第四章算术编码

1、第四章：算术编码n算术编码：越来越流行（在很多标准中被采用）n适合的场合：n小字母表：如二进制信源n概率分布不均衡n建模与编码分开n内容：n算术编码的基本思想n一些性质n实现n有限精度：区间缩放（浮点数/整数实现）n计算复杂度：用移位代替乘法二进制编码n自适应模型nQM编码器：自适应二进制编码回顾： Huffman编码n例1：信源的符号数目很少n :0.10.9XabP X0ab1a=0, b=1回顾：扩展的Huffman编码n例2：信源的符号的概率严重不对称：nA = a, b, c, P(a) = 0.95, P(b) = 0.02, P(c) = 0.03nH = 0.335 bits/

2、symbolnHuffman编码：a0b11c10nl = 1.05 bits/symboln冗余（Redundancy） = l - H = 0.715 bits/sym (213%!)n问题：能做得更好吗？10ab c01回顾：扩展的Huffman编码n基本思想：n考虑对两个字母序列而不是单个字母编码l = 1.222/2 = 0.611冗余 = 0.276 bits/symbol（27%）回顾：扩展的Huffman编码n该思想还可以继续扩展n考虑长度为n的所有可能的mn 序列 (已做了32)n理论上：考虑更长的序列能提高编码性能n实际上：n字母表的指数增长将使得这不现实n例如：对长度为3

3、的ASCII序列：2563 = 224 = 16Mn需要对长度为n的所有序列产生码本n很多序列的概率可能为0n分布严重不对称是真正的大问题：nA = a, b, c, P(a) = 0.95, P(b) = 0.02, P(c) = 0.03 nH = 0.335 bits/symbolnl1 = 1.05, l2 = 0.611, n当n = 8时编码性能才变得可接受n但此时|alphabet| = 38 = 6561 !算术编码（Arithmetic Coding）n算术编码：从另一种角度对很长的信源符号序列进行有效编码n对整个序列信源符号串产生一个唯一的标识（ tag ）n直接对序列进行

4、编码（不是码字的串联）：非分组码n不用对该长度所有可能的序列编码n标识是0,1)之间的一个数（二进制小数，可作为序列的二进制编码）概率复习 n随机变量：n将试验的输出映射到实数n用数字代替符号nX(ai) = i, 其中 ai A (A = ai, i = 1.n)n给定信源的概率模型Pn概率密度函数（probability density function, pdf）n累积密度函数（cumulative density function, cdf） iP XiP a 11iiXikkFiP XkP a产生标识n定义一一映射：nak FX(k-1), FX(k), k = 1.m, FX(0)

5、 = 0nFX(k-1), FX(k)区间内的任何数字表示 akn对2字母序列ak aj编码n对ak ，选择FX(k-1), FX(k)n然后将该区间按比例分割并选取第j个区间： 00,.1,1XXXFmiiFiF 11,111XXXXXXXXFjFjFkFkFkFkFkFkv将0, 1)分为m个区间：产生标识：例n考虑对a1a2a3编码：nA = a1, a2, a3, P = 0.7, 0.1, 0.2)n映射：a1 1, a2 2, a3 3ncdf: FX(1) = 0.7, FX(2) = 0.8, FX(3) = 1.0映射成实数nA = a1, a2, , amv对公平掷骰子的例

6、子：1, 2, 3, 4, 5, 66.161kforkXP iXPiFiXPkXPaTXikiX2112111 25. 022112XPXPTX 75. 0521541XPkXPTkX词典顺序（ Lexicographic order ）n字符串的词典顺序：n其中 表示“在字母顺序中，y在x的前面”nn 为序列的长度 ( ):12inXiiTPPy y xxyxxy 词典顺序：例n考虑两轮连续的骰子：n输出 = 11, 12, , 16, 21, 22, , 26, , 61, 62, , 66 469. 07251321121113xPxPxPTXv注意：为了产生13的标识，我们不必对产生

