Set Theoretic Estimation for Problems in Subtractive Color

Transcription

1 Set Theoretic Estiation for Probles in Subtractive Color Gaurav Shara Digital Iaging Technology Center, Xerox Corporation, MS E, 800 Phillips Rd, Webster, NY Received 25 March 1999; accepted 3 Noveber 1999 Abstract: The use of linear algebra and set theoretic estiation for probles in color science and iaging is reviewed. Through a product-space foralis, the powerful projections onto convex sets (POCS) algorith is extended to subtractive color systes satisfying convex constraints in the density doain. Several convex sets are defined, which are useful in color science and iaging, and projections onto these sets are presented. The usefulness of the new ethods is deonstrated by applying the to three practical probles: (1) odel-based scanner calibration, (2) design of color scanning filters that are color ixture curves, and (3) colorant forulation John Wiley & Sons, Inc. Col Res Appl, 25, , 2000 INTRODUCTION Several researchers have used a vector-space fraework for the description of color atching 1 3 and color systes. 4 7 The vector space fraework allows one to exploit the vast body of atheatical results fro linear algebra in the analysis and design of color systes. 7 The power of the vector space approach is further enhanced when it is cobined with set theoretic estiation, which has proven a powerful technique for solving signal and iage processing probles. 8 The vector space and set theoretic approaches were cobined and successfully applied to several probles in color science and color systes by Trussell 9 and later by other other researchers. 10 One liitation of set-theoretic schees is the lack of globally convergent estiation algoriths for general constraint sets. The ost powerful and useful algoriths are variants of the ethod of successive projections onto convex sets (POCS), 11 which requires constraint sets to be closed and convex. While additive color systes naturally yield constraint sets that are closed and convex in the Correspondence to: Gaurav Shara, Digital Iaging Technical Center, Xerox Corporation MS E, 800 Phillips Rd., Webster NY (e-ail: g.shara@ieee.org) 2000 John Wiley & Sons, Inc. spectral (reflectance/transittance/radiance) doain, this is usually not true of subtractive systes. Consequently, the exaples presented in 9,10 focused on additive systes, and while soe probles in subtractive systes were entioned, coputational ethods for solving those probles were not addressed in detail. Recent research has enlarged the class of probles for which robust set theoretic estiation schees are available. In particular, the generalized product space forulation proposed by Cobettes 12 allows general constraints to be eployed in a POCS fraework, provided each constraint can be ade convex in its own Hilbert space. A nuber of constraint sets arising in subtractive systes are convex in the optical density (logarithic) doain. However, applications typically require the use of these sets in conjunction with other constraint sets that are convex in the spectral doain, which is not possible in the conventional POCS algoriths that assue a single underlying Hilbert space. In this article, a suitable Hilbert space structure is introduced, which yields convex representations for these subtractive systes constraint sets in the spectral doain. The product space forulation entioned above then allows these sets to be cobined effectively with other constraint sets that are convex in the spectral doain under the noral Hilbert space structure in N. The utility of the new algoriths is deonstrated by applying the to three practical probles that are of interest to the color science and iaging counity: odel-based scanner calibration, the design of color scanning filters that are linear cobinations of coloratching-functions (CMFs), and colorant forulation. The rest of this article is organized as follows. The following section provides a very brief overview of the vector space description of color atching and colorietry. The description of color systes in the vector space notation is presented next. Set theoretic estiation and the generalized product space forulation are then described briefly, with particular ephasis on POCS. The convexity of several sets of interest in color science and iaging are then exained. Then an alterate Hilbert space structure is Volue 25, Nuber 5, October

2 introduced over the space of (positive) color spectra, which yields convex representations for several constraint sets in subtractive systes. Specific applications that exploit this Hilbert space (in the product space forulation) to solve probles in subtractive color are presented in the last section. Finally, the appendix describes atheatical details of projection operators that are oitted fro the ain text. VECTOR SPACE DESCRIPTION OF COLORIMETRY The color sensation produced by light incident on the huan eye depends on its power spectral distribution, i.e., the distribution of energy as a function of the wavelength. In air or vacuu, the visible region of the electroagnetic spectru corresponds to the wavelength interval fro n. Physical devices that respond to light energy ay, however, be sensitive over a different region. For coputational purposes, the spectra ay be represented by N- vectors consisting of N saples over the wavelength interval of interest. For ost color spectra, a sapling rate of 10 n provides sufficient accuracy, but a higher sapling rate or alternative approaches ay be required for applications involving fluorescent laps that have sharp spectral peaks The fact that the huan eye has three distinct color sensing cones in the retina that respond in a linear fashion to incident light* fors the basis of the vector space approach to color atching. The responses of the cones to incident light with spectral distribution specified by the N-vector f can be expressed atheatically as c S T f, (1) * Strictly speaking the cones do not respond linearly, but for a fixed state of adaptation, the linear approxiation odels the first stage of the color sensing process fairly accurately. For concise definitions of ters fro atheatical analysis used in this paper, the reader is referred to [17]. where the superscript T denotes the transpose, c [c 1, c 2, c 3 ] T is the vector of cone responses, S is an N 3 atrix whose i th colun s i is the spectral sensitivity of the i th cone. Matheatically, fro Eq. (1) it is apparent that the cone responses are the inner-products of the cone sensitivities and the incident spectru f. Hence, the cone responses can be used to deterine the projection of the spectru onto the 3 space spanned by three sensitivity functions {s i } (i.e., the colun space of S), and vice versa. This space is called the huan visual subspace (HVSS). 