Tài liệu Xử lý hình ảnh kỹ thuật số P6 pdf

141 6 IMAGE QUANTIZATION Any analog quantity that is to be processed by a digital computer or digital system must be converted to an integer number proportional to its amplitude. The conver- sion process between analog samples and discrete-valued samples is called quanti- zation. The following section includes an analytic treatment of the quantization process, which is applicable not only for images but for a wide class of signals encountered in image processing systems. Section 6.2 considers the processing of quantized variables. The last section discusses the subjective effects of quantizing monochrome and color images. 6.1. SCALAR QUANTIZATION Figure 6.1-1 illustrates a typical example of the quantization of a scalar signal. In the quantization process, the amplitude of an analog signal sample is compared to a set of decision levels. If the sample amplitude falls between two decision levels, it is quantized to a fixed reconstruction level lying in the quantization band. In a digital system, each quantized sample is assigned a binary code. An equal-length binary code is indicated in the example. For the development of quantitative scalar signal quantization techniques, let f and represent the amplitude of a real, scalar signal sample and its quantized value, respectively. It is assumed that f is a sample of a random process with known proba- bility density . Furthermore, it is assumed that f is constrained to lie in the range (6.1-1) f ˆ p f() a L fa U ≤≤ Digital Image Processing: PIKS Inside, Third Edition. William K. Pratt Copyright © 2001 John Wiley & Sons, Inc. ISBNs: 0-471-37407-5 (Hardback); 0-471-22132-5 (Electronic) 142 IMAGE QUANTIZATION where and represent upper and lower limits. Quantization entails specification of a set of decision levels and a set of recon- struction levels such that if (6.1-2) the sample is quantized to a reconstruction value . Figure 6.1-2a illustrates the placement of decision and reconstruction levels along a line for J quantization lev- els. The staircase representation of Figure 6.1-2b is another common form of description. Decision and reconstruction levels are chosen to minimize some desired quanti- zation error measure between f and . The quantization error measure usually employed is the mean-square error because this measure is tractable, and it usually correlates reasonably well with subjective criteria. For J quantization levels, the mean-square quantization error is (6.1-3) FIGURE 6.1-1. Sample quantization. 256 11111111 11111110 00100000 00000010 00000001 00000000 ORIGINAL SAMPLE DECISION LEVELS BINARY CODE QUANTIZED SAMPLE RECONSTRUCTION LEVELS 00011111 00011110 255 254 33 32 31 30 3 2 1 0 a U a L d j r j d j fd j 1 + <≤ r j f ˆ E Ef f ˆ –() 2 {} f f ˆ –() 2 pf() fdfr j –() 2 pf()fd j 0 = J 1 – ∑ = a L a U ∫ == SCALAR QUANTIZATION 143 For a large number of quantization levels J, the probability density may be repre- sented as a constant value over each quantization band. Hence (6.1-4) which evaluates to (6.1-5) The optimum placing of the reconstruction level within the range to can be determined by minimization of with respect to . Setting (6.1-6) yields (6.1-7) FIGURE 6.1-2. Quantization decision and reconstruction levels. pr j () E pr j () fr j –() 2 fd d j d j 1 + ∫ j 0 = J 1 – ∑ = E 1 3 --- pr j ()d j 1 + r j –() 3 d j r j –() 3 –[] j 0 = J 1 – ∑ = r j d j 1 – d j E r j Ed r j d ------ 0= r j d j 1 + d j + 2 ----------------------= 144 IMAGE QUANTIZATION Therefore, the optimum placement of reconstruction levels is at the midpoint between each pair of decision levels. Substitution for this choice of reconstruction levels into the expression for the quantization error yields (6.1-8) The optimum choice for