7、其他标识=但是，为了产生长度为n的字符串的标识，我们必须知道比其短的字符串的概率 这与产生所有的码字一样繁重！应该想办法来避免此事区间构造n观察n包含某个标识的区间与所有其他标识的区间不相交n基本思想n递归：将序列的下/上界视为更短序列的界的函数n上述骰子的例子：n考虑序列：3 2 2 n令u(n), l(n) 为长度为n序列的上界和下界，则u(1) = FX(3), l(1) = FX(2)u(2) = FX(2)(32), l(2) = FX(2)(31) (2)32(11).(16)(21).(26)(31)(32)XFPPPPPPxxxxxx区间构造 (2)32(11).(16)(21

8、).(26)(31)(32)XFPPPPPPxxxxxx6661212112111,iiiPkiP xk xiP xkP xiP xkwherex xxx(2)1132(1)(2)(31)(32)(2)(31)(32)XXFP xP xPPFPPxxxx1221(31)(32)(3)(1)(2)(3)(2)XPPP xP xP xP xFxx)2()3()3(1XXFFxP)2()2()3()2(32)2(XXXXXFFFFF)2()1()1()1()2(XFlulu区间构造 ) 1 ()2()3()2(31)2(XXXXXFFFFF) 1 ()1()1()1()2(XFlull()()(3)

9、( )(3)( )322 ,32133XXuFlF=()()(3)(2)(2)(2)322(31)(32)(31)(2)XXXXXFFFFF=+-()()(3)(2)(2)(2)321(31)(32)(31)(1)XXXXXFFFFF=+-)2()2()2()2()3(XFlulu) 1 ()2()2()2()3(XFlull)2()2()3()2(32)2(XXXXXFFFFF)2()1()1()1()2(XFlulu产生标识n通常，对任意序列x = x1x2xn ( )( )2nnXulTx( )(1)(1)(1)( )(1)(1)(1)()(1)kkkkkkXkXkkkulullluFx

10、Fxl只需知道信源的cdf，即信源的概率模型算术编码：编码和解码过程都只涉及算术运算（加、减、乘、除）产生标识：例n考虑随机变量X(ai) = i n对序列1 3 2 1编码：3, 1)(, 1)3(,82. 0)2(, 8 . 0) 1 (, 0, 0)(kkFFFFkkFXXXXX1, 0)0()0(ul8 . 0) 1 (0)0()0()0()0()1()0()0()0()1(XXFluluFlull18 . 0)3(656. 0)2()1()1()1()2()1()1()1()2(XXFluluFlull1377408. 0)2(7712. 0) 1 ()2()2()2()3()2()

11、2()2()3(XXFluluFlull132()()(4)(1)(3)(3)(4)(3)(3)(3)( )0.7712(1)0.7735040XXllulFululF=+-=+-=1321(4)(4)13210.7723522XulT( )(1)(1)(1)( )(1)(1)(1)()(1)kkkkkkXkXkkkulullluFxFxl解码标识nAlgorithmnInitialize l(0) = 0, u(0) = 1.1. For each i, i = 1.nCompute t*=(tag-l(k-1)/(u(k-1)-l(k-1).2. Find the xk: FX(xk-1)

12、 t* FX(xk).3. Update u(n), l(n)4. If done-exit, otherwise goto 1.解码：例3, 1)(, 1)3(,82. 0)2(, 8 . 0) 1 (, 0, 0)(kkFFFFkkFXXXXXvAlgorithmInitialize l(0) = 0, u(0) = 1.1.Compute t*=(tag-l(k-1)/(u(k-1)-l(k-1).2.Find the xk: FX(xk-1) t* FX(xk).3.Update u(k), l(k)4.If done-exit, otherwise goto 1. 8 .0)1 (0

13、)0()1 (8 .00)0(772352.0010772352.0)0()0()0()1()0()0()0()1(*XXXXFluluFlullFtFt 8 .0)3(656.0)2()3(182.0)2(96544.008 .00772352.0)1()1()1()2()1()1()1()2(*XXXXFluluFlullFtFt77408.0)2(7712.0)1 ()2(82.08 .0)1 (808.0656.08 .0656.0772352.0)2()2()2()3()2()2()2()3(*XXXXFluluFlullFtFt)1 (8 .00)0(4 .07712.077408

