DIGITAL IMAGE PROCESSING 4th phần 3 doc

TRISTIMULUS VALUE TRANSFORMATION 61 3.4. TRISTIMULUS VALUE TRANSFORMATION From Eq. 3.3-7 it is clear that there is no unique set of primaries for matching colors. If the tristimulus values of a color are known for one set of primaries, a simple coor- dinate conversion can be performed to determine the tristimulus values for another set of primaries (16). Let (P 1 ), (P 2 ), (P 3 ) be the original set of primaries with spec- tral energy distributions , , , with the units of a match determined by a white reference (W) with matching values , , . Now, consider a new set of primaries , , with spectral energy distributions , , . Matches are made to a reference white , which may be different than the reference white of the original set of primaries, by matching values , , . Referring to Eq. 3.3-10, an arbitrary color (C) can be matched by the tristimulus values , , with the original set of primaries or by the tristimulus values , , with the new set of primaries, according to the matching matrix relations (3.4-1) The tristimulus value units of the new set of primaries, with respect to the original set of primaries, must now be found. This can be accomplished by determining the color signals of the reference white for the second set of primaries in terms of both sets of primaries. The color signal equations for the reference white become (3.4-2) where . Finally, it is necessary to relate the two sets of primaries by determining the color signals of each of the new primary colors , , in terms of both primary systems. These color signal equations are (3.4-3a) (3.4-3b) (3.4-3c) where P 1 λ()P 2 λ()P 3 λ() A 1 W()A 2 W()A 3 W() P ˜ 1 ()P ˜ 2 ()P ˜ 3 () P ˜ 1 λ() P ˜ 2 λ()P ˜ 3 λ() W ˜ () A ˜ 1 W() A ˜ 2 W()A ˜ 3 W() T 1 C()T 2 C()T 3 C() T ˜ 1 C()T ˜ 2 C() T ˜ 3 C() e C() KA W()t C() K ˜ A ˜ W ˜ ()t ˜ C()== W ˜ e W ˜ () KA W()t W ˜ () K ˜ A ˜ W ˜ ()t ˜ W ˜ ()== T ˜ 1 W ˜ () T ˜ 2 W ˜ () T 3 ˜ W ˜ () 1=== P ˜ 1 ()P ˜ 2 ()P ˜ 3 () e P 1 ˜ () KA W()t P 1 ˜ () K ˜ A ˜ W ˜ ()t ˜ P 1 ˜ ()== e P 2 ˜ () KA W()t P 2 ˜ () K ˜ A ˜ W ˜ ()t ˜ P 2 ˜ ()== e P 3 ˜ () KA W()t P 3 ˜ () K ˜ A ˜ W ˜ ()t ˜ P 3 ˜ ()== t ˜ P ˜ 1 () 1 A 1 W ˜ () 0 0 = t ˜ P ˜ 2 () 0 1 A 2 W ˜ () 0 = t ˜ P ˜ 3 () 0 0 1 A 3 W ˜ () = 62 PHOTOMETRY AND COLORIMETRY Matrix equations 3.4-1 to 3.4-3 may be solved jointly to obtain a relationship between the tristimulus values of the original and new primary system: (3.4-4a) (3.4-4b) (3.4-4c) where denotes the determinant of matrix T. Equations 3.4-4 then may be written in terms of the chromaticity coordinates , , of the new set of pri- maries referenced to the original primary coordinate system. With this revision, (3.4-5) T ˜ 1 C() T 1 C() T 1 P ˜ 2 ()T 1 P ˜ 3 () T 2 C() T 2 P ˜ 2 ()T 2 P ˜ 3 () T 3 C() T 3 P ˜ 2 ()T 3 P ˜ 3 () T 1 W ˜ ()T 1 P ˜ 2 ()T 1 P ˜ 3 () T 2 W ˜ ()T 2 P ˜ 2 ()T 2 P ˜ 3 () T 3 W ˜ () T 3 P ˜ 2 () T 3 P ˜ 3 () = T ˜ 2 C() T 