BACKGROUND

[0001] Color is a concept that is understood intuitively by human beings. However, it is a subjective phenomenon rooted in the retinal and neural circuits of a human brain. A “color” is a category that is used to denote similar visual perceptions; two colors are said to be the same if they produce a similar effect on a group of one or more people. Color can be represented in a large variety of ways. For example, in one case a color may be represented by a power or intensity spectrum across a range of visible wavelengths. In another case, a color model may be used to represent a color using a small number of variables.

BRIEF DESCRIPTION OF THE DRAWINGS

[0002] Various features of the present disclosure will be apparent from the detailed description which follows, taken in conjunction with the accompanying drawings, which together illustrate certain example features, and wherein:

[0003] FIG. 1 is a flow chart representation of a method according to examples described herein;

[0004] FIG. 2 shows a set of target color values volumes corresponding to a respective set of target color values in a first color space, determined according to part of a method according to examples described herein;

[0005] FIG. 3 shows a set of reflectance functions forming a paramer set determined according to a method according to examples described herein for an example target color value shown in FIG. 2;

[0006] FIG. 4 shows a set of respective mismatch volumes of paramer sets determined by a method described herein for the target color values shown in FIG. 2;

[0007] FIG. 5 shows a plot of the magnitudes of the mismatch volumes shown in FIG. 4; [0008] FIG. 6A shows a set of mismatch volumes for a plurality of paramer sets determined according to examples described herein using a first color value criterion;

[0009] FIG. 6B shows a set of mismatch volumes for a set of paramer sets determined according to examples described herein using a second color value criterion;

[0010] FIG. 7 shows an imaging system for implementing a method according to examples described herein; and

[0011] FIG. 8 shows a non-transitory machine for implementing a method according to examples described herein.

DETAILED DESCRIPTION

[0012] In the following description, for purposes of explanation, numerous specific details of certain examples are set forth. Reference in the specification to "an example" or similar language means that a particular feature, structure, or characteristic described in connection with the example is included in at least that one example, but not necessarily in other examples.

[0013] Certain examples described herein relate to color mapping in an imaging system, such as a printer, a scanner or a camera. Color mapping is a process by which a first representation of a given color is mapped to a second representation of the same color. As mentioned above, color may be represented by a power or intensity spectrum across a range of visible wavelengths. However, this is a high dimensionality representation. To represent color at a lower dimensionality, i.e. using a lower number of variables, a color model can be used.

[0014] Within this context, a color model can define a color space. A color space in this sense may be defined as a multi-dimensional space, wherein a point in the multi-dimensional space represents a color value and dimensions of the space represent variables within the color model. For example, in a Red, Green, Blue (RGB) color space, an additive color model defines three variables representing different quantities of red, green and blue light. Other color spaces include: a Cyan, Magenta, Yellow and Black (CMYK) color space, wherein four variables are used in a subtractive color model to represent different quantities of colorant, e.g. for a printing system; the International Commission on Illumination (CIE) 1931 XYZ color space, wherein three variables (‘X’, Y’ and Z’ or tristimulus values) are used to model a color, and the CIE 1976 (L*, a*, b* — CIELAB) color space, wherein three variables represent lightness ('!_’) and opposing color dimensions (‘a’ and ‘b’). Certain color spaces, such as RGB and CMYK may be said to be device-dependent, e.g. an output color with a common RGB or CMYK value may have a different perceived color when using different imaging systems.

[0015] Colors are formed by the interaction of a surface, a light source, and an observer. A light source may be characterized by a power or intensity function defining the power or intensity of the light across the range of visible wavelengths. The power or intensity spectrum of the light source may be characterized by use of a number of spectral bands distributed across the range of visible wavelengths. For example, the spectrum for a given light source may be specified by N intensity values representing the intensity of the light source at a N different sample wavelengths distributed evenly over the range of visible wavelengths of 400 - 700nm. In some examples, N may be, for example, 16 or 31. This representation of the spectrum for the given light source may be referred to as an intensity function. The reflectance of a surface may similarly be defined by N values specifying the proportion of light reflected by the surface at the N different sample wavelengths. This representation of the reflectance of a surface may be referred to as the reflectance function, or simply the reflectance, of the surface. In such a representation as is described above, therefore, the reflectance of a surface is defined in a spectral space of N dimensions. This spectral space of N dimensions is sometimes referred to as reflectance space. In a similar manner, the sensitivity of an observer to light may vary with the wavelength of the light. Thus, an observer can be characterized by a spectral sensitivity function.