14、.07712.0772352.0*XXFtFt1131321321算术编码的唯一性和效率n上述产生的标识可以唯一表示一个序列，这意味着该标识的二进制表示为序列的唯一二进制编码n但二进制表示的精度可以是无限长：保证唯一性但不够有效n为了保证有效性，可以截断二进制表示，但如何保证唯一性？n答案：为了保证唯一性和有效性，需取小数点后l位数字作为信源序列的码字，其中n注意：P(x)为最后区间的宽度，也是该符号串的概率n符合概率匹配原则：出现概率较大的符号取较短的码字，而对出现概率较小的符号取较长的码字 1log1( )lPxx算术编码的唯一性和效率n长度为n的序列的算术编码的平均码长为： ( )1 (

15、 )log1( )1 ( ) log1 1( ) ( )log( )2( ) 22AnlPlPPPPPPPH XnH X xxxxxxxxx 2 2 nAAHnH XlnH XXlH Xn算术编码的效率高：当信源符号序列很长，平均码长接近信源的熵实现n迄今为止 0, 1);E1(x) = 2xnE2: 0. 5,1) = 0, 1);E2(x) = 2(x-0.5)n注意：在缩放过程中我们丢失了最高位n但这不成问题我们已经发送出去了n解码器n根据0/1比特并相应缩放n与编码器保持同步标识产生：带缩放的例子n考虑随机变量X(ai) = i n对序列1 3 2 1编码：3, 1)(, 1)3(,8

16、2. 0)2(, 8 . 0) 1 (, 0, 0)(kkFFFFkkFXXXXX1, 0)0()0(ul(1)(0)(0)(0)(1)(0)(0)(0)(1)(1)(1)(1)(0)0(1)0.8,)0,0.5),)0.5,1)XXllulFululFluluget next symbolInput: 1321Output: 6 . 0)5 . 08 . 0(2312. 0)5 . 0656. 0(2) 1 , 5 . 08 . 0,656. 08 . 0)3(656. 0)2()2()2()1()1()1()2()1()1()1()2(ulFluluFlullXXInput: -321Ou

17、tput: 1标识产生：带缩放的例子Input: -21Output: 1154816. 0)2(5424. 0) 1 ()2()2()2()3()2()2()2()3(XXFluluFlull6 . 0,312. 0)2()2(ul09632. 05 . 054816. 020848. 05 . 05424. 02)3()3(ulInput: -1Output: 11019264. 009632. 021696. 00848. 02)3()3(ulInput: -1Output: 110038528. 019264. 023392. 01696. 02)3()3(ulInput: -1Out

18、put: 11000标识产生：带缩放的例子nEOT:nl(n),u(n)区间内的任何一个数字n在此选 0.510 = 0.1277056. 038528. 026784. 03392. 02)3()3(ulInput: -1Output: 11000154112. 0)5 . 077056. 0(23568. 0)5 . 06784. 0(2)3()3(ulInput: -1Output: 110001504256. 0) 1 ()3568. 054112. 0(3568. 03568. 0)0()3568. 054112. 0(3568. 0)4()4(XXFuFlInput: -1Outp

19、ut: 110001Output: 11000110v注意：0.1100011 = 2-1+2-2+2-6+2-7= 0.7734375 0.7712,0.77408增量式标识解码n我们需要增量式解码n如：网络传输n问题n如何开始？n必须有足够的信息，可以对第一个字符无歧义解码 接收到的比特数应比最小标识对应的区间更小n怎样继续？n一旦有一个非歧义的开始，只要模拟编码器即可n如何结束？标识解码：例n假设码字长为6n输入：110001100000n0.1100012 = 0.76562510n第1位：1Output: 1 )3(182.0)2(9570.008 .00765625.0*XXFtF

20、t8 .0)3(656.0)2()1()1()1()2()1()1()1()2(XXFluluFlullInput: 110001100000Output: 13Input:-10001100000 (左移)6 .0)5 .08 .0(2312.0)5 .0656.0(2)2()2(ul )2(82.08 .0)1 (8155.0312.06 .0312.0546875.0*XXFtFtInput: -10001100000 (0.546875)Output: 13254816.0)2(5424.0)1 ()2()2()2()3()2()2()2()3(XXFluluFlull09632. 0