4,1,5,18 In noral huan observers, the spectral sensitivities of the three cones are linearly independent, so the HVSS is a 3-diensional subspace of the N-diensional spectral space. While the final perception of color depends on nonlinear processing of the retinal responses in the neural pathways and the brain, to a first order of approxiation, the sensation of color (under siilar conditions of adaptation) ay be specified by the responses of the cones. This is the basis of all colorietry and is iplicitly assued throughout this article. Thus, two spectra f and g atch in color if and only if (iff) S T f S T g. (2) Alternately, one can say that two spectra atch in color iff their projections onto the HVSS are identical. Since the HVSS is only a 3-diensional subspace of the N-diensional spectral space, there are ultiple spectra that atch in color. These are known as etaers. The differences between the spectral sensitivities of people with noral color vision are (relatively) sall. 19,20 [See Ref. 21, pp. 343]. Hence, the color-atching characteristics of color-noral observers can be captured through the definition of a standard HVSS. The HVSS ay be defined by using the cone sensitivity atrix S, or ore generally, any nonsingular transforation of S of the for SQ, where Q is a (possibly unknown) nonsingular 3 3 atrix. The cone responses theselves are difficult to easure/copute directly, but nonsingular transforations of the above type, known as color-atching functions (CMFs), can be readily deterined through color atching experients. (see Refs. 6, 7 for a description of color-atching experients that uses notation and terinology consistent with that used in this article). The CIE (Coission Internationale de l Éclairage) XYZ CMFs for one such set of CMFs that is used as a standard for colorietry. 22 In this article, the atrix of CIE XYZ CMFs is denoted by A. For a given irradiant color spectru f, the 3-vector A T f specifies the color of f in the CIE XYZ space and is referred to as the (CIE XYZ) tristiulus of f. Two spectra f and g atch in color if and only if their tristiuli are equal, i.e., A T f A T g. Also, the HVSS is identical to the colun space of A. DESCRIPTION OF COLOR SYSTEMS IN VECTOR SPACE NOTATION If a reflective nonluinous object with reflectance r is illuinated by an illuinant with spectru l, the spectru of the reflected light is given by Lr, where L is a diagonal atrix with l as the diagonal. The CIE XYZ tristiulus values of the reflective object under the viewing-illuinant l are, therefore, given by t A T Lr A L T r, where A L LA. The process of recording a color iage on a color caera or a scanner can also be described in a anner analogous to the cone response echanis entioned in the last section. For a K channel color recording device, the vector of recorded values can be written as t s M s T r, (3) where r is the N 1 vector of reflectance saples, M s is an N K atrix whose i th colun, i is the spectral sensitivity of the i th channel (including effects of the recording illuinant, filter transittance and the detector sensitivity), and is the K 1 easureent noise vector. In the absence of noise, the process of color recording in (3) 334 COLOR research and application

3 can be shown that (Ref. 23, Chap. 7) the optical density of the i th colorant layer, which is defined as the negation of the natural logarith* of its transittance, is given by d i n i ln u i n i n i d i, (4) FIG. 1. Subtractive color reproduction. ay be interpreted as the projection of the reflectance spectru onto the recording device s visual subspace. Additive color reproduction systes such as CRT displays and projection television produce colors through the additive cobination of priary spectra in varying aplitudes. Thus, the range of spectra producible on an M- priary additive device can be represented as S a P {Px x M,0 x i 1}, where P [p 1, p 2,...p M ] is the atrix of the priary spectra p i at their axiu aplitudes. The corresponding gaut in CIE XYZ space is given by S a T {A T Px x M, 0 x i 1} {T p x x M 0 x i 1}, where T p [t 1, t 2,...t M ] A T P is the atrix of priary tristiuli {t i } M. As an interesting aside, note that the volue of the gaut in CIE XYZ space can be expressed as i C det([t i1 t i2 t i3 ]), where C is the set of all possible three-eleent cobinations fro the index set {1, 2,...M}, det represents the deterinant, and i {i1, i2, i3} is a cobination fro C. Subtractive color reproduction systes produce colors by overlaying layers that absorb (subtract out) light in different regions of the visible spectru. Most subtractive color reproduction systes, are inherently nonlinear and cannot be odeled as easily/accurately as additive systes. Figure 1 illustrates the subtractive principle for a transissive syste. The incident light with spectral distribution l passes through a nuber of layers (three in the figure) containing colorants that absorb light in specific regions of the spectru. The spectru of the light transitted through the three layers is given by g l V u 1 V u 2 V u 3, where u i is the spectral transittance of the i th layer and V represents the ter by ter ultiplication operator for N-vectors. If the colorants are transparent (i.e., do not scatter incident light) and their absorption coefficients are assued to be proportional to their concentration (Bouguer Beer law), it where u i (n i ) is the transittance of the i th colorant layer, n i is the noralized concentration of the i th colorant, which varies between 0 and 1, and d i d i (1) is the density at axiu concentration. The spectru of the transitted 3 light can, therefore, be expressed as l V exp( n i d i ). While the above discussion focused on color spectra in transissive edia, the sae odel is valid for reflective prints on paper eploying transparent colorants, provided that the reflectance of the paper substrate is accounted for along with the spectru of incident light and the densities are doubled to account for the two-way transission through the ediu. 7 If K colorant layers are used in a subtractive syste, the set of reflectance spectra producible