Chapter 8 SCALAR QUANTIZATION
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1 Outlne Chapter 8 SCALAR QUANTIZATION Yeuan-Kuen Lee [ CU, CSIE ] 8.1 Overvew 8. Introducton 8.4 Unform Quantzer 8.5 Adaptve Quantzaton 8.6 Nonunform Quantzaton 8.7 Entropy-Coded Quantzaton Ch 8 Scalar Quantzaton 8.1 Overvew 8. Introducton In ths chapter, we begn our study of uantzaton, one of the smplest and most general deas n lossy compresson. Scalar uantzaton ( n ths chapter ), Vector uantzaton ( n the next chapter ). In many lossy compresson applcatons we are reured to represent each source output usng one of a small number of codeword. The number of possble dstnct source output values s generally much larger than the number of codewords avalable to represent them. The process of representng a large possble nfnte set of values wth a much smaller set s called uantzaton. Source ~ 10.0 Infnte number of values.47, Only 1 values { -10, -9, 0 9, 10 } A smple uantzaton scheme would be to represent each output of the source wth the nteger value closest to t. ( f the source output s eually close to two ntegers, we wll randomly pck one of them ) At the same tme we have also forever lost the orgnal value of the source output. 3?.95, 3.16, or any other of an nfnte set of values. We have lost some nformaton. - lossy compresson Ch 8 Scalar Quantzaton 3 Ch 8 Scalar Quantzaton 4
2 8. Introducton The set of nput and output of a uantzer can be scalar or vectors. Scalar Vector Scalar Quantzer Chapter 8 Vector Quantzer Chapter 9 Scalar Vector In practce, the uantzer conssts of two mappngs: Encoder mappng: (rreversble) Dvdes the range of values that the source generates nto a number of ntervals. Each nterval s represented by a dstnct codeword. When the sample value comes from an analog source, the encoder s called an analog-to-dgtal (A/D) converter. Decoder mappng: Because a codeword represents an entre nterval, and there s no way of knowng whch value n the nterval was actually generated by the source, the decoder puts out a value that, n some sense, best represents all the values n the nterval. dpont of the nterval If the reconstructon s analog, the decoder s often referred to as a dgtal-to-analog (D/A) converter. Ch 8 Scalar Quantzaton 5 Ch 8 Scalar Quantzaton 6 Codes Input Codes Output Fgure 8.1 appng for 3-bt encoder Fgure 8. appng for 3-bt D/A converter. Ch 8 Scalar Quantzaton 7 Ch 8 Scalar Quantzaton 8
3 Example Suppose a snusod 4cos(πt) was sampled every 0.05 second. The sample was dgtzed usng the A/D mappng shown n Fgure 8.1, and the reconstructed usng the D/A mappng shown n Fgure 8.. The frst few nputs, codewords, and reconstructon values are gven n Table 8.1. Table 8.1 Dgtzng a sne wave. t 4cos(πt) A/D Output D/A Output Error y 4 * cos ( πx ) Ch 8 Scalar Quantzaton 9 Ch 8 Scalar Quantzaton 10 Constructon of the ntervals (ther locaton, etc.) can be vewed as part of the desgn of the encoder. Selecton of the reconstructon values s part of the desgn of the decoder output The fdelty of the reconstructon depends on both the ntervals and the reconstructon values. 1.5 encoder decoder ( Fgure 8.1 ) ( Fgure 8. ) We call ths encoder-decoder par a uantzer nput ( Fgure 8. ) -3.5 Input-output map Fgure 8.3 Quantzer nput-output map. Ch 8 Scalar Quantzaton 11 Ch 8 Scalar Quantzaton 1
4 Dstorton Average suared dfference between the uantzer nput and output. We call ths mean suared uantzaton error (mse) and denote t by σ The rate of the uantzer s the average number of bts reured to represent a sngle uantzer output. We would lke to get the lowest dstorton for a gven rate, or the lowest rate for a gven dstorton. Suppose we have an nput modeled by a random varable X wth pdf f X (x). If we wshed to uantze ths source usng a uantzer wth ntervals, we would have to specfy + 1 endponts for the ntervals, and a representatve value for each of the ntervals. The endponts of the ntervals are known as decson boundares, whle the representatve values are called reconstructon levels Ch 8 Scalar Quantzaton 13 Ch 8 Scalar Quantzaton 14 Let us denote the decson boundares by the reconstructon level by { b } 0 { y } 1 the uantzaton operaton by Q( ) Quantzaton nose Then, Q(x) y ff b -1 < x b The mean suared uantzaton error s then gven by σ 1 b b 1 ( x Q ( x )) ( x y ) f X f X ( x ) dx ( x ) dx (8.1) (8.) (8.3) Quantzer nput + Quantzer output Quantzer output Quantzer output + Quantzaton nose The dfference between the uantzer nput x and output y Q(x), besdes beng referred to as the uantzaton error, s also called the uantzer dstorton or uantzaton nose. Fgure 8.4 Addtve nose model of a uantzer. Ch 8 Scalar Quantzaton 15 Ch 8 Scalar Quantzaton 16