[0016] A given combination of a light source having a given intensity function, a surface having a given reflectance function, and an observer having a given spectral sensitivity function induces a given color response in the observer. The observer can characterize the color response by a set of color values in a color space. An observer may represent the color response in a color space which has a lower number of dimensions that the reflectance space. For example, the observer may characterize the color by XYZ tristimulus values, thereby representing the color response in a 3-dimensional color space.

[0017] However, various different combinations of light source and surface can induce in the observer the same color response represented in the color space. For example, under the same light source, various different reflectance functions can induce the same tristimulus color values in the observer. The set of reflectance functions which induce the same color value in the observer under the same light source is referred to as the metamer set. The reflectance functions making up the metamer set are referred to as metamers.

[0018] Determining the metamer set for a given color value under given conditions may be useful in various color matching or color reproductions tasks. For example, a printer may be tasked with printing a color which induces a given XYZ color value in a given observer under a given light source. The given XYZ color value may be referred to as a target color value. If the metamer set for the target color value under the given conditions is known, then, an appropriate surface which is printable by the printer and which has a reflectance function matching one of the metamers in the metamer set may be used to print the target color value. For example, the reflectance functions of various inks and combinations of inks printable by the printer may be known. An appropriate selection of inks for printing the color may then be made from those combinations of inks which produce the target color value under the given reference conditions. That is, the reflectances which form part of the metamer set form candidate reflectances for matching the given color value. Each of the candidate reflectances may correspond to surfaces having different properties. For example, each combination of inks in the printer which has a suitable reflectance function for matching the target color value may have a different level of ink use efficiency, or may produce a surface having a different grain. Another example property of a candidate surface is the robustness of its color when viewed under conditions different to the reference conditions. For example, it may be beneficial for some applications to use a surface which matches the target color value under a wide range of light sources. Another color reproduction example in which it is useful to determine the metamer set is for reproduction of a color on a display screen. For example, candidate combinations of LED lights having a suitable spectral intensity for reproducing the target color may be determined by determining the metamer set for the target color value.

[0019] A paramer set defines a set of reflectance functions which match a target color value under given conditions but which do not necessarily match the target color value as strictly as reflectances forming a metamer set for the target color value under the given conditions. While a metamer is a reflectance which provides a color match to the target color value under a given conditions, a paramer is a reflectance which, under the given conditions, provides a match to the target color value to within a particular tolerance.

[0020] Certain examples described herein allow for the computation of a paramer set for a target color value. Certain examples described herein allow for a paramer set to be computed analytically, without imposing certain constraints which may be imposed in methods for computing metamer sets. For example, in methods of computing a metamer set, reflectance functions may be modelled using a linear model of lower dimensionality than the reflectance space. Certain example methods described herein allow for a paramer set to be computed in the full N- dimensional reflectance space.

[0021] The paramers forming the paramer set for a target color value may each have different properties, in a similar manner to as has been described above for metamer sets. Since a paramer need not match a target color as strictly as a metamer, determining a set of paramers instead of metamers may provide a larger number of candidate reflectances, each having corresponding properties. In printing applications, this can allow for a wider range of surfaces to be selected from for printing a target color value and therefore allow greater opportunity for optimizing the properties of the printing operation.

[0022] FIG. 1 shows a block diagram representation of a method 1000 of determining a paramer set according to examples described herein. The method comprises, at block 1010, determining a target color value corresponding to a point in a first color space. The target color value may also be referred to herein as a target colorimetry. The first color space may be, for example, a tristimulus color space wherein points in the color space are defined by tristimulus values, such as CIE XYZ, L* a* b* or an RGB color space. The target color defines a point in the first color space such that the target color value, in examples, is a multidimensional color value. For example, in the case where the first color space is the XYZ color space, the target color value defines 3 values respectively for X, Y and Z.

