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A critical examination of the assumptions used in dynamic allocation
J. Vis. Commun. Image R. 20 (2009) 351–363
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J. Vis. Commun. Image R. journal homepage: www.elsevier.com/locate/jvci
A critical examination of the assumptions used in dynamic allocation J.A. García, Rosa Rodriguez-Sánchez, J. Fdez-Valdivia * Departamento de Ciencias de la Computación e I. A., CITIC-UGR (Research Center on Information and Communications Technology), Universidad de Granada, 18071 Granada, Spain
a r t i c l e
i n f o
Article history: Received 24 September 2008 Accepted 12 March 2009 Available online 24 March 2009 Keywords: Image transmission Multiple quantizers Dynamic allocation Assumptions Bit allocation Knowledge Coder evaluation Compound gain Congestion control
a b s t r a c t In dynamic allocation quantizers are capable of choosing between limited allocation of bits and bit allocation without restriction. The goal of this paper is to perform a comparative analysis of the assumptions used in a transmission system which still has quantizers using restrained bit allocation in the long time and in a transmission system for which all quantizers end up using heavy bit allocation. Then, based on the validity of the assumptions derived, we will be able to predict the performance of each system in a real problem. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction If we wish that a transmission system can operate in a more efficient fashion, one way is to allow a dynamic allocation of bits to the most needed quantizers. There, quantizers monitor the level of utility of bit allocation across bit rates and, at any point of the transmission, they might decide to switch the behavior between restrained bit allocation and heavy bit allocation. Restrained bit allocation implies limitation, as on quantizer’s freedom of bit allocation; while heavy allocation means bit allocation without restriction. Thus, at any given bit rate, quantizers evaluate the perceived payoff for restrained and heavy allocation and switch to the strategy with the highest benefit. To illustrate this process, Figs. 2 and 3 show plots of the number of quantizers, noted as n1, which choose the strategy of restrained allocation at each transmission time while using the REWIC coder, [1,2], on each test image of the dataset in Fig. 1. As can be seen from these plots, initially all quantizers (n = 16) make use of restrained allocation but in the long transmission time all quantizers end up using heavy bit allocation. We have developed a modified version of the REWIC coder, called as the REWIC with Congestion Control (RCC), which implements a congestion control algorithm described in Ref. [3] (see Appendix A). Figs. 2 and 3 also display plots of the number of quantizers, n1, which * Corresponding author. Fax: +34 58 24 3317. E-mail addresses: [email protected] (J.A. García), [email protected] (R. Rodri guez-Sánchez), [email protected] (J. Fdez-Valdivia). 1047-3203/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jvcir.2009.03.006
choose the strategy of restrained allocation at each transmission time while using the RCC coder on each test image of the dataset in Fig. 1. From these figures, it can be seen that RCC still has quantizers using restrained bit allocation in the long transmission time limit. The goal of this paper is to made a comparison of assumptions behind dynamic allocation in a transmission system like RCC (which still has quantizers using restrained bit allocation in the long time limit) and in a system like REWIC in which all quantizers end up using heavy bit allocation. And based on the validity of assumptions derived we will be able to predict the performance of each transmission system in real applications. Section 2 examines the assumptions behind a transmission system like the REWIC coder in which in the long transmission time all quantizers end up using heavy bit allocation to uncover that (i) the payoffs to each quantizer are independent of what the others are doing; and (ii) there is perfect knowledge in the transmission system. In most circumstances however, perfect knowledge about the state of the transmission is not available, which often results in a degradation of the performance of the transmission system. On the contrary, Section 2 shows that, the assumption of imperfect knowledge may be true for a transmission system like the RCC coder which still has quantizers using restrained bit allocation in the long time limit. In this case, assumptions derived are more realistic, and it probably results in a better performance. Section 3 shows an objective and subjective coder evaluation, in order to investigate the performance of the RCC coder as compared with that of the REWIC coder. The main conclusions of the paper are summarized in Section 4.
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Fig. 1. Image database used in the experiments.
2. Dynamic allocation model We assume that the image can be divided into a finite number n of regions, each of which is represented by a quantizer. Quantizers are capable of choosing between restrained bit allocation and heavy bit allocation. Thus, at any time, quantizers might decide to switch their behavior among restrained and heavy allocation according to the perceived payoff. Transmission times of relevant information are often quite different for various quantizers, so only a fraction of them will consider switching between restrained and heavy bit allocation during a given interval Mt of transmission time, if Mt is small enough. Hence probabilistic dynamics can be used to provide an analytic description of this process at which quantizers may decide to switch their behavior.
Let n1(t) be the number of quantizers following restrained bit allocation at time t; and n2(t) be the number of quantizers using the strategy of heavy bit allocation at this time. Here, we assume that n2(t) = n n1(t). Assuming that the perceived payoff does not change over Mt, the probability p that a quantizer changes from restrained bit allocation to heavy bit allocation in Mt is given by:
p ¼ ð1 qÞ r Mt
ð1Þ
where q is the probability that restrained bit allocation will be perceived by a quantizer to be better than heavy bit allocation; and r is the average rate at which quantizers reevaluate their preferences regarding bit allocation.
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Image #1
Image #2
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #3
t
Image #4
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #5
t
Image #6
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #7
t
Image #8
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #9
t
Image #10
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
t
Fig. 2. Plots of the number of quantizers n1 which choose the strategy of restrained allocation at each transmission time t while using, respectively, the REWIC coder and the RCC coder (images 1–10).
Following Ref. [4], we can derive the law governing the rate, dn1 =dt, where n1 denotes the average number of quantizers following a strategy of restrained bit allocation, as given by:
n1 ¼
n X i¼0
i PC ði; tÞ
ð2Þ
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Image #11
Image #12
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #13
t
Image #14
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #15
t
Image #16
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #17
t
Image #18
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #19
t
Image #20
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
t
Fig. 3. Plots of the number of quantizers n1 which choose the strategy of restrained allocation at each transmission time t while using, respectively, the REWIC coder and the RCC coder (images 11–20).
with PC(i, t) being the probability that i quantizers follow a strategy of restrained allocation at t. The law governing the rate at which quantizers choose a strategy is then given by:
n X dn1 1 dPC i ¼r r dt dt i¼0
where
ð3Þ
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dPC PC ði; t þ MtÞ P C ði; tÞ ¼ lim Mt!0 Mt dt
ð4Þ
Following Appendix B, we have that
PC ði; t þ MtÞ PC ði; tÞ ¼ PC ði; tÞ½ið1 qÞ ðn iÞq rMt þ PC ði þ 1; tÞði þ 1Þð1 qiþ1 Þ þ PC ði 1; tÞðn i þ 1Þqi1
ð5Þ
up to terms of order Mt. C ði;tÞ In the limit of Mt ? 0, we have that PC ði;tþMtÞP becomes 1r dPdtC rMt and the corrections of order Mt vanish; therefore, from Eqs. (4) and (5), it follows that n n X X 1 dP C i iP C ði; tÞ½ið1 qÞ ðn iÞq ¼ r dt i¼0 i¼0
þ þ ¼
n X i¼0 n X
ði 1ÞPC ði; tÞ½ið1 qÞ
n X
i¼0
i¼0
PC ði; tÞ½ið1 qÞ þ
¼n
n X i¼0
qP C ði; tÞ
n X
2.1.2. (Case 1.2) Imperfect knowledge In most circumstances however decisions on what information will be transmitted first should often be made with incomplete knowledge of which strategy is best. In this case, to be congruent with the reality the assumptions behind a transmission system should be modified as follows: (i) the payoffs to each quantizer may still be independent of what the others are doing, but (ii) there exists imperfect knowledge in the transmission system, and the perceived payoff will be an inaccurate version of the actual payoff. Following [4], imperfect knowledge is modeled by assuming that the perceived payoff is normally distributed around the actual payoff Bi. That is, the probability for the perceived payoff for strategy i to be x is given by:
1 f ðx; Bi ; r2 Þ ¼ pffiffiffiffiffiffiffi expfðx Bi Þ2 =2r2 g r 2p
ði þ 1ÞPC ði; tÞ½ðn iÞq
i¼0 n X
all quantizers end up using heavy bit allocation. It corresponds to the behavior of the REWIC coder in Figs. 2 and 3.
with variance r2 and mean Bi. In a transmission system which has imperfect knowledge, r is larger than zero. From Eq. (10) we have that the probability that restrained allocation is perceived to be better that heavy allocation is given by
PC ði; tÞ½ðn iÞq
iPC ði; tÞ ¼ nq n1
ð6Þ
i¼0
q¼
Z
where q ¼ i¼0 q P C ði; tÞ. Thus, following Eqs. (3) and (6), we have that
ð7Þ
2.1. (Case 1) Payoffs to each quantizer are independent of what the others are doing At different bit rates quantizers evaluate the perceived payoff for restrained and heavy allocation and switch to the strategy with the highest benefit. Let Bi be the payoff function accrued by each quantizer using strategy i, with i = 1, 2. 2.1.1. (Case 1.1) Perfect knowledge We first consider the case in which the underlying assumptions are: (i) the payoffs Bi to each quantizer are independent of what the others are doing; and (ii) there is perfect knowledge in the transmission system, and thus the perceived payoff for each strategy (restrained or heavy bit allocation) will be an accurate version of the actual payoff. Specifically suppose that payoff B1 accrued by each quantizer using the strategy of restrained bit allocation is smaller than payoff B2 using a strategy of heavy bit allocation: B1 < B2. Then, since there is perfect knowledge we have that the probability that restrained allocation is perceived to be better that heavy allocation is zero. In this case, it follows that Eq. (7) becomes
ð8Þ
whose solution is
n1 ðtÞ ¼ n1 ð0Þert
dxf ðx; B1 ; r2 Þ
Z
x
dyf ðy; B2 ; r2 Þ
ð11Þ
1
which can be evaluated to be
Using the law given in Eq. (7), in the following we can made an examination of the basic assumptions behind dynamic allocation in a transmission system. It shall lead to a prediction of the performance of a transmission system in real applications.
