Multiresolution Signal Decomposition Transforms, Subbands, and Wavelets phần 7 potx

Figure 4.11: a Hierarchical decimation/interpolation branch arid b its lent.equiva-The advantage of this analysis in a lossless M-band filter bank structure is itsability to decompose th

4.10 ALIASING ENERGY IN MULTIRESOLUTION DECOMPOSITION 311

Figure 4.10 (continued)

Figure 4.11: (a) Hierarchical decimation/interpolation branch arid (b) its lent.

equiva-The advantage of this analysis in a lossless M-band filter bank structure is itsability to decompose the signal energy into a kind of time-frequency plane Wecan express the decomposed signal energy of branches or subbands in the form of

an energy matrix defined as (Akansu and Caglar, 1992)

Each row of the matrix E represents one of the bands or channels in the

filter bank and the columns correspond to the distributions of subband energies infrequency The energy matrices of the 8-band DCT, 8-band (3-level) hierarchicalfilter banks with a 6-tap Biriomial-QMF (BQMF), and the most regular wavelet

4.10 ALIASING ENERGY IN MULTIRESOLUTION DECOMPOSITION 318 filter (MRWF) (Daubechies) for an AR(1) source with p = 0.95 follow:

" 7.17200.05670.02580.00420.01960.00190.00450.0020

" 7.16110.05890.02620.00430.01960.00200.00470.0020

0.12110.18810.01360.00320.00320.00120.00040.00020.05670.19870.00250.00140.00190.00610.00010.00010.05890.19560.00280.00180.00200.00640.00010.0001

0.02800.15110.05690.00500.00160.00650.00010.00000.00140.05670.06400.00250.00010.00190.00140.00010.00180.05890.06280.00280.00010.00200.00170.0001

0.01570.02650.01360.02790.00320.00220.00040.00000.00050.02580.00250.02950.00130.00460.00010.00010.00060.02620.00280.02910.00140.00470.00010.0001

0.01320.01130.03450.00510.01760.00260.00530.00000.00010.00050.02580.00250.02230.00130.00450.00010.00010.00060.02620.00280.02210.00140.00470.0001

0.01570.00910.00780.00320.00320.01320.00330.00020.00050.00140.00420.00140.00130.01670.00200.00010.00060.00170.00430.00180.00140.01640.00200.0001

0.02800.01130.00460.01580.00160.00260.01180.00530.00140.00050.00610.00420.00010.00130.01620.00200.00180.00060.00640.00430.00010.00140.01600.0020

0.1211 10.02650.00780.00610.00320.00220.00330.01550.0567 "0.02580.00420.01960.00190.00450.00200.02200.0589 "0.02620.00430.01960.00200.00470.00200.0218

We can easily extend this analysis to any branch in a tree structure, as shown

in Fig 4.11 (a) We can obtain an equivalent structure by shifting the antialiasingniters to the left of the decimator and the interpolating filter to the right of theup-sampler as shown in Fig 4.11(b) The extension is now obvious

4.10.2 Nonaliasing Energy Ratio

The energy compaction measure GTC does not consider the distribution of the

band energies in frequency Therefore the aliasing portion of the band energy istreated no differently than the nonaliasing component This fact becomes im-portant particularly when all the analysis subband signals are not used for thereconstruction or whenever the aliasing cancellation in the reconstructed signal isnot perfectly performed because of the available bits for coding

From Eqs (4.78) and (4.79), we define the nonaliasing energy ratio (NER) of

an M-band orthonormal decomposition technique as

where the numerator term is the sum of the nonaliasing terms of the band energies.The ideal filter bank yields NER=1 for any M as the upper bound of this measurefor any arbitrary input signal

4.11 GTC an(i NER Performance

We consider 4-, 6-, 8-tap Binomial-QMFs in a hierarchical filter bank structure

as well as the 8-tap Smith-Barnwell and 6-tap most regular orthonormal waveletfilters, and the 4-, 6-, 8-tap optimal PR-QMFs along with the ideal filter banks forperformance comparison Additionally, 2 x 2, 4 x 4, and 8 x 8 discrete cosine, dis-crete sine, Walsh-Hadamard, and modified Hermite transforms are considered for

comparison purposes The GTC and NER performance of these different

decom-position tools are calculated by computer simulations for an AR(1) source model