[0023] At block 1020, the method comprises determining a color value criterion defining a target volume in the first color space. The color value criterion may also be referred to herein as a colorimetry criterion. The color value criterion defines a target volume forming a target region in the first color space which includes the target color value.

[0024] The target volume defines the colors in the first color space which are considered a match to the target color under the given conditions for which the paramer set is being computed. In other words, the color value criterion defines how closely the color induced by a reflectance function under reference conditions should match the target color value for the reflectance function to be a paramer of the target color value. In the example where the first color space is the 3- dimensional XYZ space, the target color is represented by a point and the target volume defines a 3-dimensional volume including the point representing the target color. In such an example, the color value criterion may define a cube, a sphere, or another type of target volume enclosing the target color value. For example, the color value criterion may define the target volume as including the points falling within a range of the target color value plus or minus a tolerance. The target volume may, for example, be a cuboid if the tolerances define the target volume as (X, Y, Z) — (Xfarget +/“ Xf ₀ |, Yfarget +/“ Yf ₀ |, Zfarget +/“ Zf ₀ |), Where Xfarget, Yfarget, and Zfarget are respectively the coordinates of the target color value in XYZ space and X _to i, Y _to i, and Z _toi are respectively tolerances in the X, Y and Z dimensions which define the target volume. For example, if in the above representation the tolerances in the X, Y and Z dimensions are equal to one another then the target volume is a cube having sides of length 2 x the tolerance with the target color at the center of the cube. In another example, the target volume in the color space may be a sphere, or an approximation of a sphere. The sphere may be centered on the target color value.

[0025] In certain examples, the target volume in the first color space may be defined according to a volume in a second color space. For example, the target volume may be defined as the volume in the first color space which maps to a particular volume in the second color space. The second color space may be a perceptually uniform color space, such as the L*a*b* color space. In one example, a sphere may be defined in L*a*b* such that points within the sphere are within a given perceptual distance to the target color volume, as defined by a point in the L*a*b* space. Each color in the L*a*b* space may be mapped to a color in the XYZ color space by a suitable process. Accordingly, the sphere in the L*a*b* space may be mapped into the XYZ space to produce the target volume. The target volume in the first, XYZ, space may therefore comprise a projection of a sphere defined in the second, L*a*b*, color space. Where a sphere is defined in L*a*b* space, a radius of the sphere may be set such that each of the colors in the sphere are within a given perceptual distance of the target color value. For example, the radius of the sphere in L*a*b* may be set as 1 delta E (DE) unit such that the sphere, and the projection of the target volume into the first color space, defines colorimetries which are within 1 DE of the target colorimetry under the reference conditions.

[0026] In examples, the target volume in the first color space may be defined by a convex hull of a polytope, e.g. where the first color space is a 3D color space, a polyhedron. A given target volume may have a shape which is an approximation of a given geometric shape defined by a convex hull of a polytope. For example, where the first color space is a 3D color space, the target volume in the first color space may be an approximation of a sphere defined by a convex hull of a polyhedron. The target volume in the first color space may, in some examples, be a projection in the first color space of a convex polytope in the second color space. For example, a convex hull of a polyhedron which approximates a sphere may be defined in the second color space and the target volume may be a projection in the first color space of that approximation of a sphere. The convex polytope in the first color space and/or the convex polytope in the second color space may be defined by a set of inequalities. In other examples, the target volume may be a polytope which is not convex. Where the target volume in the first color space is a polytope which is not convex, the target volume may be defined by a surface tessellation rather than by a set of inequalities. Such a surface tessellation may describe a surface defining the target volume using the smallest degree simplex of the color space. For example, in a 3D color space the target volume may be defined by a triangulation, while in a 4D color space the target volume may be defined by a tetrahedralization. In some examples where the target volume in the first color space is a projection of a volume in the second color space, the volume in the second color space may be convex polytope but when projected into the first color space may define a target volume which is not convex. The reverse may also be true in that the volume in the second color space may be a non-convex polytope while the target volume is a convex polytope. In such examples the non-convex polytope, whether in the first color space or the second color space, may be described by a surface tessellation.