dn1 ¼ rn1 dt
1
1
Pn
dn1 ¼ rðnq n1 Þ: dt
ð10Þ
ð9Þ
From Eq. (9), we have that, although initially all quantizers make use of restrained allocation, n1(0) = n, in the long transmission time
1 2
q ¼ 1 þ erf
B1 B2 pffiffiffi r 2
ð12Þ
with
2 erfðzÞ ¼ pffiffiffiffi
p
Z
z
2
et dt:
ð13Þ
0
Thus it follows that when error (r > 0) is small we have that q close to zero, given that B1 < B2 as above. For a transmission system which has extreme uncertainty about which strategy is most suitable, r becomes very large in Eq. (12). In this case q goes to 1/2 and will vary almost linearly with B1 B2 and the dynamical behavior of n1 ðtÞ generated by Eq. (7) can be evaluated as:
dn1 ¼ rðnq n1 Þ dt
ð14Þ
whose solution is
n1 ðtÞ ¼ nq ðnq n1 ð0ÞÞert
ð15Þ
From Eq. (15), it can be seen that, unlike the previous case of perfect knowledge, the transmission system still has quantizers using restrained bit allocation in the long transmission time limit, since n1 ðtÞ ¼ nq for long times. 2.2. (Case 2) Payoffs depend on how many quantizers are using that particular strategy Finally, we come to examine the last case in which the underlying assumptions are: (i) the payoffs Bi to each quantizer will depend on how many quantizers are using that particular strategy; and (ii) there is either perfect or imperfect knowledge in the transmission system. Then, we have that Bi will be a positive function of ni(t), and monotonically decreasing with the number of quantizers using strategy i as follows. Let R be the total capacity of bit resources available for the transmission problem at hand, and let di be the amount by which the utility of strategy i decreases as each additional quantizer
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chooses to use it for further transmission. The payoff function Bi accrued by each quantizer using strategy i is then defined by:
Bi ðtÞ ¼ R ni ðtÞ di
ð16Þ
with i = 1,2; and where 0 < d1 < d2, since d2 corresponds to the payoff decrement as one more quantizer chooses the strategy of heavy bit allocation, where as d1 corresponds to that of the strategy of restrained bit allocation. Using the law given in Eq. (7) we can now analyze the behavior of a transmission system at which quantizers choose between restrained and heavy allocation with either perfect or imperfect knowledge. Recall that decisions on what information will be transmitted first should often be made with incomplete knowledge of which strategy is best. From Eq. (12) we have that the probability that restrained allocation is perceived to be better that heavy allocation is given by
1 n d n d q ¼ 1 þ erf 2 2 pffiffiffi 1 1 : 2 r 2
ð17Þ
To evaluate the behavior of the dynamical equation, dn1 ¼ rðnq n1 Þ, governing the rate at which quantizers choose bedt tween restrained and heavy allocation in a transmission system, firstly we need to compute the average function P q ¼ ni¼0 q PC ði; tÞ, where q is given by Eq. (17). Since this evaluation is nontrivial, we will use the mean field approximation which consists in writing the average of a function as a function of the average, [5]. The main idea of the application of the mean field theory (MFT) in our problem is to focus on one quantizer and assume that the most important contribution to the interactions of such quantizer with its neighboring quantizers is determined by the mean field due to the neighboring quantizers. In this mean field approximation we have that
1 2
q ¼ 1 þ erf
n2 d2 n1 d1 pffiffiffi r 2
ð19Þ
since q close to one, whose solution is (with n1(0) being the number of quantizers that are initially using a strategy of restrained bit allocation)
ð20Þ
until it reaches the equilibrium point n1 ðtÞ ¼ nd2 =ðd1 þ d2 Þ, after which the solution is
n1 ðtÞ ¼ nd2 =ðd1 þ d2 Þ
ð21Þ
For n2 d2 < n1 d1 , or equivalently n1 > nd2 =ðd1 þ d2 Þ, Eq. (7) becomes (since q close to zero)
dn1 ¼ rn1 dt
ð22Þ
whose solution is
n1 ðtÞ ¼ n1 ð0Þert
2.2.2. (Case 2.2) Imperfect knowledge For a transmission system which has extreme uncertainty about which strategy is most suitable, r becomes very large in Eq. (18). In this case q goes to 1/2 and the dynamical behavior of n1 ðtÞ generated by Eq. (7) can be evaluated as:
dn1 ¼ rðn=2 n1 Þ dt
ð24Þ
whose solution is
n1 ðtÞ ¼ n=2 ðn=2 n1 ð0ÞÞert
ð25Þ
Thus in a system with extreme uncertainty about which strategy is most suitable (r very large) we have that the asymptotic number of quantizers (never achieved) which choose restrained bit allocation is n1 = n/2. For cases where the amount of uncertainty about which strategy is most suitable is intermediate between the two previous extreme values of error (r = 0 and r very large), the behavior of n1 ðtÞ exhibits exponential relaxation toward an asymptotic value (which is never achieved) between n1 = nd2/ (d1 + d2) and n1 = n/2. Again, the transmission system still has quantizers using restrained bit allocation in the long transmission time limit. 3. Experimental results
For n2 d2 > n1 d1 , or equivalently n1 < nd2 =ðd1 þ d2 Þ, Eq. (7) becomes
n1 ðtÞ ¼ n ðn n1 ð0ÞÞert
Thus, for a transmission system which has perfect knowledge (r = 0 in Eq. (18)), the behavior of n1 ðtÞ exhibits exponential relaxation n1 ð0Þert toward the fixed point n1 = nd2/(d1 + d2). Hence, the transmission system still has quantizers using restrained bit allocation in the long transmission time limit.
ð18Þ
2.2.1. (Case 2.1) Perfect knowledge For a transmission system which has perfect knowledge (r = 0), Eq. (18) resembles a step function: q close to one when n2 d2 > n1 d1 1 and close to zero when n2 d2 < n1 d1 . Thus dn ¼ rðnq n1 Þ becomes dt piecewise linear and the dynamical behavior of n1 ðtÞ generated by this equation can be evaluated as follows:
dn1 ¼ rðn n1 Þ dt
until it reaches n1 ðtÞ ¼ nd2 =ðd1 þ d2 Þ, after which it stays at this fixed value.
ð23Þ
In the following, we present an objective and subjective evaluation of the REWIC coder and the RCC coder in order to investigate their comparative performance. 3.1. Objective coder evaluation Tests here reported were performed on a dataset composed of twenty 512 512 grayscale test images shown in Fig. 1. For example, test image #1 is composed of four different subimages: noiselike texture, houses, text, and graphics. In this first experiment, we perform a thorough comparison of SPIHT [6], REWIC [1,2], and the RCC coder (Appendix A), by means of the CG-rate curves following Refs. [8,7]. Thus, the compound gain CG is applied to quantify the visual distinctness by means of the difference between the original image and the images reconstructed under various degrees of lossy compression. It allows us to analyze the behavior of SPIHT, REWIC, and RCC across bit rates; with the best coder in the CG sense achieving the lowest value of the compound gain CG at most bit rates. Figs. 4 and 5 show the CG-rate curves on the twenty test images given in Fig. 1. The compression ratio ranges from 20:1 to 120:1. From these plots we have that, at most bit rates, RCC decoded outputs achieve a better objective quality than both REWIC and SPIHT outputs for fifteen test images (75%): test image #1,#3, #4, #5, #6, #7, #10, #11, #13, #15, #16, #17, #18, #19, and #20. It means that the RCC coder (which still has quantizers using restrained bit allocation in the long transmission time limit) is the overall winner in this objective coder evaluation using CG-rate curves, as it was predicted by the comparison of assumptions behind dynamic allocation. Next we perform another experiment
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Image #1
Image #2
4
2⋅10 CG 2⋅104 2⋅104 2⋅104 1⋅104 4 1⋅10 1⋅104 8⋅103 6⋅103 4⋅103 20:1
5⋅10
4
CG 5⋅104 4⋅104 4⋅10
4
3⋅104 2⋅104 2⋅104
RCC SPIHT REWIC 40:1
60:1
80:1
100:1
RCC SPIHT REWIC
2⋅104 1⋅10
120:1
4
20:1
40:1
60:1
Compression Ratio
4⋅10
CG
3
4
3⋅104
4⋅103
2⋅10
4
4⋅10
3
2⋅10
4
4⋅10
3
4
3
2⋅10
3⋅10 2⋅10
3
2⋅10
3
20:1
RCC SPIHT REWIC 40:1
60:1
80:1
100:1
RCC SPIHT REWIC
1⋅104 5⋅10
120:1
3
20:1
40:1
60:1
Compression Ratio
CG
1⋅10
4
8⋅10
3
6⋅10
3
4⋅10
3
2⋅10
3
20:1
RCC SPIHT REWIC 40:1
60:1
80:1
100:1
120:1
1⋅10 CG 1⋅104 4 1⋅10 4 1⋅10 3 9⋅10 3 8⋅10 7⋅103 3 6⋅10 3 5⋅10 4⋅103 3 3⋅10 20:1
RCC SPIHT REWIC 40:1
60:1
Compression Ratio
2⋅10
CG
4⋅10
4
3⋅10
2⋅10
4
1⋅104 1⋅104
RCC SPIHT REWIC 40:1
60:1
80:1
100:1
120:1
1⋅10
4
8⋅10
3
6⋅10
3
4⋅10
3
20:1
RCC SPIHT REWIC 40:1
60:1
3⋅10
4
4
2⋅10
4
2⋅10
4
2⋅10
4
2⋅10
4
2⋅10
4
CG
RCC SPIHT REWIC
4
20:1
40:1
60:1
80:1
100:1
120:1
Image #10 4
2⋅104
80:1
Compression Ratio
4⋅10
1⋅10
120:1
CG 2⋅104
Image #9 4
100:1
Image #8
4
Compression Ratio
4⋅10
80:1
Compression Ratio
Image #7
4
2⋅10 CG 1⋅104 1⋅104 1⋅104 4 1⋅10 1⋅104 9⋅103 8⋅103 3 7⋅10 3 6⋅10 3 5⋅10 20:1
120:1
4
1⋅104 4
100:1
Image #6
4
1⋅10
80:1
Compression Ratio
Image #5 2⋅10
120:1
Image #4
3
CG 5⋅103 5⋅10
100:1
Compression Ratio
Image #3 6⋅10
80:1
100:1
RCC SPIHT REWIC
1⋅104 120:1
Compression Ratio
5⋅10
3
20:1
40:1
60:1
80:1
100:1
120:1
Compression Ratio
Fig. 4. CG-rate curves on the 1–10 test images given in Fig. 1. The compression ratio ranges from 20:1 to 120:1.