in the ediu can be expressed as S D s r r p exp Dc c K,0 c i 1, (5) where r p is the spectral reflectance of the paper substrate, D [d 1, d 2...d K ] is the atrix of colorant densities at axiu concentrations, and c is the vector of noralized colorant concentrations corresponding to the reflectance r. While the assuption of transparent layers with no scattering and no interaction between layers is soeties too siplistic (for instance, for halftone prints and/or pigented colorants), it is also fairly accurate for a nuber of useful cases including typical photographic slides and (to a lesser degree) photographic prints. The Bouguer Beer law is also applicable for hoogeneous isotropic absorption color filters with low and oderate concentrations of the absorbing solute in the filter. Therefore, these can also be represented by odel very siilar to (5). The set of filter spectral transittances that can be synthesized by using K absorbing solutes in a ediu whose thickness ay be varied between t in and t ax is given by (see Ref. 21, pp ): K S u u f 0 i te i in i i ax i ; exp t in t tax, (6) where u 0 is a transittance factor deterined by the reflections fro the surfaces of the filter ediu, e i is the extinction of the i th solute, i denotes the concentration of * Note that conventionally the logarith to the base 10 is used in defining density, but for notational siplicity the natural logarith is used throughout this article. Technically, the Kubelka Munk odel (Ref. 23, Chap. 7) should be used with the scattering ters set to zero. The atheatical details are, however, unaffected by this technicality. Volue 25, Nuber 5, October

4 the i th solute, and i in and i ax are the lower and upper liits on i, respectively. SET THEORETIC ESTIMATION AND THE GENERALIZED PRODUCT SPACE FORMULATION The goal of set theoretic estiation is to obtain a feasible solution satisfying ultiple constraints. Though set theoretic schees have been investigated in ore general settings, 8 the scope of the discussion here is liited to ethods applicable in a Hilbert space (see Ref. 17, pp. 201) setting. A Hilbert space is a linear vector space endowed with the geoetric notions of distance (nor) and orthogonality (inner-product) with the additional desirable property of copleteness. A siple exaple of a Hilbert space* is the space of N-vectors N with the following defined operations and functions: FIG. 2. Exaples of convex and nonconvex sets. Addition and subtraction operators for vectors: coponent-wise addition and subtraction. Scalar ultiplication operator: scaling all the coponents of a vector by the given scalar, i.e., real nuber. Inner-product of two-vectors: the real nuber obtained by ter-wise ultiplication and suation of the vectors N x, y x i y i. The inner-product is indicative of the alignent of the two vectors and is zero when the vectors are orthogonal (perpendicular). Nor or length of a vector: x x, x. The quantity x y represents the distance between the vectors x and y. Given a set of constraints in a Hilbert space, settheoretic estiation tries to deterine a feasible solution that satisfies all the constraints. Matheatically, the settheoretic estiation proble can be stated as follows: Given constraints { i }, Find a* S 0 S i, (7) where {S i } are the constraint sets defined by S i a a satisfies i. (8) * The purpose of this exaple is to aid intuitive understanding. For precise and coplete definitions, the reader is referred to Ref. 17. The abstract notation used above is better understood by using a concrete exaple for the involved ters. For instance, consider that the Hilbert space is the space N defined earlier. A specific constraint (represented above by, say, k ) could then represent the physical constraint that reflectance vectors are nonnegative. The corresponding constraint set S k then denotes the set of vectors in N that satisfy this nonnegativity constraint, i.e., the set of nonnegative N-vectors. The notation a* in (7), then denotes an N-vector that satisfies all the constraints { i }, and, therefore, lies in all the constraint sets {S i }, and, therefore, in the intersection S 0 of these constraint sets. The ethod of successive projections onto convex sets (POCS) and its variants are powerful algoriths for solving set theoretic estiation probles. The POCS algoriths typically require that {S i } be all closed convex sets in the Hilbert space. A set is said to be convex if, for any pair of eleents a and b and any real nuber between 0 and 1, the eleent a (1 )b also lies in the set. Intuitively speaking, a set is convex if, for any two eleents in the set, the line segent joining the eleents lies copletely in the set. Two-diensional exaples of convex and nonconvex sets are shown in Fig. 2, where for each of the nonconvex sets a broken line has been superposed on the figure. The broken line segent joins two points within the set, but contains points outside the set, thereby establishing the nonconvexity. Intuitively, a set is closed if it includes its boundary. The interval of real nubers defined as 0 a 1 is an exaple of a closed set, whereas the set 0 a 1 is an exaple of a nonclosed set (because it does not include 1, which lies at the boundary). If the sets {S i } are all closed convex sets in the Hilbert space, the POCS estiate is deterined as the liit of the sequence { y k }, which is defined recursively by y k 1 P Sn P Sn 1...P S2 P S1 y k..., (9) where y 0 is an arbitrary starting point, and P S ( z) denotes the projection of z onto the constraint set S, defined as P S z arg in x z, x S which can be described in words as the point in S closest to z. The iterative process of successive projections in (9) is 336 COLOR research and application