[0027] The color value criterion may be formulated to specify that, under a set of reference conditions, i.e. under a given illuminant and when observed by a given observer, a reflectance induces in the observer a color value within the target volume in the first color space. In some examples, the reference conditions comprise a standard illuminant and/or a standard observer, for example a standard CIE illuminant and/or a standard CIE observer. For example, the illuminant of the reference conditions may be one of the standard CIE illuminants or illuminant series A, B, C, D, E or F. Each of these standard illuminants or series of illuminants is defined to approximate the spectral power distribution of a given type of light. For example, Illuminant A is intended to represent an average incandescent light, while Illuminant B is intended to represent direct sunlight. Illuminant series D is a series of standard illuminants intended to represent natural daylight. An example standard illuminant of the Illuminant series D is the D50 illuminant. One example of a standard observer which may form a part of the reference conditions is the CIE 1931 standard observer, another is the CIE 1964 standard observer. The color value induced in the observer by a reflectance under the reference conditions may be stated as S1*ref, where: S1 defines the reference conditions including the reference illuminant and the reference observer; and ref is the reflectance. Thus, the color value criterion may comprise one of more inequalities. For example, the color value criterion where the target volume is a cube may be formulated as an inequality having the following form:

S1 ref — (Xfarget +/“ Xf ₀ |, Yfarget +/“ Xf ₀ |, Zfarget +/“ Xf ₀ |)

Other ways of defining the target volume are, for example, in the case where the target volume is convex, by a set of linear half spaces defined in the form [A, b] such that A *(X _ta rget, Ytarget, ^-target) b. In examples where the target volume is not convex, a surface tessellation, as mentioned above, may be defined that contains (Xfarget, Yfarget, Zfarget)-

[0028] The color value criterion, and hence the tolerance in color matching of the paramer set, may be determined according to the intended application of the paramer set. For example, in an image capturing system, the tolerance may be determined according to the noise (e.g. of the input device) which gives a range of possible RGB values measured by the device, or the tolerance may be determined according to the repeatability of the target device for the output of the process (e.g. the print/measure repeatability of a system). In another example, the tolerance may be selected according a maximum color difference from the target color value for a given printing application. Such examples are discussed below in more detail.

[0029] At block 1030, the method comprises determining one or more reflectance criteria. The one or more reflectance criteria are criteria to be satisfied by each reflectance function which forms a member of a paramer set for the target color. In certain examples, the one or more reflectance criteria define one or more inequalities to be satisfied by individual reflectance values which make up a given reflectance function. That is, as described above, each reflectance function defines N different reflectance values, each of the N reflectance values corresponding to a proportion of light which is reflected at different wavelengths. One or more inequalities may be defined for each of the N reflectance values.

[0030] In one example, the one or more reflectance criteria comprise a first reflectance criterion defining that no individual reflectance value may be less than 0. This represents the physical constraint that the percentage of light which is reflected by a surface at any one wavelength cannot be less than 0, i.e. at any given wavelength, no less than no light can be reflected. The first reflectance criteria, in examples, is formulated as N inequalities to be satisfied by a given reflectance function, each of which specifies that a reflectance value at a given wavelength is greater than or equal to 0.

[0031] An example second reflectance criterion defines a limit on the maximum value that individual reflectance values of the reflectance function may take. For most surfaces, an appropriate maximum reflectance value at any given wavelength is 100%. However, in some circumstances, such as that of a fluorescent surface, the maximum value of reflectance for a surface at a given wavelength may be greater than 100%. The second reflectance criterion may also be formulated as a set of inequalities to be satisfied by the individual reflectance values of a reflectance function. The maximum reflectance value may be the same for each of the wavelengths for which the reflectance function defines a reflectance value, or the maximum reflectance value may be different for different wavelengths. The maximum reflectance value may be set based on empirical knowledge of the reflectances of the type of surfaces of interest. If the paramer set is to be used in the context of fluorescent surfaces, then a maximum reflectance may be set based on known maximum reflectance values of fluorescent surfaces of interest. In one such example, where fluorescent surfaces are of interest, a maximum reflectance value is set at around 1.8 (i.e. 180% reflectance). In other such examples, the maximum reflectance value may be set at from 1.5 and 2. In certain examples, maximum reflectance values may be relaxed (e.g. set at greater than 1 ) at wavelengths at which emission peaks of known fluorescent surfaces occur. In such examples, maximum reflectance values at other wavelengths at which emission peaks are not know to occur may remain at a standard value, of e.g. 1 .