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Image #11 CG
1⋅10
4
1⋅10
4
1⋅10
4
Image #12 5⋅10
4
CG 4⋅104 4⋅10
4
3⋅10
4
1⋅10
4
2⋅10
4
2⋅10
4
9⋅10
3
RCC SPIHT REWIC
8⋅103 7⋅103 20:1
2⋅104
40:1
60:1
80:1
100:1
1⋅10
5⋅103 20:1
120:1
RCC SPIHT REWIC
4
40:1
60:1
Compression Ratio
Image #13 CG
4
1⋅104 1⋅10
4
8⋅10
3
6⋅103 RCC SPIHT REWIC 40:1
60:1
80:1
100:1
RCC SPIHT REWIC
4⋅103 2⋅10
120:1
3
20:1
40:1
60:1
4
2⋅104 2⋅104 RCC SPIHT REWIC
4
5⋅103 20:1
40:1
60:1
80:1
100:1
120:1
2⋅104 CG 2⋅104 2⋅104 4 2⋅10 4 2⋅10 1⋅104 1⋅104 1⋅104 8⋅103 6⋅103 20:1
40:1
60:1
4
5⋅10
4
4⋅104 4
2⋅10
4
2⋅104 20:1
RCC SPIHT REWIC 40:1
60:1
80:1
100:1
120:1
1⋅104 CG 1⋅104 1⋅104 9⋅103 8⋅103 7⋅103 6⋅103 5⋅103 4⋅103 3⋅103 2⋅103 20:1
40:1
60:1
Image #19 4
3⋅10
4
100:1
120:1
Image #20 3⋅104
CG 2⋅10
4
2⋅104
2⋅104
2⋅104
2⋅104
2⋅104
1⋅104
RCC SPIHT REWIC
1⋅104 5⋅103 20:1
80:1
Compression Ratio
4⋅104 4⋅10
120:1
RCC SPIHT REWIC
Compression Ratio
CG
100:1
Image #18
4⋅104
3⋅10
80:1
Compression Ratio
Image #17 6⋅104 5⋅10
120:1
RCC SPIHT REWIC
Compression Ratio
CG
100:1
Image #16
3⋅104
1⋅10
80:1
Compression Ratio
Image #15 4⋅104
2⋅10
120:1
Image #14 1⋅10
Compression Ratio
CG
100:1
Compression Ratio
4
1⋅10 CG 1⋅104 1⋅104 3 9⋅10 8⋅103 3 7⋅10 3 6⋅10 3 5⋅10 3 4⋅10 3⋅103 3 2⋅10 20:1
80:1
40:1
60:1
80:1
100:1
120:1
Compression Ratio
5⋅103 20:1
RCC SPIHT REWIC 40:1
60:1
80:1
100:1
120:1
Compression Ratio
Fig. 5. CG-rate curves on the 11–20 test images given in Fig. 1. The compression ratio ranges from 20:1 to 120:1.
J.A. García et al. / J. Vis. Commun. Image R. 20 (2009) 351–363
359
where rank orders by human observers predict visual fidelity using the compression methods under analysis. 3.2. Subjective coder evaluation This second experiment was designed to compare the subjective quality of decoded outputs using the three compression methods on seven test images: #2, #6, #12, #15, #16, #18, and #20. Figs. 6–12 show the respective decoded outputs which were used in this subjective evaluation. These figures illustrate the comparative performance of SPIHT, REWIC, and RCC. It can be seen that, in general, SPIHT reconstructions exhibit bad visual fidelity since
Fig. 7. Visual comparison in performance for image #6 at 0.08 bpp in order to illustrate the subjective quality of the SPIHT, REWIC, and RCC coders.
Fig. 6. Visual comparison in performance for image #2 at 0.08 bpp in order to illustrate the subjective quality of the SPIHT, REWIC, and RCC coders.
noise data was incorrectly prioritized before significant visual information from various areas. The RCC outputs are fair since essential details are clearly detected in the relevant sections of the picture. The visual quality of REWIC outputs is intermediate between that of SPIHT and that of RCC. Ten volunteers subjectively evaluated the reconstructed images using an ITU-R Recommendation 500-10 [9]. The ITU-R 500-10 recommends to classify the test pictures into five different quality groups: 5 = excellent, the distortions are imperceptible; 4 = good, the distortions are perceptible, but not annoying; 3 = fair, the distortions are slightly annoying; 2 = poor, the distortions are annoying; 1 = bad, the distortions are very annoying.
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Fig. 8. Visual comparison in performance for image #12 at 0.16 bpp in order to illustrate the subjective quality of the SPIHT, REWIC, and RCC coders.
The method of assessment was cyclic in that the assessor was first presented with the original picture, then with the same picture but decoded at a bit rate. For each original test image, the assessor was presented with a series of pictures decoded using SPIHT, RCC, and REWIC (at the same bit rate) in random order to be assessed. Following this she/he was asked to rank order the three decoded outputs, keeping the original in mind. Table 1 summarizes the subjective rankings for each test image: the first, second, and third position in the rank orders given by each observer. Table 2 gives the total number of first, second and third positions in the subjective rank orders. As can be seen from these tables, subjective rankings by human observers predict that RCC decoded outputs exhibit a better visual fidelity than SPIHT and
Fig. 9. Visual comparison in performance for image #15 at 0.157 bpp in order to illustrate the subjective quality of the SPIHT, REWIC, and RCC coders.
REWIC reconstructions: the number of first positions in the rank orders is of 47 for RCC, whereas the number of first positions is of 17 for REWIC, and 6 for SPIHT. 4. Conclusions 1 ¼ rðnq n1 Þ, governs the rate at The dynamical equation, dn dt which quantizers choose between restrained and heavy bit allocation in a transmission system with n quantizers; where r is the average rate at which quantizers reevaluate their preferences regarding bit allocation, and q is the probability that restrained bit allocation will be perceived by a quantizer to be better than
J.A. García et al. / J. Vis. Commun. Image R. 20 (2009) 351–363
361
Fig. 10. Subjective comparison of decoded outcomes at 0.089 bpp in Image #16 using SPIHT, REWIC, and RCC coders.
Fig. 11. Subjective comparison of decoded outcomes at 0.2 bpp in Image #18 using SPIHT, REWIC, and RCC coders.
heavy bit allocation. Using this law we have examined the assumptions behind a transmission system in which quantizers are capable of choosing between restrained bit allocation and heavy bit allocation. A basic assumption of a transmission system for which in the long transmission time all quantizers end up using heavy bit allocation is that there exists perfect knowledge and thus the perceived payoff for each strategy (either restrained or heavy bit allocation) is an accurate version of the actual payoff. On the contrary, the assumption of imperfect knowledge may be true for a transmission system which still has quantizers using restrained bit allocation in the long time limit.
In most real applications, perfect knowledge about the state of the transmission is not available, therefore decisions on what information will be transmitted first should often be made with incomplete knowledge of which strategy is best. Then, based on the validity of the assumptions derived, we can now predict the performance of the transmission system in a real problem: the assumption of imperfect knowledge for a system which still has quantizers using restrained allocation in the long limit is more realistic, which probably results in a better performance than in the case of a system for which all quantizers end up using heavy allocation, whose assumption of perfect knowledge will often be incorrect.
