CS681 Computational Colorimetry

Transcription

1 CS681 Computational Colorimetry Min H. Kim KAIST School of Computing Energy on a Lambertian surface An ideal surface that provides uniform diffusion of the incident radiation such that its luminance is the same in the all directions (Q) You have a Lambertian surface that is 100% reflective. If you got a reading of 1.0 cd/sqm, how much illuminance would you have? (A) lux 1

2 E = = = Energy on a Lambertian surface Why π lux (illuminance) on the Lambertian surface from 1.0 cd/m (luminance)? ò ò de p p 0 0 p ( ò ) 1 0 ( Q ) L cosqdw Figure 1 ò ò ( Q ) Lcosq sinq dqdf Figure Figure. Unit surface (1) azimuth p æ ö = df ç sin cos 0 ò q q dq L 0 è ø 1 d sinq = plò x dx ( Q x = sin q, dx = cosqdq) Q = cosq 0 dq éx ù = p L ê ú ë û = p L E F \ L = = p Ap () zenith r sinq rsinq df q r dq Figure 1. Cosine law df r dq q q L Lcosθ lengthof thearc = radius angle rsinq df r dq Unit surface ( r sinqdf)( rdq) dw = = sinqdqdf r 3 COLOR CALIBRATION & CHARACTERIZATION 4

3 What is Calibration? Calibration is the setting up of a device or process so that it gives repeatable data. [Johnson, 1996] Using calibration and characterization: Determine the state of medium, or alter the medium so as to be in the desired state (e.g., to conform to standard) Characterize medium If the medium is likely to change since it was characterized, calibrate it before using characterization model with it. What is Characterization? Characterization defines the relationship between the [medium] color space and the CIE system of colorimetry [Johnson, 1996] A characterization model is a model that can predict outputs from inputs and inputs from outputs for a color imaging device or medium A characterization model is only valid when the medium it describes is in the state in which it was when it was characterized. 3

4 What is Characterization? Forward characterization model Given device color space coordinates (e.g., monitor RGB), a forward characterization model can be used for predicting the colorimetry of the stimulus Input: RGB, Output: XYZ Inverse characterization model Given a colorimetric description of a stimulus, an inverse characterization model can be used for predicting what device color space coordinates (e.g., monitor RGB) will result in it Input: XYZ, Output: RGB Color Reproduction Pipeline Original medium encoding Forward device model Inverse device model Original XYZ Original gamut in perception Gamut mapping Reproduced medium encoding Forward device model Inverse device model Reproduced XYZ Reproduced gamut in perception 4

5 Characterization Objectives Colorimetric accuracy Visual acceptability Computational simplicity The minimum number of measurements Analytical invertibility (for output devices) CAMERA CHARACTERIZATION 10 5

6 Additive Color: Displays An additive color model is based on the linear summation of light from separate primaries, assuming it is seen in the dark Displays utilize the additive color model by mixing the primary phosphors of red, green, and blue light Intermediate colors are generated by modulating the intensity of the light output from RGB channels in varying proportions (signals 0 55) G W B R K Polynomial Regression Model Solving a system of linear equations: general form a 1,1 x 1 + a 1, x a 1,n x n = b 1 a,1 x 1 + a, x a,n x n = b a 3,1 x 1 + a 3, x a 3,n x n = b 3! a m,1 x 1 + a m, x a m,n x n = b n e.g., a m,1 = a m 0,a m, = a m 1,a m,3 = a m,!,a m,n = a m n b n is the expected value of a dependent variable of XYZ. a m,n is the n-th polynomial basis-function of an independent variable of RGB (m: measurement). x n is the unknown parameters that describe the camera response function 6

7 Polynomial Regression Model Can be written in matrix form: A = So that a 1,1 a 1,! a 1,n a,1 a,! a,n " " " a m,1 a m,! a m,n Ax = b, x = Min H. Kim x is (KAIST) the unknown parameters that CS681: describe Computational the camera Colorimetryresponse function x 1 x! x n ˆx = argmin x, b = b is the expected value of a dependent variable of XYZ. A is the polynomial basis-functions of an independent variable of RGB. b 1 b! b n. Ax b Polynomial Regression Model Such that the row of A is longer than its width (m>n) We could solve x Ax = b A T Ax = A T b x = (A T A) 1 A T b Pseudo inverse Here x includes coefficients for the polynomial expansion of the training set data 7

8 Camera Characterization Measure spectrum and compute XYZ of a training set data à b Measure camera response RGB à A Linear blending of RGB to XYZ à x x can be used for prediction of a new color via least-squares solving Camera Characterization Camera characterization yields Relative scale of irradiance reading on detector Often called radiometric calibration Since it is a smooth curve and monotonic (each color is independent) we could estimate this characterization model without radiometric measurement à high dynamic range (HDR) imaging (we will learn later) 8

9 Camera Characterization In general, higher-order models can improve the modeling accuracy However, higher-order models end up with the over-fitting problem Lower-order polynomial will be a better option PRINTER CHARACTERIZATION 18 9

10 Subtractive Color: Printers Produce desired spectrum by subtracting from white light Photographic media (slides, prints) work this way Assuming the light and medium base is white M K C Y W Main Approaches to Characterization Physical models e.g. Neugebauer, Kubelka-Munk, Murray-Davies Numerical models e.g. polynomial regression, neural network Table look-up (with/without linear interpolation) In practice, real-world implementations often use a combination of these approaches 10

11 Neugebauer Model 1 Neugebauer Model Computational model of subtractive color printing systems (CMY) Predict the color produced by a combination of halftones printed in cyan, magenta, and yellow inks. 11