[0032] In some examples, minimum and/or maximum reflectance values, as defined by the first and second reflectance criteria, may be set to account for noise. For example, instead of the first reflectance criteria defining a minimum reflectance of 0, the minimum reflectance value may be set at 0 - E . Similarly, the maximum reflectance value may be set at 1 + Eu, or at a set reflectance value + Eu, e.g. where the set reflectance value is set to account for fluorescence, as described above. Here, E and Eu are tolerances to account for noise at, respectively, lower and upper ends of the range of possible reflectance values.

[0033] In some examples, one or more further reflectance criteria are used in addition to the first and the second reflectance criteria. One example of a further reflectance criterion is a naturalness constraint. A naturalness constraint imposes the condition that any reflectance function which is to form a member of the paramer set has a particular property related to the physical realizability of the reflectance function. That is, a naturalness constraint is intended to limit the reflectance functions for inclusion in the paramer set to reflectance functions which correspond to real-world surfaces. For example, the naturalness constraint may specify that any reflectance function which is to form a member of the paramer set varies smoothly over the range of visible wavelengths. Accordingly, in one example, a set of representative physically-realizable reflectances may be defined. A given reflectance may be considered to satisfy the naturalness constraint if the reflectance is an additive combination of one or more of the reflectances included in the set of representative physically-realizable reflectances. The naturalness constraint may be formulated such that it is satisfied by reflectances which fall within the convex hull of the representative set of physically-realizable reflectance functions and not by reflectances which fall outside this convex hull. In an example, the naturalness constraint, like the first and second reflectance criteria, may take the form of a set of linear inequalities placed on the reflectance function with one or more linear inequalities placing constraints on each of the N values of the reflectance function. The naturalness constraint may be formulated as defined in the following paper, the entirety of which is incorporated herein by reference: Peter Morovic and Graham D. Finlayson, "Metamer-set-based approach to estimating surface reflectance from camera RGB," J. Opt. Soc. Am. A 23, 1814-1822 (2006).

[0034] At block 1040, the method comprises computing, based on the one or more reflectance criteria and the color value criterion, a boundary, in the reflectance space, defining points which satisfy the color value criterion and the one or more reflectance criteria. The boundary may define a bounding box in the N-dimensional reflectance space of points satisfying the color value criterion and the one or more reflectance criteria. In some examples, the boundary is a convex hull defining the points which satisfy the color value criterion and the one or more reflectance criteria. Computing the convex hull in reflectance space provides a set of reflectance functions forming the extreme vertices of the convex hull. Each of these reflectance functions satisfies the color value criteria and the one or more reflectance criteria. The reflectance functions so obtained form the paramer set for the target color value. The paramer set comprises one or more reflectance functions. The convex hull may be computed using a program such as Qhull, such as is described in the following paper, the entirety of which is incorporated herein by reference: Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., "The Quickhull algorithm for convex hulls," ACM Trans, on Mathematical Software, 22(4):469-483, Dec 1996, http://www.qhull.org. [0035] In some examples, the convex hull may be computed to determine a paramer set which comprises reflectances having a certain property. If the certain property can be expressed as a convex function in reflectance space, then convex programming, e.g. linear or quadratic programming, may be employed to compute the convex hull. The paramer set may therefore be computed such that each of the paramers making up the paramer set has the certain property. One or more paramers may make up the paramer set.

[0036] An example of the method described above with reference to FIG. 1 will now be described with reference to FIG. 2 to FIG. 6B.