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J.A. García et al. / J. Vis. Commun. Image R. 20 (2009) 351–363 Table 1 Subjective rankings for each test image: the first, second, and third position in the rank orders given by each human observer. Decoded outputs
Coder
Rank order given by 10 observers #1
#2
#3
#4
#5
#6
#7
#8
#9
#10
Image #2 at 0.08 bpp
SPIHT RCC REWIC
3 1 2
3 2 1
3 2 1
3 1 2
3 1 2
3 2 1
3 1 2
3 1 2
3 1 2
3 2 1
Image #6 at 0.08 bpp
SPIHT RCC REWIC
3 1 2
3 2 1
3 1 2
3 1 2
3 2 1
3 2 1
3 1 2
3 1 2
3 2 1
3 1 2
Image #12 at 0.16 bpp
SPIHT RCC REWIC
1 2 3
3 1 2
3 1 2
3 2 1
1 2 3
3 1 2
1 2 3
3 2 1
1 2 3
3 1 2
Image #15 at 0.1568 bpp
SPIHT RCC REWIC
3 1 2
3 2 1
2 1 3
3 1 2
3 1 2
3 2 1
3 1 2
3 2 1
3 1 2
3 1 2
Image #16 at 0.0889 bpp
SPIHT RCC REWIC
3 1 2
3 1 2
3 1 2
3 2 1
3 1 2
3 1 2
3 2 1
3 1 2
1 2 3
2 1 3
Image #18 at 0.2 bpp
SPIHT RCC REWIC
3 1 2
3 1 2
3 2 1
3 1 2
1 2 3
3 1 2
3 1 2
3 2 1
3 1 2
3 1 2
Image #20 at 0.1333 bpp
SPIHT RCC REWIC
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
Table 2 Total number of first, second, and third positions in the subjective rank orders given by human observers. Coder
SPIHT RCC REWIC
Ranking given by human observers # First positions
# Second positions
# Third positions
6 47 17
2 23 45
62 0 8
Appendix A. Congestion control algorithm
Fig. 12. Subjective comparison of decoded outcomes at 0.133 bpp in Image #20 using SPIHT, REWIC, and RCC coders.
Experimental results have demonstrated the validity of this prediction of superior performance of the RCC coder which still has quantizers using restrained bit allocation in the long time, as compared with the REWIC coder in which all quantizers end up using heavy bit allocation.
Acknowledgments This research was sponsored by the Spanish Board for Science and Technology (CICYT) under Grant TEC2007-60450. We are deeply indebted to the referees for suggesting several good ways to improve the quality of the initial manuscript.
In dynamic allocation, quantizers monitor the level of utility of bit allocation across bit rates and, at any point of the transmission, they can decide to switch the behavior between restrained bit allocation and heavy bit allocation. The problem is that, without central control, all quantizers might switch from restrained to heavy allocation at higher bit rates, leading to a congestion of the transmission, if total demand for bit allocation exceeds the available capacity. Following Ref. [3], here we give a brief description of the congestion control algorithm which is based on the behavior of the 1 ¼ rðnq n1 Þ. In this algorithm we have to dynamical equation, dn dt distinguish the basic transmission model and the congestion avoidance. The basic transmission model assumed follows a rational embedded zerotree encoding, [1]. Thus, in a first stage, a wavelet transform is applied to the original data. The second stage of the coding scheme is composed of two key elements: (i) quantizer formation where coefficients of the wavelet transform are partitioned into a number of quantizers; and (ii) a prioritization protocol which is used to choose among competitor quantizers, at each truncation time in the progressive transmission, in order to deliver a subsequence of the completely embedded bit stream of the selected quantizer. In a third stage, adaptive arithmetic encoding exploits both inter-band and intra-band relationships that still remain after the wavelet transformation and zerotree encoding.
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We can now show the application of congestion control to this basic transmission model as follows: the maximum number of quantizers which can choose heavy bit allocation is n2 = n nd2/ (d1 + d2); with d1 being the restrained allocation limit, d2 being the heavy allocation limit, and n being the total number of quantizers. For each quantizer Qi we select the bit stream BSti , candidate to be transmitted at time t, by using a window size WS to segment data when the quantizer is coded bit-plane by bit-plane through successive-approximation quantization. Amongst all candidates BSti , we choose a bit stream BStm achieving the highest expected increase in utility per coding bit, at time t. Initially all quantizers are using the strategy of restrained allocation, and thus the number of quantizers using the strategy of heavy bit allocation at t = 0 is given by n2 = 0. The congestion avoidance is used when the number of bits of BStm , noted as #ðBStm Þ, is greater than the restrained allocation limit d1: #ðBStm Þ > d1 . In this case, if #ðBStm Þ 6 d2 and n2 < n nd2/ (d1 + d2), then BStm may still be sent to the decoder and the number of quantizers n2 which choose heavy allocation is increased by one. But if #ðBStm Þ > d2 or n2 P n nd2/(d1 + d2), then the window size WS to segment data is divided by 2 and we recalculate (using this new window size) the bit stream BSti candidate to be transmitted for each quantizer Qi. The software and documentation can be accessed in: http://decsai.ugr.es/cvg/rewic_revic/ Appendix B. Derivation of probabilities We have that the probability P1(x; n1) for x quantizers to change from restrained bit allocation to heavy bit allocation, given that there are initially n1 quantizers using the strategy of restrained allocation, is defined by:
P1 ðx; n1 Þ ¼
n1 x p ð1 pÞn1 x x
ð26Þ
with p, as given in Eq. (1), being the probability that a quantizer currently engaged in restrained bit allocation changes to heavy bit allocation. Similarly, we can define the probability P2(y; n2) for y quantizers to change from heavy to restrained allocation, given there are initially n2 quantizers that follow the strategy of heavy allocation, using the probability q that a quantizer changes from heavy to restrained allocation in Mt:
P2 ðy; n2 Þ ¼
n2 y
qy ð1 qÞn2 y
ð27Þ
where q = q r Mt. In order to derive PC(i, t + Mt), recall that for sufficiently small Mt interval, most quantizers will not be making a choice about to switch between strategies and thus there are only three possible values for the net change from heavy to restrained allocation (with a net change being the difference between the number that go from heavy to restrained allocation and those doing the opposite): 0, 1, and 1. Here a net change from heavy to restrained allocation of size 1 is given by y = 1 quantizers that change from heavy to re-
strained allocation. A net change from heavy to restrained allocation of size-1 is given by x = 1 quantizers that change from restrained to heavy allocation. It follows that
PC ði; t þ MtÞ ¼ PC ði; tÞ½1 P1 ðx ¼ 1; iÞ P 2 ðy ¼ 1; n iÞ þ P C ði þ 1; tÞP1 ðx ¼ 1; i þ 1Þ þ PC ði 1; tÞP2 ðy ¼ 1; n i þ 1Þ
ð28Þ
where 1 P1(x = 1; i) P2(y = 1; n i) is the probability that during Mt there is a net change of 0 from heavy to restrained allocation, given that there are initially i quantizers using a strategy of restrained bit allocation; with P1(x = 1; i + 1) being the probability that during Mt there is a net change of 1 from heavy to restrained allocation, given that there are initially i + 1 quantizers using a strategy of restrained bit allocation; and P2(y = 1;n i + 1) being the probability that during Mt there is a net change of 1 from heavy to restrained bit allocation, given that there are initially i 1 quantizers using a strategy of restrained allocation. From Eqs. (26) and (27), we have that Eq. (28) becomes, up to terms of order Mt:
PC ði; t þ MtÞ ¼ PC ði; tÞ½1 ið1 qÞrMt ðn iÞqrMt þ PC ði þ 1; tÞði þ 1Þð1 qiþ1 ÞrMt þ PC ði 1; tÞðn i þ 1Þqi1 rMt
ð29Þ
with qi+1 and qi1 being the probability q evaluated at i + 1 and i 1, respectively. From Eq. (29), it follows that:
PC ði; t þ MtÞ PC ði; tÞ ¼ P C ði; tÞ½ið1 qÞ ðn iÞq rMt þ P C ði þ 1; tÞði þ 1Þð1 qiþ1 Þ þ P C ði 1; tÞðn i þ 1Þqi1 up to terms of order Mt. References [1] J.A. Garcia, Rosa Rodriguez-Sanchez, J. Fdez-Valdivia, Xose R. Fdez-Vidal, Rational systems exhibit moderate risk aversion with respect to gambles on variable-resolution compression, Optical Engineering 41 (9) (2002) 2216–2237. [2] J.A. Garcia, Rosa Rodriguez-Sanchez, J. Fdez-Valdivia, An embedded coder providing better image quality at very low bit rates, Optical Engineering 43 (3) (2004) 615–627. [3] J.A. Garcia, Rosa Rodriguez-Sanchez, J. Fdez-Valdivia, Congestion Control in Dynamic Allocation. DECSAI Technical Report, University of Granada, Spain, 2008. [4] Bernardo A. Huberman, Tad Hogg, The behaviour of computational ecologies, in: B.A. Huberman (Ed.), The Ecology of Computation, Elsevier Science Publishers B.V., 1988, pp. 77–115. [5] G. Parisi, Statistical Field Theory, Westview Press, 1998. [6] A. Said, W.A. Pearlman, A new fast and efficient image codec based on set partitioning in hierarchical trees, IEEE Transactions on Circuit and System for Video Technology 6 (3) (1996) 243–250. [7] J.A. Garcia, R. Rodriguez-Sanchez, J. Fdez-Valdivia, Progressive Image Transmission: The Role of Rationality, Cooperation and Justice, SPIE Press, PM-140, Bellingham, Washington, USA, 2004, p. 230. [8] J.A. Garcia, J. Fdez-Valdivia, Xose R. Fdez-Vidal, Rosa Rodriguez-Sanchez, Information theoretic measure for visual target distinctness, IEEE Transactions on Pattern Analysis and Machine Intelligence 23 (4) (2001) 362–383. [9] ITU-R Recommendations, Broadcasting service (television): Recommendation 500-10, Supplement 3 to vol. 1997, BT Series, 2000 Edition.