Table 4.15 displays GTC and NER performance of the techniques considered with

M = 2,4,8

It is well known that the aliasing energies become annoying, particularly at lowbit rate image coding applications The analysis provided in this section explainsobjectively some of the reasons behind this observation Although the ratio of thealiasing energies over the whole signal energy may appear negligible, the misplacedaliasing energy components of bands may be locally significant in frequency andcause subjective performance degradation

While larger M indicates better coding performance by the GTC measure, it is

known that larger size transforms do not provide better subjective image codingperformance The causes of this undesired behavior have been mentioned in theliterature as intercoefficient or interband energy leakages, bad time localization,

etc The NER measure indicates that the larger M values yield degraded

perfor-mance for the finite duration transform bases and the source models considered.This trend is consistent with those experimental performance results reported in

the literature This measure is therefore complementary to GTC: which does not

consider aliasing

4.12 QUANTIZATION EFFECTS IN FILTER BANKS 315

M=4

G TC (NER)

5.7151 (0.9372)3.9106 (0.8532)3.7577 (0.8311)5.2173 (0.9356)6.4322 (0.9663)6.7665 (0.9744)6.9076 (0.9784)6.9786 (0.9813)6.7255 (0.9734)7.0111 (0.9831)6.9899 (0.9834)6.8624 (0.9776)6.8471 (0.9777)6.4659 (0.9671)6.4662 (0.9672)7.230 (1.000)

M=8

GTC (NER)

7.6316 (0.8767)4.8774 (0.7298)4.4121 (0.5953)6.2319 (0.8687)8.0149 (0.9260)8.5293 (0.9427)8.7431 (0.9513)8.8489 (0.9577)8.4652 (0.9406)8.8863 (0.9615)8.8454 (0.9623)8.6721 (0.9497)8.6438 (0.9503)8.0693 (0.9278)8.0700 (0.9280)9.160 (1.000)

*This optimal QMF is based on energy compaction

**This optimal QMF is based on minimized aliasing energy

Table 4.15: Performance of several orthonormal signal decomposition techniques

for AR(1), p — 0.95 source.

4.12 Quantization Effects in Filter Banks

A prime purpose of subband filter banks is the attainment of data rate sion through the use of pdf-optimized quantizers and optimum bit allocation foreach subband signal Yet scant consideration had been given to the effect of codingerrors due to quantization Early studies by Westerink et al (1992) and Vanden-dorpe (1991) were followed by a series of papers by Haddad and his colleagues,Kovacevic (1993), Gosse and Duhamel (1997), and others This section provides adirect focus on modeling, analysis, and optimum design of quantized filter banks

compres-It is abstracted from Haddad and Park (1995)

We review the gain-plus-additive noise model for the pdf-optimized quantizeradvanced by Jayant and Noll (1984) Then we embed this model in the time-domain filter bank representation of Section 3.5.5 to provide an M-band quanti-zation model amenable to analysis This is followed by a description of an optimum

two-band filter design which incorporates quantization error effects in the designmethodology.

4.12.1 Equivalent Noise Model

The quantizer studied in Section 2.2.2 is shown in Pig 4.12(a) We assume that

the random variable input x has a known probability density function (pdf) with

zero mean If this quantizer is pdf-optirnized, the quantization error ? is zero

mean and orthogonal to the quantizer output x (Prob.2.9), i.e.,

But the quantization error x is correlated with the input so that the variance of

the quantization is (Prob 4.24)

where a 2 refers to the variance of the respective zero mean signals Note that forthe optimum quantizer, the output signal variance is less than that of the input.Hence the simple input-independent additive noise model is only an approximation

to the noise in the pdf-optirnized quantizer

Figure 4.12: (a) pdf-optimized quantizer; (b) equivalent noise model

Figure 4.12(b) shows a gain-plus-additive noise representation which is tomodel the quantizer In this model, we can impose the conditions in Eq (4.82)

and force the input x and additive noise r to be uncorrelated The model eters are gain a and variance of With x — ax + r, the uncorrelated requirement

param-becomes

4.12 QUANTIZATION EFFECTS IN FILTER BANKS 317

Equating cr| in these last two equations gives one condition Next, we equate

E{xx} for model and quantizer From the model,

and for the quantizer,

These last two equations provide the second constraint Solving all these gives

For the model, r and x are uncorrelated and the gain a and variance a^, are

input-signal dependent

Figure 4.13: /3(R), a(R) versus R for AR(1) Gaussian input at p=0.95.