21、5 . 054816. 020848. 05 . 05424. 02)3()3(ulInput: -001100000 (左移)Input: -0001100000 (左移)(1)(1)0(10)000(10)0.80.8lu此时解码最后一个符号标识解码：例19264. 009632. 021696. 00848. 02)3()3(ulInput: -01100000 (左移)38528. 019264. 023392. 01696. 02)3()3(ulInput: -1100000 (左移)77056. 038528. 026784. 03392. 02)3()3(ulInput: -10

22、0000 (左移)504256. 0) 1 ()3568. 054112. 0(3568. 03568. 0)0()3568. 054112. 0(3568. 0)4()4(XXFuFlInput: -100000*2*1000000.5,(0)00.8(1)XXtFtFOutput: 1321第三种情况：E3n回忆三种情况1.l(n), u(n) 0, 0.5): E1: 0, 0.5) = 0, 1); E1(x) = 2x2.l(n), u(n) 0.5, 1): E2: 0. 5, 1) = 0, 1); E2(x) = 2(x-0.5)3.l(n) 0, 0.5), u(n) 0.5

23、, 1): E3(x) = ?nE3 缩放nE3: 0.25, 0.75) = 0, 1); E3(x) = 2(x-0.25)n编码nE1 = 0, E2 = 1, E3 = ?n注意： nE3 E3 E1 = E1 E2 E2 nE3 E3 E2 = E2 E1 E1 n规则：记录E3 连续的次数，并在输出下一个E2/E1 之后发送该次数证明请见文献： Witten, Radford, Neal, Cleary, “Arithmetic Coding for Data Compression” Communications of the ACM, vol. 30, no. 6, pp. 5

24、20-540, June 1987. 第三种情况：E3整数实现n假设码字长度为n，则nni = 长度为TotalCount（TC） 的序列中字符i出现的次数n累积计数：CC timestimes0,1000111nn 1times0.5100nTCkCCkFnkCCXkii)()()(1( )(1)(1)(1)( )(1)(1)(1)()(1)kkkkkkXkXkkkulullluFxFxl整数实现nMSB(x) = Most Significant Bit of xnLSB(x) = Least Significant Bit of xnSB(x, i) = ith Significant

25、Bit of xnMSB(x) = SB(x, 1); LSB(x) = SB(x, m)nE3(l, u) = (SB(l, 2) = 1 & SB(u, 2) = 0)l=000, u=111, e3_count=0repeat x=get_symbol l=l+ (u-l+1)CC(x-1)/TC / lower bound update u=l+ (u-l+1)CC(x)/TC -1 / upper bound update while(MSB(u)=MSB(l) OR E3(u,l) / MSB(u)=MSB(l)=0 E1 rescaling if(MSB(u)=MSB(l) /

26、MSB(u)=MSB(l)=1 E2 rescaling send(MSB(u) l = (l1)+0 / shift left, set LSB to 0 u = (u0) send(!MSB(u) / encode accumulated E3 rescalings e3_count- endwhile endif if(E3(u,l) / perform E3 rescaling & remember l = (l1)+0 u = (u 28 x 50/1= 最小 n = 8 (28 = 256)l=000, u=111, e3_count=0repeat x=get_symbol l=

27、l+ (u-l+1)CC(x-1)/TC u=l+ (u-l+1)CC(x)/TC -1 while(MSB(u)=MSB(l) OR E3(u,l) if(MSB(u)=MSB(l) send(MSB(u) l = (l1)+0 u = (u0) send(!MSB(u) e3_count- endwhile endif if(E3(u,l) l = (l1)+0 u = (u1)+1 complement MSB(u) and MSB(l) e3_count+ endif endwhileuntil done整数编码器：例l=000, u=111, e3_count=0repeat x=g

28、et_symbol l=l+ (u-l+1)CC(x-1)/TC u=l+ (u-l+1)CC(x)/TC -1 while(MSB(u)=MSB(l) OR E3(u,l) if(MSB(u)=MSB(l) send(MSB(u) l = (l1)+0 u = (u0) send(!MSB(u) e3_count- endwhile endif if(E3(u,l) l = (l1)+0 u = (u1)+1 complement MSB(u) and MSB(l) e3_count+ endif endwhileuntil done2)0(2)0()11111111(255)0000000