Contents lists available at ScienceDirect
J. Vis. Commun. Image R. journal homepage: www.elsevier.com/locate/jvci
A critical examination of the assumptions used in dynamic allocation J.A. García, Rosa Rodriguez-Sánchez, J. Fdez-Valdivia * Departamento de Ciencias de la Computación e I. A., CITIC-UGR (Research Center on Information and Communications Technology), Universidad de Granada, 18071 Granada, Spain
a r t i c l e
i n f o
Article history: Received 24 September 2008 Accepted 12 March 2009 Available online 24 March 2009 Keywords: Image transmission Multiple quantizers Dynamic allocation Assumptions Bit allocation Knowledge Coder evaluation Compound gain Congestion control
a b s t r a c t In dynamic allocation quantizers are capable of choosing between limited allocation of bits and bit allocation without restriction. The goal of this paper is to perform a comparative analysis of the assumptions used in a transmission system which still has quantizers using restrained bit allocation in the long time and in a transmission system for which all quantizers end up using heavy bit allocation. Then, based on the validity of the assumptions derived, we will be able to predict the performance of each system in a real problem. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction If we wish that a transmission system can operate in a more efficient fashion, one way is to allow a dynamic allocation of bits to the most needed quantizers. There, quantizers monitor the level of utility of bit allocation across bit rates and, at any point of the transmission, they might decide to switch the behavior between restrained bit allocation and heavy bit allocation. Restrained bit allocation implies limitation, as on quantizer’s freedom of bit allocation; while heavy allocation means bit allocation without restriction. Thus, at any given bit rate, quantizers evaluate the perceived payoff for restrained and heavy allocation and switch to the strategy with the highest benefit. To illustrate this process, Figs. 2 and 3 show plots of the number of quantizers, noted as n1, which choose the strategy of restrained allocation at each transmission time while using the REWIC coder, [1,2], on each test image of the dataset in Fig. 1. As can be seen from these plots, initially all quantizers (n = 16) make use of restrained allocation but in the long transmission time all quantizers end up using heavy bit allocation. We have developed a modified version of the REWIC coder, called as the REWIC with Congestion Control (RCC), which implements a congestion control algorithm described in Ref. [3] (see Appendix A). Figs. 2 and 3 also display plots of the number of quantizers, n1, which * Corresponding author. Fax: +34 58 24 3317. E-mail addresses: [email protected] (J.A. García), [email protected] (R. Rodri guez-Sánchez), [email protected] (J. Fdez-Valdivia). 1047-3203/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jvcir.2009.03.006
choose the strategy of restrained allocation at each transmission time while using the RCC coder on each test image of the dataset in Fig. 1. From these figures, it can be seen that RCC still has quantizers using restrained bit allocation in the long transmission time limit. The goal of this paper is to made a comparison of assumptions behind dynamic allocation in a transmission system like RCC (which still has quantizers using restrained bit allocation in the long time limit) and in a system like REWIC in which all quantizers end up using heavy bit allocation. And based on the validity of assumptions derived we will be able to predict the performance of each transmission system in real applications. Section 2 examines the assumptions behind a transmission system like the REWIC coder in which in the long transmission time all quantizers end up using heavy bit allocation to uncover that (i) the payoffs to each quantizer are independent of what the others are doing; and (ii) there is perfect knowledge in the transmission system. In most circumstances however, perfect knowledge about the state of the transmission is not available, which often results in a degradation of the performance of the transmission system. On the contrary, Section 2 shows that, the assumption of imperfect knowledge may be true for a transmission system like the RCC coder which still has quantizers using restrained bit allocation in the long time limit. In this case, assumptions derived are more realistic, and it probably results in a better performance. Section 3 shows an objective and subjective coder evaluation, in order to investigate the performance of the RCC coder as compared with that of the REWIC coder. The main conclusions of the paper are summarized in Section 4.
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Fig. 1. Image database used in the experiments.
2. Dynamic allocation model We assume that the image can be divided into a finite number n of regions, each of which is represented by a quantizer. Quantizers are capable of choosing between restrained bit allocation and heavy bit allocation. Thus, at any time, quantizers might decide to switch their behavior among restrained and heavy allocation according to the perceived payoff. Transmission times of relevant information are often quite different for various quantizers, so only a fraction of them will consider switching between restrained and heavy bit allocation during a given interval Mt of transmission time, if Mt is small enough. Hence probabilistic dynamics can be used to provide an analytic description of this process at which quantizers may decide to switch their behavior.
Let n1(t) be the number of quantizers following restrained bit allocation at time t; and n2(t) be the number of quantizers using the strategy of heavy bit allocation at this time. Here, we assume that n2(t) = n n1(t). Assuming that the perceived payoff does not change over Mt, the probability p that a quantizer changes from restrained bit allocation to heavy bit allocation in Mt is given by:
p ¼ ð1 qÞ r Mt
ð1Þ
where q is the probability that restrained bit allocation will be perceived by a quantizer to be better than heavy bit allocation; and r is the average rate at which quantizers reevaluate their preferences regarding bit allocation.
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J.A. García et al. / J. Vis. Commun. Image R. 20 (2009) 351–363
Image #1
Image #2
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #3
t
Image #4
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #5
t
Image #6
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #7
t
Image #8
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #9
t
Image #10
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
t
Fig. 2. Plots of the number of quantizers n1 which choose the strategy of restrained allocation at each transmission time t while using, respectively, the REWIC coder and the RCC coder (images 1–10).
Following Ref. [4], we can derive the law governing the rate, dn1 =dt, where n1 denotes the average number of quantizers following a strategy of restrained bit allocation, as given by:
n1 ¼
n X i¼0
i PC ði; tÞ
ð2Þ
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J.A. García et al. / J. Vis. Commun. Image R. 20 (2009) 351–363
Image #11
Image #12
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #13
t
Image #14
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #15
t
Image #16
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #17
t
Image #18
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
Image #19
t
Image #20
16
16
n1(t) 14
n1(t) 14
12 10 8 6 4 2 0
12 10 8 6 4 2 0
REWIC RCC 0
transmission time
t
REWIC RCC 0
transmission time
t
Fig. 3. Plots of the number of quantizers n1 which choose the strategy of restrained allocation at each transmission time t while using, respectively, the REWIC coder and the RCC coder (images 11–20).
with PC(i, t) being the probability that i quantizers follow a strategy of restrained allocation at t. The law governing the rate at which quantizers choose a strategy is then given by:
n X dn1 1 dPC i ¼r r dt dt i¼0
where
ð3Þ
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dPC PC ði; t þ MtÞ P C ði; tÞ ¼ lim Mt!0 Mt dt
ð4Þ
Following Appendix B, we have that
PC ði; t þ MtÞ PC ði; tÞ ¼ PC ði; tÞ½ið1 qÞ ðn iÞq rMt þ PC ði þ 1; tÞði þ 1Þð1 qiþ1 Þ þ PC ði 1; tÞðn i þ 1Þqi1
ð5Þ
up to terms of order Mt. C ði;tÞ In the limit of Mt ? 0, we have that PC ði;tþMtÞP becomes 1r dPdtC rMt and the corrections of order Mt vanish; therefore, from Eqs. (4) and (5), it follows that n n X X 1 dP C i iP C ði; tÞ½ið1 qÞ ðn iÞq ¼ r dt i¼0 i¼0
þ þ ¼
n X i¼0 n X
ði 1ÞPC ði; tÞ½ið1 qÞ
n X
i¼0
i¼0
PC ði; tÞ½ið1 qÞ þ
¼n
n X i¼0
qP C ði; tÞ
n X
2.1.2. (Case 1.2) Imperfect knowledge In most circumstances however decisions on what information will be transmitted first should often be made with incomplete knowledge of which strategy is best. In this case, to be congruent with the reality the assumptions behind a transmission system should be modified as follows: (i) the payoffs to each quantizer may still be independent of what the others are doing, but (ii) there exists imperfect knowledge in the transmission system, and the perceived payoff will be an inaccurate version of the actual payoff. Following [4], imperfect knowledge is modeled by assuming that the perceived payoff is normally distributed around the actual payoff Bi. That is, the probability for the perceived payoff for strategy i to be x is given by:
1 f ðx; Bi ; r2 Þ ¼ pffiffiffiffiffiffiffi expfðx Bi Þ2 =2r2 g r 2p
ði þ 1ÞPC ði; tÞ½ðn iÞq
i¼0 n X
all quantizers end up using heavy bit allocation. It corresponds to the behavior of the REWIC coder in Figs. 2 and 3.
