Symbol) and LPS (Less Probable Symbol). Table 1 Interval division table Probability interval division

Transcription

1 Proceedings of the IIEEJ Image Electronics and Visual Computing Workshop 0 Kuching, alaysia, November -4, 0 AN ADAPTIVE BINARY ARITHETIC CODER USING A STATE TRANSITION TABLE Ikuro Ueno and Fumitaka Ono Graduate School, Tokyo Polytechnic University ABSTRACT We propose a fast and simple binary arithmetic coder, STT-coder, in which arithmetic operation of dividing the probability interval can be done using a state transition table. In order to simplify the state transition table by omitting to retain the both of the bottom address of the valid interval and renormalized interval, renormalization is mostly executed together with the flush operation of the register. For the prevention of the loss of the coding efficiency caused by flushing, the PS/LPS interval to be renormalized is further divided into subintervals having no code space loss in flushing, by assigning them the combination of the current and succeeding symbols. We then propose an adaptive control method of probability estimation in STT-coder for unknown or non-stationary information sources. The probability estimation is also executed based on another state transition table to simplify the operation. In this paper, we describe the concept of STT-coder taking the case of quite short register size, and evaluate its coding efficiency compared with well-known Q-coder. We further introduce out study to extend the register size in order to improve the performance of STT-coder for high PS probability sources.. INTRODUCTION Arithmetic coding has flexible adaptability to various information sources, and provides high coding efficiency. Because of its advantage, it is extensively used in many image and video coding standards such as JPEG000 and PEG-4 AVC. On the other hand, the disadvantage of arithmetic coding is its complexity of operations. The studies on arithmetic coding have hitherto been mainly tuned for the simplification of probability interval calculation such as Q-Coder[] and Q-Coder[,]. In this paper, we propose an adaptive binary arithmetic coder, STT-coder, which can be realized by simple and fast arithmetic operations using state transition table. For that purpose, renormalization is mostly executed together with flush operation, and in order to keep its high coding efficiency, source extension idea is newly introduced. We introduce the implementation method of STT-coder with a -bit interval register, and propose an adaptive probability estimation scheme for unknown information sources. We further study the longer interval register designing case by extending the register size to 6 bits and show the coding performance for stationary information sources.. ARITHETIC CODER USING A STATE TRANSITION TABLE.. STT-coder block diagram STT-coder can be divided into two main blocks, probability estimation and probability interval division as shown in Figure. In this section, we explain those two blocks using a system example with a -bit interval register. Note that this arithmetic coder deals with binary symbols as the combination of PS (ore Probable Figure STT-coder block diagram