29、0(0ulfalse32)1(2)1(),()()11001011(203150402560)00000000(05002560EuMSBlMSBulInput: 1321Output: Input: -321Output: 1 1)()()11001011(203150502040)10100111(167504120402)2(2)2(uMSBlMSBul整数编码器：例l=000, u=111, e3_count=0repeat x=get_symbol l=l+ (u-l+1)CC(x-1)/TC u=l+ (u-l+1)CC(x)/TC -1 while(MSB(u)=MSB(l) O

30、R E3(u,l) if(MSB(u)=MSB(l) send(MSB(u) l = (l1)+0 u = (u0) send(!MSB(u) e3_count- endwhile endif if(E3(u,l) l = (l1)+0 u = (u1)+1 complement MSB(u) and MSB(l) e3_count+ endif endwhileuntil doneInput: -21Output: 110 1, 1)()()10010100(1481504114828)10010010(1465040148282)3(2)3(e3_countuMSBlMSBul175)10

31、000000(11)10010111(281000000001)01001110(151)10010111(11)11001011(78)01001110(01)10101011(22)2(2)2(322)2(22)2(xorxortrueulEule3_count = 1整数编码器：例l=000, u=111, e3_count=0repeat x=get_symbol l=l+ (u-l+1)CC(x-1)/TC u=l+ (u-l+1)CC(x)/TC -1 while(MSB(u)=MSB(l) OR E3(u,l) if(MSB(u)=MSB(l) send(MSB(u) l = (

32、l1)+0 u = (u0) send(!MSB(u) e3_count- endwhile endif if(E3(u,l) l = (l1)+0 u = (u1)+1 complement MSB(u) and MSB(l) e3_count+ endif endwhileuntil doneInput: -1Output: 1100 0)()(41)00101001(11)10010100(36)00100100(1)10010010(22)3(22)3(uMSBlMSBulInput: -1Output: 11000 0)()(83)01010011(11)00101001(72)01

33、001000(1)00100100(22)3(22)3(uMSBlMSBulInput: -1Output: 110001 1)()(167)10100111(11)01010011(144)10010000(1)01001000(22)3(22)3(uMSBlMSBul整数编码器：例l=000, u=111, e3_count=0repeat x=get_symbol l=l+ (u-l+1)CC(x-1)/TC u=l+ (u-l+1)CC(x)/TC -1 while(MSB(u)=MSB(l) OR E3(u,l) if(MSB(u)=MSB(l) send(MSB(u) l = (l

34、1)+0 u = (u0) send(!MSB(u) e3_count- endwhile endif if(E3(u,l) l = (l1)+0 u = (u1)+1 complement MSB(u) and MSB(l) e3_count+ endif endwhileuntil doneInput: -1Output: 1100010 0)()(79)01001111(11)10100111(32)00100000(1)10010000(22)3(22)3(uMSBlMSBulInput: -1191)10000000(11)10011111(01000000001)01000000(

35、),()(159)10011111(11)01001111(64)01000000(1)00100000(22)3(2)3(322)3(22)3(xorxortrueulEuMSBlMSBule3_count = 1整数编码器：例l=000, u=111, e3_count=0repeat x=get_symbol l=l+ (u-l+1)CC(x-1)/TC u=l+ (u-l+1)CC(x)/TC -1 while(MSB(u)=MSB(l) OR E3(u,l) if(MSB(u)=MSB(l) send(MSB(u) l = (l1)+0 u = (u0) send(!MSB(u) e

36、3_count- endwhile endif if(E3(u,l) l = (l1)+0 u = (u1)+1 complement MSB(u) and MSB(l) e3_count+ endif endwhileuntil doneInput: -1Output: 1100010 false32)4(2)4(),()()10011000(152150401920)00000000(05001920EuMSBlMSBulv结束通常，发送l(4): (00000000)2但是 e3_count = 1，因此在发送l(4)的第一个0后发送1最后 output: 110001001000000