5 Let i, J i, i, i denote the addition, scalar ultiplication, scalar product, and the nor in the Hilbert space i. Consider the 2-set convex feasibility proble in the product Hilbert space, Find a* S W, (11) (12) with the addition, scalar ultiplication, inner product, and nor defined in the standard coponent-wise fashion for product spaces, 17 where S S 1 S 2... S FIG. 3. Siple POCS exaple. guaranteed to converge* to a point in the intersection S 0, provided that the intersection is nonepty. 11,24 The sae convergence result holds for several variants of the basic ethod that use relaxation/parallelization to speed up convergence. 25,8 A siple exaple of the POCS algorith with three constraint sets is shown in Fig. 3. The region inside the ellipse represents the first constraint set S 1, the region inside the circle represents the second constraint set S 2, and the line segent represents the third constraint set S 3. The intersection of the three sets is the region of the line segent that lies both within the circle and the ellipse. One iteration of the POCS algorith in Eq. (9) starting fro the point y 0 is shown in the figure, where the point y 0 is projected successively onto S 1, S 2, and S 3 by finding, sequentially for each set, the point that is closest to the current point and using that as the current point for the next set. For this choice of initial point, constraint sets, and sequence of projections, the single iteration converges to a point in the intersection of the constraint sets. Typically, however, ultiple POCS iterations are required. The requireent of convexity of the constraint sets is a ajor liitation of ost of the algoriths and restricts their applicability to ore general probles. The recently developed generalized product space foralis 12 helps in partly overcoing this liitation by allowing the use of ultiple Hilbert spaces, such that each constraint i is convex in its Hilbert space i. Let { i } be (not necessarily distinct) Hilbert spaces chosen so that the i th constraint yields a closed convex set in i. Define S i a i a satisfies i,1 i. (10) Then each of the sets S i is closed and convex in its corresponding Hilbert space i. * Actually, only weak convergence 17 is assured for general Hilbert spaces, but for ost cases of practical interest, the Hilbert space is finite diensional and the notions of weak and strong convergence coincide. is the product constraint set, and W a, a,...,a a i is the diagonal subspace in. By definition, eleents of the product Hilbert space are -tuples with the i th coponent as an eleent of i. Likewise, an eleent of the product constraint set S is an -tuple whose i th coponent is an eleent of i and satisfies the constraint S i. The diagonal subspace W is the set of -tuples fro whose coponents are identical. The solution a* to the above 2-set convex feasibility proble lies in W and can be, therefore, written as a* (a*, a*,...,a*); a* i. Since a* also lies in S, it follows that a* S or equivalently a* Si. Hence, the above feasibility proble is equivalent to the convex feasibility proble (7). Since (11) represents a 2-set convex feasibility proble in the product space, it can be solved using the ethod of successive projections onto convex sets (POCS). Fro the definition of the product Hilbert space, it directly follows that the projection of z ( z 1, z 2,...z ) onto S is P S (z) (P S1 ( z 1 ), P S2 ( z 2 ),...P S ( z )), i.e., the coponent-wise projection onto the corresponding convex sets in their respective Hilbert spaces. If the Hilbert spaces { i } are all identically N with the usual inner product and Euclidean nor, the projection of z ( z 1, z 2,...z ) onto W is siply P w (z) ( z a, z a,...z a ), where z a 1/ z i, i.e., the average of the -coponents of z. One iteration of the POCS ethod consisting of a projection onto S followed by a projection onto W is then equivalent to projecting onto the individual constraint sets (in parallel) followed by averaging of the projections, which ay be viewed as a parallel POCS schee. For the general case, when the Hilbert spaces { i } are not identical, the atheatical expression for the projection onto W is given in the appendix. This projection operation can be interpreted as the averaging of the coponents of a vector in using the corresponding distance etric in. Hence, the product space fraework can be interpreted as a generalized parallel POCS algorith, in which at each iteration the estiate is obtained by averaging the projections onto the constraint sets {S i } Volue 25, Nuber 5, October

6 in their respective Hilbert spaces { i }. 12,26 It ay also be noted that even if the constraints are inconsistent, i.e., S 0, the product space forulation deterines an optial estiate in the sense that the estiate iniizes the su of squares of distances fro the estiate to the constraint sets, where the distances are coputed in the respective Hilbert spaces. 26 CONVEXITY OF SETS IN COLOR SYSTEMS The vector space description of colorietry and color systes naturally leads to a nuber of sets that are of interest in color science and iaging. Soe of these sets have been defined earlier. Several other sets are defined here and their convexity is exained, to evaluate their suitability for POCS based set theoretic estiation schees. Consider the abstract set definition, S l B, y, x N B T x y, (13) where y is soe vector in M, B is an N M atrix, denotes the Euclidean vector nor, and 0 is soe nonnegative real constant. It can be readily established that S l (B, y, ) is a closed convex set in the Hilbert space N. Several sets that are of interest in color probles can be shown to be specific instances of the set S l (B, y, ). Exaples include the set of radiant spectra having a specified tristiulus value t (etaers), S l A, t, 0 f A T f t, the set of irradiant spectra whose tristiuli are close to (within a specified distance ) of the target tristiulus t, S l A, t, f A T f t, the set of reflectance spectra that produce a tristiulus close to t under the illuinant l, i.e., S l LA, t, r A T Lr t, the set of reflectance spectra with specified chroaticity values ( x, y) under the illuinant l, T S l LA 1 x x x y 1 y y, t, 0 r 1 x x x y 1 y y A T Lr 0, and the set of reflectance spectra that could give rise to scanner easureent t s, S l M s, t s, r M s T r t s, where is a paraeter that ay be deterined fro the statistics of the noise at an appropriate confidence level. 27 More often the value of is epirically set equal to the noise variance. Fro the fact that the generic set S l (B, y, ) is a closed convex set, it follows that all these spectral sets are closed convex sets in N. The utility of the above sets in set theoretic estiation is further enhanced when they are used in conjunction with other physical constraints that apply to color spectra. Two useful constraints in this category are nonnegativity and soothness. A soothness constraint set can also be expressed in the for of the generic set (13) as S l (H T, 0, ), where H is a high-pass operator and is an upper bound on the high-pass energy (as deterined by H) that ensures a desired degree of soothness. 