Modeling of the operational rate-distortion characteristics of a signal can significantly reduce the computational complexity of an optimal bit allocation algorithm. In this report, such models are studied

Rajpoot, Nasir M. (Nasir Mahmood)

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http://wrap.warwick.ac.uk/            Original citation: Rajpoot, Nasir M. (Nasir Mahmood) (2004) Model based optimal bit allocation. University of Warwick. Department of Computer Science. (Department of Computer Science Research report).  CS-RR-397  Permanent WRAP url: http://wrap.warwick.ac.uk/61393            Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions.  Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners.  To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.  Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge.  Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.  A note on versions: The version presented in WRAP is the published version or, version of record, and may be cited as it appears here.For more information, please contact the WRAP Team at: publications@warwick.ac.uk  Model Based Optimal Bit AllocationNasir M. RajpootDepartment of Computer ScienceUniversity of WarwickCoventry CV4 7ALUnited Kingdomemail: nasir@dcs.warwick.ac.ukJanuary, 2004AbstractModeling of the operational rate-distortion characteristics of a signal cansignificantly reduce the computational complexity of an optimal bit allocationalgorithm. In this report, such models are studied.1 IntroductionIn recent years, the problem of bit allocation has been a subject of investigationby many researchers working in the areas of audio, image and video coding. Theproblem is conventionally defined as follows: given a signal x of size N divided into nnon-overlapping subsignals xi of size Ni, where∑ni=1 Ni = N , find the most efficientdistribution of a given bit budget R among a number of available quantizers for thesubsignals. The bit allocation algorithm of Shoham & Gersho [1] was to be the firstone in a series of algorithms which provide an optimal solution to the problem for anarbitrary set of quantizers. However, this algorithm and its other derivations suchas [2] have high computational complexity due to a need to compute the operationalrate-distortion (R-D) characteristics for all available quantizers in order to find anoptimal solution. The complexity of such algorithms can be greatly reduced if theoperational R-D characteristics could be approximated efficiently and with as lesserror as possible. Two types of bit allocation algorithms that approximate the R-Dbehaviour of a source can be found in the literature: algorithms which use polynomialfunctions (such as splines [3]) to fit the operational R-D curve, and algorithms whichuse analytical models (such as the average distortion-rate function for high bit rates[4, 5]) to approximate the empirical R-D curve. The scope of former class of algorithmsis somewhat limited by their requirement to specify the control points on the empiricalR-D curve. Moreover, their ability to extrapolate the R-D characteristics beyond the1range of control points remains questionable. The latter class of algorithms overcomeboth of these problems by estimating the parameters of an anlytical model and using itfor generate an arbitrary point on the operational rate-distortion curve. An attractivefeature of such algorithms is that some of the models used may lend themselves to aconvenient mathematical analysis of the problem.In this report, we investigate further the problem of optimal bit allocation usinganalytical models for operational R-D characteristics. The average distortion-ratefunction D(R) = σ22−2R has been used in [4] and [5] for optimal bit allocation invideo and image coding applications respectively. However, as pointed in [6], there aretwo