Color Printer Characterization in MATLAB

M. J. Vrhel H. J. TrussellColor Savvy Systems Ltd. Dept. of Electrical and Computer Engineering724 Pleasant Valley Drive North Carolina State UniversitySpringboro, OH 45066 Raleigh, NC 27695-7911

AbstractMany researchers are working on various aspects of

color imaging. However, it is apparent from the pa-pers published that it is difficult for many of them toproduce high quality color reproductions. While ourprevious work outlined the steps needed to produce cal-ibrated color images, the problem of creating the pro-grams to implement the process locally seems to be aserious problem. To surmount this final barrier, wehave developed a web site that has the MATLAB pro-grams necessary to characterize printers and producehigh quality images.

1 IntroductionOur previous work [1], provided a high level math-

ematical description of the problem of producing ac-curate color. There appear to be two significant diffi-culties for many researchers: 1) the cost in time andenergy to implement the software solutions describedin [1], and 2) the access to a color measurement de-vice, e.g., colorimeter or spectrophotometer. The goalof this work is to provide MATLAB routines that solvethe first problem. Users must still obtain a measure-ment device if accurate color is desired. Researcherscan use this software to characterize their color print-ers, and hopefully improve the quality of their colorprinting. We will first describe the problem mathe-matically, provide one solution, and end with an ex-ample.

2 The ProblemThe printer characterization problem is to deter-

mine a mapping that takes a CIE colorimetric valueand finds the “best” printer control value (RGB value)to create that color. Similar methods can be used foruse with softcopy. Ideally, the mapping is determinedas follows:

Let F be the mapping from printer control values(RGB values) to CIEL*a*b* values, and assume thatthis mapping is known. The problem, is to determinea mapping Goptim, from CIEL*a*b* to RGB, such that

Goptim = arg(minG

E{||F(G(t)) − t||}) (1)

where t is a CIEL*a*b* value, and the expectationvalue is taken over these colorimetric values. Often,the mapping F is referred to as the forward mappingand G is referred to as the inverse mapping

3 A Solution

The methods for characterization can be dividedinto parametric and non-parametric based approaches.The parametric based methods require the estimationof model parameters using data collected from a col-orimeter. The non-parametric based methods requirethe estimation of look-up-table (LUT) entries or pos-sibly neural net coefficients.

Printer characterization is difficult due to the in-herent nonlinearity of the printing process, and thewide variety of methods used for color printing (e.g.lithography, inkjet, dye sublimation, etc.). Becauseof these difficulties, printing devices are often profiledwith a multi-dimension LUT (MLUT) and interpola-tion [2, 3, 4]. In other words, the continuum of valuesis found by interpolating between points in the MLUT.For particular printing methods, model based methodsare also used for characterizing printers [5, 6].

In this paper, we will describe an MLUT approachto printer characterization. The approach is as follows:

• Print an N×N×N RGB sequence on the printer.The term N × N × N RGB sequence denotes asequence of N3 values that give a good coverage ofthe 3-D printer RGB color space. As an example,a 3×3×3 RGB sequence would be given by (fromleft to right and then top to bottom – black to

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white)

0, 0, 0 0, 0, 127 0, 0, 2550, 127, 0 0, 127, 127 0, 127, 2550, 255, 0 0, 255, 127 0, 255, 255127, 0, 0 127, 0, 127 127, 0, 255127, 127, 0 127, 127, 127 127, 127, 255127, 255, 0 127, 255, 127 127, 255, 255255, 0, 0 255, 0, 127 255, 0, 255255, 127, 0 255, 127, 127 255, 127, 255255, 255, 0 255, 255, 127 255, 255, 255

• Measure each of the N3 printed samples with acolorimeter and obtain the CIEL*a*b* values.

• Approximate the forward mapping F by the con-struction of an N ×N ×N MLUT.

• Build an M ×M ×M MLUT in which each gridpoint is obtained by solving Equation(1), therebyproviding an approximation to the inverse map-ping G.

The most time-consuming part of the process isthe measurement and recording of the sample val-ues. The only computationally difficult problem inthe above steps is the construction of the mapping G(the CIEL*a*b* to RGB mapping). To simplify thedescription of our construction process, we will use thefollowing notation:

• Given an N ×N ×N RGB to CIEL*a*b* MLUTF , let the CIEL*a*b* value in the table at index[i, j, k] be denoted by

FN ([i, j, k]) i, j, k ∈ [0, 1, ..., N − 1] (2)

Note that FN is a 3-vector, representing L∗, a∗

and b∗ values.

