pigmented colorants: dependence on media and time - Cornell ...

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idj− jdi i 2 Sdx = −2aj i From the Quotient rule, we observe that Setting r = j i then yields 203 j2 + +1 (A.5) i2 j d 2 i j j = −2a + +1 (A.6) Sdx i i and therefore via rearrangement and integration dr Sdx = r2 − 2ar +1 (A.7) dr r2 = S − 2ar +1 dx = Sx (A.8) Since we have assumed the paint is homogeneous, the scattering coefficient S is constant throughout the material and can be brought outside the integral on the right hand side. Our goal is to find the value of the change in r as the thickness varies from zero to some thickness x. At a thickness of zero, the reflectance is simply the reflectance of the substrate R0. At thickness x, the reflectance is some Rx. Hence, we are interested in evaluating the integral on the left-hand side of Equation A.8 over the range R0 to Rx. To simplify the integral, we factor the integral by writing b = √ a 2 − 1 and integrate via partial fractions: Rx R0 dr r2 Rx 1 = − 2ar +1 2b R0 Rx dr 1 − r − (a + b) 2b R0 = 1 2b ln (Rx − a − b)(R0 − a + b) (Rx − a + b)(R0 − a − b) dr r − (a − b) (A.9)
Equation A.8 is now: Rearranging yields: 1 2b ln (Rx − a − b)(R0 − a + b) (Rx − a + b)(R0 − a − b) 204 = Sx (A.10) ln (Rx − a − b)(R0 − a + b) =2Sxb (Rx − a + b)(R0 − a − b) (A.11) (Rx − a − b)(R0 − a + b) e2Sxb =(Rx− a + b)(R0 − a − b) (A.12) Assume that the paint is applied so thickly that the substrate is not visible. Then, x →∞so R0 = 0 and we make the substitution Rx = R∞. Since e ∞ →∞, the left-hand side of A.12 goes to zero. We now have: Solving for the reflectance R∞, we find (R∞ − a + b)(−a − b) = 0 (A.13) R∞ = a − b = 1 a + b Recall that a =1+ K S and b = √ a 2 − 1, hence we get 1 R∞ = a + √ a2 − 1 1 = 1+ K S + 1+ K 2 − 1 S (A.14) (A.15) Equation A.15 represents the solution to the most basic Kubelka-Munk dif- ferential equations as they were originally presented [KM31]. Fishkin describes the evolution of K-M theory through several years of improvements by a series of researchers [Fis83], which we only summarize here.
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PIGMENTED COLORANTS: DEPENDENCE ON
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ABSTRACT We present a physically ba
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Acknowledgements This work would no
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Table of Contents 1 Introduction 1
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List of Tables 2.1 Average pigment
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3.15 Photoreceptor absorption .....
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B.15 Variation of spectral reflecta
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corrode the surfaces of materials.
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Figure 1.1: Detail of the Azor-Sado
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However, since mural artwork is typ
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at a very high quality (albeit limi
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heterogeneous sum of very different
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which linseed oil derives (the most
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oration of a rigidly painted layer
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ing will hit the ground and reflect
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tion of paint tubes in 1841 and the
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Figure 2.8: Natural pigments of ant
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ecome more common recently. The Ame
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A V = O(r2 ) O(r3 ) = 1 O(r) 24 (2.
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Table 2.2: Pigment specific gravity
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provide the underlying color to a p
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though the speed of light in air is
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In general, when light travels from
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materials while lit from behind. Li
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Table 2.4: Index of refraction η f
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to completely congeal. Hence, these
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ond together to form proteins. Exam
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to its original state by any means.