problems with using this function: first, it satisfies the so-called high resolutionhypothesis and so is useful only for bit rates of over 1 bits/pixel in the context ofimage coding; second, it assumes that the signal is a realization of a Gaussian source,an assumption which often does not hold for natural images or their transformations.This should be motivation enough to look for other models for the empirical R-Dbehaviour.Two major contributions of this work are: (1) a comparative study of differentmodels that can be used to approximate the operational R-D characteristics of asource, and (2) proposal of two new models which appear to provide better fit forthe empirical R-D curve in most cases, and a study of some of their properties. Theremainder of this report is organised as follows. In the next section, two new modelsare proposed, and comparative results for empirical R-D curve fitting are providedfor these models, the aforementioned average distortion-rate function, and the lowbit rate model of Mallat & Falzon [6]. Section 3 describes how these models can beused for optimal bit allocation. Experimental results are presented in Section 4 andthe report ends with concluding remarks and some directions for future work.2 The R-D ModelThe average distortion-rate function D(1)(R) = σ22−2R can also be written asD(1)(R) = eaR+b (1)where a, b∈< and a < 0, b > 0. The above equation also shows there to be a linearrelationship between R and logD(1). Motivated by the limitations of model in (1),Mallat & Falzon analyzed the coefficients of wavelet and block DCT transforms ofimages and developed the relationship D(2)(R) = R1−2γ, where γ is of the order of 1for most images. In general, their model can be written as follows,logD(2)(R) = a logR + b (2)One apparent problem with this model is that the distortion does not approach vari-ance (ie, maximum distortion) as R approaches zero. Noting that the distortion variesmuch more rapidly at low bit rates than at high bit rates, we propose the followingtwo models,D(3)(R) = eeαR+β(3)2where α < 0, β > 0, andD(4)(R) = eaR2+bR+c (4)where a, b, c∈<.Property 1 (Convexity) The distortion-rate functions given by (3) and (4) areconvex⋃functions for D≥1, α≥0 and D≥0, a > 0 respectively.The proof of this property follows directly from the fact that ∂2D(3)/∂R2 = α2(D(3)logD(3))+αlogD(3) is non-negative only when D≥1 and α≥0, and ∂2D(4)/∂R2 = D(4)[(2aR +b)2 + 2a] is non-negative when D≥0, a > 0. For the mean square error distortionmeasure, we can safely say that D≥1 at low bit rates. Also, slope of the straightline obtained by plotting log(log(D(3))) against R is always going to be non-negative,assuming the distortion will always go up when the allocated bit rate is reduced.Property 2 (Lower Bound on the Average Distortion) For the distortion-ratefunction given by (3), the average distortion is bounded from below by the distortionat average bit rate.The above property is proved by considering the fact that due to its being convex⋃,the distortion-rate function of (3) satisfies the Jensen’s inequalityE{D(R)} ≥ D(E{R}) (5)Figure 1 shows empirical R-D curves for three images and their wavelet transforms,with the results of fitting the curves with all four of the above models. From theseresults, we observe that the empirical R-D model D(4)(R) represented by (4) outper-forms all the other models in almost all the cases. It is also to be noted that the modelD(1)(R) given in (1) produces the worst fit for both images in the transform domain(whose operating points in our examples belong to the low bit rate regime). On theother hand, Mallat & Falzon’s model performs badly in the high bit rate regime (ascan be seen from the R-D curves of the original images). The two proposed modelsseem to be able to adapt to both low and high bit rate regimes.2.1 Solving