[0037] In this example, the first color space is the CIE XYZ color space. The reference conditions are a CIE D50 illuminant and a 1931 XYZ observer. FIG. 2 shows 24 target color values corresponding to the 24 colorimetries of the MacBeth ColorChecker Chart, which is a standard calibration chart. 24 respective target volumes having the above properties are shown in FIG. 2, respectively surrounding the 24 target colorimetries. Each of the 24 target volumes is a cube in the XYZ color space, wherein each cube has sides of length 1. A first target cube 210 corresponding to one of the 24 target color values is labelled in FIG. 2. Each target cube, including the first target cube 210, may be represented as follows (X = X _ta rget +/- 0.5, Y — Yfarget +/“ 0.5, Z — Zfarget +/“ 0.5).

[0038] FIG. 3 shows an example of a set of reflectances forming a paramer set determined according to a method described herein. The set of reflectances shown in FIG. 3 form a paramer set for the first target cube 210 shown in FIG. 2. The reflectance functions in this example are represented in a 16-dimensional reflectance space. Accordingly, 16 values corresponding to the proportion of light reflected at 16 different wavelengths across the wavelength range 400nm to 700nm define each reflectance function. It can be seen from FIG. 3 that the reflectance functions comprise sharp peaks, i.e. for a particular reflectance function the difference between the reflectance value at a given sample wavelength and the reflectance value at a neighboring sample wavelength may be large. This is a consequence of the reflectances forming the paramer sets being determined without constraints being placed on their smoothness.

[0039] Certain examples of the method described herein involve determining properties of a paramer set determined in the manner described above. One such property is the mismatch volume of a paramer set. The mismatch volume represents the potential mismatch between colors induced in an observer by paramers of the paramer set when those paramers are viewed by an observer under a different illuminant to the reference illuminant with which the paramer set was computed. That is, reflectances forming a paramer set induce in an observer a matching color value, to within a given tolerance, under reference conditions. However, if viewed under conditions different to the reference conditions then reflectances which were determined under the reference conditions to match to a given degree may no longer match to the same degree. A mismatch volume provides a measure of this degree of mismatch in the color values induced by a paramer set under different conditions. The mismatch volume may, for example, be computed by projecting the paramer set onto a different illuminant or a different observer and plotting the projection in a color space. This can be done by taking the reflectances forming the paramer set and computing colorimetries, in e.g. CIE XYZ, resulting when the reflectances are illuminated with a different illuminant to the illuminant used to compute the paramer set and/or when the reflectances are viewed by a different observer than the observer used to compute the paramer set.

[0040] FIG. 4 shows a representation of respective mismatch volumes of 24 sets of paramers determined by a method described herein for the 24 target colorimetries shown in FIG. 2. FIG. 4 shows the projection of the paramer sets, which were computed using the reference conditions of a CIE D50 illuminant and a 1931 XYZ observer, as described above, onto a different illuminant which is the CIE standard Illuminant A. The mismatch volumes are represented in the L*a*b* color space. As can be seen from FIG. 4, the mismatch volumes are of different shapes and sizes for different paramer sets. [0041] FIG. 5 shows a plot in the CIE XYZ color space of the magnitudes of the mismatch volumes of the paramer sets shown in FIG. 4. In FIG. 5, a larger circle represents a paramer set having a larger mismatch volume. For example, a magnitude of a first mismatch volume 510 can be seen to be smaller than a magnitude of a second mismatch volume 520. An indication of the theoretical metamerism of a given colorimetry can be given by the volumes of the paramer sets shown in FIG. 5. These volumes can be computed by computing the volume of the convex hull volume of the resulting reflectances. An approximation of the volume can be found by projecting the resulting convex hull reflectances onto a linear model basis and computing the volume of the resulting polytope, or by projecting onto a different set of ilium inant/observer.

