This paper describes an efficient one dimensional wavelet adaptive coding algorithm for the compression of audio signals. The new algorithm has the property ...
Signal Compression using a new Adaptive Wavelet Technique Nivin A. E.Ghamry*, Hamed A. Elsimary*, Serag E.-D. Habib** *Electronics Research Institute, Giza , Egypt. ** Electronics and Communications Dept., Faculty of Engineering Cairo University, Egypt
The organization of this paper is as follows. We begin in Section II with a brief overview of the wavelet transform framework. Section III summarizes the coding algorithm and offers the new adaptive coding procedure. Section VI provides simulation results on the tested signals and finally Section V concludes the paper.
ABSTRACT This paper describes an efficient one dimensional wavelet adaptive coding algorithm for the compression of audio signals. The new algorithm has the property that the most appropriate wavelet filter is selected from a library of wavelet filters according to a proposed switching criterion. This algorithm yields a system that can adapt to the time variance of a signal. Simulation results are demonstrated for several audio signals to show the superior performance of the new method.
Index Terms- 1-D biorthogonal filters.
2. 1-D WAVELET TRANSFORM FRAMEWORK An efficient algorithm for implementing discrete wavelet transform using filters was developed in 1988 by S. Mallat [4]. The Mallat algorithm is in fact a classical scheme known in the signal processing community as a two-channel subband coder. In wavelet analysis, we often speak of approximations and details. The approximations are the high-scale, low frequency components of the signal. The details are the low-scale, high frequency components. Let H(z) and G(z) represent the wavelets low pass and high pass digital filters, respectively. Assume that Cj represents the low pass component of the original signal at the jth resolution and D j represents the corresponding high pass component. Then, the output of the wavelet filters at (j+1) th resolution can be written using matrices as:
wavelet transform, adaptive coding,
1. INTRODUCTON The most significant use of DWT has been in data compression, where its natural multi resolution capability, perfect reconstruction property and the absence of blocking artifacts pose it as a good transform for still images and video signals. Much research work is geared in both directions, to meet the demands of the emerging applications. Related work is given in [1-2]. Sinha and Tewfik[3] described a scheme in which the wavelet decomposition of audio signals is obtained and then quantized to keep the distortion below the masking threshold for the signal. In our current work we provide an efficient and simple adaptive coding method that is capable of dealing with slow and fast varying audio signals. The wavelet filter type is dynamically adapted to the time variance of a 1-D signal. The basic idea behind the adaptive method provided here is simple but powerful. For intervals that include very little changes, we use short filter, i.e. a small number of filter coefficients, as most of the actual signal energy resides in the low frequency bands. On the other hand, when the signal is of a large time variance then we need to code much of the change information by using long filters. A simple criterion is needed to automatically decide whether the current interval is of small or large time variance. In our scheme, we use the significance of the second level detail coefficients as the criterion for switching between the different filter types. The concept of adaptive coding defines implicitly a general coding scheme, which relies on the partitioning of the signal and the choice of the associated wavelet filter. The filters considered here are the 5- and 16-coefficient biorthogonal filters. The proposed coding scheme is tested at different 1-D signals and is benchmarked against the traditional 1-D wavelet coding methods to assess its efficiency. This combination of short filters in slow varying intervals and long filters in the fast varying intervals improves the tradeoff between the compression ratio and the reconstructed signal quality.
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2002 IEEE
Cj+1 = H * Cj Dj+1 = G * Cj
CD1
Gt Ht
Gt G
CD2 CA2
Ht H Fig. 1: Tree-structured 2-level one-dimensional Wavelet Decomposition
The above two operations are usually followed by dyadic downsampling the resulting coefficients. Given those coefficients at the first level of approximation, the decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal
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is broken down into many lower-resolution components by the same digital filtering scheme. This is called the wavelet decomposition tree. The complete one-dimensional treestructure decomposition of the proposed coding system is shown in Fig.1.The signal decomposition is carried out using 5or 16-tap biorthogonal filters. The choice between the two filter types is achieved according to an adaptive criterion explained in Section III.
3. THE ADAPTIVE CODING ALGORITHM In our work we developed an adaptive algorithm that is capable of selecting the optimum filter to be used in the decomposition of the input signal. The algorithm is developed to deal especially with signals that include large time invariance. This type of signals is expected to have a spectrum that does not decay fast with increasing frequency. Generally, these properties manifest themselves in the wavelet transform of that signals. Rapid changes in a certain interval correspond to the high frequency subband of the signal, and are expected to give significant values in the detail coefficients. These significant values appear clearly in the first level detail coefficients. If the time variance in that specific interval is large enough, then the detail coefficients still contain significant values in the second level or even in higher analysis levels. On the other hand, little changes can be detected as having negligible detail coefficient values, as most of the energy of the signal is concentrated in the low frequency regions. In our adaptive coding method we start the l-D analysis process by applying short low- and high pass filters on the input signal to obtain the first level approximation coefficients CA1 and CD1. The CA1 subband is further decomposed into the second level approximation coefficients CA2 and CD2. The low frequency subband CA 2t contains much of the information of the original data. The subband CD2, which corresponds to the high frequency subband, is of a particular interest here, since it acts as a change detector, which means that by accurately coding this subband, we are encoding much of the variation information. The values the detail coefficients CD2 are tested for significance by comparing them with a suitable threshold T1. The number of significant values is registered. If that number is greater than another threshold T2, then that means than the rate of change in the signal has increased and large variations are expected in the coming intervals. In this case, the system turns to using the longer filter to follow up that rapid change. Otherwise, the coder deals with the signal as being nearly stationary in that interval and keeps using the short filter. The adaptive coding scheme keeps switching between the short and long filters according to the time variance of the signal. The coding procedure is detailed below. We used the terms HS, GS and HL, GL to denote the temporal short and long filters, respectively.