From rate distortion theory (Berger 1971), the quantization error variance <r|for the pdf-optimized quantizer is

The parameter (3(R) in Eq (4.89) depends only on the pdf of the unit variance

signal being quantized and on J?,, the number of bits assigned to the quantizer

It does not depend on the autocorrelation of the input signal Earlier approaches

treated (3(R) as a constant for a particular pdf We show the plot of (3 versus R for a Gaussian input in Fig 4.13 Jayant and Noll reported {3=2,7 for a Gaussian

input, the asymptotic value indicated by the dashed line in Fig 4.13 From Eqs

(4.88) and (4.89) the nonlinear gain a can be evaluated as

Figure 4.13 also shows a vs R using Eq (4.90) As R gets large, j3 approaches its asymptotic value, and a approaches unity Thus, the gain-plus additive noise model parameters a and d^ are determined once R and the signal pdf are specified.

Note that a different plot and different asymptotic value result for differing signalpdfs

4.12.2 Quantization Model for M-Band Codec

The maximally decimated M-band filter bank with the bank of pdf-optimizedquantizers and a bank of scalar compensators (dotted lines) are shown in Fig.4.14(a) Each quantizer is represented by its equivalent noise model, and theanalysis and synthesis banks by the equivalent polyphase structures This givesthe equivalent representation of Fig 4.14(b), which, in turn, is depicted by thevector-matrix equivalent structure of Fig 4.14(c) Thus, by moving the samplers

to the left and right of the filter banks, and focusing on the slow-clock-rate signals,the system to be analyzed is time-invariant, but nonlinear because of the presence

of the signal dependent gain matrix A.

By construction the vectors t>[n] and r[n] are uricorrelated, and A, S are

diag-onal gain and compensation matrices, respectively, where

This representation well now permits us to calculate explicitly the total meansquare quantization error in the reconstructed output in terms of analysis and syn-thesis filter coefficients, the input signal autocorrelation, the scalar compensators.and implicitly in terms of the bit allocation for each band

4.12 QUANTIZATION EFFECTS IN FILTER BANKS 319

Figure 4.14: (a) M-band filter bank structure with compensators, (b) polyphaseequivalent structure, (c) vector-matrix equivalent structure

We define the total quantization error as the difference

where the subscript "o" implies the system without quantizers and compensators.From Fig 4.14(c) we see that

where B - S - /, and V(z) = H p (z)£(z) and C(z) = G' p (z)B, T>(z) = Q' p (z)S We note that v(n) and r(n) are uncorrelated by construction.

For a time-invariant system with M x 1 input vector x and output vector y,

we define M x M power spectral density (PSD) and correlation matrices as

Using these definitions and the fact that v(n) and r(n) are uncorrelated, we can calculate the PSD S nqnq (z] and covariance R nq n q [fn\ for the quantization error r) q (n).

It is straightforward to show (Prob 4.24) that

where C(z] «-» Ck and T>(z) +-* D^ are Z transform pairs.

At fc=0, this becomes

From Fig 4.14(b), we can demonstrate that R rm (o] is the covariance of the Mth

block output vector

4.12 QUANTIZATION EFFECTS IN FILTER BANKS

Consequently,

321

Note that this is cyclostationary; the covariance matrix of the next block of M

outputs will also equal /^[O] Each block of M output samples will thus have

same sum of variances We take the MS value of the output as the average of thediagonal elements of Eq (4.101),

