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The Mathematics of Color Calibration

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The Mathematics of Color Calibration The Mathematics of Color Calibration M. J. Vrhel H. J. Trussell Color Savvy Systems Ltd. Dept. of Electrical and Computer Engineering 305 South Main Street North Carolina State University Springboro, OH 45066 Raleigh, NC 27695-7911 Abstract contain the sampled CIE XYZ color matching func- The mathematical formulation of calibrating color tions and the CIE XYZ value of the spectrum Lr is T image reproduction and recording devices is presented. given by t = A Lr. This formulation provides a foundation for future re- A device independent color space is dened as any search in areas of characterization of devices and dis- space that has a one-to-one mapping onto the CIE play of color images. The procedure outlined in this XYZ color space. Device independent values describe paper should become standard for displaying color im- color for the standard CIE observer. ages for the image processing community. By denition, a device dependent color space cannot have a one-to-one mapping onto the CIE XYZ color 1 Introduction space. In the case of a recording device, the device With the advent of low cost color printers and dependent values describe the response of that partic- scanners, there is increased interest in the image pro- ular device to color. For a reproduction device, the cessing community to develop techniques for enhanc- device dependent values describe only those colors the ing, restoring, and reproducing color images [1]. The device can produce. demonstration of the performance of methods for color The use of device dependent descriptions of color image processing has presented a problem due to an in- presents a problem in the world of networked com- ability to control the nal printing process, misunder- puters and printers. The same RGB vector can result standings with regard to color spaces and what RGB in dierent colors on dierent monitors and printers. really means, and improper or poorly managed scan- Similarly, a specied CMYK value can result in dier- ning to name a few. Problems can be clearly seen ent colors on dierent printers. in comparing the various “unprocessed” color images A solution is to dene images in terms of a CIE of Lena in [1] (a recent special issue on color imag- color space and then transform this data to device de- ing) with the actual original Lena image. A simu- pendent descriptors for the device on which the image lation of the dierences is available at the web site is to be reproduced. This solution requires knowledge F ftp://ftp.eos.ncsu/pub/hjt/prole. The dierence be- of a function device which will provide a mapping tween the images demonstrates the need for a stan- from device dependent control values to a CIE color dard image dened in terms of CIE values with which space. In the case of a printer, it is necessary to deter- F 1 to demonstrate color image processing algorithms, in mine a transformation device (which may or may not F addition to standards for displaying and porting pro- exist). For a monitor, the transformations device and F 1 cessed images. device are both needed since the monitor is used as both a source of image data, and as a display device. 2 Background Modern printers and display devices are limited in We will use a vector space notation in which the vis- the colors they can produce. This limited set of colors ible spectrum is mathematically sampled at N wave- is dened as the gamut of the device. If cie is the lengths. An illuminant spectrum is represented by an range of numerical values in the selected CIE color N N diagonal matrix and the spectral reectance space and print is the numerical range of the device of an object is represented by an N element vector r. control values then the set The radiant spectrum reected from the object with G = { t ∈ |∃c ∈ where F (c)=t } spectral reectance r under the illuminant L can be cie print device expressed as Lr. The columns of the Nx3 matrix A denes the gamut of the color output device. Simi- larly, the complement set 4 Scanners Mathematically, the recording process of a scanner c { ∈ |6∃ ∈ F } G = t cie c print where device(c)=t can be expressed as denes colors outside the device gamut. For colors in T zi = H(M ri) the gamut, there will exist a mapping between the de- vice dependent control values and the CIE XYZ color where the matrix M contains the spectral sensitiv- space. Colors which are in Gc cannot be reproduced ity (including the scanner illuminant) of the three (or and must be gamut-mapped to a color which is within more) bands of the device, ri is the spectral reectance G. The gamut mapping algorithm D is a mapping at spatial point i, H models any nonlinearities in the from cie to G, that is D(t) ∈ G ∀t ∈ cie. scanner (invertible in the range of interest), and zi is F F 1 D the vector of recorded values. The mappings device, device, and make up what is dened as a device prole. These mappings The calibration problem is to determine a continu- describe how to transform between a CIE color space ous mapping Fscan which will transform the recorded and the device control values. The International Color values to a CIE color space, i.e. Commission (ICC) has suggested a standard format t = AT Lr = F (z) for describing a prole[7]. scan ∈ 3 Monitors for all r r, the space of reection values. Look-up-tables, nonlinear and linear models for A monitor is often used to provide softcopy preview F for the printing process and is a common source for scan have been used to calibrate color scanners [3, 4]. user generated images. Monitor calibration is almost In all of these approaches, the rst step is to select a always based on a physical model of the device [5]. A collection of color patches which span the colors of in- typical model is t = H[r0,g0,b0]T where terest. Ideally these colors should not be metameric in terms of the scanner sensitivities or to the standard 0 r r =[(r r0)/(rmax r0)] observer under the illuminant for which the calibration is being produced. This constraint is easily obtained 0 g g =[(g g0)/(gmax g0)] (1) and assures a one-to-one mapping between the scan 0 b values and the device independent values across these b =[(b b0)/(bmax b0)] samples. where t is the CIE value produced by driving the mon- The reectance spectra of these Mq color patches itor with control value d =[r, g, b]T , and the param- { } will be denoted by q k for 1 k Mq . These eters r, g, b, r0, g0, b0, rmax, gmax, bmax, and H patches are measured using a spectrophotometer or a are dened in the prole. colorimeter which will provide the device independent Creating a prole for a monitor involves the deter- values mination of these parameters where rmax, gmax, bmax T are the maximum values of the control values (e.g. {tk = A qk} for 1 k Mq. 255). To determine the parameters, a series of color These values {t } could represent any colorimetric or patches is displayed on the monitor and measured with k device independent values, e.g. CIELAB, CIELUV a colorimeter which will provide pairs of CIE values in which case {t = L(AT q )} where L()isthe {t } and control values {d } k =1, ..., M. k k k k transformation from CIEXYZ to the appropriate color Values for r, g, b, r0, g0, and b0 are determined 0 0 0 space. The patches are also measured with the scan- such that the elements of [r ,g ,b] are linear with re- ner to be calibrated providing {z = H(MT q )} for spect to the elements of XY Z and scaled between the k k 1 k M . range [0,1] (cf. Eq. 1). The matrix H is determined q Mathematically, the calibration problem is: nd a using F transformation scan where 0 X XRmax XGmax XBmax r M 0 Xq Y = YRmax YGmax YBmax b F = arg(min ||F(z ) t ||2) 0 scan F i i Z ZRmax ZGmax ZBmax g i=1 T 2 where [XRmax,YRmax,ZRmax] is the CIE XYZ tris- and ||.|| is the error metric in the CIE color space. timulus value of the red phosphor for control value Other metrics may be used if desired. In practice, it T d =[rmax, 0, 0] . The other phosphors are dened may be necessary and desirable to incorporate con- similarly. straints on Fscan. 5 Printers that all the colors can be reproduced. For detailed Printer calibration is dicult due to the nonlin- gamut mapping experiments, the reader is referred to earity of the printing process, and the wide variety [9, 8]. of methods used for color printing (e.g. lithography, 7 Constraints inkjet, dye sublimation etc.). Because of these di- The formulation for the scanner calibration and culties, printing devices are often calibrated using an the printer calibration are mathematically similar. In LUT and interpolation [2, 3]. many practical cases, there are constraints which the To produce a prole of a printer, a subset of values mappings should satisfy. For example, in the printer spanning the space of allowable control values for the prole there are several factors such as ink limit and printer is rst selected. Denote these device dependent undercolor removal which often come into play. values by c for 1 k M .
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