ENHANCED SCALABLE TO LOSSLESS AUDIO CODING SCHEME

Haiyan Shu, Haibin Huang, Te Li and Susanto Rahardja

Signal Processing DepartmentInstitute for Infocomm Research

Agency for Science, Technology & ResearchSingapore 138632

ABSTRACT

Scalable to lossless (SLS) audio coding is a state-of-art au-dio coding technique that has been adopted as MPEG scal-able audio coding tool. To realize bit-plane refinement, thistechnique employs bit-plane arithmetic coding for lossless en-tropy coding, and Laplacian distribution is used to model theinput data to realize high compression efficiency. In this pa-per, bit-plane probability is analyzed when generalized Gaus-sian distribution is used to model the input data. Based on theresult of bit-plane probability for generalized Gaussian distri-bution, a low cost bit-plane arithmetic coding method is pre-sented. This scheme is implemented in the SLS audio codingplatform. With the same computational complexity, the pro-posed algorithm presents higher compression efficiency thanSLS.

Index Terms— Scalable to lossless audio coding, arith-metic coding, bit-plane probability, generalized Gaussian dis-tribution.

1. INTRODUCTION

Lossless coding of data is a very challenging problem formany applications. Additionally, frequent delays in transmis-sions require progressive coding in order to provide the endusers with useful successive refinements of the final data. Em-bedded progressive coding refers to the encoding of data intoa bitstream that can be parsed efficiently to obtain lower rateand lower quality descriptions of the data. It enables the gran-ularity quality improvement for different environment con-straints and applications.

Bit-plane arithmetic coding is an efficient solution toreach embedded progressive coding. It encodes the data intomultiple bit-plane layers. Each bit-plane may refine the finalquality with different contributions. When all the bit-planesare correctly received, lossless quality is achieved. Thesebit-planes in the generated bitstream can also be arbitrarilytruncated when certain constraint is applied. This fulfillsthe requirements of embedded coding. Bit-plane arithmeticcoding is widely used for data compression [1, 2].

In bit-plane arithmetic coding process, it is necessary toget the actual bit-plane probability (BPP) for efficient com-pression. In [3], scalable to lossless (SLS) audio coding ispresented. This technique utilizes the bit-plane arithmeticcoding scheme to realize the scalability of the bitstream.An bit-plane probability estimation method, named bit-planeGolomb code (BPGC), was introduced in SLS. Based on thistechnology, SLS improves the coding efficiency of scalableaudio significantly. It is for this reason, this audio codingscheme was adopted as MPEG-4 scalable audio coding toolin 2006 [4]. In BPGC, the distribution of the input data isassumed to be Laplacian distribution. It presents a simple butefficient estimation.

However, not all data follows Laplacian distribution.When the input data is not Laplacian distributed, the codingefficiency will be degraded. In [5], approximated bit-planeprobability for generalized Gaussian distribution has beenproposed. This approximation gives an analytic solution toestimate the bit-plane probability for generalized Gaussiandistribution. However, when applying for real application,it still requires additional overhead bits. In this paper, wepropose to apply this approximated solution for scalable tolossless audio coding on the SLS platform. The proposedsolution can further improve the coding efficiency of SLSwithout additional overhead information transmission. Inaddition, the proposed scheme can keep the computationalcomplexity as low as SLS.

This paper is organized as follows. The scalable to loss-less audio coding and BPGC is first reviewed in Section 2.Based on the result from the approximated BPP of gener-alized Gaussian distribution, enhanced bit-plane arithmeticcoding for SLS is proposed in Section 3. Experimental re-sults are presented in Section 4, and the conclusion is drawnin Section 5.

2. SCALABLE TO LOSSLESS AUDIO CODING

MPEG-4 SLS audio coding [4] was released as a standard au-dio coding tool in June 2006. It allows the fully scaling up tolossless with a wide range of intermediate bitrate representa-

377978-1-4244-4296-6/10/$25.00 ©2010 IEEE ICASSP 2010

tions.In SLS, the input integer PCM data is first transformed

into the frequency domain using integer modified discrete Co-sine transform (intMDCT) [6]. The output intMDCT coeffi-cients are passed to the AAC encoder to generate the corelayer AAC bitstream, and the coding residue is then codedusing BPGC to generate the scalable enhancement layer bit-stream.