37、0 Initialize l, u, t / t = first n bitsrepeat k=0 while( (t-l+1)TC-1)/(u-l+1) CC(k) k+ x = decode_symbol(k) l=l+ (u-l+1)CC(x-1)/TC u=l+ (u-l+1)CC(x)/TC -1 while(MSB(u)=MSB(l) OR E3(u,l) if(MSB(u)=MSB(l) / Perform E1/E2 rescaling of l,u,t l = (l1)+0 / add 0 to the right of l u = (u1)+1 / add 1 to the

38、 right of u t = (t1)+next_bit endif if(E3(u,l) / Perform E3 rescaling of l,u,t l = (l1)+0 u = (u1)+1 t = (t1)+next_bit complement MSB(u), MSB(l), MSB(t) endif endwhileuntil done整数解码器整数解码器：例Initialize l, u, t repeat k=0 while( (t-l+1)CC(x-1)/TC CC(k) k+ x = decode_symbol(k) l=l+ (u-l+1)CC(x-1)/TC u=l

39、+ (u-l+1)CC(x)/TC -1 while(MSB(u)=MSB(l) OR E3(u,l) if(MSB(u)=MSB(l) l = (l1)+0 u = (u1)+1 t = (t1)+next_bit endif if(E3(u,l) l = (l1)+0 u = (u1)+1 t = (t1)+next_bit complement MSB(u), MSB(l), MSB(t) endif endwhileuntil doneInput: 1100010010000000 222)11000100(196)11111111(255)00000000(0tul113825615

40、01970*xktOutput: 1false322),()()11001011(20350402560)00000000(05002560EuMSBlMSBul33482031501970*xktOutput: 13整数解码器：例Input: 1100010010000000 )()()11001011(203150502040)10100111(16750412040)10001001(222uMSBlMSBultInput: 1100010010000000 true322222),()()10010111(11)11001011()01001110(1)10100111()000100

41、10(EuMSBlMSBultInput: 1100010010000000 falsexorxorxor3222222),()(175)10111001(1000000011)10010111(28)00011100(100000001)01001110(146)10010010(10000000)00010010(EuMSBlMSBultInitialize l, u, t repeat k=0 while( (t-l+1)CC(x-1)/TC CC(k) k+ x = decode_symbol(k) l=l+ (u-l+1)CC(x-1)/TC u=l+ (u-l+1)CC(x)/TC

42、 -1 while(MSB(u)=MSB(l) OR E3(u,l) if(MSB(u)=MSB(l) l = (l1)+0 u = (u1)+1 t = (t1)+next_bit endif if(E3(u,l) l = (l1)+0 u = (u1)+1 t = (t1)+next_bit complement MSB(u), MSB(l), MSB(t) endif endwhileuntil done整数解码器：例2240) 128175(150) 128146*xktOutput: 1322228148 40 50146(10010010)28148 41 501148(10010

43、100)( )( )5luMSB lMSB u ，共 次v结束已解码接收到了预定数目的字符，或接收到EOTInitialize l, u, t repeat k=0 while( (t-l+1)CC(x-1)/TC CC(k) k+ x = decode_symbol(k) l=l+ (u-l+1)CC(x-1)/TC u=l+ (u-l+1)CC(x)/TC -1 while(MSB(u)=MSB(l) OR E3(u,l) if(MSB(u)=MSB(l) l = (l1)+0 u = (u1)+1 t = (t1)+next_bit endif if(E3(u,l) l = (l1)+0

44、 u = (u1)+1 t = (t1)+next_bit complement MSB(u), MSB(l), MSB(t) endif endwhileuntil done22*(01000000)(10011111)0 1tlutx 最后，Output: 1321算术编码 vs. Huffman编码nn = 序列长度n平均码长：n算术：n扩展Huffman：n观察n二者的积渐近性质相同nHuffman有一个微小的边际n扩展的Huffman要求巨大数量的存储和编码mnn而算术编码不用 实际应用中可以对算术编码假设较大的长度n 但 Huffman不可 我们期望算术编码表现更好（除了当概率为2