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While it has been used since the 19
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organic binder found in more conven
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3.1.1 Visible Light Spectrum Light
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Figure 3.2: Hue, brightness and sat
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measures how much light is reflecte
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Figure 3.6: Additive color mixing o
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Figure 3.9: The physical amount of
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The vascular tunic lies on the wall
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Figure 3.11: Cross section of the r
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Not all areas have the same sensiti
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identify and distinguish between va
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that range between zero and full in
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spectrum. This insures that the res
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Figure 3.19: Graphical representati
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X x = (X + Y + Z) Y y = (X + Y + Z)
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color into the appropriate RGB colo
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components in the image (the smalle
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haviors of subtractive mixing and i
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of perceptually equal sizes. Unfort
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enabling one to transform between t
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andwidth as converted signals are s
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Figure 3.28: Retinal ganglion recep
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example shown in Figure 3.28. Under
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Figure 3.31: The work of Josef Albe
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sensations for each trial and picke
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Chapter 4 Previous Work There is no
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different location. Subsurface scat
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describes the scattering and absorp
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only accounts for a small percentag
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σ ′ t = σa + σ ′ s and α
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104 Figure 4.4: A model of an alaba
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effects or are too computationally
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108 Figure 4.6: Results using Kubel
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specifying pigments works adequatel
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color representation, it is not fea
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time. A metallic surface patina is
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the system consisted of wetness and
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or darkening enables the museum per
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in front of a displayed work. Felle
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light exposure. This effect is espe
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over time. Perceptually, these hues
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The rate of change of the concentra
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over white grounds) suffer the grea
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protects the deeper remaining color
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light source. This method is used w
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chemical smog and have been identif
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Chapter 5 Preparation & Measurement
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tending transparent pigments used i
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140 Figure 5.1: Spectral reflectanc
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the requirements of the work at han
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144 Figure 5.2: Left: Magnified vie
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Table 5.2: Relevant data regarding
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historically one of the best grades
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created far too strong of a binder,
Page 165 and 166: too brittle and cracking. While wor
Page 167 and 168: 154 Figure 5.4: Reflectance values
Page 169 and 170: 5.1.5 Painting After the layers of
Page 171 and 172: The gray card reflected 18% of the
Page 173 and 174: 160 Figure 5.8: The optical setup f
Page 175 and 176: Note that harmonics reflect at the
Page 177 and 178: Note that the calibration factors f
Page 179 and 180: Chapter 6 Experimental Results 6.1
Page 181 and 182: 168 Figure 6.1: Lapis Lazuli in dif
Page 183 and 184: (such as CIE standards C, D50, orD6
Page 185 and 186: 172 Table 6.2: Perceptual differenc
Page 187 and 188: 174 Figure 6.3: Munsell plots of La
Page 189 and 190: and j has a saturation of red rsat,
Page 191 and 192: 178 Figure 6.4: Variation of spectr
Page 193 and 194: visible spectrum are increasing at
Page 195 and 196: 182 Figure 6.5: Munsell plots of La
Page 197 and 198: trying to match color ca from the p
Page 199 and 200: as it ages. This is in direct contr
Page 201 and 202: teractive viewer. In our applicatio
Page 203 and 204: 190 Figure 7.1: Our simulated canva
Page 205 and 206: where n is the normal at the curren
Page 207 and 208: In summary, our implementation prov
Page 209 and 210: Chapter 8 Conclusion We have studie
Page 211 and 212: inks are beginning to find their wa
Page 213 and 214: Appendix A Derivation of K-M theory
Page 215: ∆i − =(K + S) idx 202 ∆j −
Page 219 and 220: and if the paint is infinitely thic
Page 221 and 222: One concatenates the layers into a
Page 223 and 224: 210 Figure B.1: Cold Glauconite in
Page 225 and 226: Table B.1: Color conversions from t
Page 227 and 228: 214 Figure B.4: Munsell plots of Co
Page 229 and 230: 216 Figure B.5: Burnt Sienna in dif
Page 233 and 234: 220 Figure B.8: Munsell plots of La
Page 235 and 236: 222 Figure B.9: Red Ochre Tint in d
Page 239 and 240: 226 Figure B.12: Munsell plots of C
Page 241 and 242: 228 Figure B.13: Variation of spect
Page 243 and 244: 230 Figure B.15: Variation of spect
Page 245 and 246: Table B.13: Color conversions of Bu
Page 247 and 248: 234 Figure B.18: Munsell plots of C
Page 249 and 250: 236 Figure B.20: Munsell plots of R
Page 251 and 252: References [Alb71] Josef Albers. In
Page 253 and 254: 240 [GLL + 04] Michael Goesele, Hen
Page 255 and 256: 242 [KM31] Paul Kubelka and Franz M
Page 257 and 258: [Rat72] Floyd Ratliff. Contour and
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