with variance r2 and mean Bi. In a transmission system which has imperfect knowledge, r is larger than zero. From Eq. (10) we have that the probability that restrained allocation is perceived to be better that heavy allocation is given by
PC ði; tÞ½ðn iÞq
iPC ði; tÞ ¼ nq n1
ð6Þ
i¼0
q¼
Z
where q ¼ i¼0 q P C ði; tÞ. Thus, following Eqs. (3) and (6), we have that
ð7Þ
2.1. (Case 1) Payoffs to each quantizer are independent of what the others are doing At different bit rates quantizers evaluate the perceived payoff for restrained and heavy allocation and switch to the strategy with the highest benefit. Let Bi be the payoff function accrued by each quantizer using strategy i, with i = 1, 2. 2.1.1. (Case 1.1) Perfect knowledge We first consider the case in which the underlying assumptions are: (i) the payoffs Bi to each quantizer are independent of what the others are doing; and (ii) there is perfect knowledge in the transmission system, and thus the perceived payoff for each strategy (restrained or heavy bit allocation) will be an accurate version of the actual payoff. Specifically suppose that payoff B1 accrued by each quantizer using the strategy of restrained bit allocation is smaller than payoff B2 using a strategy of heavy bit allocation: B1 < B2. Then, since there is perfect knowledge we have that the probability that restrained allocation is perceived to be better that heavy allocation is zero. In this case, it follows that Eq. (7) becomes
ð8Þ
whose solution is
n1 ðtÞ ¼ n1 ð0Þert
dxf ðx; B1 ; r2 Þ
Z
x
dyf ðy; B2 ; r2 Þ
ð11Þ
1
which can be evaluated to be
Using the law given in Eq. (7), in the following we can made an examination of the basic assumptions behind dynamic allocation in a transmission system. It shall lead to a prediction of the performance of a transmission system in real applications.
dn1 ¼ rn1 dt
1
1
Pn
dn1 ¼ rðnq n1 Þ: dt
ð10Þ
ð9Þ
From Eq. (9), we have that, although initially all quantizers make use of restrained allocation, n1(0) = n, in the long transmission time
1 2
q ¼ 1 þ erf
B1 B2 pffiffiffi r 2
ð12Þ
with
2 erfðzÞ ¼ pffiffiffiffi
p
Z
z
2
et dt:
ð13Þ
0
Thus it follows that when error (r > 0) is small we have that q close to zero, given that B1 < B2 as above. For a transmission system which has extreme uncertainty about which strategy is most suitable, r becomes very large in Eq. (12). In this case q goes to 1/2 and will vary almost linearly with B1 B2 and the dynamical behavior of n1 ðtÞ generated by Eq. (7) can be evaluated as:
dn1 ¼ rðnq n1 Þ dt
ð14Þ
whose solution is
n1 ðtÞ ¼ nq ðnq n1 ð0ÞÞert
ð15Þ
From Eq. (15), it can be seen that, unlike the previous case of perfect knowledge, the transmission system still has quantizers using restrained bit allocation in the long transmission time limit, since n1 ðtÞ ¼ nq for long times. 2.2. (Case 2) Payoffs depend on how many quantizers are using that particular strategy Finally, we come to examine the last case in which the underlying assumptions are: (i) the payoffs Bi to each quantizer will depend on how many quantizers are using that particular strategy; and (ii) there is either perfect or imperfect knowledge in the transmission system. Then, we have that Bi will be a positive function of ni(t), and monotonically decreasing with the number of quantizers using strategy i as follows. Let R be the total capacity of bit resources available for the transmission problem at hand, and let di be the amount by which the utility of strategy i decreases as each additional quantizer
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chooses to use it for further transmission. The payoff function Bi accrued by each quantizer using strategy i is then defined by:
Bi ðtÞ ¼ R ni ðtÞ di
ð16Þ
with i = 1,2; and where 0 < d1 < d2, since d2 corresponds to the payoff decrement as one more quantizer chooses the strategy of heavy bit allocation, where as d1 corresponds to that of the strategy of restrained bit allocation. Using the law given in Eq. (7) we can now analyze the behavior of a transmission system at which quantizers choose between restrained and heavy allocation with either perfect or imperfect knowledge. Recall that decisions on what information will be transmitted first should often be made with incomplete knowledge of which strategy is best. From Eq. (12) we have that the probability that restrained allocation is perceived to be better that heavy allocation is given by
1 n d n d q ¼ 1 þ erf 2 2 pffiffiffi 1 1 : 2 r 2
ð17Þ
To evaluate the behavior of the dynamical equation, dn1 ¼ rðnq n1 Þ, governing the rate at which quantizers choose bedt tween restrained and heavy allocation in a transmission system, firstly we need to compute the average function P q ¼ ni¼0 q PC ði; tÞ, where q is given by Eq. (17). Since this evaluation is nontrivial, we will use the mean field approximation which consists in writing the average of a function as a function of the average, [5]. The main idea of the application of the mean field theory (MFT) in our problem is to focus on one quantizer and assume that the most important contribution to the interactions of such quantizer with its neighboring quantizers is determined by the mean field due to the neighboring quantizers. In this mean field approximation we have that
1 2
q ¼ 1 þ erf
n2 d2 n1 d1 pffiffiffi r 2
ð19Þ
since q close to one, whose solution is (with n1(0) being the number of quantizers that are initially using a strategy of restrained bit allocation)
ð20Þ
until it reaches the equilibrium point n1 ðtÞ ¼ nd2 =ðd1 þ d2 Þ, after which the solution is
n1 ðtÞ ¼ nd2 =ðd1 þ d2 Þ
ð21Þ
For n2 d2 < n1 d1 , or equivalently n1 > nd2 =ðd1 þ d2 Þ, Eq. (7) becomes (since q close to zero)
dn1 ¼ rn1 dt
ð22Þ
whose solution is
n1 ðtÞ ¼ n1 ð0Þert
2.2.2. (Case 2.2) Imperfect knowledge For a transmission system which has extreme uncertainty about which strategy is most suitable, r becomes very large in Eq. (18). In this case q goes to 1/2 and the dynamical behavior of n1 ðtÞ generated by Eq. (7) can be evaluated as:
dn1 ¼ rðn=2 n1 Þ dt
ð24Þ
whose solution is
n1 ðtÞ ¼ n=2 ðn=2 n1 ð0ÞÞert
ð25Þ
Thus in a system with extreme uncertainty about which strategy is most suitable (r very large) we have that the asymptotic number of quantizers (never achieved) which choose restrained bit allocation is n1 = n/2. For cases where the amount of uncertainty about which strategy is most suitable is intermediate between the two previous extreme values of error (r = 0 and r very large), the behavior of n1 ðtÞ exhibits exponential relaxation toward an asymptotic value (which is never achieved) between n1 = nd2/ (d1 + d2) and n1 = n/2. Again, the transmission system still has quantizers using restrained bit allocation in the long transmission time limit. 3. Experimental results
For n2 d2 > n1 d1 , or equivalently n1 < nd2 =ðd1 þ d2 Þ, Eq. (7) becomes
n1 ðtÞ ¼ n ðn n1 ð0ÞÞert
Thus, for a transmission system which has perfect knowledge (r = 0 in Eq. (18)), the behavior of n1 ðtÞ exhibits exponential relaxation n1 ð0Þert toward the fixed point n1 = nd2/(d1 + d2). Hence, the transmission system still has quantizers using restrained bit allocation in the long transmission time limit.
ð18Þ
2.2.1. (Case 2.1) Perfect knowledge For a transmission system which has perfect knowledge (r = 0), Eq. (18) resembles a step function: q close to one when n2 d2 > n1 d1 1 and close to zero when n2 d2 < n1 d1 . Thus dn ¼ rðnq n1 Þ becomes dt piecewise linear and the dynamical behavior of n1 ðtÞ generated by this equation can be evaluated as follows:
dn1 ¼ rðn n1 Þ dt
until it reaches n1 ðtÞ ¼ nd2 =ðd1 þ d2 Þ, after which it stays at this fixed value.