decision levels may be found by minimization of in Eq. 6.1-8 by the method of Lagrange multipliers. Following this procedure, Panter and Dite (1) found that the decision levels may be computed to a good approximation from the integral equation (6.1-9a) where (6.1-9b) for j = 0, 1, ., J. If the probability density of the sample is uniform, the decision lev- els will be uniformly spaced. For nonuniform probability densities, the spacing of decision levels is narrow in large-amplitude regions of the probability density func- tion and widens in low-amplitude portions of the density. Equation 6.1-9 does not reduce to closed form for most probability density functions commonly encountered in image processing systems models, and hence the decision levels must be obtained by numerical integration. If the number of quantization levels is not large, the approximation of Eq. 6.1-4 becomes inaccurate, and exact solutions must be explored. From Eq. 6.1-3, setting the partial derivatives of the error expression with respect to the decision and recon- struction levels equal to zero yields (6.1-10a) (6.1-10b) E 1 12 ------ pr j ()d j 1 + d j –() 3 j 0 = J 1 – ∑ = E d j a U a L –()pf()[] 13⁄ – fd a L a j ∫ pf()[] 13⁄ – fd a L a U ∫ ----------------------------------------------------------------= a j ja U a L –() J ------------------------- a L += E∂ d j ∂ ------- d j r j –() 2 pd j () d j r j 1 – –() 2 pd j ()– 0== E∂ r j ∂ ------ 2 fr j –()pf()fd d j d j 1 + ∫ 0== SCALAR QUANTIZATION 145 Upon simplification, the set of equations (6.1-11a) (6.1-11b) is obtained. Recursive solution of these equations for a given probability distribution provides optimum values for the decision and reconstruction levels. Max (2) has developed a solution for optimum decision and reconstruction levels for a Gaus- sian density and has computed tables of optimum levels as a function of the number of quantization steps. Table 6.1-1 lists placements of decision and quantization lev- els for uniform, Gaussian, Laplacian, and Rayleigh densities for the Max quantizer. If the decision and reconstruction levels are selected to satisfy Eq. 6.1-11, it can easily be shown that the mean-square quantization error becomes (6.1-12) In the special case of a uniform probability density, the minimum mean-square quantization error becomes (6.1-13) Quantization errors for most other densities must be determined by computation. It is possible to perform nonlinear quantization by a companding operation, as shown in Figure 6.1-3, in which the sample is transformed nonlinearly, linear quanti- zation is performed, and the inverse nonlinear transformation is taken (3). In the com- panding system of quantization, the probability density of the transformed samples is forced to be uniform. Thus, from Figure 6.1-3, the transformed sample value is (6.1-14) where the nonlinear transformation is chosen such that the probability density of g is uniform. Thus, FIGURE 6.1-3. Companding quantizer. r j 2d j r j 1 – –= r j fp f()fd d j d j 1 + ∫ pf()fd d j d j 1 + ∫ -------------------------------= pf() E min f 2 pf() fdr j 2 pf()fd d j d j 1 + ∫ – d j d j 1 + ∫ j 0 = J 1 – ∑ = E min 1 12J 2 ------------= gTf{}= T · {} 146 IMAGE QUANTIZATION TABLE 6.1-1. Placement of Decision and Reconstruction Levels for Max Quantizer Uniform Gaussian Laplacian Rayleigh Bits d i r i d i r i d i r i d i r i 1 –1.0000 –0.5000 – –0.7979 – –0.7071 0.0000 1.2657 0.0000 0.5000 0.0000 0.7979 0.0000 0.7071 2.0985 2.9313 1.0000 – 2–1.0000 –0.7500 – –1.5104 –1.8340 0.0000 0.8079 –0.5000 –0.2500 –0.9816 –0.4528 –1.1269 –0.4198 1.2545 1.7010 –0.0000 0.2500 0.0000 0.4528 0.0000 0.4198 2.1667 2.6325 0.5000 0.7500 0.9816 1.5104 1.1269 1.8340 3.2465 3.8604 1.0000 3 –1.0000 –0.8750 – –2.1519 – –3.0867 0.0000 