27 A general for of the nonnegativity constraint on the spectra can be defined by the constraint set S n B, y x N B T x y, (14) where B is an arbitrary N M atrix and y is a M 1 vector of upper bounds. The set S n (B, y) is a closed convex set in N and can be used to express several useful constraints including the set S n ( I, 0) of nonnegative spectra, the set S n (I, 1) of spectral transittances and reflectances that are bounded above by unity, and the set of transittances/reflectance spectra that are concave over a specified spectral region, which is obtained by setting B T equal to the first derivative operator and y The last set was used in Ref. 27 to approxiate a uniodality constraint on filter transittances. For general values of B, the projection onto S n (B, y) cannot be evaluated analytically. However, the set can be eaningfully decoposed into the intersection of several sets as M S n B, y S n b i, y i, (15) where b i is the i th row of B (an N 1 row-vector), and y i is the i th eleent of y. Therefore, the single constraint set S n (B, y) ay be replaced by the M sets {S n (b i, y i )} M. In the design of filters for three channel-color recording devices such as colorieters and scanners, it is useful to have sensitivities that are linear transforations of the CMFs. Hence, the set of device spectral sensitivities that are a linear cobination of the CMFs is a useful set in filterdesign applications. This set is the specific instance of the colun space (or range) of a atrix, which ay be expressed generically as S R B y M y Bx, x N, where B is an M N atrix and y and x are vectors in M and N, respectively. For any arbitrary M, N, and B; S R (B) is a closed convex set in M. The set of linear cobinations of CMFs is then S R (A). Note that instead of requiring an exact color ixture curve, the design requireent ay be relaxed soewhat to allow filter transittances that are fairly close to being color ixture curves. This set can be expressed as S l ((I P A ), 0, ), where P A is the orthogonal projection atrix that projects onto the colun space of A (the HVSS in this case) and is a suitably sall upper bound for the filter transittance energy outside this space. Such a set was also proposed in Ref. 27 for use in scanner spectral characterization. For additive systes, it can be readily seen that both the set of producible spectra S P a and the gaut in tristiulus space S T a are convex closed sets in the Hilbert spaces N and 338 COLOR research and application

7 FIG. 4. Exaple deonstrating nonconvexity of S D s. s * It can be seen that S D is convex if the dyes have nonoverlapping spectral bands, such as dyes that satisfy the block-dye assuption. 6 However, for ost practical cases, S s D is not convex in N. 3, respectively. It can be readily seen that S a P S R (P) and S a T S R (T), hence as a first approxiation these colun spaces ay be used instead of the gaut sets. This approxiation is particularly appropriate and useful in situations where the liits on the priary aplitudes are unknown. The projection onto the generic sets S l (B, y, ), S n (b, y), and S R (B) can be coputed analytically using the ethod of Lagrange ultipliers and the Kuhn Tucker conditions for constrained optiization (see Ref. 28, p ) and these are tabulated in the appendix. The projections onto the specific instances of these sets that are relevant in color applications are readily coputed fro these tabulations. For the sets S a P and S a T, analytic expressions for the projections cannot be deterined and the projections ust be deterined coputationally. For subtractive color systes, the set S s D of producible spectra is generally* not convex in N. A siple exaple deonstrating the nonconvexity of S s D in N is shown in Fig. 4, where a set of three idealized cyan (C), agenta (M), and yellow (Y) dyes are considered. Each dye has a spectrally flat absorptance over its absorption band and zero absorptance outside of this band; and the printing substrate is assued to be a perfect reflector. The absorption bands for C, M, and Y are the wavelength intervals [550, ] n, [500, 600] n, and [0, 500] n, respectively (note that the spectral absorptance bands for C and M overlap). Figure 4 shows the reflectances for C, M, and Yat concentrations and also the average of the C and M reflectances (at concentration). Note that the average of the C and M reflectances is the idpoint of the line-segent joining these reflectances and should lie in the set S s D of realizable reflectances, if the set is convex in N. With a little logic, it can be seen that no cobination of the C, M, and Y dyes in accordance with the s odel used in S D can produce the average of C and M reflectances shown in Fig. 4. This deonstrates that the s the set S D is nonconvex in N for these dye densities. Therefore, it is clear that the constraint of reproducibility on a subtractive edia cannot be used in the traditional POCS algorith. In the next section, a Hilbert space is presented which akes this constraint convex allowing it to be used in the generalized product space fraework entioned in the previous section. HILBERT SPACE THAT MAKES CERTAIN CONSTRAINTS IN SUBTRACTIVE COLOR CONVEX Several constraints in subtractive color can be expressed as convex constraints in the density doain. Note that both the s set S D of spectral reflectances producible on a given subtractive ediu defined in (5) and the set S f of realizable filter transittances in (6) can be treated in a unified fashion, by defining the generic set of spectra Volue 25, Nuber 5, October