the Optimization ProblemThe optimal bit allocation problem can be formulated as an optimization problemwhich aims to minimize the overall distortion while remaining within an upper limiton the bit budget. It was proved by Everett [7] that an optimal solution to thisproblem can be found by solving an equivalent Lagrangian optimization problem.For the sake of completeness, the proof is provided below.Lemma 1 (Unconstrained Problem) The optimal solution r∗ to the constrainedproblemmin∑idi(ri)subject to ∑iri ≤ R30.6 0.8 1 1.2 1.4 1.6 1.8 2 2.205001000150020002500Rate (bits/pixel)Distortion (MSE)Rate−Distortion curve and different fitsOriginalD(1)(R)D(2)(R)D(3)(R)D(4)(R)0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.530405060708090100110120130Rate (bits/pixel)Distortion (MSE)Rate−Distortion curve and different fitsOriginalD(1)(R)D(2)(R)D(3)(R)D(4)(R)(a) (b)0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.605001000150020002500Rate (bits/pixel)Distortion (MSE)Rate−Distortion curve and different fitsOriginalD(1)(R)D(2)(R)D(3)(R)D(4)(R)0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.950100150200250300Rate (bits/pixel)Distortion (MSE)Rate−Distortion curve and different fitsOriginalD(1)(R)D(2)(R)D(3)(R)D(4)(R)(c) (d)0.6 0.8 1 1.2 1.4 1.6 1.8 20200400600800100012001400160018002000Rate (bits/pixel)Distortion (MSE)Rate−Distortion curve and different fitsOriginalD(1)(R)D(2)(R)D(3)(R)D(4)(R)0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.56080100120140160180Rate (bits/pixel)Distortion (MSE)Rate−Distortion curve and different fitsOriginalD(1)(R)D(2)(R)D(3)(R)D(4)(R)(e) (f)Figure 1: Empirical R-D behaviour and results of fitting(a) 512×512 Lena image, and (b) 5-level DWT of Lena;(c) 512×512 Barbara image, and (d) 5-level DWT of Barbara;(e) 512×512 Goldhill image, and (f) 5-level DWT of Goldhill;4is also optimal solution of the unconstrained problemmin∑i(di(ri) + λri)for the partcilar case where R = R(λ∗) =n∑i=1r∗i .Proof:Consider the Lagrangian function given byI(λ) =n∑i=1di(ri) + λn∑i=1ri.orI(λ) = D(r) + λn∑i=1ri,where D(r) =n∑i=1di(ri) denotes the overall distortion. Let λ∗ denotes the value of λcorresponding to the optimal solution r∗. Then clearly,I(λ∗) ≤ I(λ), ∀λorD(r∗) + λ∗n∑i=1r∗i ≤ D(r) + λn∑i=1riorD(r∗)−D(r) ≤ λ(n∑i=1ri −n∑i=1r∗i )orD(r∗)−D(r) ≤ λ(n∑i=1ri −R)We also know thatn∑i=1ri≤R. This implies directly that for all values of λ≥0,D(r∗) ≤ D(r).Hence the lemma.3 Bit AllocationLemma 2 (Optimality Condition) Given that the distortion-rate function di(ri)is a convex function for each segment xi of the signal, the optimal solution to theproblemmin∑idi(ri)subject to ∑iri ≤ R5is given by r∗=(r∗1, r∗2, . . . , r∗n) which satisfies the following condition∂di∂ri=∂dj∂rj∀ i, j i 6=j.Proof:We follow a simplex approach to solve the given optimization problem. Let us denotethe sum of distortions di(ri), for all the segments xi of the signal, byD(r) =D(r1, r2, . . . , rn).The constraint∑iri ≤ R corresponds to an area on and under the surface S which isgiven by ∑iri = R ∀ ri ≥ 0.There are only two possibilites as to where the optimal solution r∗ can lie: eitherunder the surface S, or on its boundary. In the former case (ie, when ∑ir∗i < R), theoptimality condition would be∂di∂ri= 0, ∀ i.But given that the function di(ri) is convex (for all values of i), it can easily bededuced that the optimal distortion D(r∗) corresponds to a solution r∗ which lieson boundary of the surface S. This implies that the gradient of overall distortionfunction D(r) should be normal to boundary of the surface S. In other words,∇D(r) = (∂d1∂r1,∂d2∂r2, . . . ,∂dn∂rn) ⊥ ∂Sor∇D(r) = (∂d1∂r1,∂d2∂r2, . . . ,∂dn∂rn) = µ(1, 1, . . . , 1)for some constant µ. This proves the lemma.Corollary 1 (Fixed λ) : From the above lemmas, it is clear that the optimal solu-tion r∗ satisfies the condition ∂di/∂ri = −λ, ∀ i. Therefore, the optimal solution canbe found by restricting the value of λ to be the same for each segment xi of the signal.From the above result, an algorithm for finding an optimal value of λ can be devisedusing a gradient descent method, as explained in [1]. The expressions ∂Di∂Riusing allthe four models are given by∂D(1)i∂Ri= aD(1)i ,∂D(2)i∂Ri= D(2)i (b+a/Ri),∂D(3)i∂Ri= αD(3)i logD(3)i ,∂D(4)i∂Ri= D(4)i (2aRi+b).