1 P ˜ 1 ()T 1 C() T 1 P ˜ 3 () T 2 P ˜ 1 ()T 2 C() T 2 P ˜ 3 () T 3 P ˜ 1 ()T 3 C() T 3 P ˜ 3 () T 1 P ˜ 1 ()T 1 W ˜ ()T 1 P ˜ 3 () T 2 P ˜ 1 ()T 2 W ˜ ()T 2 P ˜ 3 () T 3 P ˜ 1 ()T 3 W ˜ ()T 3 P ˜ 3 () = T ˜ 3 C() T 1 P ˜ 1 ()T 1 P ˜ 2 ()T 1 C() T 2 P ˜ 1 ()T 2 P ˜ 2 ()T 2 C() T 3 P ˜ 1 ()T 3 P ˜ 2 ()T 3 C() T 1 P ˜ 1 ()T 1 P ˜ 2 ()T 1 W ˜ () T 2 P ˜ 1 ()T 2 P ˜ 2 ()T 2 W ˜ () T 3 P ˜ 1 ()T 3 P ˜ 2 ()T 3 W ˜ () = T t i P ˜ 1 ()t i P ˜ 2 ()t i P ˜ 3 () T ˜ 1 C() T ˜ 2 C() T ˜ 3 C() m 11 m 12 m 13 m 21 m 22 m 31 m 31 m 32 m 33 T 1 C() T 2 C() T 3 C() = COLOR SPACES 63 where and Thus, if the tristimulus values are known for a given set of primaries, conversion to another set of primaries merely entails a simple linear transformation of coordinates. 3.5. COLOR SPACES It has been shown that a color (C) can be matched by its tristimulus values , , for a given set of primaries. Alternatively, the color may be specified by its chromaticity values , and its luminance Y(C). Appendix 2 presents formulas for color coordinate conversion between tristimulus values and chromatic- ity coordinates for various representational combinations. A third approach in speci- fying a color is to represent the color by a linear or nonlinear invertible function of its tristimulus or chromaticity values. m ij Δ ij Δ i = Δ 1 T 1 W ˜ ()Δ 11 T 2 W ˜ ()Δ 12 T 3 W ˜ ()Δ 13 ++= Δ 2 T 1 W ˜ ()Δ 21 T 2 W ˜ ()Δ 22 T 3 W ˜ ()Δ 23 ++= Δ 3 T 1 W ˜ ()Δ 31 T 2 W ˜ ()Δ 32 T 3 W ˜ ()Δ 33 ++= Δ 11 t 2 P ˜ 2 ()t 3 P ˜ 3 ()t 3 P ˜ 2 ()t 2 P ˜ 3 ()–= Δ 12 t 3 P ˜ 2 ()t 1 P ˜ 3 ()t 1 P ˜ 2 ()t 3 P ˜ 3 ()–= Δ 13 t 1 P ˜ 2 ()t 2 P ˜ 3 ()t 2 P ˜ 2 ()t 1 P ˜ 3 ()–= Δ 21 t 3 P ˜ 1 ()t 2 P ˜ 3 ()t 2 P ˜ 1 ()t 3 P ˜ 3 ()–= Δ 22 t 1 P ˜ 1 ()t 3 P ˜ 3 ()t 3 P ˜ 1 ()t 1 P ˜ 3 ()–= Δ 23 t 2 P ˜ 1 ()t 1 P ˜ 3 ()t 1 P ˜ 1 ()t 2 P ˜ 3 ()–= Δ 31 t 2 P ˜ 1 ()t 3 P ˜ 2 ()t 3 P ˜ 1 ()t 2 P ˜ 2 ()–= Δ 32 t 3 P ˜ 1 ()t 1 P ˜ 2 ()t 1 P ˜ 1 ()t 3 P ˜ 2 ()–= Δ 33 t 1 P ˜ 1 ()t 2 P ˜ 2 ()t 2 P ˜ 1 ()t 1 P ˜ 2 ()–= T 1 C() T 2 C()T 3 C() t 1 C()t 2 C() 64 PHOTOMETRY AND COLORIMETRY In this section, several standard and nonstandard color spaces for the representa- tion of color images are described. They are categorized as colorimetric, subtractive, video or nonstandard. Figure 3.5-1 illustrates the relationship between these color spaces. The figure also lists several example color spaces. Natural color images, as opposed to computer-generated images, usually origi- nate from a color scanner or a color video camera. These devices incorporate three sensors that are spectrally sensitive to the red, green and blue portions of the light spectrum. The color sensors typically generate red, green and blue color signals that are linearly proportional to the amount of red, green and blue light detected by each sensor. These signals are linearly proportional to the tristimulus values of a color at each pixel. As indicated in Figure 3.5-1, linear RGB images are the basis for the gen- eration of the various color space image representations. 