In this process, bit-plane arithmetic coding is adopted tocode each bit-plane symbol with its corresponding probabilitybased on the calculation from BPGC. In the following, wefirst review the BPP estimation scheme in BPGC.

Considering an intMDCT coefficient x, it can be repre-sented by a set of binary symbols bj ∈ {0, 1} in the followingbinary format

x = (2s− 1)M∑

j=0

bj · 2j (1)

where s is the sign symbol and M is the maximum bit-planenumber.

Since the probability of sign symbol Pr(s = 0) = Pr(s =1) = 0.5, only the non-negative part of x is discussed in thefollowing. In [3], BPGC assumes that the input intMDCTcoefficients follow zero mean Laplacian distribution:

f(x; σ) =

√1

2σ2e−

√2

σ2 |x| (2)

where σ is the standard deviation of x. Then the bit-planesymbol probability of bj taking value 1 is given by

Pj ={ 1

1+22j−L , j ≥ L12 , j < L

. (3)

Here, L is the code parameter. It is used to index the fam-ily of BPGC. The bit-plane symbols are then coded with anarithmetic coder using the probability assignment given by(3). For the sign symbols, they are simply coded with proba-bility assignment 1/2.

In [3], the code parameter L is defined as

L = min

{L′ ∈ Z|2L′+1 ≥

∑ |x|N

}(4)

where∑ |x| is the summation of the input data, N is the num-

ber of input data, and Z is the set of integer.

3. ENHANCED SLS AUDIO CODING WITHGENERALIZED GAUSSIAN ASSUMPTION

The bit-plane Golomb coding has been successfully imple-mented in SLS. It assumes that the input data are Laplaciandistributed. However, when the input data are not Laplaciandistributed, bit-plane probability calculated by BPGC may

not be optimum. Generalized Gaussian distribution is a mostcommonly used description, and has been widely observed inmany practical applications. This inspires us to investigatethe coding efficiency when generalized Gaussian distributionis assumed for the input data to further improve the codingefficiency of SLS.

In [5], an approximated analytic solution for the bit-planeprobability of generalized Gaussian distribution was pro-posed. In this section, we propose a low cost scheme thatutilizes the result from [5] to enhance the performance ofbit-plane arithmetic coding for MPEG-4 SLS.

3.1. Bit-plane probability of generalized Gaussian distri-bution

Assuming the intMDCT coefficient x has a zero mean gener-alized Gaussian distribution, the probability density functionof y = |x| is given by

g(y; σ, γ) =γ · η(γ, σ)

Γ( 1γ )

· e−[η(γ,σ)·y]γ , y ≥ 0. (5)

where σ > 0 is the standard deviation, γ > 0 is the shapeparameter, and

η(γ, σ) =

[Γ( 3

γ )

σ2 · Γ( 1γ )

]1/2

, (6)

Γ(z) =∫ ∞

0

tz−1e−t dt. (7)

In [5], the approximated solution of bit-plane probabilityPj for generalized Gaussian distribution was presented as

Pj(σ, γ) =1

1 + eγ1√γ ·(η(γ,σ)·2j)

√γ

. (8)

This is an analytic description. It is verified in [5] that thissolution presents bit-plane probability very close to the actualbit-plane probability.

3.2. Practical bit-plane probability representation

Referring to (8), we can estimate the BPP of the input signalwith generalized Gaussian distribution. However, to get anaccurate estimation, we need to calculate the parameter σwhich requires high computation. In addition, many bits isrequired to transmit this non-integer parameter. To reducethe overhead information to be transmitted and lower thecomputational complexity, a practical solution is designed toimprove the coding efficiency without introducing additionalcomputation.