0.5016 –0.7500 –0.6250 –1.7479 –1.3439 –2.3796 –1.6725 0.7619 1.0222 –0.5000 –0.3750 –1.0500 –0.7560 –1.2527 –0.8330 1.2594 1.4966 –0.2500 –0.1250 –0.5005 –0.2451 –0.5332 –0.2334 1.7327 1.9688 0.0000 0.1250 0.0000 0.2451 0.0000 0.2334 2.2182 2.4675 0.2500 0.3750 0.5005 0.7560 0.5332 0.8330 2.7476 3.0277 0.5000 0.6250 1.0500 1.3439 1.2527 1.6725 3.3707 3.7137 0.7500 0.8750 1.7479 2.1519 2.3796 3.0867 4.2124 4.7111 1.0000 4 –1.0000 –0.9375 – –2.7326 – –4.4311 0.0000 0.3057 –0.8750 –0.8125 –2.4008 –2.0690 –3.7240 –3.0169 0.4606 0.6156 –0.7500 –0.6875 –1.8435 –1.6180 –2.5971 –2.1773 0.7509 0.8863 –0.6250 –0.5625 –1.4371 –1.2562 –1.8776 –1.5778 1.0130 1.1397 –0.5000 –0.4375 –1.0993 –0.9423 –1.3444 –1.1110 1.2624 1.3850 –0.3750 –0.3125 –0.7995 –0.6568 –0.9198 –0.7287 1.5064 1.6277 –0.2500 –0.1875 –0.5224 –0.3880 –0.5667 –0.4048 1.7499 1.8721 –0.1250 –0.0625 –0.2582 –0.1284 –0.2664 –0.1240 1.9970 2.1220 0.0000 0.0625 0.0000 0.1284 0.0000 0.1240 2.2517 2.3814 0.1250 0.1875 0.2582 0.3880 0.2644 0.4048 2.5182 2.6550 0.2500 0.3125 0.5224 0.6568 0.5667 0.7287 2.8021 2.9492 0.3750 0.4375 0.7995 0.9423 0.9198 1.1110 3.1110 3.2729 0.5000 0.5625 1.0993 1.2562 1.3444 1.5778 3.4566 3.6403 0.6250 0.6875 1.4371 1.6180 1.8776 2.1773 3.8588 4.0772 0.7500 0.8125 1.8435 2.0690 2.5971 3.0169 4.3579 4.6385 0.8750 0.9375 2.4008 2.7326 3.7240 4.4311 5.0649 5.4913 1.0000 ∞∞ ∞∞∞ ∞∞ ∞∞∞ ∞∞ ∞∞∞ ∞∞ ∞∞∞ PROCESSING QUANTIZED VARIABLES 147 (6.1-15) for . If f is a zero mean random variable, the proper transformation func- tion is (4) (6.1-16) That is, the nonlinear transformation function is equivalent to the cumulative proba- bility distribution of f. Table 6.1-2 contains the companding transformations and inverses for the Gaussian, Rayleigh, and Laplacian probability densities. It should be noted that nonlinear quantization by the companding technique is an approxima- tion to optimum quantization, as specified by the Max solution. The accuracy of the approximation improves as the number of quantization levels increases. 6.2. PROCESSING QUANTIZED VARIABLES Numbers within a digital computer that represent image variables, such as lumi- nance or tristimulus values, normally are input as the integer codes corresponding to the quantization reconstruction levels of the variables, as illustrated in Figure 6.1-1. If the quantization is linear, the jth integer value is given by (6.2-1) where J is the maximum integer value, f is the unquantized pixel value over a lower-to-upper range of to , and denotes the nearest integer value of the argument. The corresponding reconstruction value is (6.2-2) Hence, is linearly proportional to j. If the computer processing operation is itself linear, the integer code j can be numerically processed rather than the real number . However, if nonlinear processing is to be performed, for example, taking the loga- rithm of a pixel, it is necessary to process as a real variable rather than the integer j because the operation is scale dependent. If the quantization is nonlinear, all process- ing must be performed in the real variable domain. In a digital computer, there are two major forms of numeric representation: real and integer. Real numbers are stored in floating-point form, and typically have a large dynamic range with fine precision. Integer numbers can be strictly positive or bipolar (negative or positive). The two's complement number system is commonly pg() 1= 1 2 --- – g 1 2 --- ≤≤ Tf{} pz() z 1 2 ---–d ∞ – f ∫ = jJ1–() fa L – a U a L – ------------------ N = a L a U · [] N r j a U a L – J ------------------ j a U a L – 2J ------------------ a L ++= r j r j r j 148 TABLE 6.1.