3.5.1. Colorimetric Color Spaces The class of colorimetric color spaces includes all linear RGB images and the stan- dard colorimetric images derived from them by linear and nonlinear intercomponent transformations. FIGURE 3.5-1. Relationship of color spaces. nonstandard colorimetric linear subtractive CMY/CMYK colorimetric nonlinear video gamma RGB colorimetric linear RGB video gamma luma/chroma YCC linear intercomponent transformation nonlinear intercomponent transformation linear intercomponent transformation linear and nonlinear intercomponent transformation linear point transformation nonlinear intercomponent transformation nonlinear point transformation COLOR SPACES 65 R C G C B C Spectral Primary Color Coordinate System. In 1931, the CIE developed a standard primary reference system with three monochromatic primaries at wave- lengths: red = 700 nm; green = 546.1 nm; blue = 435.8 nm (11). The units of the tris- timulus values are such that the tristimulus values R C , G C , B C are equal when matching an equal-energy white, called Illuminant E, throughout the visible spectrum. The primary system is defined by tristimulus curves of the spectral colors, as shown in Figure 3.5-2. These curves have been obtained indirectly by experimental color- matching experiments performed by a number of observers. The collective color- matching response of these observers has been called the CIE Standard Observer. Figure 3.5-3 is a chromaticity diagram for the CIE spectral coordinate system. FIGURE 3.5-2. Tristimulus values of CIE spectral primaries required to match unit energy throughout the spectrum. Red = 700 nm, green = 546.1 nm and blue = 435.8 nm. FIGURE 3.5-3. Chromaticity diagram for CIE spectral primary system. 66 PHOTOMETRY AND COLORIMETRY R N G N B N NTSC Receiver Primary Color Coordinate System. Commercial televi- sion receivers employ a cathode ray tube with three phosphors that glow in the red, green and blue regions of the visible spectrum. Although the phosphors of commercial television receivers differ from manufacturer to manufacturer, it is com- mon practice to reference them to the National Television Systems Committee (NTSC) receiver phosphor standard (14). The standard observer data for the CIE spec- tral primary system is related to the NTSC primary system by a pair of linear coordi- nate conversions. Figure 3.5-4 is a chromaticity diagram for the NTSC primary system. In this system, the units of the tristimulus values are normalized so that the tristimulus values are equal when matching the Illuminant C white reference. The NTSC phosphors are not pure monochromatic sources of radiation, and hence the gamut of colors producible by the NTSC phosphors is smaller than that available from the spectral primaries. This fact is clearly illustrated by Figure 3.5-3, in which the gamut of NTSC reproducible