For generalized Gaussian distribution, we define the codeparameter L as

L =⌊log2

1η(γ, σ)

⌋. (9)

378

Table 1. Comparison of compression ratio of different SLS codecs in non-core mode.Lossless coders SLS-BPGC Proposed scheme improvement

Set 1 (15 sequences) 2.2308 2.2342 0.15%Set 2 (15 sequences) 1.5804 1.5821 0.1%Set 3 (15 sequences) 2.1277 2.1297 0.1%Set 4 (7 sequences) 2.5511 2.5539 0.11%

and α as

α = log21

η(γ, σ)−

⌊log2

1η(γ, σ)

⌋. (10)

where the value of α varies between 0 and 1 (i.e. α ∈ [0, 1)).So, we have following relationship:

η(γ, σ) = 2−L−α. (11)

Then the BPP of generalized Gaussian distribution pre-sented in (8) can be written as

Pj(σ, γ) =1

1 + eγ1√γ ·(2j−L−α)

√γ

. (12)

According to the probability density function of y definedin (5), the expectation of y is derived as

E{y} =∫ ∞

0

y · g(y; σ, γ)dy

=∫ ∞

0

y · γ · η(γ, σ)Γ( 1

γ )· e−[η(γ,σ)·y]γ dy

ω=[η(γ,σ)·y]γ

=γ · η(γ, σ)

Γ( 1γ )

·∫ ∞

0

ω1γ · e−ω

η2(γ, σ) · γ · ω1γ−1dω

= η−1(γ, σ) ·Γ( 2

γ )

Γ( 1γ )

. (13)

According to 11, the following result is derived:

L = min

{L′ ∈ Z|2L′+1 ≥ E{y} · Γ( 1

γ )

Γ( 2γ )

}. (14)

In this equation, E{y} is expectation of the input y.∑

yN

is its unbiased estimator, where N is the number of samples.In practical calculation,

∑y

N is used to replace E{y}. Codeparameter L can be determined with little computational cost.

However, additional computationwill be introduced in theestimation of code parameter L when comparing with SLS.To solve this problem, we rewrite (12) as

Pj(σ, γ) =1

1 + eγ1√γ ·(Γ(2/γ)

Γ(1/γ) ·2j−L−α̂)√

γ. (15)

In such a way, the code parameter L in (15) is defined thesame way as that in (4).

3.3. Analysis of complexity

In MPEG-4 SLS, the bit-plane probability is pre-calculated.Only parameter L is calculated online and transmitted to thedecoder. The proposed solution calculates the parameter Lin the same way as SLS. In addition, the BPP table is pre-calculated by (15) and stored the same way as MPEG-4 SLS.This process requires no additional storage and introduces noadditional computation.

4. EXPERIMENTAL RESULTS

In this section, the performance of the proposed enhancedSLS audio coding scheme will be evaluated.

First, we examine the performance of BPP for general-ized Gaussian distribution in the format given in (15). Theprobability calculated from (15) is compared with the actualBPP value based on some parameter sets (γ, σ). In this sim-ulation, two different shape parameters (γ) are chosen. Theirvalues are 0.4 and 1.3, respectively. For each γ, three differentstandard deviations (σ = 20, 80, 640) are selected. The com-parison results of two (approximated and actual) BPP valuesbased on the different parameter sets (γ, σ) are illustrated inFig. 1. It can be observed that, the approximated values arequite close to the actual one, especially when the BPP valuefalls into [0.05, 0.4]. This shows that the format given in (15)can be used as an approximation of the actual BPP value ofgeneralized Gaussian distribution.

Next, we apply our proposed scheme in the SLS plat-form to verify its performance. In the experiment, the MPEGtest sequences are coded. The test sequences consist of 15different types of music with difference sampling frequencyand resolution. According to the sampling frequency andresolution, they are grouped into four sets: 48kHz/16bit,48kHz/24bit, 96kHz/24bit, and 192kHz/24bit, respec-tively. Each waveform lasts 30 seconds. The performanceof the proposed scheme is compared with the SLS Bit-PlaneGolomb Coding mode (SLS-BPGC) [4].