Adaptive Coding Procedure Step1:
/*Two-level 1-D wavelet decomposition is performed */ H = HS G = GS Output={second level detail coefficients CD2}; /* Reset Counter C */ C=0;
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Step 2:
/* The number of significant values in registered */
CD2 is
IF Abs (CD2 (i)) > T1, /* Increment Counter */ THEN C = C + 1; Step 3:
IF C > T2, THEN H = H L ELSE H = HS /* Reset Counter */ C=0; Return to Step 1. END;
G = GL G = GS
When the adaptive coding procedure is completed, the efficient zerotree algorithm is applied, to exploit the self-similarities across the different subbands[5]. The zerotree algorithm is based on the simple hypothesis that if a coefficient value at a course level is insignificant, then coefficients in the same spatial orientation at finer levels are most likely to be insignificant. These values can be discarded without causing severe degradation in the reconstructed signal.
4. SIMULATION RESULTS In our simulation the 5- and 16-coefficient biorthogonal filters perform the wavelet decomposition according to the adaptive coding algorithm explained in the previous section. The simulations are carried out using the Matlab Wavelet Toolbox. The signals used for testing the proposed adaptive coding algorithm are three well known audio signals: “The Microsoft Sound”, “Start”, “Recycle” and a high frequency sinusoidal signal called “linchirp”[6]. To obtain a common framework work of comparisons, all signals are encoded using the short filter alone and the long filters alone for the entire signal as a benchmark, and then the adaptive coding method is applied. The PSNRs and compression ratios R of the four signals are tabulated in Table 1. For the “The Microsoft Sound” the PSNR value using the 16-coefficient filter is about 3 dB higher than that of the 5-coefficient filter. According to our adaptive criterion explained before the coder turned to use the long filter for 11% of the signal. This gave an increase of about 0.5 dB in the PSNR with respect to the 5-coefficient filter PSNR. For the signals “Start” and “Recycle” the long filter was applied for 39% and 51%of the samples, respectively. Consequently, the PSNR value was improved by 1.2-1.4 dB. For the “linchirp” signal, the long filter was applied for 60% of the signal and the PSNR was increased by 1.1 dB. Fig.2 shows the original “Recycle” signal as well as the reconstructed signals using the three coding methods. The objective results given in Table 1 show clearly that the increase in the PSNR is associated with a drop in the compression ratio. They also indicate that the long filter is preferred for large- and the short filter for small time variance. Hence, using the adaptive coding method, which switches between both types of filters, can maintain the quality level more consistent.
Table 1 Comparisons of the mean PSNRs in dB and Compression Ratios R The Microsoft Sound
Start
Recycle
linshirp
Filter Type
PSNR
R
PSNR
R
PSNR
R
PSNR
R
5-tap
43.39
1.54
34.28
2.76
42.92
1.67
55.63
2
16-tap
46.62
1.51
37.01
2.72
45.42
1.56
57.48
1.74
Adaptive
43.73
1.536
35.34
2.744
44.15
1.61
56.7
1.844
PSNR for fast signals as compared to the traditional 1-D wavelet coding methods. 300
300
200
200
100
100
0
6. REFERENCES [1] M. Antonini and I. Daubechies, “Image
coding using wavelet transform’, IEEE Trans. on IP, vol.1, 1992.
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[2] S. Ranganath and A. Kassim, “ Highly scalable waveletbased video codec for very low bit-rate environment”, IEEE Journal on Selected Areas in Communications, Vol. 16, Jan 1998. [3] D. Sinha and A. Tewfik, “Low bit rate transparent audio compression using adapted wavelets”, IEEE Trans. on Signal Processing, Dec. 1993.
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100
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100
Fig. 2: “Recycle”(a) Original Signal (b) Reconstructed Signal with 5-Coefficient Filter (c) Reconstructed Signal with 16-Coefficient Filter (d) Reconstructed Signal with Adaptive Filter
[4] S. Mallat, “ A theory of multiresolution decomposition: the wavelets representation”, IEEE Trans. on PAMI-11, No.7, July 1989. [5] J. M. Shapiro, “ An embedded Wavelet Hierarchical Image Coder”, in Proc. IEEE Int. Conf. Acoustic, Speech, Signal Processing (ICASSP), Mar 1992. [6] Matlab Toolbox User’Guide, The MathWorks .Inc., 1997.
5. CONCLUSION This paper presents a 1-D adaptive wavelet coder that targets the coding of signals of a large time variance. The structure of the proposed coder employs the classical one-dimensional wavelet analysis. A new adaptive coding algorithm is used to efficiently encode the input signal. According to the time variance of the signal the filter type is dynamically adapted. The switching between the short and long filters is carried out according to the significance of the detail coefficients. Simulations of the proposed coding algorithm are performed on several signals and compared with the traditional 1-D wavelet subband coding methods. The simulation results obtained show the ability of the adaptive algorithm to strike an improvement balance between the compression ratios and
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