colors is plotted in the spectral primary chromaticity diagram (11). In modern practice, the NTSC chromaticities are combined with Illuminant D65. R E G E B E EBU Receiver Primary Color Coordinate System. The European Broad- cast Union (EBU) has established a receiver primary system whose chromatici- ties are close in value to the CIE chromaticity coordinates, and the reference white is Illuminant C (17). The EBU chromaticities are also combined with the D65 illuminant. R R G R B R CCIR Receiver Primary Color Coordinate Systems. In 1990, the Inter- national Telecommunications Union (ITU) issued its Recommendation 601, which FIGURE 3.5-4. Chromaticity diagram for NTSC receiver phosphor primary system. COLOR SPACES 67 specified the receiver primaries for standard resolution digital television (18). Also, in 1990, the ITU published its Recommendation 709 for digital high-definition television systems (19). Both standards are popularly referenced as CCIR Rec. 601 and CCIR Rec. 709, abbreviations of the former name of the standards committee, Comité Consultatif International des Radiocommunications. R S G S B S SMPTE Receiver Primary Color Coordinate System. The Society of Motion Picture and Television Engineers (SMPTE) has established a standard receiver primary color coordinate system with primaries that match modern receiver phosphors better than did the older NTSC primary system (20). In this coordinate system, the reference white is Illuminant D65. XYZ Color Coordinate System. In the CIE spectral primary system, the tristimulus values required to achieve a color match are sometimes negative. The CIE has developed a standard artificial primary coordinate system in which all tristimulus values required to match colors are positive (4). These artificial primaries are shown in the CIE primary chromaticity diagram of Figure 3.5-3 (11). The XYZ sys- tem primaries have been chosen so that the Y tristimulus value is equivalent to the luminance of the color to be matched. Figure 3.5-5 is the chromaticity diagram for the CIE XYZ primary system referenced to equal-energy white (4). The linear trans- formations between R C G C B C and XYZ are given by FIGURE 3.5-5. Chromaticity diagram for CIE XYZ primary system. 68 PHOTOMETRY AND COLORIMETRY (3.5-1a) (3.5-1b) The color conversion matrices of Eq. 3.5-1 and those color conversion matrices defined later are quoted to eight decimal places (21,22). In many instances, this quo- tation is to a greater number of places than the original specification. The number of places has been increased to reduce computational errors when concatenating trans- formations between color representations. The color conversion matrix between XYZ and any other linear RGB color space can be computed by the following algorithm. 