Similarly, if we define y q (ri) as the quantization error in the reconstructed output

then the total mean square quantization error (MSE) at the system output is

Next, by substituting Eq (4.99) into Eq (4.104), we obtain

The first term, <rj, of Eq (4.105) is the component of the MSE due to the nonlinear

gain matrix A and compensation matrix S The second term a^ accounts for the

additive fictitious random noise r(n) These terms <rj, <r^ are called the signaldistortion and random noise components of the MSE, respectively Under PR

constraints, <jj measures the deviation from perfect reconstruction due to the

quantizer and compensator This decomposition of the total MSE enables us toanalyze each component error separately This is the main theoretical consequence

of the gain-plus-additive noise quantizer model where the signals v(n) and random

noise r(n) are uncorrelated

The MSE in Eq (4.105) can be written in an explicit closed form time-domainexpression in te;rms of the analysis and synthesis filter coefficients This is achieved

by expanding the polyphase coefficient matrices in terms of the synthesis filtercoefficients via

and substituting into Eq (4.105) The results are rather messy and are not sented here The interested reader can refer to the reference for details The last

pre-step In our formulation requires a further breakdown of R vv [m] in Eq (4.105) Prom Fig 4.14(a) R ViVj [m] can be represented as

By defining the correlation function pji(m) — hi(m) * hj(-rri) we have

This concludes the formulation of the output MSE in terms of the

analy-sis/synthesis filter coefficients /ij(n), gi(ri), the input autocorrelation function RXX[™}-> the nonlinear gain c^, and compensator Si.

Some simplifying assumptions on R^k) can be argued First, we note that the

decimated signals ('t^(n)} occupy frequency bands that can be made to overlap

slightly Hence, {vi(n}} and {VJ(H + m)} tend to be weakly correlated The random errors {n(n)} due to each quantizer are, by design, uncorrelated with the respective {vi(n}} Therefore, as a simplifying assumption we can say that

.12 UANTIZATION EFFECTS IN FILTER BANKS 823 E[ri(n}rj(n j r-m)} ~ 0 This makes Rrr[n} a diagonal matrix Next, it is often true

that the quantization error for a given signal swing (as measured by crjj sweeps

over several quantization levels When this is true, E[ri(n}ri(n + m)] = of ,<S(m).

Then, the random component of reduces to a simpler form

but a^ remains messy

From the foregoing, several observations regarding compensators can be noted:

(i) By setting Si=l, we have no compensation and a\ in Eq (4.105), and of in

Eq (4.109) constitute the MSE in the uncompensated structure As we shall see inthe next section, 5^=1 is the optimized selection when paraunitary PR constraintsare imposed on the non-quantized system

(ii) By choosing Si = 1/c^, the "null compensation," we can eliminate

com-pletely the signal distortion term o~§, leaving only the noise term

(iii) However, this solution is not optimal at the stated operating conditions.

The quantizer gain c^ < 1 and Eq (4.110) show that we can expect a largerrandom component than that of the uncompensated structure In fact, for theuncompensated structure, this random component is dominant Increasing thiscomponent by the null condition is decidedly not optimal

(iv) However, when the input statistics change from nominal values, the nullcompensation is found to be superior to the "optimal" one, which is, in fact,

optimal only at the nominal values of p In this account, we minimize the total MSE by minimizing jointly the sum of o\ and o\ subject to defined PR constraints.

4.12.3 Optimal Design of Bit-Constrained, pdf-Optimized Filter

Banks

The design problem is the determination of the optimal FIR filter coefficients,compensators, and integer bit allocation that minimize the MSE subject to con-straints of filter length, average bit rate, and PR in the absence of quantizers, for

an input signal with a given autocorrelation function

For the paraimitary case, the orthogonality properties eliminate the correlation between analysis channels, which is implicit in the crj component of

cross-Eq (4.105) The MSB in this case reduces to

It is now easy to show that the optimized compensator for this paraunitary

condi-tion is s\ — 1 Then the uncompensated system is optimal for the pdf-optimized paraunitarjr FB (On the other hand, si = 1 is not optimal for the biorthogona)

structure because of the cross-correlation between analysis channels.)

