142 IMAGE QUANTIZATION Let represent the upper bound of x(i) and the lower bound. Then each quantization cell has dimension (5.3-6) Any color with color component x(i) within the quantization cell will be quantized to the color component value . The maximum quantization error along each color coordinate axis is then (5.3-7) FIGURE 5.3-6. Chromaticity shifts resulting from uniform quantization of the smpte_girl_linear color image. a U i() a L i() qi() a U i() a L i()– 2 Bi() = x ˆ i() ε i() xi() x ˆ i()– a U i() a L i()– 2 Bi() 1+ == REFERENCES 143 Thus, the coordinates of the quantized color become (5.3-8) subject to the conditions . It should be observed that the values of will always lie within the smallest cube enclosing the color solid for the given color coordinate system. Figure 5.3-6 illustrates chromaticity shifts of various colors for quantization in the R N G N B N and Yuv coordinate systems (12). Jain and Pratt (12) have investigated the optimal assignment of quantization deci- sion levels for color images in order to minimize the geodesic color distance between an original color and its reconstructed representation. Interestingly enough, it was found that quantization of the R N G N B N color coordinates provided better results than for other common color coordinate systems. The primary reason was that all quantization levels were occupied in the R N G N B N system, but many levels were unoccupied with the other systems. This consideration seemed to override the metric nonuniformity of the R N G N B N color space. Sharma and Trussell (13) have surveyed color image quantization for reduced memory image displays. REFERENCES 1. P. F. Panter and W. Dite, “Quantization Distortion in Pulse Code Modulation with Non- uniform Spacing of Levels,” Proc. IRE, 39, 1, January 1951, 44–48. 2. J. Max, “Quantizing for Minimum Distortion,” IRE Trans. Information Theory, IT-6, 1, March 1960, 7–12. 3. V. R. Algazi, “Useful Approximations to Optimum Quantization,” IEEE Trans. Commu- nication Technology, COM-14, 3, June 1966, 297–301. 4. R. M. Gray, “Vector Quantization,” IEEE ASSP Magazine, April 1984, 4–29. 5. W. M. Goodall, “Television by Pulse Code Modulation,” Bell System Technical J., January 1951. 6. R. L. Cabrey, “Video Transmission over Telephone Cable Pairs by Pulse Code Modula- tion,” Proc. IRE, 48, 9, September 1960, 1546–1551. 7. L. H. Harper, “PCM Picture Transmission,” IEEE Spectrum, 3, 6, June 1966, 146. 8. F. W. Scoville and T. S. Huang, “The Subjective Effect of Spatial and Brightness Quanti- zation in PCM Picture Transmission,” NEREM Record, 1965, 234–235. 9. I. G. Priest, K. S. Gibson, and H. J. McNicholas, “An Examination of the Munsell Color System, I. Spectral and Total Reflection and the Munsell Scale of Value,” Technical Paper 167, National Bureau of Standards, Washington, DC, 1920. 10. J. H. Ladd and J. E. Pinney, “Empherical Relationships with the Munsell Value Scale,” Proc. IRE (Correspondence), 43, 9, 1955, 1137. 11. C. E. Foss, D. Nickerson and W. C. Granville, “Analysis of the Oswald Color System,” J. Optical Society of America, 34, 1, July 1944, 361–381. x ˆ i() xi() εi()±= a L i() x ˆ i() a U i()≤≤ x ˆ i() 144 IMAGE QUANTIZATION 12. A. K. Jain and W. K. Pratt, “Color Image Quantization,” IEEE Publication 72 CH0 601-5-NTC, National Telecommunications Conference 1972 Record, Houston, TX, December 1972. 13. G. Sharma and H. J. Trussell, “Digital Color Imaging,” IEEE Trans. Image Processing, 6, 7, July 1997, 901–932. PART 3 DISCRETE TWO-DIMENSIONAL PROCESSING Part 3 of the book is concerned with a unified analysis of discrete two-dimensional processing operations. Vector-space methods of image representation are developed for deterministic and stochastic image arrays. Several forms of discrete two- dimensional superposition and convolution operators are developed and related to one another. Two-dimensional transforms, such as the Fourier, Hartley, cosine and Karhunen–Loeve transforms, are introduced. Consideration is given to the utilization of two-dimensional transforms as an alternative means of achieving convolutional processing more efficiently. 