1. Compute the colorimetric weighting coefficients a(1), a(2), a(3) from (3.5-2a) where x k , y k , z k are the chromaticity coordinates of the RGB primary set. 2. Compute the RGB-to-XYZ conversion matrix. (3.5-2b) The XYZ-to-RGB conversion matrix is, of course, the matrix inverse of . Table 3.5-1 lists the XYZ tristimulus values of several standard illuminants. The XYZ chromaticity coordinates of the standard linear RGB color systems are presented in Table 3.5-2. From Eqs. 3.5-1 and 3.5-2 it is possible to derive a matrix transformation between R C G C B C and any linear colorimetric RGB color space. The book’s CD con- tains a file that lists the transformation matrices (22) between the standard RGB color coordinate systems and XYZ and UVW, defined below. X Y Z 0.49018626 0.30987954 0.19993420 0.17701522 0.81232418 0.01066060 0.00000000 0.01007720 0.98992280 R C G C B C = R C G C B C 2.36353918 0.89582361– 0.46771557– 0.51511248– 1.42643694 0.08867553 0.00524373 0.01452082– 1.00927709 X Y Z = a 1() a 2() a 3() x R x G x B y R y G y B z R z G z B 1– x W y W ⁄ 1 z W y W ⁄ = M 11,()M 12,()M 13,() M 21,()M 22,()M 23,() M 31,()M 32,()M 33,() x R x G x B y R y G y B z R z G z B a 1() 00 0 a 2() 0 00a 3() = M COLOR SPACES 69 UVW Uniform Chromaticity Scale Color Coordinate System. In 1960, the CIE adopted a coordinate system, called the Uniform Chromaticity Scale (UCS), in which, to a good approximation, equal changes in the chromaticity coordinates result in equal, just noticeable changes in the perceived hue and saturation of a color. The V component of the UCS coordinate system represents luminance. The u, v chromaticity coordinates are related to the x, y chromaticity coordinates by the relations (23). TABLE 3.5-1. XYZ Tristimulus Values of Standard Illuminants Illuminant X 0 Y 0 Z 0 A 1.098700 1.000000 0.355900 C 0.980708 1.000000 1.182163 D50 0.964296 1.000000 0.825105 D65 0.950456 1.000000 1.089058 E 1.000000 1.000000 1.000000 TABLE 3.5-2. XYZ Chromaticity Coordinates of Standard Primaries Standard xy z CIE R C 0.640000 0.330000 0.030000 G C 0.300000 0.600000 0.100000 B C 0.150000 0.06000 0.790000 NTSC R N 0.670000 0.330000 0.000000 G N 0.210000 0.710000 0.080000 B N 0.140000 0.080000 0.780000 SMPTE R S 0.630000 0.340000 0.030000 G S 0.310000 0.595000 0.095000 B S 0.155000 0.070000 0.775000 EBU R E 0.640000 0.330000 0.030000 G E 0.290000 0.60000 0.110000 B E 0.150000 0.060000 0.790000 CCIR R R 0.640000 0.330000 0.030000 G R 0.30000 0.600000 0.100000 B R 0.150000 0.060000 0.790000 70 PHOTOMETRY AND COLORIMETRY (3.5-3a) (3.5-3b) (3.5-3c) (3.5-3d) Figure 3.5-6 is a UCS chromaticity diagram. The tristimulus values of the uniform chromaticity scale coordinate system UVW are related to the tristimulus values of the spectral coordinate primary system by (3.5-4a) (3.5-4b) FIGURE 3.5-6. Chromaticity diagram for CIE uniform chromaticity scale primary system. u 4x 2x–12y 3++ = v 6y 2x–12y 3++ = x 3u 2u 8v–4– = y 2v 2u 8v–4– = U V W 0.32679084 0.20658636 0.13328947 0.17701522 0.81232418 0.01066060 0.02042971 1.06858510 0.41098519 R C G C B C = R C G C B C 2.84373542 0.50732308 0.93543113– 0.63965541– 1.16041034 0.17735107 1.52178123 3.04235208– 2.01855417 U V W = [...]