Sample designs and simulations for a six-coefficient paraunitary two-band

structure for an AR(1) input with p — 0.95 are shown in Table 4.13 MSE refers

to the theoretical calculations and MSEsjm, the simulation results Table 4.13demonstrates that the optimal filter coefficients are quite insensitive to changes in

the average bit rate R and in input correlation p Figure 4.15(a) shows explicitly

the distortion and random components of the total MSE The simulation resultsclosely match the theoretical ones The random noise cr^ is clearly the dominantcomponent of the MSE Figure 4.15(b) compares the optimally compensated with

the null compensated (si — l/cti) paraunitary systems designed for p — 0.95 The

null compensated is more robust for changing input statistics and performs better

than the fixed optimally compensated one when p changes from its design value

of p = 0.95

Similar designs and simulations were executed for the biorthogonal two-bandcase with equal length (6 taps) analysis and synthesis filters For the same operat-ing conditions, the biorthogonal structure is superior to the paraunitary in terms

of the output MSE However, the biorthogonal filter coefficients are very sensitive

to R> the average number of bits, and to the value of p The paraunitary design

is far more robust and emerges as the preferred design when p is uncertain.

4.13 Summary

This chapter is dedicated to the description, evaluation, and design of practicalQMFs We described and compared the performance of several known paraunitarytwo-band PR-QMF families These were shown to be special cases of a filter designphilosophy based on Bernstein polynomials

We described a new approach to the optimal design of filters using extendedperformance criteria This route provides new directions for filter bank designswith particular applications in visual signal processing

4,13 SUMMARY 325

R

11.522.53

,9=0.95

#0

12345

Ri

11111

MSB0.35330.11820.03870.01510.0086

MSE s ,; m

0.35220.11830.03910.01540.0087

M2)0.4345170.4281420.4281430.4281460.428146

M3)-0.122522-0.140852-0.140852-0.140851-0.140851

M4)-0.117625-0.106698-0.106698-0.106696-0.106699

M5)5.2485e-25.1677e-25.1677e-25.1677e-25.1677e-2

(b)

Table 4.13: Optimum designs for the paraunitary FB at p = 0.95 (a) optimum

bits and MSE; (b) optimum filter coefficients

rigure 4.lo(aj: -theoretical and simulation results ol trie total output Mblii withdistortion and random components for the paraunitary FB at p=0.95 (b) MSE

of optimally compensated, s^—1, and null compensated, Si — l/a^ structures signed for p—0.95) versus p for paraunitary FB with AR(1) signal input, 0^=1,

(de-RQ=$, R]—\.

Figure 4.15(b): Theoretical and simulation results of the total output MSE withdistortion and random components for the paraunitary FB at p=0.95 (b) MSE

of optimally compensated, 5^=1, and null compensated, si — 1/cti structures signed for p—0.95) versus p for paraunitary FB with AR(1) signal input, cr^.—l.

(de-O (de-O P 1

itO—O, It]—1.

Aliasing energy in a subband tree structure was defined and analyzed alongwith a new performance measure, the nonaliasing energy ratio (NER) These mea-sures demonstrate that filter banks outperform block transforms for the examplesand signal sources under consideration On the other hand, the time and frequencycharacteristics of functions or filters are examined and comparisons made betweenblock transforms, hierarchical subband trees, and direct M-band paraunitary filterbanks

We presented a methodology for rigorous modeling and optimal compensationfor quantization effects in M-band codecs, and showed how an MSE metric can

be minimized subject to paraunitary constraints

We will present the theory of wavelet transforms in Chapter 6 There we willsee that the two-band paraunitary PR-QMF is the basic ingredient in the design

of the orthonormal wavelet kernel, and that the dyadic subband tree can provide

the fast algorithm for wavelet transform with proper initialization The

Binornial-QMF developed in this chapter is the unique maximally flat magnitude squaretwo-band unitary filter In Chapter 6, it will be identified as a wavelet filter andthus provides a specific example linking subbands and orthonormal wavelets

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Of special interest are nonstationary signals, that is, signals whose salientfeatures change with time For such signals, we will demonstrate that classicalFourier analysis is inadequate in highlighting local features of a signal.