[0042] The magnitude and the shape of the respective mismatch volumes of the paramer sets determined according to examples described herein is determined at least in part by the color value criterion used in the method of determining the paramer set. For example, defining a larger tolerance about the target XYZ color value results in a larger target volume. A larger target volume may result in a larger mismatch volume. However, different target volumes of the same size in the XYZ color space may result in different mismatch volumes, depending on their colorimetry. For example, each of the mismatch volume magnitudes represented in FIG. 5 result from the same size target volume in XYZ space under the reference conditions. However, when projected onto a different illuminant, different magnitudes of mismatch volume result. Thus, the first mismatch volume magnitude 510 which corresponds to a first target colorimetry is smaller than the second mismatch volume magnitude 520 which corresponds to a second target colorimetry.

[0043] FIG. 6A and FIG. 6B show examples of sets of mismatch volumes for paramer sets computed under the reference conditions described above of a CIE D50 illuminant and a 1931 XYZ observer. However, the paramer sets represented in FIG. 6A and FIG. 6B were computed using different sizes of target volume in the XYZ color space. The paramer sets represented in FIG. 6A were computed using a target volume in the XYZ color space defining cube of tolerance 0.05 in each direction about the target color value. The paramer sets represented in FIG. 6B were computed using a target volume in the XYZ color space defining cube of tolerance 0.95 in each direction about the target color value. The mismatch volumes of the paramer sets shown in FIG. 6B, computed using the larger tolerance about the target color value, can be seen to be larger than those shown in FIG. 6A which were computed using the smaller tolerance about the color value.

[0044] While examples above directly obtain reflectances forming a paramer set, in other examples according to the present disclosure a paramer set may be computed on a linear model basis. For example, the above formulation can be applied to solving for linear model weights instead of direct reflectances. This may be done by multiplying all left-hand-side inequalities defining the color value criterion and the one or more reflectance criteria with a matrix B that represents the linear model basis in N dimensions. For example, the matrix B is a 16xN matrix if the original sampling is 16 and N is < 16.

[0045] Certain examples described herein allow for paramer sets to be determined for arbitrary spectral stimuli, including stimuli produced by reflectance from surfaces which are fluorescent and/or which are non-Lambertian in in some other sense. Methods described herein allow for paramer sets to be determined in these and other circumstances without introducing certain constraints on the reflectances forming the paramer set, such as a smoothness constraint or a modelling of the reflectances by a lower-dimensional linear model. Accordingly, example methods described herein allow for paramer sets to be computed in circumstances where such constraints are not applicable. For example, fluorescent inks have reflectance functions which are not accurately modelled by other approaches. By applying the methods described herein, a paramer set for a fluorescent ink or other printing fluids can be computed.

[0046] Certain examples described herein allow for tolerances for the matching of candidate reflectances to a given target color value to be incorporated into the process of determining the candidate reflectances. Therefore, reflectances which have a particular attribute and which induce a color value matching, to a particular degree, the target color value can be determined. By computing a paramer set rather than a metamer set, a reflectance having a more suitable value for a given attribute, such as ink efficiency or robustness may be determined, since the constraint that a given reflectance exactly matches the target color value is relaxed.

[0047] FIG. 7 shows an imaging system 700. Certain examples described herein may be implemented within the context of this imaging system. In the example of FIG. 7, image data 710 is passed to an image processor 720. The image processor 720 process image data 710. The image data 710 may comprise color data as represented in an input color space, such as pixel representations in an RGB color space or in an XYZ color space. The image processor 720 determines one or more paramer sets in reflectance space for the color data, e.g. one paramer set per pixel or one paramer set per different RGB or XYZ color value, according to methods described above.