8 S s d 0, D, c in, c ax x N x exp d 0 Dc, c i in c i c i ax, (16) where is the set of positive real nubers, d 0 N, D is an N M atrix, and c in, c ax, c M. Its equivalent in the density doain is the corresponding set of producible densities S d s d 0, D, c in, c ax d d d 0 Dc, c i in c i c i ax. (17) It is readily seen that S d s (d 0, D, c in, c ax ) is a convex closed set in N. However, since this constraint set is defined in the density doain, it cannot be used in conjunction with the other sets defined earlier, which were convex in the reflectance/transittance doain in the original POCS forulation. To see how the product space foralis can be exploited to ake the set S s (d 0, D, c in, c ax ) convex, define the Hilbert space over the field of real nubers by the set of vectors x n x i 0 (18) and the addition and scalar product operators, and J, respectively, as x y exp ln x ln y x y (19) and the inner-product J x exp ln x x, (20) x, y ln x T ln y. (21) Then it can be readily verified that (,, J) defines an Hilbert space with the inner product, and the nor defined as x x, x ln x. (22) It can also be seen that the set S s (d 0, D, c in, c ax )isa closed convex set in the Hilbert space. The projection onto S s (d 0, D, c in, c ax )in can be deterined only nuerically for general values of D, c in, and c ax. However, two special cases for which the projection can be deterined analytically are of practical interest. The first case is the one in which there are no liits on the densities, i.e., c in, c ax, and the second in which the atrix D has orthonoral coluns. The latter case is of interest in applications in which the densities constituting D have been indirectly deterined through a principal coponents analysis of the spectra. 29 The projections for these two cases are listed in the appendix. Note that the distance etric proposed above is siply the Euclidean distance in logarithic (density) space. Alternate definitions of the inner product and nor can be readily obtained by introducing a positive spectral weighting function for the nor and inner product. Exaples of applications where such weighting functions have been successfully used (though not in a set theoretic fraework) can be found in Ref. 30. APPLICATIONS As outlined above, several constraint sets in subtractive color are convex in the Hilbert space (,, J). The generalized product space forulation can, therefore, be used to cobine these sets with the other sets described previously that are convex in the spectral reflectance/transittance doain. The application of such an approach to specific probles in color science and iaging is deonstrated in this section. Three illustrative exaples are considered here: odel based scanner calibration, design of color scanning filters, and colorant forulation for transparent colorants. Model-Based Spectral Scanner Calibration The goal of scanner calibration is to provide a transforation fro the scanner easureents (typically, RGB values in three channel scanners) to a device-independent color space (such as CIE XYZ space) or to spectral reflectance (fro which tristiuli can be readily coputed). Typically, this transforation takes the for of a look-up table or regression polynoial, 31 and is deterined by scanning a calibration target and establishing the correspondence between the scanner output and independently easured colorietric/spectral data fro the target. For a noiseless scanner, it can be readily seen that exact CIE XYZ tristiulus values (under a specified viewing illuinant) can be obtained fro the scanner easureents by eans of a linear transforation, if the product of the viewing illuinant and the CIE XYZ CMFs are linear cobinations of the scanner sensitivities Since ost present day scanners do not satisfy this condition, independent calibrations for different input edia (for instance, photographic/xerographic/lithographic prints) yield significantly better results than a single calibration over ultiple edia. Therefore, in order to obtain accurate colors fro a scanner for a given input ediu, the scanner should be calibrated with a calibration target with spectral characteristics siilar to that of the input ediu. The largest single class of scanner inputs is probably photographic prints. A nuber of anufacturers of photo-processing products are offering photographic scanner calibration targets. 35 However, the targets typically correspond to a single type and batch of photographic paper and dyes, and can vary considerably in their spectral characteristics fro each other and fro photographic prints fro the sae and other anufacturers. As a result, scanner calibration targets are often unavailable for the specific ediu on which the input iages are produced. This is a fundaental liitation of the easureent-based scanner calibration schee described above, and odel-based calibration is therefore an attractive alternative. The idea behind odel-based scanner calibration is to exploit odels for the ediu and the scanner to obtain a calibration transforation. First, fro direct easureents or indirect estiation ethods, spectral odels are obtained for the scanner and for the ediu of interest. The calibra- 340 COLOR research and application

9 tion is then perfored by deterining for each set of scanner easureents a feasible reflectance spectru for the given ediu that would give rise to specified scanner easureents. The calibration process thus defines a transforation fro scanner easureents to spectra (on the given ediu) that result in those easureents. The spectra can be readily used to obtain tristiuli under any desired illuinant and, therefore, the spectral calibration has an advantage over schees that transfor scanner easureents into tristiuli under a particular viewing illuinant. To illustrate the idea of odel-based scanner calibration, the specific case of scanning iages on photographic edia is used in the reainder of this section, and results fro siulations and actual experients are presented. This application is also presented in greater detail in Refs. 36, 37. As entioned earlier, the Bouguer Beer subtractive odel of (5) holds fairly well for photographic edia; where cyan, agenta, and yellow dyes are used for obtaining the color prints. Since pure cyan, agenta, and yellow tone prints are not norally available in iages, the densities corresponding to the dyes cannot be directly easured. Note, however, that the spectra in the odel of (5) can be rewritten as 3 ln r ln r p c i d i. (23) The left-hand side of the above equation represents the density corresponding to the reflectance r relative to the white paper reflectance r p. Fro the above equation, it is clear that these paper-relative spectral densities are linear 3 cobinations of the densities {d i }. Hence, the paperrelative spectral densities lie in a three-diensional space (excluding noise effects), and principal coponents analysis can be used to deterine an orthonoral set of basis vectors for this space. This set of vectors serves as principal dye densities and is a linearly transfored version of the actual dye densities. This idea of utilizing principal coponents analysis in the density doain has been used earlier. 