What is needed is a kernel capable of concentrating its strength over segments

in time and segments in frequency so as to allow localized feature extraction.The short-time Fourier (or Gabor) transform and the wavelet transform have thiscapability for continuous-time signals

In this chapter, we focus on the description and evaluation of techniques forachieving time-frequency localization on discrete-time signals We hope to providethe reader with an exposure to current literature on the subject and to serve as aprelude to the wavelet and applications chapters which follow

First we review the classical analog uncertainty principle and the short-timeFourier transform Then we develop the discrete-time counterparts to these andshow how the binomial sequences emulate the continuous-time Gaussian func-tions Following this introduction, we define, calculate, and compare localization

331

features of filter banks and standard block transforms and explore the role oftree-structured filter banks in achieving desired time-frequency resolution Then

we conclude with a section on achieving arbitrary "tiling" of the time-frequencyplane using block transforms and demonstrate the utility of this approach withapplications to signal compaction and to interference excision in spread spectrumcommunications systems

A word on the notation used in this chapter is in order The terms Z, R and

R + denote the set of integers, real numbers, and positive real numbers,

respec-tively; L' 2 (R) denotes the Hilbert space of measurable, square-integrable functions, i.e., the space of what are termed finite energy signals /(£), or sequences f ( n ) sat-

isfying

All one-dimensional functions dealt with in this chapter are assumed to havefinite energy Also, the inner product of two functions is denoted by

5.2 Analog Background —

Time Frequency Resolution

A basic objective in signal analysis is to devise an operator capable of extractinglocal features of a signal in both time- and frequency-domains This requires akernel whose extent or spread is simultaneously narrow in both domains That is,

the transformation kernel <j)(t) arid its Fourier transform $(O) should have narrow spreads about selected points £&, &<k in the time-frequency plane However, theuncertainty principle described below bounds the simultaneous realization of thesedesiderata Narrowness in one domain necessarily implies a wide spread in theother

Standard Fourier analysis decomposes a signal into frequency components and

determines the relative strength of each component It does not tell us when the

5.2 ANALOG BACKGROUND TIME FREQUENCY RESOLUTION 333

signal exhibited the particular frequency characteristic, since the Fourier kernel

e:?fit is spread out evenly in time It is not time-limited

If the frequency content of the signal were to vary substantially from interval

to interval as in a musical scale, the standard Fourier transform

would sweep evenly over the entire time axis and wash out any local anomalies ofthe signal (e.g., short duration bursts of high-frequency energy) It is clearly notsuitable for nonstationary signals

Confronted with this challenge, Gabor (1946) resorted to the windowed, time Fourier transform (STFT), which moves a fixed-duration window over thetime function and extracts the frequency content of the signal within that interval.This would be suitable, for example, for speech signals which generally are locallystationary but globally nonstationary

short-The STFT positions a window g(t) at some point r on the time axis and

calculates the Fourier transform of the signal contained within the spread of thatwindow, to wit

When the window g(t) is Gaussian, the STFT is called the Gabor transform (Gabor, 1946) The STFT basis functions are generated by modulation and trans- lation of the window function g(t) by parameters il and r, respectively Typical

Gabor basis functions and their associated transforms are shown in Fig 5.1

The window function is also called a prototype function, or sometimes, a mother function As T increases this mother function simply translates in time keeping

the time-spread of the function constant Similarly, as seen in Fig 5.1, as themodulation parameter H^ increases, the transform of the mother function also,simply, translates in frequency, keeping a constant bandwidth

The difficulty with the STFT is that the fixed-duration window g(t) is

accom-panied by a fixed frequency resolution and thus allows only a fixed time-frequencyresolution This is a consequence of the classical uncertainty principle (Papoulis,1977) This theorem asserts that for any function 0(£) with Fourier transform

$(O), (and with Vt(f)(t) —> 0, as t —> =F oo) it can be shown that

where O~T and a$i are, respectively, the RMS spreads of 4>(t) and &(Q) around the

center values That is,

Figure 5.1: Typical basis functions for STFTs and their Fourier transforms

where E is the energy in the signal,

5.2 ANALOG BACKGROUND-TIME FREQUENCY RESOLUTION 335

Figure 5.1: (continued)

Figure 5.1: (continued)