45、的整数次幂的情况）()( )2AH XlH xn()( ) 1HH XlH xn算术编码 vs. Huffman编码n增益为字母表大小和分布的函数n较大的字母表（通常）较适合 Huffmann不均衡的分布更适合算术编码n很容易将算术编码扩展到多个编码器nQM编码器n很容易将算术编码适应到统计变化模型（自适应模型、上下文模型）n不必对树重新排序n可以将建模与编码分开应用：图像压缩自适应算术编码：对像素值自适应算术编码：对像素差分自适应算术编码n统计编码技术需要利用信源符号的概率，获得这个概率的过程称为建模。不同准确度（通常也是不同复杂度）的模型会影响算术编码的效率。n建模的方式：n静态建模：在编

46、码过程中信源符号的概率不变。但一般来说事先知道精确的信源概率是很难的，而且是不切实际的。n自适应动态建模：信源符号的概率根据编码时符号出现的频繁程度动态地进行修改。当压缩消息时，我们不能期待一个算术编码器获得最大的效率，所能做的最有效的方法是在编码过程中估算概率。n算术编码很容易与自适应建模相结合。自适应算术编码n自适应算术编码：n在编码之前，假设每个信源符号的频率相等（如都等于1），并计算累积频率n从输入流中读入一个字符，并对该符号进行算术编码n更新该符号的频率，并更新累积频率n由于在解码之前，解码器不知道是哪个信源符号，所以概率更新应该在解码之后进行n对应的，编码器也应在编码后进行概率更新

47、自适应算术编码n为了提高解码器的搜索效率，信源符号的频率、累积频率表按符号出现频率降序排列n在自适应算术编码中，可以用平衡二叉树来快速实现对频率和累积频率的更新n平衡二叉树可用数组实现（类似Huffman编码中的堆）n最可能出现的符号安排在根附近，从而平均搜索的次数最小n详见数据压缩原理与应用第2.15节自适应算术编码n例：信源符号及其出现频率为：数组下标：1 2 3 4 5 6 7 8 9 10 信源符号：频率：辅助变量：辅助变量为该节点左子树的总计数，用于计算累积频率例：计算a6 的累积频率1.得到从节点10（a6）到根节点的路径： 10 5 2 1 a6 a1 a2a8 2. 令 af=

48、0, 沿树枝从根节点向节点10（a6） 1) 取根节点的左分支到子节点a2，af不变 2) 取节点a2的右分支到到节点a1 ， 将该节点的两个数值加到af af = af + 12 + 16 = 28 2)取节点a1的左分支到到节点a6后 ，将其左子树计数0加至af，最后af=28为累积频率区间的起点自适应算术编码n 数组下标：1 2 3 4 5 6 7 8 9 10 信源符号：频率：辅助变量：频率、累积频率表：自适应算术编码n在平衡二叉树上更新频率：n例：读进符号a9，将其频率计数从12变为13n在数组中寻找符号a9的左边、离a9最远的、频率小于a9的元素，同时更新左子树计数值总结n直接对序

49、列进行编码（不是码字的串联）：非分组码n可证明是唯一可译码n渐近接近熵界n对不均衡的分布，比Huffman更有效n只产生必要的码字n但是实现更复杂n允许将建模和编码分开，容易与自适应模型和上下文模型结合n对错误很敏感，如果有一位发生错误就会导致整个消息译码错误下节课内容n作业：nSayood 3rd, pp.114-115n必做：5, 6, 7, 8n下节课内容：HistorynThe idea of encoding by using cumulative probability in some ordering, and decoding by comparison of magnitud

50、e of binary fraction was introduced in Shannons celebrated paper 1948. nThe recursive implementation of arithmetic coding was devised by Elias (another member in Fanos first information theory class at MIT). This unpublished result was first introduced by Abramson as a note in his book on informatio

51、n theory and coding 1963. nThe result was further developed by Jelinek in his book on information theory 1968. nThe growing precision problem prevented arithmetic coding from practical usage, however. The proposal of using finite precision arithmetic was made independently by Pasco 1976 and Rissanen 1976.HistorynPractical arithmetic coding was developed by several independent groups Rissanen 1979, Rubin 1979, Guazzo 1980. nA well-known tutorial paper on ar