unbalanced lifting structure

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Bilkent University Institutional Repository

ZENODO

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY