IMAGE SAMPLING AND RECONSTRUCTIONIn digital image processing systems, one usually deals with arrays of numbers obtained by spatially sampling points of a physical image. After processing, another array of numbers is produced, and these numbers are then used to reconstruct a continuous image for viewing. Image samples nominally represent some physical measurements of a continuous image field, for example, measurements of the image intensity or photographic density. Measurement uncertainties exist in any physical measurement apparatus....
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Xử lý hình ảnh kỹ thuật số P4 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)4IMAGE SAMPLING ANDRECONSTRUCTIONIn digital image processing systems, one usually deals with arrays of numbersobtained by spatially sampling points of a physical image. After processing, anotherarray of numbers is produced, and these numbers are then used to reconstruct a con-tinuous image for viewing. Image samples nominally represent some physical mea-surements of a continuous image field, for example, measurements of the imageintensity or photographic density. Measurement uncertainties exist in any physicalmeasurement apparatus. It is important to be able to model these measurementerrors in order to specify the validity of the measurements and to design processesfor compensation of the measurement errors. Also, it is often not possible to mea-sure an image field directly. Instead, measurements are made of some functionrelated to the desired image field, and this function is then inverted to obtain thedesired image field. Inversion operations of this nature are discussed in the sectionson image restoration. In this chapter the image sampling and reconstruction processis considered for both theoretically exact and practical systems.4.1. IMAGE SAMPLING AND RECONSTRUCTION CONCEPTSIn the design and analysis of image sampling and reconstruction systems, inputimages are usually regarded as deterministic fields (1–5). However, in somesituations it is advantageous to consider the input to an image processing system,especially a noise input, as a sample of a two-dimensional random process (5–7).Both viewpoints are developed here for the analysis of image sampling andreconstruction methods. 9192 IMAGE SAMPLING AND RECONSTRUCTION FIGURE 4.1-1. Dirac delta function sampling array.4.1.1. Sampling Deterministic FieldsLet F I ( x, y ) denote a continuous, infinite-extent, ideal image field representing theluminance, photographic density, or some desired parameter of a physical image. Ina perfect image sampling system, spatial samples of the ideal image would, in effect,be obtained by multiplying the ideal image by a spatial sampling function ∞ ∞ S ( x, y ) = ∑ ∑ δ ( x – j ∆x, y – k ∆y ) (4.1-1) j = –∞ k = – ∞composed of an infinite array of Dirac delta functions arranged in a grid of spacing( ∆x, ∆y ) as shown in Figure 4.1-1. The sampled image is then represented as ∞ ∞ F P ( x, y ) = FI ( x, y )S ( x, y ) = ∑ ∑ FI ( j ∆x, k ∆y )δ ( x – j ∆x, y – k ∆y ) (4.1-2) j = –∞ k = –∞where it is observed that F I ( x, y ) may be brought inside the summation and evalu-ated only at the sample points ( j ∆x, k ∆y) . It is convenient, for purposes of analysis,to consider the spatial frequency domain representation F P ( ω x, ω y ) of the sampledimage obtained by taking the continuous two-dimensional Fourier transform of thesampled image. Thus ∞ ∞ F P ( ω x, ω y ) = ∫–∞ ∫–∞ FP ( x, y ) exp { –i ( ωx x + ωy y ) } dx dy (4.1-3) IMAGE SAMPLING AND RECONSTRUCTION CONCEPTS 93By the Fourier transform convolution theorem, the Fourier transform of the sampledimage can be expressed as the convolution of the Fourier transforms of the idealimage F I ( ω x, ω y ) and the sampling function S ( ω x, ω y ) as expressed by 1 F P ( ω x, ω y ) = -------- F I ( ω x, ω y ) * S ( ω x, ω y ) - (4.1-4) 2 4πThe two-dimensional Fourier transform of the spatial sampling function is an infi-nite array of Dirac delta functions in the spatial frequency domain as given by(4, p. 22) 2 ∞ ∞ 4π - S ( ω x, ω y ) = -------------- ∆x ∆y ∑ ∑ δ ( ω x – j ω xs, ω y – k ω ys ) (4.1-5) j = –∞ k = –∞where ...