(6)It may be noted here that expression only for∂D(1)i∂Riand∂D(3)i∂Rilead to plausible ana-lytical and efficient numerical solutions, as they involve only distortion.6Proposition 1 (Equivalence) Let r∗ = (r1, r2, . . . , rn) be the optimal budget dis-tribution for a given bit budget R bits per pixel, where each ri, ∀ i = 1, 2, . . . , ncorresponds to the ith subsignal having αi and βi as its R-D model parameters. Letxi = αiri+βi, ∀ i = 1, 2, . . . , n. Then xi = AiR+Bi (where Ai, Bi ∈ < are constants,∀ i) holds if and only if x1 = x2 = . . . = xn.Proof:We know from Theorem 2 that the optimality condition to distribute the given budgetR into the n subsignals is∂di∂ri=∂dj∂rj∀ i, j i6=jorα1(d1logd1) = α2(d2logd2) = . . . = αn(dnlogdn).The above set of equations can be re-written as the following n − 1 independentequations.x1 + ex1 = x2 + ex2 +K1x2 + ex2 = x3 + ex3 +K2...xn−1 + exn−1 = xn + exn +Kn−1(7)whereKi = log (αi+1αi),andxi = αiri + βi (8)for all i = 1, 2, . . . , n− 1. We also know thatr1 + r2 + . . .+ rn = nRPutting the values of ri from equation (8) into the above equation, we getn∑i=1aixi = nR +n∑i=1βiαi(9)where ai = 1/αi, ∀ i = 1, 2, . . . , n. The set of equations 7 can also be written as,φ(xi) = φ(xi+1) +Ki, ∀ i = 1, 2, . . . , n− 1 (10)whereφ(x) = x+ ex.Equation (10) implies that given the value of xi, xi+1 can be found by solving thisequation. Although it does not have an obvious analytical solution, let us assumethat xi+1 is directly related to xi by the following function,xi+1 = fi(xi) (11)7Re-arranging the terms of equation (9), we get the following relation between xi+1and xi,xi+1 =n∑j=1j 6=i+1bjxj +naiR +1ain∑j=1(βjαj) (12)where bj = −aj/ai. Now let us suppose that xi = AiR + Bi, ∀ i = 1, 2, . . . , n suchthat∂xi∂R= Ai 6= 0. (13)Comparing equations (11) and (12), we obtainfi(xi) =n∑j=1j 6=i+1bjxj +naiR +1ain∑j=1(βjαj)Differentiating above equation with respect to R, we have∂fi∂xi·∂xi∂R=n∑j=1j 6=i+1bj·∂xj∂R+naiorfi′(xi) = Mi (14)where Mi =1Ai∑nj=1j 6=i+1bjAj +naiAiis a constant. It is also clear from (14) thatfi′′(xi) = 0. From (10) and the implicit function theorem, we obtainf ′i(xi) =φ′(xi)φ′(fi(xi))=1 + exi1 + exi+1(15)Differentiating (15) and equating f ′′i (xi) to zero, we obtaine−xi+1 + exi+1 = e−xi + exi .The function (e−x+ex) = 2cosh(x) is an even function and thus the above equation hasone of the two possible solutions: either xi+1 = xi or xi+1 = −xi, ∀ i = 1, 2, . . . , n−1.The latter of these solutions can clearly be ruled out. Thus we havex1 = x2 = . . . = xnConversely, if xi+1 = xi, ∀ i = 1, 2, . . . , n − 1, then xi = AiR + Bi holds for someconstants Ai and Bi by the definition of xi (ie, xi = αiri + βi, ∀ i = 1, . . . , n).Corollary 2 (Linear Variation) From the above proposition, it is obvious that theoptimal bit budget distributions ri, ∀ i = 1, 2, . . . , n vary linearly with the target bitbudget R according to the following relationri =ααiR + (β − βiαi)8Bit Budget Uniform Allocation var-based Allocation Optimal Allocation(bpp.) bpp. PSNR (dB) bpp. PSNR (dB) bpp. PSNR (dB)0.5 37.24 0.68 39.24 0.36 35.250.5 41.50 0.51 41.61 0.26 38.110.5 34.89 0.35 32.73 0.33 32.450.5 31.41 0.37 29.75 0.40 30.100.5 33.13 0.43 32.34 0.32 30.980.5 37.66 0.38 36.05 0.21 32.500.5 28.61 0.47 28.18 0.67 30.320.5 29.22 0.53 29.52 0.70 31.590.5 0.5 28.64 0.71 30.48 0.60 29.600.5 33.31 0.53 33.81 0.33 30.880.5 26.04 0.69 27.68 