147 6 Digital Image Processing: PIKS Scientific Inside, Fourth Edition, by William K. Pratt Copyright © 2007 by John Wiley & Sons, Inc. DISCRETE IMAGE MATHEMATICAL CHARACTERIZATION Chapter 1 presented a mathematical characterization of continuous image fields. This chapter develops a vector-space algebra formalism for representing discrete image fields from a deterministic and statistical viewpoint. Appendix 1 presents a summary of vector-space algebra concepts. 6.1. VECTOR-SPACE IMAGE REPRESENTATION In Chapter 1, a generalized continuous image function F(x, y, t) was selected to represent the luminance, tristimulus value, or some other appropriate measure of a physical imaging system. Image sampling techniques, discussed in Chapter 4, indicated means by which a discrete array F(j, k) could be extracted from the contin- uous image field at some time instant over some rectangular area , . It is often helpful to regard this sampled image array as a element matrix (6.1-1) for where the indices of the sampled array are reindexed for consistency with standard vector-space notation. Figure 6.1-1 illustrates the geometric relation- ship between the Cartesian coordinate system of a continuous image and its matrix array of samples. Each image sample is called a pixel. J– jJ≤≤ K– kK≤≤ N 1 N 2 × F Fn 1 n 2 ,()[]= 1 n i N i ≤≤ 148 DISCRETE IMAGE MATHEMATICAL CHARACTERIZATION For purposes of analysis, it is often convenient to convert the image matrix to vector form by column (or row) scanning F, and then stringing the elements together in a long vector (1). An equivalent scanning operation can be expressed in quantita- tive form by the use of a operational vector and a matrix defined as (6.1-2) Then the vector representation of the image matrix F is given by the stacking oper- ation (6.1-3) In essence, the vector extracts the nth column from F and the matrix places this column into the nth segment of the vector f. Thus, f contains the column-scanned FIGURE 6.1-1. Geometric relationship between a continuous image and its matrix array of samples. N 2 1× v n N 1 N 2 ⋅ N 2 × N n v n 0 0 1 0 0 = … … 1 n 1– n n 1+ N 2 … … N n 0 0 1 0 0 = …… 1 n 1– n n 1+ N 2 …… fN n Fv n n 1= N 2 ∑ = v n N n GENERALIZED TWO-DIMENSIONAL LINEAR OPERATOR 149 elements of F. The inverse relation of casting the vector f into matrix form is obtained from (6.1-4) With the matrix-to-vector operator of Eq. 6.1-3 and the vector-to-matrix operator of Eq. 6.1-4, it is now possible easily to convert between vector and matrix representa- tions of a two-dimensional array. The advantages of dealing with images in vector form are a more compact notation and the ability to apply results derived previously for one-dimensional signal processing applications. It should be recognized that Eqs 6.1-3 and 6.1-4 represent more than a lexicographic ordering between an array and a vector; these equations define mathematical operators that may be manipulated ana- lytically. Numerous examples of the applications of the stacking operators are given in subsequent sections. 