2 Symbol) and LPS (Less Probable Symbol)... Probability interval division In the probability interval division of STT-coder, the interval division table, which outputs a codeword or a succeeding probability interval for each binary input symbol, according to the size of the current probability interval, is used to execute the probability interval division as shown in Figure, instead of calculating the interval width as in usual arithmetic coders. To realize simple and fast operation, renormalization of the probability interval is executed together with flush operation in many cases, omitting the operations to store the bottom address and the probability interval after renormalization. If a flush operation will cause a loss of the coding efficiency, the PS/LPS interval is further subdivided into two areas combining succeeding symbols to keep higher coding efficiency. Since this procedure can be considered as source extension, we hereafter call this flush procedure expanded symbol flush. We show an example of the interval division table of STT-coder with a -bit interval register in Table, and all of its probability interval division patterns in Figure... Probability estimation... Estimation by probability state transition In order to execute an adaptive control of probability estimation to select appropriate coding parameters in STT-coder for unknown information sources, we adopt probability estimation system also by using another state transition table. State transition table type probability estimation is useful for fast and simple operation, as used in many coders such as Q-coder. In the state transition table type probability estimation, each context will move from a state to another state according to the occurred symbols, and the symbol probability assigned to the current state is used as the estimated symbol probability. In this paper, we at first employ 0 states, S 0 to S 9 as shown in Table. For STT-coder having a -bit interval register, the states are defined by the combination of the probability interval ( to 8) and the LPS width ( to 4). At the state S 0, the LPS width of one is always selected for all probability intervals, while at the state S 9 the LPS width is set to the maximum width possible to each probability interval. We call these states as probability states, hereafter. In Table, border PS probabilities P Bi between neighboring probability states S i and S i+ are shown. Also occurrence probabilities of probability intervals are shown in Table, which are calculated on the assumption that PS probabilities will distribute uniformly between 0.5 and.0 for the multi-context source model. Table Interval division table Input output bit bit bit bit bit bit Prob LPS PS interval Code / Prob index width /LPS interval index Code length outbit (R8) 0(R7) 0(R6) 00(R5) 4 0(N) - 00(R) - 00(R4a) 00(R5a) 000(R6a) 4 0(R7) 00 - L - 0(R6) 00 - L 0-00(R5) 00 - L 000(R6a) 0 (R8) 0 0 L (R8) 0 0(R6) 00 - L 0-00(R5) 00 - L 00(R4a) 0 (R8) 0 0 L 0(R6) 0 00(R5) 00 - L 0 - (R8) 0 0 L 0 0-0(R6) 0 0 L 00(R) 00 - (R8) 0 0 L 00-0(N) 00 - L L L 0-00(R) 00 - L L (R8) 0 L 0-00(R) 00 - L 0-00(R5a) 00 - L 00 - (R8) 0 L (R6a) 0 L 0(N) 00 - To encode a binary symbol, probability state is given by referring to the probability state table which stores the current probability state of each context. Then the LPS width is decided from the probability state and the probability interval index by referring to the LPS width table, and finally the probability interval is divided into an appropriate PS/LPS width.

3 R8 (prob interval=8) R7 (prob interval=7) R6 (prob interval=6) R5 (prob interval=5) L L 0 L 0 L R6a L L 0 0 L +R4a 0 L0 0 +R6 L 0 L R 00 L00 L LL0 0 N R6 L LPS width 4 R6a (prob interval=6) R5a (prob interval=5) R4a (prob interval=4) 0 0 L 0 0 +R6a 0 0 R5a LL00 00 R 00 R 0 N 0 L0 L0 0 L0 L 0 L0 L 0 00 L L LPS width LPS width Figure Probability interval division patterns : PS L : LPS Numerals after /L denote codeword or the next probility interval index after renormalization.... Dynamic adaptive control In order to estimate the occurrence probability of each binary symbol for each context, the content of probability state table should be updated by the following dynamic adaptive control algorithm as in Figure. () Define probability states from S 0 to S 9. () If an LPS happens, update the state for the current context from S i to S i+. If the current state is S J- corresponding to the minimum PS probability, we do not change the probability state, but exchange the sense of PS with that of LPS. () If an PS to force the state transition (which we call T-PS) happens, update the state for the current context from S i to S i-. If a non-transit PS (NT-PS) happens, we do not change the probability state. In S 0 which corresponds to the maximum PS probability, any PS will be NT- PS. How to define T-PS will be described later. If an PS probability P i will give the best coding efficiency for the coding parameter used at a given probability state S i, the two probabilities, transition caused by an LPS and transition caused by T-PS, shall be equal at the PS probability P i. This balance will give the most desirable transition, since it guarantees to stay at the best probability state longer than other states. Although we can change the probability state whenever an LPS happens, we need to judge the PS, whether we shall change the probability state, since an PS has larger probability than that of an LPS. To balance the transition, the following equation has to be satisfied. -P i =r i P i Figure Transition of probability states In this equation, r i is the ratio of T-PS and total PS (T-PSR, 0<r i <=). Therefore, the T-PSR r i is given by r i = (-P i )/ P i T-PSR value r i being calculated by the equation above is shown in Table for each probability state. Note that we use the just center value of the upper and lower border values of PS probabilities as the optimum PS probability P i for each probability state. For the selection of T-PSs, we shall control the value of T-PSR according to each probability state. As the distribution of the probability intervals to encounter is common to all probability states, we can control the