29,38 The principal dye densities can be deterined fro a sall nuber of spectral easureents fro the iages to be scanned and, therefore, do not require a calibration target with unifor patches. Also note that, while the concentrations corresponding to the real dye densities in (5) were subject to siple upper and lower bounds, siilar bounds cannot be obtained for the virtual dyes obtained fro the principal coponents analysis and the inforation in the bounds is, therefore, lost. If O [o 1, o 2, o 3 ]isthe atrix of the (orthonoral) virtual dye densities obtained through the principal coponents analysis, the constraint set S s (ln(r p ), O,, ) can be used to describe producible spectra. Now consider the process of scanning a photographic print characterized by the above edia odel on a color scanner that is accurately represented by the odel of (3). A specific scanner easureent vector t s can arise fro reflectances in the set S l (M s, t s, ). In addition, fro the FIG. 5. Densities for the three principal dyes. odel for the ediu, it is known that the input spectra lie in S s (ln(r p ), O,, ). Since S l (M s, t s, ) is a convex closed set in N and S s (ln(r p ), O,, ) is a convex closed set in, the ethod of POCS can be used in the generalized product space fraework to obtain feasible spectra that agree with both the scanner odel and the odel for the input ediu, thereby yielding a spectral scanner calibration. The results of applying this ethod to the calibration of a siulated scanner and to an actual scanner are described in the next two sections. In both cases, the Kodak IT8 photographic target 35 is used for testing the odel-based calibration schee. The reflectance spectra for the 264 patches in the Kodak IT8 target were easured independently using a spectrophotoeter. The reflectance of the white patch in the gray-wedge on the target is used as the reflectance of the paper substrate r p in coputing paperrelative spectral densities in (23). The first three principal coponents of the 264 densities account for 97.2% of the signal energy in density space, and are used as the (orthonoral) densities o 1, o 2, o 3 of three principal dyes constituting the prints. These densities are shown in Fig. 5. The constraint that spectra obey the edia odel is represented by the set S s (ln(r p ), O,, ), where O [o 1, o 2, o 3 ]. Siulation Results. A three-channel color scanner is synthesized by defining sensitivities for its channels as the cobination of the Wratten 39 WR-26 red, WR-49 green, and WR-52 blue filters with a cool white fluorescent lap (the scanning illuinant). The resulting scanner sensitivities for the three channels are shown in Fig. 6. To test the odel-based calibration schee, scanner RGB values t s are generated using the odel of (3) and the easured reflectance r for each patch on the Kodak IT8 target. For the purpose of the siulations, the noise ter was set to zero. A spectral reflectance estiate rˆ corresponding to the scanner easureent t s is then obtained by coputing a feasible spectru lying in the constraint sets S s (ln(r p ), O,, ) and S l (M s, t s, 0). For this purpose, the POCS algorith was applied in the generalized product space fraework. An Volue 25, Nuber 5, October

10 outline of the coplete algorith is given in Table I. Note that in this algorith the projection onto the diagonal subspace has been replaced by a sipler averaging step, the otivation behind which is described in the appendix. To estiate the accuracy of the odel-based calibration schee, the coputed spectru rˆ is copared with the actual spectru r. Two etrics are used for this coparison: (1) the noralized ean squared spectral error (NMSSE) defined (in db) as NMSSE 10 log 10 E r rˆ 2 E r 2, (24) where E{ } denotes the average over the spectral enseble (in this case the Kodak IT8 target spectra), and (2) the E* ab color-difference 22 under CIE D50 daylight illuinant, which approxiates the perceived agnitude of the color difference and is frequently used in coparison of calibrations. For the siulated odel-based scanner calibration, over the 264 patches in the Kodak IT8 target, the NMSSE is db and the E* ab error has an average value of 0.62 and a axiu value of The nuber of iterations required in the algorith of Table I depends on the chosen tolerance for the convergence test in step 7. For the ipleentation of this section, an average of 53 iterations per saple were required to achieve convergence. Note, however, that the rate of convergence of the algorith is not a serious concern, because in a practical application the algorith would be utilized to build a look-up table for the scanner calibration, which would then be used to transfor the data fro the scanner. Details on the use of the ethod in a practical application can be found in Refs. 36 & 37. Linear Media Model. The odel-based calibration schee presented above utilizes a linear odel for the scanner and a nonlinear odel for the spectra producible on the edia, which are cobined in a POCS estiation schee using the generalized product space fraework. A TABLE I. Algorith for odel-based scanner calibration. 1. Set M s as the spectral sensitivity of the scanner, r p as the white paper reflectance of the scanned ediu, O as the atrix of orthonoral principal dye densities for the scanned ediu, and t s as the set of scanner RGB values for which a calibration is desired. 2. Initialize i 0; and spectral estiate r i (M T s ) t s, where indicates the pseudo-inverse. Set coponents of r i that are zero or negative to a sall positive value and values above unity to Deterine projection onto the set of spectra that produce the given scanner easureent in N (see the appendix for details on the projection): x P Sl Ms,ts,0 r i. 4. Deterine projection onto the set of spectra producible on the given ediu in (see the appendix for details on the projection): y P S ln rp,o,, r s i. 5. Constrain to lie in the diagonal subspace W in N r x y /2. Set coponents of r i 1 that are zero or negative to a sall positive value. 6. Update iteration count: i i Check for convergence: If r i is not in S s (ln(r p ), O,, ) S l (M s, t s, 0) return to 3; otherwise, proceed to Set rˆ r i. This is the estiate of the reflectance on the given ediu corresponding to the scanner easureent t s. sipler alternative is to utilize a linear odel for the spectra producible on the edia. Since the scanner has three channels, if reflectance spectra producible on the edia are represented as linear cobinations of three basis vectors (whose scanner responses are linearly independent), the scanner easureents can be used to estiate the linear cobination of the basis vectors that results in the given scanner response. 