-2. Companding Quantization Transformations Probability Density Forward Transformation Inverse Transformation Gaussian Rayleigh Laplacian where erf {x} and pf() 2πσ 2 () 1 – 2⁄ f 2 2σ 2 ---------–    exp= g 1 2 ---erf f 2σ -----------    = f ˆ 2 σ erf 1 – 2 g ˆ {}= pf() f σ 2 ------ f 2 2σ 2 ---------–    exp= g 1 2 --- f 2 2σ 2 ---------–    exp–= f ˆ 2σ 2 ln 1 1 2 --- g ˆ –   ⁄    12⁄ = pf() α 2 --- α f–{}exp= g 1 2 ---1 αf–{}exp–[]f 0≥= g 1 2 ---1 αf{}exp–[]f 0<–= f ˆ 1 α ---– ln 1 2– g ˆ {} g ˆ 0≥= f ˆ 1 α --- l n 1 2 g ˆ +{} g ˆ 0<= 2 π ------- y 2 –{}exp yd 0 x ∫ ≡ α 2 σ -------= PROCESSING QUANTIZED VARIABLES 149 used in computers and digital processing hardware for representing bipolar integers. The general format is as follows: S.M 1 ,M 2 , .,M B-1 where S is a sign bit (0 for positive, 1 for negative), followed, conceptually, by a binary point, M b denotes a magnitude bit, and B is the number of bits in the com- puter word. Table 6.2-1 lists the two's complement correspondence between integer, fractional, and decimal numbers for a 4-bit word. In this representation, all pixels are scaled in amplitude between –1.0 and . One of the advantages of TABLE 6.2-1. Two’s Complement Code for 4-Bit Code Word Code Fractional Va l u e Decimal Val u e 0.111 + +0.875 0.110 + +0.750 0.101 + +0.625 0.100 + +0.500 0.011 + +0.375 0.010 + +0.250 0.001 + +0.125 0.000 0 0.000 1.111 – –0.125 1.110 – –0.250 1.101 – –0.375 1.100 – –0.500 1.011 – –0.625 1.010 – –0.750 1.001 – –0.875 1.000 – –1.000 1.0 2 B 1 – () – – 7 8 --- 6 8 --- 5 8 --- 4 8 --- 3 8 --- 2 8 --- 1 8 --- 1 8 --- 2 8 --- 3 8 --- 4 8 --- 5 8 --- 6 8 --- 7 8 --- 8 8 --- 150 IMAGE QUANTIZATION this representation is that pixel scaling is independent of precision in the sense that a pixel is bounded over the range regardless of the number of bits in a word. 6.3. MONOCHROME AND COLOR IMAGE QUANTIZATION This section considers the subjective and quantitative effects of the quantization of monochrome and color images. 6.3.1. Monochrome Image Quantization Monochrome images are typically input to a digital image processor as a sequence of uniform-length binary code words. In the literature, the binary code is often called a pulse code modulation (PCM) code. Because uniform-length code words are used for each image sample, the number of amplitude quantization levels is determined by the relationship (6.3-1) where B represents the number of code bits allocated to each sample. A bit rate compression can be achieved for PCM coding by the simple expedient of restricting the number of bits assigned to each sample. If image quality is to be judged by an analytic measure, B is simply taken as the smallest value that satisfies the minimal acceptable image quality measure. For a subjective assessment, B is lowered until quantization effects become unacceptable. The eye is only capable of judging the absolute brightness of about 10 to 15 shades of gray, but it is much more sensitive to the difference in the brightness of adjacent gray shades. For a reduced number of quantization levels, the first noticeable artifact is a gray scale contouring caused by a jump in the reconstructed image brightness between quantization levels in a region where the original image is slowly changing in brightness. The minimal number of quantization bits required for basic PCM coding to prevent gray scale contouring is dependent on a variety of factors, including the linearity of the image display and noise effects before and after the image digitizer. Assuming that an image sensor produces an output pixel sample proportional to the image intensity, a question of concern then is: Should the image intensity itself, or some function of the image intensity, be quantized? Furthermore, should the quantiza- tion scale be linear or nonlinear? Linearity or nonlinearity of the quantization scale can Fjk,() 1.0– Fjk,()1.0<≤ L 2 B =