0.81 28.580.5 24.59 0.71 27.00 0.97 27.010.5 38.02 0.22 35.05 0.22 35.050.5 36.96 0.31 35.42 0.23 34.500.5 25.98 0.32 24.35 0.66 27.530.5 26.00 0.79 28.71 0.93 29.89Table 1: Results of bit allocation for coding sixteen 128× 128 blocks of Barbarawhereα =nn∑i=11αiandβ =αnn∑i=1βiαi.4 Experimental ResultsThe optimal bit allocation algorithm outlined in the previous section was tested onblock partitions of an image in order to efficiently distribute a given bit budget be-tween the image blocks. The model parameters were determined by quantizing withfive different step sizes the wavelet transform coefficients of each image block. Forthe purposes of experimentation, third model for the empirical R-D curve was used.Results of budget distribution between 16 equal sized partitions of the Barbara imageare given in Table 1. For comparison, results are also provided for a relatively naiveactivity based bit allocation algorithm which works by analyzing the variance of eachof the blocks, and for a uniform distribution of the bit budget. The bit allocationalgorithm assigns more bits to an image block which contains regions of high activity.95 ConclusionsThis report addressed the issue of model based optimal bit allocation. Two newmodels were proposed for representing the empirical R-D curves. It was demonstratedthat while D(4) provides the best fit for operational R-D characteristics, D(3) lendsitself to tractable analysis of the bit allocation problem. Future work may includea generalization of the bit allocation algorithm to all models, and an investigationinto the relationship of these models with the distribution of transform coefficientsfor subband image coding.6 AcknowledgementsThe author would like to thank Artur Sowa and Franc¸ois Meyer for their ideas andmany motivating discussions. Part of this work was carried out when the author wasvisiting the department of Mathematics, Yale University, with generous support fromProfessor Ronald Coifman.References[1] Y. Shoham and A. Gersho. Efficient bit allocation for an arbitrary set of quantiz-ers. IEEE Transactions on Acoustics, Speech, and Signal Processing, 36(9):1445–1453, September 1988.[2] K. Ramchandran and M. Vetterli. Best wavelet packet bases in a rate-distortionsense. IEEE Transactions on Image Processing, 2(2):160–175, April 1993.[3] L-J. Lin and A. Ortega. Bit-rate control using piecewise approximated rate-distortion characteristics. IEEE Transactions on Circuits and Systems for VideoTechnology, 8(4):446–459, August 1998.[4] W. Y. Lee and J. B. Ra. Fast algorithm for optimal bit allocation in rate-distortionsense. IEE Electronic Letters, 32:1871–1896, October 1996.[5] J. Li, P-Y. Cheng, and C-C. Kuo. Embedded wavelet packet image coder withfast rate-distortion optimized decomposition. In SPIE: Visual Communicationand Image Processing, volume 3204, pages 1077–1088, February 1997.[6] S. Mallat and F. Falzon. Analysis of low bit rate image transform coding. IEEETransactions on Signal Processing, April 1998.[7] H. Everett. Generalized Lagrange multiplier method for solving problems of op-timum allocation of resources. Operations Research, 11:399–417, 1963.10

Analysis of low bit rate image transform coding.

Best wavelet packet bases in a rate-distortion sense.

Bit-rate control using piecewise approximated ratedistortion characteristics.

Ecient bit allocation for an arbitrary set of quantizers.

Embedded wavelet packet image coder with fast rate-distortion optimized decomposition.

Fast algorithm for optimal bit allocation in rate-distortion sense.

Generalized Lagrange multiplier method for solving problems of optimum allocation of resources.

Model based optimal bit allocation

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Model based optimal bit allocation

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