ð23Þ
In the following, we present an objective and subjective evaluation of the REWIC coder and the RCC coder in order to investigate their comparative performance. 3.1. Objective coder evaluation Tests here reported were performed on a dataset composed of twenty 512 512 grayscale test images shown in Fig. 1. For example, test image #1 is composed of four different subimages: noiselike texture, houses, text, and graphics. In this first experiment, we perform a thorough comparison of SPIHT [6], REWIC [1,2], and the RCC coder (Appendix A), by means of the CG-rate curves following Refs. [8,7]. Thus, the compound gain CG is applied to quantify the visual distinctness by means of the difference between the original image and the images reconstructed under various degrees of lossy compression. It allows us to analyze the behavior of SPIHT, REWIC, and RCC across bit rates; with the best coder in the CG sense achieving the lowest value of the compound gain CG at most bit rates. Figs. 4 and 5 show the CG-rate curves on the twenty test images given in Fig. 1. The compression ratio ranges from 20:1 to 120:1. From these plots we have that, at most bit rates, RCC decoded outputs achieve a better objective quality than both REWIC and SPIHT outputs for fifteen test images (75%): test image #1,#3, #4, #5, #6, #7, #10, #11, #13, #15, #16, #17, #18, #19, and #20. It means that the RCC coder (which still has quantizers using restrained bit allocation in the long transmission time limit) is the overall winner in this objective coder evaluation using CG-rate curves, as it was predicted by the comparison of assumptions behind dynamic allocation. Next we perform another experiment
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Image #1
Image #2
4
2⋅10 CG 2⋅104 2⋅104 2⋅104 1⋅104 4 1⋅10 1⋅104 8⋅103 6⋅103 4⋅103 20:1
5⋅10
4
CG 5⋅104 4⋅104 4⋅10
4
3⋅104 2⋅104 2⋅104
RCC SPIHT REWIC 40:1
60:1
80:1
100:1
RCC SPIHT REWIC
2⋅104 1⋅10
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4
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40:1
60:1
Compression Ratio
4⋅10
CG
3
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RCC SPIHT REWIC 40:1
60:1
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RCC SPIHT REWIC
1⋅104 5⋅10
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3
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60:1
Compression Ratio
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1⋅10
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RCC SPIHT REWIC 40:1
60:1
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1⋅10 CG 1⋅104 4 1⋅10 4 1⋅10 3 9⋅10 3 8⋅10 7⋅103 3 6⋅10 3 5⋅10 4⋅103 3 3⋅10 20:1
RCC SPIHT REWIC 40:1
60:1
Compression Ratio
2⋅10
CG
4⋅10
4
3⋅10
2⋅10
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1⋅104 1⋅104
RCC SPIHT REWIC 40:1
60:1
80:1
100:1
120:1
1⋅10
4
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3
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3
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RCC SPIHT REWIC 40:1
60:1
3⋅10
4
4
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RCC SPIHT REWIC
4
20:1
40:1
60:1
80:1
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120:1
Image #10 4
2⋅104
80:1
Compression Ratio
4⋅10
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CG 2⋅104
Image #9 4
100:1
Image #8
4
Compression Ratio
4⋅10
80:1
Compression Ratio
Image #7
4
2⋅10 CG 1⋅104 1⋅104 1⋅104 4 1⋅10 1⋅104 9⋅103 8⋅103 3 7⋅10 3 6⋅10 3 5⋅10 20:1
120:1
4
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100:1
Image #6
4
1⋅10
80:1
Compression Ratio
Image #5 2⋅10
120:1
Image #4
3
CG 5⋅103 5⋅10
100:1
Compression Ratio
Image #3 6⋅10
80:1
100:1
RCC SPIHT REWIC
1⋅104 120:1
Compression Ratio
5⋅10
3
20:1
40:1
60:1
80:1
100:1
120:1
Compression Ratio
Fig. 4. CG-rate curves on the 1–10 test images given in Fig. 1. The compression ratio ranges from 20:1 to 120:1.
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Image #11 CG
1⋅10
4
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Image #12 5⋅10
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8⋅103 7⋅103 20:1
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Image #13 CG
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4
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2⋅104 CG 2⋅104 2⋅104 4 2⋅10 4 2⋅10 1⋅104 1⋅104 1⋅104 8⋅103 6⋅103 20:1
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1⋅104 CG 1⋅104 1⋅104 9⋅103 8⋅103 7⋅103 6⋅103 5⋅103 4⋅103 3⋅103 2⋅103 20:1
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CG 2⋅10
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80:1
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4⋅104
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Image #17 6⋅104 5⋅10
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RCC SPIHT REWIC
Compression Ratio
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100:1
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3⋅104
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80:1
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Image #15 4⋅104
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Image #14 1⋅10
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100:1
Compression Ratio
4
1⋅10 CG 1⋅104 1⋅104 3 9⋅10 8⋅103 3 7⋅10 3 6⋅10 3 5⋅10 3 4⋅10 3⋅103 3 2⋅10 20:1
80:1
40:1
60:1
80:1
100:1
120:1
Compression Ratio
5⋅103 20:1
RCC SPIHT REWIC 40:1
60:1
80:1
100:1
120:1
Compression Ratio
Fig. 5. CG-rate curves on the 11–20 test images given in Fig. 1. The compression ratio ranges from 20:1 to 120:1.
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where rank orders by human observers predict visual fidelity using the compression methods under analysis. 3.2. Subjective coder evaluation This second experiment was designed to compare the subjective quality of decoded outputs using the three compression methods on seven test images: #2, #6, #12, #15, #16, #18, and #20. Figs. 6–12 show the respective decoded outputs which were used in this subjective evaluation. These figures illustrate the comparative performance of SPIHT, REWIC, and RCC. It can be seen that, in general, SPIHT reconstructions exhibit bad visual fidelity since
Fig. 7. Visual comparison in performance for image #6 at 0.08 bpp in order to illustrate the subjective quality of the SPIHT, REWIC, and RCC coders.
Fig. 6. Visual comparison in performance for image #2 at 0.08 bpp in order to illustrate the subjective quality of the SPIHT, REWIC, and RCC coders.
noise data was incorrectly prioritized before significant visual information from various areas. The RCC outputs are fair since essential details are clearly detected in the relevant sections of the picture. The visual quality of REWIC outputs is intermediate between that of SPIHT and that of RCC. Ten volunteers subjectively evaluated the reconstructed images using an ITU-R Recommendation 500-10 [9]. The ITU-R 500-10 recommends to classify the test pictures into five different quality groups: 5 = excellent, the distortions are imperceptible; 4 = good, the distortions are perceptible, but not annoying; 3 = fair, the distortions are slightly annoying; 2 = poor, the distortions are annoying; 1 = bad, the distortions are very annoying.
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Fig. 8. Visual comparison in performance for image #12 at 0.16 bpp in order to illustrate the subjective quality of the SPIHT, REWIC, and RCC coders.
The method of assessment was cyclic in that the assessor was first presented with the original picture, then with the same picture but decoded at a bit rate. For each original test image, the assessor was presented with a series of pictures decoded using SPIHT, RCC, and REWIC (at the same bit rate) in random order to be assessed. Following this she/he was asked to rank order the three decoded outputs, keeping the original in mind. Table 1 summarizes the subjective rankings for each test image: the first, second, and third position in the rank orders given by each observer. Table 2 gives the total number of first, second and third positions in the subjective rank orders. As can be seen from these tables, subjective rankings by human observers predict that RCC decoded outputs exhibit a better visual fidelity than SPIHT and
Fig. 9. Visual comparison in performance for image #15 at 0.157 bpp in order to illustrate the subjective quality of the SPIHT, REWIC, and RCC coders.
REWIC reconstructions: the number of first positions in the rank orders is of 47 for RCC, whereas the number of first positions is of 17 for REWIC, and 6 for SPIHT. 4. Conclusions 1 ¼ rðnq n1 Þ, governs the rate at The dynamical equation, dn dt which quantizers choose between restrained and heavy bit allocation in a transmission system with n quantizers; where r is the average rate at which quantizers reevaluate their preferences regarding bit allocation, and q is the probability that restrained bit allocation will be perceived by a quantizer to be better than
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Fig. 10. Subjective comparison of decoded outcomes at 0.089 bpp in Image #16 using SPIHT, REWIC, and RCC coders.
Fig. 11. Subjective comparison of decoded outcomes at 0.2 bpp in Image #18 using SPIHT, REWIC, and RCC coders.
heavy bit allocation. Using this law we have examined the assumptions behind a transmission system in which quantizers are capable of choosing between restrained bit allocation and heavy bit allocation. A basic assumption of a transmission system for which in the long transmission time all quantizers end up using heavy bit allocation is that there exists perfect knowledge and thus the perceived payoff for each strategy (either restrained or heavy bit allocation) is an accurate version of the actual payoff. On the contrary, the assumption of imperfect knowledge may be true for a transmission system which still has quantizers using restrained bit allocation in the long time limit.
In most real applications, perfect knowledge about the state of the transmission is not available, therefore decisions on what information will be transmitted first should often be made with incomplete knowledge of which strategy is best. Then, based on the validity of the assumptions derived, we can now predict the performance of the transmission system in a real problem: the assumption of imperfect knowledge for a system which still has quantizers using restrained allocation in the long limit is more realistic, which probably results in a better performance than in the case of a system for which all quantizers end up using heavy allocation, whose assumption of perfect knowledge will often be incorrect.