... 13 = m 23 m 33 K3 (3. 5-22b) where the transformation matrix with general term m ij composed of the eigenvectors of the RGB covariance matrix with general term u ij The transformation matrix satisfies the relation m 11 m 12 m 13 u 11 u 12 u 13 m 11 m 21 m 21 m 22 m 23 u 12 u 22 u 23 m 12 m 22 m 32 m 31 m 32 m 33 u 13 u 23 u 33 m 13 m 23 λ1 m 31 m 33 = 0 0 0 λ2 0 0 0 3 (3. 5- 23) where λ 1 , λ 2 , λ 3. .. Yo ⎛ - ⎞ ⎝ 25 ⎠ (3. 5-9a) 3 (3. 5-9b) 12 – 3u′ – 20v′ Z = Y -4v′ (3. 5-9c) u∗ u′ = - + u′ o 13L∗ (3. 5-9d) v∗ v' = - + u′ o 13L∗ (3. 5-9e) where Figure 3. 5-7 shows the linear RGB components of an NTSC receiver primary color image This color image is printed in the color insert If printed properly, the color image and its monochromatic component images will appear to be... became a CIE standard in 1976 It is defined as Y ⎞1 ⁄ 3 ⎧ ⎛ – 16 L∗ = ⎨ 25 ⎝ 100 - ⎠ Yo ⎩ Y for - ≥ 0.008856 (3. 5-8a) Y L∗ = 9 03. 3 Yo Y for - < 0.008856 (3. 5-8b) Yo Yo u∗ = 13L∗ ( u′ – u′ ) o (3. 5-8c) v∗ = 13L∗ ( v′ – v′ ) o (3. 5-8d) 4X u′ = -X + 15Y + 3Z (3. 5-8e) 9Y v′ = X + 15Y + 3Z (3. 5-8f) where COLOR SPACES 73 and u′ and v′ are obtained by substitution of the... 980–992 33 D B Judd, “Standard Response Functions for Protanopic and Deuteranopic Vision,” J Optical Society of America, 35 , 3, March 1945, 199–221 PART 2 DIGITAL IMAGE CHARACTERIZATION Digital image processing is based on the conversion of a continuous image field to equivalent digital form This part of the book considers the image sampling and quantization processes that perform the analog image to digital. .. PHOTOMETRY AND COLORIMETRY (a) Y, 0.000 to 0.994 (b) l, −0.276 to 0 .34 7 (c) Q, = 0.147 to 0.169 FIGURE 3. 5-14 YIQ components of the gamma corrected dolls_gamma color image The I and Q signals are related to the U and V signals by a simple rotation of coordinates in color space: I = – U sin 33 ° + V cos 33 ° Q = U cos 33 ° + V sin 33 ° (3. 5-17a) (3. 5-17b) It should be noted that the U and V components of the... specified in Table 3. 5 -3 for the COLOR SPACES 79 general case and for conversion to the SMPTE, CCIR and CIE lightness components Figure 3. 5-12 is a plot of the gamma correction curve for the CCIR Rec 709 primaries TABLE 3. 5 -3 Gamma Correction Constants General SMPTE CCIR CIE L* c1 1.00 1.1115 1.099 116.0 c2 0.45 0.45 0.45 c3 0.00 −0.1115 −0.099 c4 0.00 4.0 4.5 b 0.00 0.0228 0.018 0 .33 33 −16.0 9 03. 3 0.008856... 3 are the eigenvalues of the covariance matrix and 2 u 11 = E { ( R – R ) } 2 u 22 = E { ( G – G ) } 2 (3. 5-24a) (3. 5-24b) u 33 = E { ( B – B ) } (3. 5-24c) u 12 = E { ( R – R ) ( G – G ) } (3. 5-24d) u 13 = E { ( R – R ) ( B – B ) } (3. 5-24e) u 23 = E { ( G – G ) ( B – B ) } (3. 5-24f) In Eq 3. 