Xử lý hình ảnh kỹ thuật số P6

IMAGE QUANTIZATIONAny 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 conversion process between analog samples and discrete-valued samples is called quantization. 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.
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Xử lý hình ảnh kỹ thuật số P6 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)6IMAGE QUANTIZATIONAny analog quantity that is to be processed by a digital computer or digital systemmust 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 quantizationprocess, which is applicable not only for images but for a wide class of signalsencountered in image processing systems. Section 6.2 considers the processing ofquantized variables. The last section discusses the subjective effects of quantizingmonochrome and color images.6.1. SCALAR QUANTIZATIONFigure 6.1-1 illustrates a typical example of the quantization of a scalar signal. In thequantization process, the amplitude of an analog signal sample is compared to a setof decision levels. If the sample amplitude falls between two decision levels, it isquantized to a fixed reconstruction level lying in the quantization band. In a digitalsystem, each quantized sample is assigned a binary code. An equal-length binarycode is indicated in the example. For the development of quantitative scalar signal quantization techniques, let fand ˆ represent the amplitude of a real, scalar signal sample and its quantized value, frespectively. It is assumed that f is a sample of a random process with known proba-bility density p ( f ) . Furthermore, it is assumed that f is constrained to lie in the range aL ≤ f ≤ a U (6.1-1) 141142 IMAGE QUANTIZATION 256 11111111 255 11111110 254 33 00100000 32 00011111 31 00011110 30 3 00000010 2 00000001 1 00000000 0 ORIGINAL DECISION BINARY QUANTIZED RECONSTRUCTION SAMPLE LEVELS CODE SAMPLE LEVELS FIGURE 6.1-1. Sample quantization.where a U and a L represent upper and lower limits. Quantization entails specification of a set of decision levels d j and a set of recon-struction levels r j such that if dj ≤ f < dj + 1 (6.1-2)the sample is quantized to a reconstruction value r j . Figure 6.1-2a illustrates theplacement 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 ofdescription. Decision and reconstruction levels are chosen to minimize some desired quanti-zation error measure between f and ˆ . The quantization error measure usually femployed is the mean-square error because this measure is tractable, and it usuallycorrelates reasonably well with subjective criteria. For J quantization levels, themean-square quantization error is J–1 2 aU 2 2 E = E{( f – ˆ ) } = f ∫a ( f – ˆ ) p ( f ) df = f ∑ ( f – rj ) p ( f ) df (6.1-3) L j=0 SCALAR QUANTIZATION 143 FIGURE 6.1-2. Quantization decision and reconstruction levels.For a large number of quantization levels J, the probability density may be repre-sented as a constant value p ( r j ) over each quantization band. Hence J –1 dj + 1 2 E = ∑ p ( r j ) ∫d j ( f – r j ) df (6.1-4) j= 0which evaluates to J–1 1 3 3 E = -- 3 - ∑ p ( rj ) [ ( dj + 1 – rj ) – ( dj – rj ) ] (6.1-5) j= 0The optimum placing of the reconstruction level r j within the range d j – 1 to d j canbe determined by minimization of E with respect to r j . Setting dE ------ = 0 (6.1-6) dr jyields dj + 1 + d j r j = ---------------------- (6.1-7) 2144 IMAGE QU ...