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J.A. García et al. / J. Vis. Commun. Image R. 20 (2009) 351–363 Table 1 Subjective rankings for each test image: the first, second, and third position in the rank orders given by each human observer. Decoded outputs
Coder
Rank order given by 10 observers #1
#2
#3
#4
#5
#6
#7
#8
#9
#10
Image #2 at 0.08 bpp
SPIHT RCC REWIC
3 1 2
3 2 1
3 2 1
3 1 2
3 1 2
3 2 1
3 1 2
3 1 2
3 1 2
3 2 1
Image #6 at 0.08 bpp
SPIHT RCC REWIC
3 1 2
3 2 1
3 1 2
3 1 2
3 2 1
3 2 1
3 1 2
3 1 2
3 2 1
3 1 2
Image #12 at 0.16 bpp
SPIHT RCC REWIC
1 2 3
3 1 2
3 1 2
3 2 1
1 2 3
3 1 2
1 2 3
3 2 1
1 2 3
3 1 2
Image #15 at 0.1568 bpp
SPIHT RCC REWIC
3 1 2
3 2 1
2 1 3
3 1 2
3 1 2
3 2 1
3 1 2
3 2 1
3 1 2
3 1 2
Image #16 at 0.0889 bpp
SPIHT RCC REWIC
3 1 2
3 1 2
3 1 2
3 2 1
3 1 2
3 1 2
3 2 1
3 1 2
1 2 3
2 1 3
Image #18 at 0.2 bpp
SPIHT RCC REWIC
3 1 2
3 1 2
3 2 1
3 1 2
1 2 3
3 1 2
3 1 2
3 2 1
3 1 2
3 1 2
Image #20 at 0.1333 bpp
SPIHT RCC REWIC
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
3 1 2
Table 2 Total number of first, second, and third positions in the subjective rank orders given by human observers. Coder
SPIHT RCC REWIC
Ranking given by human observers # First positions
# Second positions
# Third positions
6 47 17
2 23 45
62 0 8
Appendix A. Congestion control algorithm
Fig. 12. Subjective comparison of decoded outcomes at 0.133 bpp in Image #20 using SPIHT, REWIC, and RCC coders.
Experimental results have demonstrated the validity of this prediction of superior performance of the RCC coder which still has quantizers using restrained bit allocation in the long time, as compared with the REWIC coder in which all quantizers end up using heavy bit allocation.
Acknowledgments This research was sponsored by the Spanish Board for Science and Technology (CICYT) under Grant TEC2007-60450. We are deeply indebted to the referees for suggesting several good ways to improve the quality of the initial manuscript.
In dynamic allocation, quantizers monitor the level of utility of bit allocation across bit rates and, at any point of the transmission, they can decide to switch the behavior between restrained bit allocation and heavy bit allocation. The problem is that, without central control, all quantizers might switch from restrained to heavy allocation at higher bit rates, leading to a congestion of the transmission, if total demand for bit allocation exceeds the available capacity. Following Ref. [3], here we give a brief description of the congestion control algorithm which is based on the behavior of the 1 ¼ rðnq n1 Þ. In this algorithm we have to dynamical equation, dn dt distinguish the basic transmission model and the congestion avoidance. The basic transmission model assumed follows a rational embedded zerotree encoding, [1]. Thus, in a first stage, a wavelet transform is applied to the original data. The second stage of the coding scheme is composed of two key elements: (i) quantizer formation where coefficients of the wavelet transform are partitioned into a number of quantizers; and (ii) a prioritization protocol which is used to choose among competitor quantizers, at each truncation time in the progressive transmission, in order to deliver a subsequence of the completely embedded bit stream of the selected quantizer. In a third stage, adaptive arithmetic encoding exploits both inter-band and intra-band relationships that still remain after the wavelet transformation and zerotree encoding.
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We can now show the application of congestion control to this basic transmission model as follows: the maximum number of quantizers which can choose heavy bit allocation is n2 = n nd2/ (d1 + d2); with d1 being the restrained allocation limit, d2 being the heavy allocation limit, and n being the total number of quantizers. For each quantizer Qi we select the bit stream BSti , candidate to be transmitted at time t, by using a window size WS to segment data when the quantizer is coded bit-plane by bit-plane through successive-approximation quantization. Amongst all candidates BSti , we choose a bit stream BStm achieving the highest expected increase in utility per coding bit, at time t. Initially all quantizers are using the strategy of restrained allocation, and thus the number of quantizers using the strategy of heavy bit allocation at t = 0 is given by n2 = 0. The congestion avoidance is used when the number of bits of BStm , noted as #ðBStm Þ, is greater than the restrained allocation limit d1: #ðBStm Þ > d1 . In this case, if #ðBStm Þ 6 d2 and n2 < n nd2/ (d1 + d2), then BStm may still be sent to the decoder and the number of quantizers n2 which choose heavy allocation is increased by one. But if #ðBStm Þ > d2 or n2 P n nd2/(d1 + d2), then the window size WS to segment data is divided by 2 and we recalculate (using this new window size) the bit stream BSti candidate to be transmitted for each quantizer Qi. The software and documentation can be accessed in: http://decsai.ugr.es/cvg/rewic_revic/ Appendix B. Derivation of probabilities We have that the probability P1(x; n1) for x quantizers to change from restrained bit allocation to heavy bit allocation, given that there are initially n1 quantizers using the strategy of restrained allocation, is defined by:
P1 ðx; n1 Þ ¼
n1 x p ð1 pÞn1 x x
ð26Þ
with p, as given in Eq. (1), being the probability that a quantizer currently engaged in restrained bit allocation changes to heavy bit allocation. Similarly, we can define the probability P2(y; n2) for y quantizers to change from heavy to restrained allocation, given there are initially n2 quantizers that follow the strategy of heavy allocation, using the probability q that a quantizer changes from heavy to restrained allocation in Mt:
P2 ðy; n2 Þ ¼
n2 y
qy ð1 qÞn2 y
ð27Þ
where q = q r Mt. In order to derive PC(i, t + Mt), recall that for sufficiently small Mt interval, most quantizers will not be making a choice about to switch between strategies and thus there are only three possible values for the net change from heavy to restrained allocation (with a net change being the difference between the number that go from heavy to restrained allocation and those doing the opposite): 0, 1, and 1. Here a net change from heavy to restrained allocation of size 1 is given by y = 1 quantizers that change from heavy to re-
strained allocation. A net change from heavy to restrained allocation of size-1 is given by x = 1 quantizers that change from restrained to heavy allocation. It follows that
PC ði; t þ MtÞ ¼ PC ði; tÞ½1 P1 ðx ¼ 1; iÞ P 2 ðy ¼ 1; n iÞ þ P C ði þ 1; tÞP1 ðx ¼ 1; i þ 1Þ þ PC ði 1; tÞP2 ðy ¼ 1; n i þ 1Þ
ð28Þ
where 1 P1(x = 1; i) P2(y = 1; n i) is the probability that during Mt there is a net change of 0 from heavy to restrained allocation, given that there are initially i quantizers using a strategy of restrained bit allocation; with P1(x = 1; i + 1) being the probability that during Mt there is a net change of 1 from heavy to restrained allocation, given that there are initially i + 1 quantizers using a strategy of restrained bit allocation; and P2(y = 1;n i + 1) being the probability that during Mt there is a net change of 1 from heavy to restrained bit allocation, given that there are initially i 1 quantizers using a strategy of restrained allocation. From Eqs. (26) and (27), we have that Eq. (28) becomes, up to terms of order Mt:
PC ði; t þ MtÞ ¼ PC ði; tÞ½1 ið1 qÞrMt ðn iÞqrMt þ PC ði þ 1; tÞði þ 1Þð1 qiþ1 ÞrMt þ PC ði 1; tÞðn i þ 1Þqi1 rMt
ð29Þ
with qi+1 and qi1 being the probability q evaluated at i + 1 and i 1, respectively. From Eq. (29), it follows that:
PC ði; t þ MtÞ PC ði; tÞ ¼ P C ði; tÞ½ið1 qÞ ðn iÞq rMt þ P C ði þ 1; tÞði þ 1Þð1 qiþ1 Þ þ P C ði 1; tÞðn i þ 1Þqi1 up to terms of order Mt. References [1] J.A. Garcia, Rosa Rodriguez-Sanchez, J. Fdez-Valdivia, Xose R. Fdez-Vidal, Rational systems exhibit moderate risk aversion with respect to gambles on variable-resolution compression, Optical Engineering 41 (9) (2002) 2216–2237. [2] J.A. Garcia, Rosa Rodriguez-Sanchez, J. Fdez-Valdivia, An embedded coder providing better image quality at very low bit rates, Optical Engineering 43 (3) (2004) 615–627. [3] J.A. Garcia, Rosa Rodriguez-Sanchez, J. Fdez-Valdivia, Congestion Control in Dynamic Allocation. DECSAI Technical Report, University of Granada, Spain, 2008. [4] Bernardo A. Huberman, Tad Hogg, The behaviour of computational ecologies, in: B.A. Huberman (Ed.), The Ecology of Computation, Elsevier Science Publishers B.V., 1988, pp. 77–115. [5] G. Parisi, Statistical Field Theory, Westview Press, 1998. [6] A. Said, W.A. Pearlman, A new fast and efficient image codec based on set partitioning in hierarchical trees, IEEE Transactions on Circuit and System for Video Technology 6 (3) (1996) 243–250. [7] J.A. Garcia, R. Rodriguez-Sanchez, J. Fdez-Valdivia, Progressive Image Transmission: The Role of Rationality, Cooperation and Justice, SPIE Press, PM-140, Bellingham, Washington, USA, 2004, p. 230. [8] J.A. Garcia, J. Fdez-Valdivia, Xose R. Fdez-Vidal, Rosa Rodriguez-Sanchez, Information theoretic measure for visual target distinctness, IEEE Transactions on Pattern Analysis and Machine Intelligence 23 (4) (2001) 362–383. [9] ITU-R Recommendations, Broadcasting service (television): Recommendation 500-10, Supplement 3 to vol. 1997, BT Series, 2000 Edition.