To be compatible with the SLS coding platform, the pa-rameter γ is fixed for the proposed algorithm. There is noneed to estimate the value of γ. Table 1 summarizes the ex-perimental results for the tested coders in the non-core modewhere γ = 1.04 for the proposed algorithm, and Table 2shows the results when there is an AAC core coder running at128kbps and γ = 1.34 for the proposed algorithm. Compared

379

0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Bit−Plane (j)

Pro

babi

lity

γ = 0.4

Actual(σ = 20)Approx(L = −1)

Actual(σ = 80)Approx(L = 1)Actual(σ = 640)Approx(L = 4)

(a)

0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Bit−Plane (j)

Pro

babi

lity

γ = 1.3

Actual(σ = 20)Approx(L = 4)

Actual(σ = 80)Approx(L = 6)Actual(σ = 640)Approx(L = 9)

(b)

Fig. 1. Comparison between the actual bit-plane probability and the approximated bit-plane probability based on referencebit-plane (L) with different standard deviation (σ) (a) γ = 0.4; (b) γ = 1.3.

Table 2. Comparison of different SLS codecs in 128kbps AAC core mode.Lossless coders SLS-BPGC Proposed scheme improvement

Set 1 (15 sequences) 2.1291 2.1333 0.20%Set 2 (15 sequences) 1.5498 1.5514 0.1%Set 3 (15 sequences) 2.1004 2.1025 0.1%Set 4 (7 sequences) 2.5268 2.5305 0.14%

to SLS-BPGC, the proposed scheme presents 0.12% improve-ments in noncore mode and 0.13% improvement in 128kbpsAAC core mode. In fact, SLS-BPGC is a special case of theproposed BPP approximation which assumes Laplacian dis-tribution (γ = 1).

These comparisons show that the compression ratio of theproposed scheme is higher than that of SLS-BPGC. In addi-tion, the computational complexity of the proposed scheme iskept as low as SLS-BPGC.

5. CONCLUSION

In this paper, enhanced scalable to lossless audio codingbased on generalized Gaussian distribution was proposed. Inorder to improve the coding efficiency of SLS, we further as-sumed that the input intMDCT data are generalized Gaussiandistributed. Based on this assumption, the bit-plane symbolprobability has been estimated. To be compatible with theexisting SLS platform, the bit-plane symbol probability waspresented based on the code parameter L. In this way, no ad-ditional computation and overhead information is introduced.Experimental results show that, with the same computationalcomplexity, the proposed scheme presents better compressionratio than the SLS BPGC mode. This scheme can also be in-corporated into other multimedia coding scheme to furtherimprove the compression efficiency.

6. REFERENCES

[1] Y. V. Ramana Rao and C. Eswaran, “A New Algorithmfor BTC Image Bit Plane Coding,” IEEE Trans. Commu-nications, vol. 43, no. 6, pp. 2010–2011, Jun. 1995.

[2] Majid Rabbani and Paul W. Melnychuck, “ConditioningContexts for the Arithmetic Coding of Bit Planes,” IEEETrans. Communications, vol. 40, no. 1, pp. 232–236, Jan.1992.

[3] Rongshan Yu, Susanto Rahardja, Xiao Lin, andChi Chung Ko, “A fine granular scalable to lossless audiocoder,” IEEE Trans. Audio Speech Language Processing,vol. 14, no. 4, pp. 1352–1363, Jul. 2006.

[4] ISO/IEC 14496-3:2005/Amd 3:2006, “Scalable LosslessCoding (SLS),” June 2006.

[5] Haiyan Shu, Haibin Huang, Te Li, and Susanto Rahardja,“Bit-Plane Coding for Source with Generalized GaussianDistribution,” Accepted by IEEE International Sympo-sium on Multimedia, 2009.

[6] Haibin Huang, Susanto Rahardja, Rongshan Yu, and XiaoLin, “A Fast Algorithm of Integer MDCT for LosslessAudio Coding,” Proc. IEEE Int. Conf. Acoustics, Speech,and Signal Processing, vol. 4, pp. 177– 180, 2004.

380