• Define a cubic collection of table indices as

C[i,j,k] = {FN ([i, j, k]), FN ([i + 1, j, k]),FN ([i, j + 1, k]), FN ([i, j, k + 1]),FN ([i + 1, j + 1, k]), FN ([i + 1, j, k + 1]),FN ([i, j + 1, k + 1]), FN ([i + 1, j + 1, k + 1])}

• Given a cubic collection of table indices C[i,j,k]

and an RGB value v that is within that cube,the MLUT interpolated CIEL*a*b* value will bedetermined by the function

H(v, Ci,j,k, [i, j, k], N,RANGE(v)) (3)

where RANGE(v) returns the minimum andmaximum values possible for v, and the interpola-tion function can be any desired multi-dimensioninterpolation function (e.g. trilinear, tetrahedral,pyramid, etc) [3].

Using the above notation, the construction of theinverse MLUT table G is performed as follows:

• Take an arbitrary grid point p (a CIEL*a*b*value) of the (M3) points that we wish to fill.The goal is to determine what RGB value shouldbe associated with that CIEL*a*b* grid point.

• Of the N3 CIEL*a*b* values that were measured,find the closest one to p in terms of Euclideandistance (∆Ea∗b∗ distance), i.e. find

[I, J,K]T = arg(mini,j,k

||FN ([i, j, k]T ) − p||2) (4)

where i, j, k ∈ [0, 1, ..., (N − 1)]

• For the indices [I, J,K] determine the eight(or fewer if an edge point) cubes that sur-round the point in the N × N × N nota-tion of the data. Assuming 1 < I, J,K <N − 2, these eight cubes would consist of thecubes C[I−1,J−1,K−1], C[I−1,J−1,K], C[I−1,J,K−1],C[I,J−1,K−1], C[I,J,K−1], C[I,J−1,K], C[I−1,J,K],C[I,J,K]. For ease of notation, denote these eightcubes simply as Cl l = 1, ..., 8.

• For each cube, solve the following optimizationproblem

voptim,l = arg(minv

||p−H(v, Cl, [i, j, k], N,RANGE(v))||2)(5)

using a nonlinear programming method.

• If the RGB vector voptim,l is within the cube Cl1,

then stop the optimizations and fill the grid pointp that we are trying to fill with the RGB valuevoptim,l.

• If the RGB vectors voptim,l are all outside of theirrespective cubes, then the CIEL*a*b* value p isoutside of the printer gamut (in other words theprinter cannot create that color). In this case, thefollowing optimization problems are solved

δe(l) = (minv

||p−H(v, Cl, [i, j, k], N,RANGE(v))||2)

where l = 1, ..., 8 and with the constraint thatthe RGB vector v must remain in the cube Cl.The optimal constrained arguments voutgamut,l

are saved with each computation. Once the δe(l)values are determined and the minimum error

L = arg(minl

δe(l))

1This test is easily done by comparing voptim,l to the RGBgrid points in Cl

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is found, the gamut mapped RGB vectorvoutgamut,L is placed in the grid point p.

Note that many different approaches to gamut map-ping could be used in the above algorithm.

4 The SubroutinesThe MATLAB code to implement the above op-

erations is containedat NCSU FTP site: ftp.eos.ncsu.edu. The directoryis pub/hjt/icip2002. It can be accessed via browserat ftp://ftp.eos.ncsu.edu/pub/hjt/icip2002/ The pri-mary routines are as follows:

ChartImage=MakeChart(N,SquareSize): WhereChartImage is an image of the N × N × N RGB se-quence to be printed and then measured with the col-orimeter. The variable SquareSize sets the dimensionof each square to be measured in terms of number ofpixels. The sampling step along each axis in the 3 di-mensional RGB space is computed as k ∗ 255/(N − 1)where k = 0, 1, ..., N−1. Note that the samples shouldbe measured left to right and top to bottom (black towhite) as the image is created.

F=Buildrgb2lab(labs,N): Where labs is a vector ofthe N3 chart samples that were measured with a col-orimeter, N is the size of the table that you wish togenerate, and F is the RGB to CIEL*a*b* MLUT forthe printer. This function performs no mathematicaloperations on the measured samples. The samples aresimply reordered into a MATLAB multidimensionalarray of size 3 ×N ×N ×N .

G=Buildlab2rgb(F,M,N): Where M is the desiredsize of the table you wish to generate, and G is theCIEL*a*b* MLUT for the printer. This routine solvesthe cost functions in Equations (4) and (5) using tri-linear interpolation for the function H.

printimage=MLUTinterp(labimage,G,M): Wherelabimage is the input image in CIEL*a*b* space andprintimage is the image that should be printed on theprinter. This function uses trilinear interpolation toapply the MLUT to the input image labimage.