[0048] Once the paramer sets for the input color values are determined, the image processor 720 can use the paramer sets for each given input color value to determine how to map each given input color value to another color representation. For example, the image processor 720 may select one of the paramers, from the paramer set for a given input color value, to correspond to the given input color value. The paramer may be selected based on its properties. For example, a paramer with the lowest color error value under multiple illuminants may be selected. That is, the paramer which has the lowest difference in color as measured in a particular color space, such as the XYZ color space, under multiple illuminants (e.g., D50, A, TL84) may be selected to represent the input color value. In another example, a paramer may be selected based on an optimization (e.g. a minimization) of a plurality of metrics, each metric being associated with a different property of the imaging device’s output. An example here is ink use efficiency in a printing system, where the paramer using the least quantity of ink could be selected. Another example is robustness to variation, where the paramer whose repeated printing and measurement would result in least color variation could be chosen. This paramer is used by the color processor to construct a color mapping between the input color space and reflectance space. A paramer may be selected based on the ability of the imaging system 700 to reproduce the paramer. A color mapping based on the selected paramer may then be sent to an output device 730 which represents the input color with use of the selected paramer. For example, where the output device 730 is part of a printer, the image processor 720 may select a paramer from the determined paramer set for a given input color value may be selected based on the combination of inks used by the printer 730 to print a surface having a reflectance corresponding to the paramer. In another example, where the output device 730 is a display device for displaying an image, the image processor 720 may select the paramer for representing a given input color on the display device 730 based on ability of the display device 730 to produce a spectral intensity distribution corresponding to the paramer.

[0049] In some examples, the image processor 720 may be arranged to compute the paramer sets for the input color values as described in certain examples above. In other examples, the image processor 720 may determine paramers for the input color values from pre-computed paramer sets which are precomputed for given color values to correspond to given reference conditions. In such examples, the paramer sets may be pre-computed by the image processor 720 or pre-computed elsewhere and supplied to the image processor 720. The paramer sets may be determined using reflectance criteria and/or a color value criterion which is suited to the particular color mapping application in which the paramer sets will be used. For example, where the method is used to compute paramer sets to be used to map colors for a printing operation, the color value criterion may be selected based on the properties of the printing operation. For example, the color value criterion may be selected based on a color error metric for the printing operation. For example, the color value criterion may be set such that each of the paramers matches the target color value to a given degree which is suitable for the particular printing application. Different metrics may define a degree to which the printed color should match the target color value under different conditions, e.g. under different illuminants. The color value criterion may be selected to provide a set of paramers which matches one or more of such metrics. The reflectance criteria may be selected based on the properties of the printing operation. For example, as described above, where the method is to applied in the context of a method of printing fluorescent inks, the reflectance criteria may be set accordingly.

[0050] Metamer mismatch volumes give an indication of color change with change of ilium inant and therefore about how color inconstant a particular surface is under different illuminants. Computing metamer mismatch volumes for different alternatives allows, for example, one of the alternatives to be selected that differs less under different illuminants and therefore is more constant in appearance. For example, metamer mismatch volumes may be computed for different inks for different illuminants and an ink selected which have a particular level of color constancy which may be selected to be suitable for a printing operation in which the ink is to be printed. In another example, metamer mismatch volumes may be computed to evaluate illuminants. For example, a printed image may be designed for viewing under a CIE D50 daylight illuminant. However, it may be desired to display the printer image, e.g. in a museum or the like, under a different type of light. Determining paramer sets and their mismatch volumes can allow a prediction of which of the printed colors in the image will change more under the different type of light. A modified version of the printer image can therefore be determined which is more suitable for the different illuminant. For example, printer colors which change by more than a given amount due to the change in illuminant may be substituted for paramers for those colors whose appearance is more constant under the change in illuminant.

[0051] FIG. 8 shows an example of a non-transitory machine 800. Certain methods and systems as described herein may be implemented by a processor 810 that processes computer program code that is retrieved from a non-transitory storage medium 820. In some examples, the processor 810 and computer readable storage medium 820 are comprised within the non-transitory machine 800. Machine-readable medium 820 can be any medium that can contain, store, or maintain programs and data for use by or in connection with an instruction execution system. Machine-readable media can comprise any one of many physical media such as, for example, electronic, magnetic, optical, electromagnetic, or semiconductor media. More specific examples of suitable machine-readable media include, but are not limited to, a hard drive, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory, or a portable disc. In FIG. 8, the machine-readable storage medium comprises instructions 830 which, when executed by the processor 810, cause the processor 810 to perform the method described above.

[0052] The preceding description has been presented to illustrate and describe examples of the principles described. This description is not intended to be exhaustive or to limit these principles to any precise form disclosed. Many modifications and variations are possible in light of the above teaching.