4 value of T-PSR by selected combination of probability intervals to encounter. For each probability state, we select the combination of probability intervals, to satisfy that the total occurrence probability is equal to its T-PSR, and probability state transition will be executed when an PS happens at the selected probability intervals for transition. For instance, for the probability state S (T-PSR, r =0.4), we classify PSs occurred at the probability interval 5 or as T- PSs, and PSs occurred at the other probability intervals (6,7, 8 and 4) as NT-PSs since the total occurrence probability of probability intervals of 5 and amounts to 0.45 which gives nearest value to the theoretical T-PSR value shown in Table.. EVALUATION OF CODING EFFICIENCY Assuming the target source model is composed of multicontext sources having PS probabilities uniformly distributed between 0.5 to.0, we evaluate the coding efficiency of STT-coder. Figure 4 shows coding efficiency calculated by computer simulation. To evaluate the effect of expanded symbol flush in STT-coder, we show the efficiency of a pure arithmetic coder (denoted as () in Figure 4) having a - bit register, in which renormalization is performed theoretically whenever the probability interval becomes less than 5, as a reference. The proposed STT-coder with ideal probability estimation (denoted as () in Figure 4), shows a little lower coding efficiency than pure arithmetic coder for high PS probabilities, because of the effect of the restriction of interval subdivision even though the expanded symbol flush is introduced. However, as for lower PS probabilities, STT-coder obtains coding efficiency pretty close to that of the pure arithmetic coder. It can be said that redundancy induced by flushing the register is efficiently removed for low PS probabilities by expanded symbol flush. We also show the coding efficiency of STT-coder with adaptive probability estimation, having 0 probability states, which is denoted as () in Figure 4. Its coding efficiency degradation to the STT-coder with ideal probability estimation () is found to be or %, but it obtains same or better coding efficiency than Qcoder except high PS probability sources. The efficiency of STT-coder () tends to become lower as the PS probability gets close to 0.5. To improve the efficiency, we tried to add an additional probability state after S 9. It is expected that this modification improves the coding efficiency, since it would work to stay longer at the probability states representing the lowest PS probability. Also we added one more state before the state S 0 to improve the efficiency at high PS probabilities. Adding these states above, we designed the STT-coder with states, and calculated its coding efficiency. It is shown as (4) in Figure 4. You can see that this modification improves the coding efficiency at high and low PS probabilities, though efficiency around the middle PS probability could be degraded very little. It will be necessary to design the transition system according to the statistics of the targeted information sources. Coding Efficiency () pure coder (no flush) () STT-coder: ideal prob estimation (static) 0.86 () STT-coder: prob estimation (0 states) 0.84 (4) STT-coder: prob estimation ( states) 0.8 (5) Q-coder PS probability Figure 4 Coding efficiency Probability state Table LPS width of each probability state for each probability interval (LPS width table in Fig.) Occurrence probability Probability interval PS prob. border P Bi T-PSR r i Prob. intervals for T-PS (sum of prob.) S S 0.4 5, (0.45) 0.79 S 0.9 6,4, (0.78) S ,5 (0.8) S ,5, (0.447) S ,4, (0.48) S ,5 (0.585) 0.6 S ,5, (0.650) S ,7,6 (0.745) 0.56 S ,6,5,4,(0.86)