6.2. GENERALIZED TWO-DIMENSIONAL LINEAR OPERATOR A large class of image processing operations are linear in nature; an output image field is formed from linear combinations of pixels of an input image field. Such operations include superposition, convolution, unitary transformation and discrete linear filtering. Consider the element input image array . A generalized linear operation on this image field results in a output image array as defined by (6.2-1) where the operator kernel represents a weighting constant, which, in general, is a function of both input and output image coordinates (1). For the analysis of linear image processing operations, it is convenient to adopt the vector-space formulation developed in Section 6.1. Thus, let the input image array be represented as matrix F or alternatively, as a vector f obtained by column scanning F. Similarly, let the output image array be represented by the matrix P or the column-scanned vector p. For notational simplicity, in the subsequent discussions, the input and output image arrays are assumed to be square and of dimensions and , respectively. Now, let T FN n T fv n T n 1= N 2 ∑ = N 1 N 2 × Fn 1 n 2 ,() M 1 M 2 × Pm 1 m 2 ,() Pm 1 m 2 ,() Fn 1 n 2 ,()On 1 n 2 m 1 m 2 ,;,() n 2 1= N 2 ∑ n 1 1= N 1 ∑ = On 1 n 2 m 1 m 2 ,;,() Fn 1 n 2 ,() Pm 1 m 2 ,() N 1 N 2 N== M 1 M 2 M== 150 DISCRETE IMAGE MATHEMATICAL CHARACTERIZATION denote the matrix performing a linear transformation on the input image vector f yielding the output image vector (6.2-2) The matrix T may be partitioned into submatrices and written as (6.2-3) From Eq. 6.1-3, it is possible to relate the output image vector p to the input image matrix F by the equation (6.2-4) Furthermore, from Eq. 6.1-4, the output image matrix P is related to the input image vector p by (6.2-5) Combining the above yields the relation between the input and output image matrices, (6.2-6) where it is observed that the operators and simply extract the partition from T. Hence (6.2-7) If the linear transformation is separable such that T may be expressed in the direct product form (6.2-8) M 2 N 2 × N 2 1× M 2 1× pTf= MN× T mn T T 11 T 12 … …… … T 1N T 21 T 22 … …… … T 2N T M1 T M2 … T MN = … … … pTN n Fv n n 1= N ∑ = PM m T pu m T m 1= M ∑ = PM m T TN n ()Fv n u m T () n 1= N ∑ m 1= M ∑ = M m N n T mn PT mn Fv n u m T () n 1= N ∑ m 1= M ∑ = TT C T R ⊗= GENERALIZED TWO-DIMENSIONAL LINEAR OPERATOR 151 where and are row and column operators on F, then (6.2-9) As a consequence, (6.2-10) Hence the output image matrix P can be produced by sequential row and column operations. In many image processing applications, the linear transformations operator T is highly structured, and computational simplifications are possible. Special cases of interest are listed below and illustrated in Figure 6.2-1 for the case in which the input and output images are of the same dimension, . 1. Column processing of F: (6.2-11) where is the transformation matrix for the jth column. FIGURE 6.2-1. Structure of linear operator matrices. T R T C T mn T R mn,()T C = PT C F T R mn,()v n u m T n 1= N ∑ m 1= M ∑ T C FT R T == MN= T diag T C1 T C2 … T CN ,,,[]= T Cj [...]... covariance function of the image array is K ( n 1, n 2 ; n 3 , n 4) = E { [ F ( n 1, n 2 ) – E { F ( n 1, n 2 ) } ] [ F∗ ( n 3, n 4 ) – E { F∗ ( n 3, n 4 ) } ] } (6.3 -4) Finally, the variance function of the image array is obtained directly from the covariance function as 2 σ ( n 1, n 2 ) = K ( n 1, n 2 ; n 1, n 2 ) (6.3-5) 1 54 DISCRETE IMAGE MATHEMATICAL CHARACTERIZATION If the image array is represented... ; m 3, m 4 ) = N2 N1 N2 ∑ ∑ ∑ ∑ RF ( n 1, n 2, n 3, n 4 )O ( n 1, n 2 ; m1, m 2 ) n1 = 1 n2 = 1 n 3 = 1 n4 = 1 × O∗ ( n 3, n 3 ; m3, m 4 ) (6.5-3) where R F ( n 1, n 2 ; n 3 , n 4 ) represents the correlation function of the input image array In a similar manner, the covariance function of the output image is found to be N1 KP ( m 1, m 2 ; m 3, m 4 ) = N2 N1 N2 ∑ ∑ ∑ ∑ KF ( n 1, n 2, n 3, n 4 )O ( n... 1 2 3 4 5 6 7 W K Pratt, “Vector Formulation of Two Dimensional Signal Processing Operations,” Computer Graphics and Image Processing, 4, 1, March 1975, 1– 24 J O Eklundh, “A Fast Computer Method for Matrix Transposing,” IEEE Trans Computers, C-21, 7, July 1972, 801–803 R E Twogood and M P Ekstrom, “An Extension of Eklundh's Matrix Transposition Algorithm and Its Applications in Digital Image Processing, ”... border region FINITE-AREA SUPERPOSITION AND CONVOLUTION 0. 