Since the above routine uses a CIEL*a*b* image asinput, the user must either have a CIEL*a*b* image orconvert an image to CIEL*a*b*. Because CIEL*a*b*images are currently rare, we include a routine to con-vert an RGB image to CIEL*a*b*. Note that someimage processing software, such as, Photoshop, canconvert files to CIEL*a*b*. To transform an RGBimage to CIEL*a*b*, it is necessary to know what thegiven RGB values mean colorimetrically. Since sRGBis becoming more common [7], we also provide a func-tion that transforms from sRGB to CIEL*a*b*.

labimage=Srgb2lab(sRGBimage): Where sRGBim-age is the input image.

To provide some speed-up , the web site containsC-MEX files for some of the above routines. In thosecases, the routine name has a prefix C added (e.g.CMLUTinterp for MLUTinterp )

5 An ExampleAs an example of using the above routines, we char-

acterized an EPSON 5000 color ink-jet printer. Theprocess was performed as follows:

• A 9 × 9 × 9 RGB chart was created using theMakeChart function. This chart was then printedon the EPSON 5000 printer.

• A Gretag SpectroScan instrument was used to ob-tain the 729 CIEL*a*b* values. Any reflectancecolorimeter can be used, however the characteri-zation is going to be valid only for the illuminantassociated with the white point of the CIEL*a*b* that is used for the recorded values.

• The 729 values were read into MATLAB as thevariable labs.

• The function F=Buildrgb2lab(labs,9) was exe-cuted.

• The function F=Buildlab2rgb(F,31,9) was exe-cuted. Depending upon your processing powerthis function takes a long time at a table size ofM=31. However, this function will need to be runonly once to characterize the printer for as longas it printing is stable. If the printer is subject to“drift”, the characterization will have to be doneas often as necessary.

• An sRGB version of the Lena image (scanned bythe authors and available at the web site) wasread into MATLAB using MATLAB’s commandsRGBimage=imread(’LenaSrgb.tiff’).

• ThesRGB image was converted to CIEL*a* b* usingthe command labimage=CSrgb2lab(sRGBimage).

• The CIEL*a*b* image labimage was converted tothe printer RGB space using the MLUT inter-polation functionprintimage=CMLUTinterp(labimage,G,31).

• The image printimage was printed on the printerproviding a good match to the original Lena im-age.

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Figure 1: sRGB version of Lena

The sRGB version of the Lena image is shown inFigure 1. If the reader is viewing this paper withan sRGB calibrated monitor or printed a copy on ansRGB calibrated printer, then this image representshow the Lena image should appear when properly dis-played. Figure 2 shows the RGB image printimage,which when printed on our Epson 5000 printer willcreate the Lena image as it should properly appear.Note that the two images do not match, which is anindication of the magnitude of error that would occurif the sRGB image was printed directly on the Epson5000 printer. Using the above routines to create andapply the transformation G to the sRGB image en-abled us to obtain the proper image from our printer.

6 ConclusionThe goal of this work was to provide researchers

with a tool to enable accurate printing of color im-ages. It is hoped that researchers will also take theseroutines and improve upon them in terms of their ac-curacy and efficiency.

References[1] M. J. Vrhel and H. J. Trussell, “Color Device Cal-

ibration: A Mathematical Formulation,” IEEETransactions on Image Processing, Vol. 8, No. 12,Dec. 1999.

[2] M. C. Stone, W. B. Cowan, and J. C. Beatty,“Color Gamut Mapping and the Printing of Digi-

Figure 2: Printer RGB version of Lena

tal Color Images,” ACM Transactions on Graph-ics, Vol. 7, No. 4, Oct. 1988.

[3] H. R. Kang, Color Technology for electronicdevices, SPIE Press, Bellingham WA, 1997.

[4] R. Balasubramanian, “Reducing the cost oflookup table based color transformations,” Proc.IS&T/SID Seventh Color Imaging Conference:Color Science, Systems and Applications, pp. 65-68, November, 1999.

[5] M. Xia, E. Saber, G. Sharma, and A. M. Tekalp,“End-to-End Color Printer Calibration by TotalLeast Squares Regression,” IEEE Trans. on Im-age Proc., vol. 8, no. 5, pp. 700-716, May 1999.

[6] R. Balasubramanian, “Colorimetric Modeling ofBinary Color Printers,” Proc. IEEE Intl. Confer-ence on Image Proc.,, Vol. 2, pp. 327-330, Nov.1995.

[7] M. Anderson et. al. , “ Proposal for a StandardDefault Color Space for the Internet,” IS&T/SID4th Color Imaging Conference, pp. 238-246, 1996.

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