5 If we use fxed-length codewords to represent the uantzer output, However, f we are allowed to used varable-length codes, such as then the sze of the output alphabet mmedately specfes the rate. Huffman codes or arthmetc codes, along wth the sze of alphabet, the If the number of uantzer output s, then the rate gven by selecton of the decson boundares wll also affect the rate of the log (8.4) R For example, f 8, then R 3. In ths case, we can pose the uantzer desgn problem as follows: Gven an nput pdf f X (x) and the number of levels n the uantzer, fnd the decson boundares { b } and the reconstructon levels { y } so as to mnmze the mean suared uantzaton error gven by Euaton (8.3) uantzer. Table 8. Codeword assgnment for an eght-level uantzer. y 1 y y 3 y 4 y 5 y 6 y 7 y The rate wll depend on the probablty of occurrence of the output. If l s the length of the codeword correspondng to the output y, and P(y ) s the probablty of occurrence of y,then the rate s gven by: R l P ( y ) (8.5) 1 Ch 8 Scalar Quantzaton 17 Ch 8 Scalar Quantzaton 18 However, the probablty { P(y ) } depend on the decson boundares { b }. For example, the probablty of y occurrng s gven by P ( y ) b b 1 f X ( x ) dx Therefore, the rate s a functon of decson boundares and gven by the expresson R R l P ( y ) (8.5) 1 1 l b b 1 f X ( x ) dx (8.6) For a gven source nput : The partton we select and the representaton for these parttons wll determne the dstorton ncurred durng the uantzaton process. The parttons we select and the bnary codes for the parttons wll determne the rate for the uantzer. Thus, the problem of fndng the optmum parttons, codes, and representaton levels are all lnked. Ch 8 Scalar Quantzaton 19 Ch 8 Scalar Quantzaton 0
6 8.4 Unform Quantzaton In lght of ths nformaton, we can restate our problem statement: Gven a dstorton constran σ D fnd the decson boundares, reconstructon levels, and bnary codes that mnmze the rate gven by Euaton (8.6), whle satsfyng Euaton (8.7). Or, Gven a rate constran * R R fnd the decson boundares, reconstructon levels, and bnary codes that mnmze the dstorton gven by Euaton (8.3), whle satsfyng Euaton (8.8). * (8.7) (8.8) The smplest type of uantzer s the unform uantzer. All ntervals are the same sze n the unform uantzer, except possbly for the outer ntervals. (.e., the decson boundares are spaced evenly.) The reconstructon values are also spaced evenly, wth the same spacng as decson boundares; n the nner ntervals, they are the mdponts of the ntervals. The constant spacng s usually referred to as the step sze and s denoted by. The uantzer shown n Fgure 8.3 s a unform uantzer wth 1. Ch 8 Scalar Quantzaton 1 Ch 8 Scalar Quantzaton 8.4 Unform Quantzaton 8.4 Unform Quantzaton drse uantzer: Fgure output does not have zero as one of ts representaton levels..0 dtread uantzer: Fgure 8.5 have zero as one of ts representaton levels. Usually, we use a mdrse uantzer f the number of levels s even and a mdtread uantzer f the number of level s odd nput -3.0 Fgure 8.5 A mdtread uantzer. Ch 8 Scalar Quantzaton 3 Ch 8 Scalar Quantzaton 4
7 8.4 Unform Quantzaton Unform Quantzaton of a Unformly Dstrbuted Source 8.4 Unform Quantzaton x-q(x) Suppose we want to desgn an -level unform uantzer for an nput that s unformly dstrbuted n the nterval [ -X max, X max ]. Step sze s gven by The dstorton becomes X max / / 3 4 Fgure 8.6 Quantzaton error for a unform mdrse uantzer wth a unformly dstrbuted nput. / x - / σ / 1 ( 1 ) 1 ( x ) 1 X max dx 1 The mean suared uantzaton error s the second moment of a random varable unformly dstrbuted n the nterval [ - /, / ]: σ 1 / / d 1 Ch 8 Scalar Quantzaton 5 Ch 8 Scalar Quantzaton Unform Quantzaton 8.4 Unform Quantzaton Example Image Compresson σ / / ( ) d ( ) ( ) 3 4 / / Fgure 8.7 Left: Orgnal Sena mage; Rght: 3-bt/pxel Ch 8 Scalar Quantzaton 7 Ch 8 Scalar Quantzaton 8
8 8.4 Unform Quantzaton Example Image Compresson Fgure 8.7 Left: 1-bt/pxel Rght: 1-bt/pxel (Cont.) Ch 8 Scalar Quantzaton 9
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