4 In this section, a scanner calibration schee based on this idea is described and copared with the ethod of the last section. This schee, based on purely linear odels, is of interest because a large body of research has explored linear odels for object reflectance spectra If the spectra on the scanned edia are expressed as linear cobinations of three basis spectra, the set of producible spectra can be expressed as S R (G), where G is the N 3 atrix with the basis spectra as its coluns. The optial (in the ean-squared error sense) set of basis vectors for representing a given enseble of spectra can be obtained through principal coponents analysis. 43,42 A spectral reflectance estiate rˆ corresponding to the scanner easureent t s can then be obtained by coputing a feasible spectru lying in the constraint sets S R (G) and S l (M s, t s, 0) representing, respectively, the spectra producible on the edia and the spectra capable of producing the given scanner easureent. For this siple case, however, the POCS ethod is not required, because the proble can be solved analytically. The estiated spectru corresponding to a scanner easureent t s is given by FIG. 6. Sensitivities for the siulated scanner channels. rˆ G M s T G 1 t s. (25) 342 COLOR research and application

11 Note that (25) is derived on the preise that the atrix M s T G is nonsingular, which is atheatically equivalent to the requireent that the scanner responses corresponding to the basis vectors are linearly independent. This corresponds to the requireent for colorietric independence for priaries used in a color-atching experient. 7 The previous siulation was repeated using the scanner calibration schee based on the purely linear odels. The first three principal coponents for the Kodak IT8 target spectra were deterined and used as the basis G for the spectra producible on the edia. As previously scanner RGB values corresponding to the reflectances of the Kodak IT8 target were generated using the scanner sensitivities shown in Fig. 6. Fro each scanner RGB triplet t s, the input spectral reflectance was estiated using (25). The accuracy of the calibration was then coputed using the etrics introduced earlier. For the scanner calibration schee based on purely linear odels, NMSSE is db and the E* ab error has an average value of 3.60 and a axiu value of over the 264 patches in the Kodak IT8 target. The errors are significantly larger than the corresponding errors in the previous siulation and the accuracy of the calibration is unacceptable for ost applications. The calibration error is largely attributable to the use of the threediensional linear odel for input reflectance spectra, which does not provide sufficient accuracy. To obtain accuracy coparable to the previous nonlinear odel, i.e., an average E* ab error under 0.62, for the reflectances in the Kodak IT8 target, 6 or ore principal coponents are required. Note, however, that such a linear odel cannot be used in the odel-based calibration for a scanner with only three channels. The nonlinear odel for the edia used previously provides a ore accurate and parsionious representation for the spectra producible on the edia, which is ore appropriate and provides uch greater accuracy, albeit through the use of a ore coplicated algorith for the odel-based calibration. Experiental Results. The odel-based spectral scanner calibration was also tested on an actual three-channel UMAX color scanner with 10 bits per channel. Since the scanner sensitivities for the red, green, and blue channels were not directly available, these were first estiated by the principal eigenvector technique* described in Ref. 27. The estiated sensitivities are shown in Fig. 7. The odel-based calibration was perfored for the Kodak IT8 in a anner identical to that eployed for the siulation and the sae etrics coputed for evaluation of the calibration accuracy. The NMSSE was db, and the average and axiu E* ab errors were 1.76 and 7.15, respectively. For coparison, the scanner was also calibrated directly using a neural-network based technique with the coplete Kodak IT8 target as the training set. The average and axiu E* ab errors (over the Kodak IT8 target) for the direct calibration were 1.05 and 4.40, respectively. * Inforation on the internal scanner atrixing was not available and, therefore, the POCS technique described in Ref. 27 could not be eployed. FIG. 7. Estiated sensitivities for the UMAX scanner. One ay note that the errors in the calibration of the UMAX scanner are larger than those obtained in the siulations. One probable cause of the larger errors is the estiation error in the scanner sensitivity. An inspection of the Fig. 7 shows that the estiated sensitivities for the UMAX scanner have a nuber of negative and positive lobes, which are highly iprobable in the actual scanner. Other potential causes include easureent noise and deviations fro the scanner odel of (3) due to fluorescence, stray light, and other nonlinear effects. 44 To verify that the increased error is indeed due to these factors, the first siulation was repeated using the estiated scanner sensitivity of Fig. 7. For this siulation, the odel-based scanner calibration yielded a NMSSE of db and average and axiu E* ab errors of 0.76 and 5.37, respectively. These values are consistent with those obtained for the first siulation. One ay also note that odel-based spectral scanner calibration for photographic edia was used as an illustrative exaple in this section and the sae ethod can be used in several other applications. In particular, the odelbased calibration fraework can be used to obtain spectrophotoetric data fro densitoeters, colorieters, and color caeras, which can also be represented by the odel in (3). The algoriths can also be readily odified to incorporate the use of ore scanner channels and ore than three principal dyes for edia that have ore physical dyes or do not obey the Bouguer Beer law exactly. The set of producible spectra could also be ade ore precise by including bounds on the concentrations of the principal dyes that are deterined statistically. Design of Color-Mixture-Curve Filters As entioned earlier, in designing filters for three-channel color recording devices such as colorieters, caeras, and scanners, it is useful to have transittances that are linear cobinations of the CMFs, which are coonly referred to as color ixture curves (CMCs). The set of Volue 25, Nuber 5, October