040 0.080 0.120 0.160 0.200 0.200 0.200 0.080 0.160 0. 240 0.320 0 .40 0 0 .40 0 0 .40 0 171 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.200 0 .40 0 0.600 0.800 1.000 1.000 1.000 0.000 0.000 0.000 0.000 0.200 0 .40 0 0.600 0.800 1.000 1.000 1.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 0.200 0 .40 0 0.600 0.800 1.000 1.000 1.000 0.000 0.000 1.000 1.000... r, θ ) = QT (6 .4- 9) IMAGE PROBABILITY DENSITY MODELS 161 FIGURE 6 .4- 1 Histograms of the red, green and blue components of the smpte_girl _linear color image where NS ( j 1, j2 ) denotes the number of pixel pairs for which F ( n1, n 2 ) = r j1 and F ( n 3, n 4 ) = r j The factor QT in the denominator of Eq 6 .4- 9 represents the total 2 number of pixels lying in an image region for which the... (6.5-1b) n1 = 1 n2 = 1 The correlation function of the output image array is RP ( m 1, m 2 ; m 3 , m4 ) = E { P ( m 1, m 2 )P∗ ( m 3, m 4 ) } (6.5-2a) or in expanded form ⎧ ⎪ R P ( m 1, m2 ; m 3 , m 4 ) = E ⎨ ⎪ ⎩ N1 N2 ∑ ∑ F ( n 1, n 2 )O ( n 1, n 2 ; m 1, m 2 ) × n 1 = 1 n 2= 1 N1 ∑ ⎫ ⎪ F∗ ( n 3, n 4 )O∗ ( n 3, n 3 ; m 3, m 4 ) ⎬ ∑ ⎪ n4 = 1 ⎭ N2 n3 = 1 (6.5-2b) After multiplication of the series,... General N4 Column processing N3 Row processing N3 Row and column processing Separable row and column processing matrix form 2N3– N2 2N3 153 IMAGE STATISTICAL CHARACTERIZATION line With this technique, however, it is necessary to transpose the result of the column transforms in order to perform the row transforms References 2 and 3 describe algorithms for line storage matrix transposition 6.3 IMAGE STATISTICAL... 3, n 4 )O ( n 1, n 2 ; m 1, m 2 ) n1 = 1 n2 = 1 n 3 = 1 n4 = 1 × O∗ ( n 3, n 3 ; m3, m 4 ) (6.5 -4) If the input and output image arrays are expressed in vector form, the formulation of the moments of the transformed image becomes much more compact The mean of the output vector p is η p = E { p } = E { Tf } = TE { f } = Tη f (6.5-5) 1 64 DISCRETE IMAGE MATHEMATICAL CHARACTERIZATION and the correlation... discrete image function is a matrix of the form E { F } = [ E { F ( n 1, n 2 ) } ] (6.3-1) If the image array is written as a column-scanned vector, the mean of the image vector is N2 ηf = E { f } = ∑ N n E { F }v n (6.3-2) n= 1 The correlation function of the image array is given by R ( n 1, n 2 ; n 3 , n 4 ) = E { F ( n 1, n 2 )F∗ ( n 3, n 4 ) } (6.3-3) where the ni represent points of the image array... DISCRETE IMAGE MATHEMATICAL CHARACTERIZATION 2 Identical column processing of F: T = diag [ T C, T C, …, T C ] = T C ⊗ I N (6.2-12) 3 Row processing of F: T mn = diag [ T R1 ( m, n ), T R2 ( m, n ), …, T RN ( m, n ) ] (6.2-13) where T Rj is the transformation matrix for the jth row 4 Identical row processing of F: T mn = diag [ T R ( m, n ), T R ( m, n ), …, T R ( m, n ) ] (6.2-14a) T = IN ⊗ TR (6.2-14b) . 1= N 2 ∑ == Rn 1 n 2 n 3 n 4 ,;,()EFn 1 n 2 ,()F ∗ n 3 n 4 ,(){}= n i Kn 1 n 2 n 3 n 4 ,;,()EFn 1 n 2 ,()EFn 1 n 2 ,(){}–[]F ∗ n 3 n 4 ,()EF ∗ n 3 n 4 ,(){}–[]{}= σ 2 n 1 n 2 ,()Kn 1 n 2 n 1 n 2 ,;,()= 1 54 DISCRETE IMAGE. possible to relate the output image vector p to the input image matrix F by the equation (6.2 -4) Furthermore, from Eq. 6.1 -4, the output image matrix P is related to the input image vector p by (6.2-5) Combining. convolutional processing more efficiently. 147 6 Digital Image Processing: PIKS Scientific Inside, Fourth Edition, by William K. Pratt Copyright © 2007 by John Wiley & Sons, Inc. DISCRETE IMAGE