5- 23, E { · } is the expectation operator and the overbar denotes the mean value of a random variable Retinal... FIGURE 3. 5-15 IHS components of the dolls_gamma color image possible to derive an orthogonal coordinate system, in which the components are uncorrelated, by a Karhunen–Loeve (K-L) transformation of the RGB tristimulus values The K-L color transform is defined as K1 K2 K3 m 11 m12 = m 13 R m 21 m 22 m 23 G m 31 m 32 m 33 B (3. 5-22a) 86 PHOTOMETRY AND COLORIMETRY R G m 11 m 21 B m 31 K1 m 12 m 22 m 32 K2... The color coordinates are Y 1 3 L∗ = 116 ⎛ - ⎞ – 16 ⎝Y ⎠ o Y L∗ = 9 03. 3 Y o Y for - > 0.008856 (3. 5-6a) Y for 0.0 ≤ - ≤ 0.008856 (3. 5-6b) Yo Yo ⎧X⎫ ⎧Y⎫ a∗ = 500 f ⎨ - ⎬ – f ⎨ - ⎬ ⎩ X o ⎭ ⎩ Yo ⎭ (3. 5-6c) ⎧X⎫ ⎧Z⎫ b∗ = 200 f ⎨ - ⎬ – f ⎨ - ⎬ ⎩ Xo ⎭ ⎩ Zo ⎭ (3. 5-6d) where f(w) = w 1 3 f ( w ) = 7.787 ( w ) + 0. 137 9 for w > 0.008856 (3. 5-6e) for 0.0 ≤ w ≤ 0.008856 (3. 5-6f) 72 PHOTOMETRY AND COLORIMETRY... 0.168 736 00 – 0 .33 126400 0.50000000 ˜ RS ˜ G – 0.08 131 200 ˜ BS 1.40168676 Y Y 0.29900000 Cb Cr ˜ RS ˜ G S ˜ BS 83 0.50000000 – 0.4186680 1.00000000 – 0.0009264 = 1.00000000 – 0 .34 369 538 – 0.71416904 1.00000000 1.77216042 0.00099022 S Cb (3. 5-18a) (3. 5-18b) Cr where the tilde denotes that the component has been gamma corrected Photo YCC Color Coordinate System Eastman Kodak company has developed an image . 0.20658 636 0. 133 28947 0.17701522 0.81 232 418 0.01066060 0.02042971 1.06858510 0.41098519 R C G C B C = R C G C B C 2.8 437 3542 0.50 732 308 0. 935 431 13 0. 639 65541– 1.16041 034 0.17 735 107 1.521781 23 3.04 235 208–. = Δ 1 T 1 W ˜ ()Δ 11 T 2 W ˜ ()Δ 12 T 3 W ˜ ()Δ 13 ++= Δ 2 T 1 W ˜ ()Δ 21 T 2 W ˜ ()Δ 22 T 3 W ˜ ()Δ 23 ++= Δ 3 T 1 W ˜ ()Δ 31 T 2 W ˜ ()Δ 32 T 3 W ˜ ()Δ 33 ++= Δ 11 t 2 P ˜ 2 ()t 3 P ˜ 3 ()t 3 P ˜ 2 ()t 2 P ˜ 3 ()–= Δ 12 t 3 P ˜ 2 ()t 1 P ˜ 3 ()t 1 P ˜ 2 ()t 3 P ˜ 3 ()–= Δ 13 t 1 P ˜ 2 ()t 2 P ˜ 3 ()t 2 P ˜ 2 ()t 1 P ˜ 3 ()–= Δ 21 t 3 P ˜ 1 ()t 2 P ˜ 3 ()t 2 P ˜ 1 ()t 3 P ˜ 3 ()–= Δ 22 t 1 P ˜ 1 ()t 3 P ˜ 3 ()t 3 P ˜ 1 ()t 1 P ˜ 3 ()–= Δ 23 t 2 P ˜ 1 ()t 1 P ˜ 3 ()t 1 P ˜ 1 ()t 2 P ˜ 3 ()–= Δ 31 t 2 P ˜ 1 ()t 3 P ˜ 2 ()t 3 P ˜ 1 ()t 2 P ˜ 2 ()–= Δ 32 t 3 P ˜ 1 ()t 1 P ˜ 2 ()t 1 P ˜ 1 ()t 3 P ˜ 2 ()–= Δ 33 t 1 P ˜ 1 ()t 2 P ˜ 2 ()t 2 P ˜ 1 ()t 1 P ˜ 2 ()–= T 1 C() T 2 C()T 3 C() t 1 C()t 2 C() 64. = Δ 1 T 1 W ˜ ()Δ 11 T 2 W ˜ ()Δ 12 T 3 W ˜ ()Δ 13 ++= Δ 2 T 1 W ˜ ()Δ 21 T 2 W ˜ ()Δ 22 T 3 W ˜ ()Δ 23 ++= Δ 3 T 1 W ˜ ()Δ 31 T 2 W ˜ ()Δ 32 T 3 W ˜ ()Δ 33 ++= Δ 11 t 2 P ˜ 2 ()t 3 P ˜ 3 ()t 3 P ˜ 2 ()t 2 P ˜ 3 ()–= Δ 12 t 3 P ˜ 2 ()t 1 P ˜ 3 ()t 1 P ˜ 2 ()t 3 P ˜ 3 ()–= Δ 13 t 1 P ˜ 2 ()t 2 P ˜ 3 ()t 2 P ˜ 2 ()t 1 P ˜ 3 ()–= Δ 21 t 3 P ˜ 1 ()t 2 P ˜ 3 ()t 2 P ˜ 1 ()t 3 P ˜ 3 ()–= Δ 22 t 1 P ˜ 1 ()t 3 P ˜ 3 ()t 3 P ˜ 1 ()t 1 P ˜ 3 ()–= Δ 23 t 2 P ˜ 1 ()t 1 P ˜ 3 ()t 1 P ˜ 1 ()t 2 P ˜ 3 ()–= Δ 31 t 2 P ˜ 1 ()t 3 P ˜ 2 ()t 3 P ˜ 1 ()t 2 P ˜ 2 ()–= Δ 32 t 3 P ˜ 1 ()t 1 P ˜ 2 ()t 1 P ˜ 1 ()t 3 P ˜ 2 ()–= Δ 33 t 1 P ˜ 1 ()t 2 P ˜ 2 ()t 2 P ˜ 1 ()t 1 P ˜ 2 ()–= T 1 C() T 2 C()T 3 C() t 1 C()t 2 C() 64