12 Neugebauer Model Halftone dots and their possible overlaps give rise to eight possible colors, assuming local randomness in the dot structure A is area of halftone, R is reflectance R = A w R w + A c R c + A m R m + A y R y + A b R b + A g R g + A r R r + A k R k w i = [(1 c)(1 m)(1 y), c(1 m)(1 y), m(1 c)(1 y), y(1 c)(1 m), cm(1 y), cy(1 m), my(1 c), cmy] 8 X cmy = w i X i, Y cmy = w i Y i, Z cmy = w i Z i, i=1 Three unknowns and three equations à solvable! 8 i=1 8 i=1 Kubelka-Munk Model Formulated, based on two light channels traveling in opposite directions for modeling translucent and opaque media Here the light is absorbed and scattered in only two directions, up and down. A background is presented at the bottom of the medium to provide the upward light reflection. 1

13 Kubelka-Munk Model R = 1+ K S 1+ K S 1 1, K(λ) S(λ) = (1 R(λ)) R(λ) R is the reflection factor for a sample of infinite thickness K is the absorption coefficient, S is the scattering coefficient Coefficients K and S are determined empirically. The ratio of these coefficients is commonly used. Why CMY + K in Printers? The additivity of C+M+Y fails in dark neutrals Pure black doesn t happen by C+M+Y Gray Component Replacement (GCR) K is the amount of black; f is a function of the Ink Level minimum colorant amount 5 K = f(min( C, M, Y)) Cyan Magenta Yellow 13

14 Gray Component Replacement D rcmyk D gcmyk D bcmyk = D rcmy D gcmy D bcmy + D rcmyk, D gcmyk, D bcmyk are the colorimetric densities of the four-color print D rcmy, D gcmy, D bcmy are the colorimetric densities of the three-color components D rk D gk D bk D rcmy D rk k r D gcmy D gk k g D bcmy D bk k b 3D Color Lookup Table (CLUT) Three steps in processing color lookup tables: 1. Packing (partition). Extraction (find) 3. Interpolation (computation) 14

15 3D Color LUT: Packing 9 3D Color LUT: Packing 3x3x3 LUT Each node stores an input value and a corresponding output value. The input space should be uniformly spaced. In practice, D LUT dimension is usually either n or n +1, where n is some power of This allows computationally simple methods of extraction or interpolation to be used [Kang 1997] 15

16 3D Color LUT: Extraction Given a color P, whose input value is bounded at the lower corner by [c 1 c c 3 ], where c 1:3 are the indices of the entries in the uniform 3D table T of the number of level n: p 1 = T (c 1,c,c 3 ) LUT row of lower corner of bounding cube is given by: p 1 = c 1 n + c n + c 3 3: B : G 1: R p 1 (0) < p(0) p 1 (1) < p(1) p 1 () < p() 31 3D Color LUT: Extraction If the LUT row of lower corner of bounding cube is given by: p 1 = c 1 n + c n + c 3 LUT row numbers of the bounding cube are then given by: p 1 + [0, 1, n, n +1, n,n +1, n + n, n + n +1] 3 16

17 3D Color LUT: Interpolation Bounding cube: tri-linear interpolation or sequential linear interpolation or Bounding tetrahedron: tetrahedral or barycentric interpolation [Kang 1997] 1D Linear Interpolation Linear interpolation (1D): p c (x) = p(x 0 ) + [(x x 0 ) / (x 1 x 0 )][ p(x 1 ) p(x 0 )]. Interpolation error: e = p(x) p c (x) [Kang 1997] 17

18 D Linear Interpolation Bilinear interpolation (D): p 0 (x) = p 00 + [(x x 0 ) / (x 1 x 0 )]( p 10 p 00 ). p 1 (x) = p 01 + [(x x 0 ) / (x 1 x 0 )]( p 11 p 01 ). p(x, y) = p 0 + [( y y 0 ) / ( y 1 y 0 )]( p 1 p 0 ). [Kang 1997] D Linear Interpolation Bilinear interpolation (D): p(x, y) = p 00 + [(x x 0 ) / (x 1 x 0 )]( p 10 p 00 ) + [( y y 0 ) / ( y 1 y 0 )]( p 01 p 00 ) + [(x x 0 ) / (x 1 x 0 )][( y y 0 ) / ( y 1 y 0 )] + ( p 11 p 01 p 10 + p 00 ) [Kang 1997] 18

19 3D Linear Interpolation Trilinear interpolation (3D): p(x, y,z) = c 0 + c 1 Δx + c Δy + c 3 Δz + c 4 ΔxΔy + c 5 ΔxΔz c 6 ΔyΔz + c 7 ΔxΔyΔz p 0 + [0,1,n,n +1,n, n +1,n + n,n + n +1] The dimension of CLUT is (n+1)x(n+1)x(n+1) [Kang 1997] 3D Linear Interpolation where Δx = x x 0 c 1 = ( p 100 p 000 ) / (x 1 x 0 ) Δy = y y 0 c = ( p 010 p 000 ) / ( y 1 y 0 ) Δz = z z 0 c 3 = ( p 001 p 000 ) / (z 1 z 0 ) c 0 = p 000 c 4 = ( p 110 p 010 p p 000 ) / [(x 1 x 0 )( y 1 y 0 )] c 5 = ( p 101 p 001 p p 000 ) / [(x 1 x 0 )(z 1 z 0 )] c 6 = ( p 011 p 001 p p 000 ) / [( y 1 y 0 )(z 1 z 0 )] c 7 = ( p 111 p 011 p 101 p p p p 010 p 000 ) / [(x 1 x 0 )( y 1 y 0 )(z 1 z 0 )]. 19

20 3D Color LUT: Interpolation Prism interpolation Pyramid interpolation Tetrahedral interpolation [Kang 1997] 0