5 4. EXTENSION OF INTERVAL REGISTER As shown in Figure 4, the coding efficiency of STTcoder with a -bit register declines as the PS probability becomes higher, due to the short register size. In this section, we explain a study method to design the interval division table of a coder with a more precise register in order to improve the coding efficiency for high PS probability. To design the interval division table, we should consider three parameters, probability interval A L, LPS width A L and offset D, which vary along the probability interval division for arithmetic coding, where offset is the distance between the bottom of the whole probability interval and the currently transmitted address of the valid interval as depicted in Figure 5. As the bit length of the interval register becomes longer, the number of possible combinations of A L, A and H would become so larger. So, we tried to restrict the combination under the following manner. Also we assumed that the LPS interval can be assigned either the upper or lower side in the interval. In this section, we deal with a 6-bit interval register case. To improve the coding efficiency for high PS probability, LPS width A L = has to be prepared for every probability interval and A L might be decreased one by one if PSs occur consecutively in high PS probability sources. Therefore every possible value of A L, from to 64, needs to be considered to remain. We restricted offset D to 8 values; 0, 8, 6, 0, 4, 6, 8 and 0, although it may take all values between 0 and, since these 8 offset values occur more frequently than other possible values, especially if we allow small LPS widths for each probability interval. As for LPS width A L, we tried to restrict the values by employing probability states by the similar manner as in section.. Also here, we assign the PS probability range to each probability state, and let P i denote the best PS probability to perform best efficiency for each probability state S i. The range of the sources to be covered by a probability state is determined so that theoretical coding efficiency by the coding method tuned for the PS probability P i, would be larger than Table LPS width of probability state for each probability interval (6-bit register, offset D=0) Probability state Probability interval A L Best PS Prob. P i S S S S S S S S (Underlined LPS width denotes that LPS interval is assigned to the lower side of the probability interval.) LPS PS Probability interval A L Offset D LPS width A L Figure 5 Probability interval, LPS width and offset Coding Efficiency Coding Efficiency S7 S6 S5 S4 S S PS probability Figure 6 Coding efficiency of the coding method tuned for each probability state () STT-coder (-bit, static) (6) STT-coder (6-bit, static) PS probability Figure 7 Coding efficiency

6 a predetermined minimum coding efficiency. Figure 6 shows the coding efficiency for each probability state with the predetermined minimum coding efficiency of After the set of probability states or set of P i is determined, we chose LPS width A L in A L, for all probability states. Table shows LPS width of each probability state for each A L, in the case of offset D=0. Note that ideal LPS width for given probability state will not increase as the probability interval decreases. However, you can see an example of inversion in the case of A L = at A L =4, and A L = at A L =, for S in Table. This phenomenon is caused by the restriction of offset values. If you allow more values for offset, such inversion may disappear and you can expect some improvement of the coding efficiency, but at the same time you have to grant larger RO size of interval division table. To evaluate the performance of the coder with a 6- bit interval register, we calculated the static coding efficiency (namely under the ideal probability estimation) with a 6-bit register. Compared with the coder with a -bit register (denoted as () in Figure 7, also () in Figure 4) the coding efficiency of the 6-bit coder (denoted as (6) in Figure 7) improves for high PS probability, while it maintains comparable efficiency to the -bit coder for low and medium PS probability. The designing of dynamic STT-Coder with 6-bit register will be done as computer-simulation basis and will be published later. 6. REFERENCES [] W. B. Pennebaker. J. L. itchell., G. G. Langdon, Jr., R. B. Arps An overview of the basic principles of Q-Coder, IB J. Res. Develop., vol., No.6 (Nov.988) [] ISO/IEC 449 Lossy/lossless coding of bi-level images (00) [] ISO/IEC JPEG 000 image coding system (000) 5. CONCLUSION Aiming at fast and simple arithmetic coding, we have proposed an arithmetic coder using a state transition table, STT-coder, in which renormalization and flush of the coder are executed at the same timing in most cases. To prevent the loss of the coding efficiency at the flush procedure, we introduced the idea of source extension to avoid the waste of codeword in available code space, and we introduced the practical design case of very short register size to show that STT-coder can be realized using a simple state transition table. We also proposed dynamic adaptive control of probability estimation, which is also based on the state transition table, noticing the balancing of the transition caused by an LPS and that by selected PS. It was found that STT-coder with this adaptive probability estimation provides comparable or better performance to Q-coder except higher PS probability sources. To improve the performance for high PS probability sources, we also studied the design of STTcoder with a 6-bit interval register and confirmed the improvement of the coding efficiency for stationary information sources.