Efficient Computation of Display Gamut Volumes in Perceptual Spaces Carlos Eduardo Rodr´ıguez-Pardo(1), Gaurav Sharma(1), Jon Speigle(2), Xiao-Fan Feng(2), Ibrahim Sezan(2) (1) ECE Dept. Univ. of Rochester, Rochester, NY 14627-0126 (2) Sharp Laboratories of America, Camas, WA 98607 Abstract larly focusing on multiprimary display systems where the defini- Gamut volume computations in perceptual spaces are use- tion of the display gamut does not naturally lead to such a repre- ful for optimizing designs of color displays. We develop a useful sentation. We then consider different methods for numerical com- representation of the gamut of an additive display that facilitates putation of display gamut volume in perceptual color spaces, all efficient numerical computation of the gamut volume. For three based on the gamut representations that we develop for this pur- primary systems, our representation coincides with the obvious pose. We evaluate the accuracy vs computation time trade-offs for representation of a three-primary additive gamut, while for multi- the different methods. primary systems, the representation we develop provides a par- The paper is organized as follows. Section 2 presents the tition of the device gamut as a disjoint union of displaced three mathematical framework for modeling display gamuts in addi- primary gamuts thereby facilitating a computation of the overall tive color spaces and characterizes the display gamut volume in gamut volume as the sum of these individual three primary gamut additive color spaces for both three primary and multi-primary volumes. Based on our representation, we develop and evaluate systems. Section 3 considers the computation of gamut volumes, several alternative numerical schemes for gamut volume compu- first outlining the computation in additive color spaces and then tations in perceptual spaces, comparing their accuracy and com- addressing the computation in perceptual color spaces. Section 4 putational requirements. presents results from tests conducted to evaluate the methods, comparing the accuracy and computation time requirements for 1 Introduction the different methods. Section 5 concludes the paper by summa- A fundamental choice when designing displays is the selection rizing the main findings. of the color primaries [1, 2]. In practice, this selection represents a multi-way trade off between the color gamut, dynamic range, 2 Display Gamuts for Three Primary and power consumption, and material cost and environmental con- Multiprimary Displays cerns. For optimizing display design, modeling and analysis of A color produced by a display system with primaries vectors , , display gamuts is therefore of significant interest. p1 p2 p3, specified in an additive color space, is obtained by a Because most displays are additive devices, their gamut in linear combination of the primaries, and can be represented by additive color spaces, such as the CIE XYZ color space [3], can be the tristimulus value modeled in a relatively straightforward manner [4–6]. For three α α, primary display systems, computation of the gamut volume in the tP( )=P (1) additive spaces is also straightforward. However, the gamut vol- , , α ume in a linear additive color space correlates very poorly with an where P =[p1 p2 p3] is the matrix of primaries,and = α ,α ,α T observer’s perceptual assessment of the gamut volume. Conse- [ 1 2 3] is the vector that determines the relative proportion quently, for the purposes of display optimization, the gamut vol- for the combination of the primaries, whose entries satisfy the ≤ α ≤ , , , ume in a perceptually uniform color space such as CIELAB [3] is constraints 0 i 1 i = 1 2 3. much more useful. Unfortunately, it is not feasible to analytically An important characterization for display systems is given compute the gamut volume in perceptually uniform color spaces in term of the gamut, or the set of colors the device is able to and numerical computation is therefore utilized in practice [7]. reproduce. Adopting for simplicity the notation of the well known 1 The difficulty of such numerical computations is com- CIEXYZ color space [3] , the gamut of the display with matrix G XYZ pounded by the fact that in perceptual color spaces, display of primaries P, denoted by P ,isdefinedas, gamuts are usually nonconvex. For multiprimary systems, an ad- G XYZ α |α ∈ , 3 ditional challenge arises from the fact that for the computation P = tP( ) [0 1] of gamut volume, the gamut representation does not immediately Pα|α ∈ , 3 indicate an obvious way for dealing with the degeneracy arising = [0 1] (2) from multiple metameric options available for a given colorime- The maximum luminance combination, usually referred as the try. white point, is obtained when α = 1, for i = 1,2,3. Note that In this paper, we first develop representations for display i gamuts in additive color spaces that facilitate computations of 1Although we use the CIE XYZ space, all definitions and results de- gamut volumes, both in additive and perceptual spaces, particu- scribed here can be applied to other additive color spaces. 132 ©2011 Society for Imaging Science and Technology from the expression in (2), it is possible to conclude that the defi- j P j Coordinates of β j T nition of gamut corresponds to the definition of the parallelepiped 1 [p1,p2,p3][0,0,0,0] T that contain the origin and in the primary vectors as four of its 2 [p1,p2,p4][0,0,1,0] T vertexes. In fact, the parallelepiped can be interpreted as the lin- 3 [p1,p3,p4][0,0,0,0] T ear transformation represented by the matrix of primaries P of the 4 [p2,p3,p4][1,0,0,0] unitary cube [0,1]3, denoted for simplicity by C 3, and referred as Table 1. Parallelepiped decomposition for the gamut for the the primary space. display with primaries in (5). A system with K primaries can be similarly represented by 100 140 90 120 a3× K matrix P =[p1,p2,...,pK],wherep1,p2,...pK are the 80 70 100 60 coordinates of the primaries in CIEXYZ space. The gamut in 80 Y 50 Y 60 CIEXYZ space can then be obtained in a manner exactly analo- 40 30 40 20 gous to (2) as 20 10 0 0 100 120 120 100 120 80 100 80 100 60 80 60 80 40 60 40 60 20 40 20 40 G XYZ α α|α ∈ C k . 0 0 20 0 0 20 P = tP( )=P (3) Z X Z X where the primary space C K corresponds to the unitary hypercube of dimension K. While the mathematical representation of the gamut for a multiprimary system in (3) closely mirrors the representation for the three primary display gamut in (2), unlike the former repre- sentation, the latter representation is not particularly well suited for computation of gamut volumes, as we shall subsequently see Figure 1. In the top left, the gamut of a system with three primaries that in Section 3. For this purpose, an alternative and more useful def- match the chromaticity coordinates for the standard REC-709 and have a inition is based on the fact that the gamut can be partitioned into white point luminance of 100 (See configuration C in Table 2). In the top parallelepipeds. Specifically, for a system described by a 3 × K 1 right, a multiprimary system obtained by adding a primary with CIEXYZ co- matrix of primaries P we have that, . , . , . T ordinates [2 02 20 02 20 02] . The gamut for the multiprimary case is com- G XYZ P , posed by the union of disjoint parallelepipeds that are shown in the bottom P = j (4) j of the image (not to scale) K where the summation over j ranges over the 3 possibilities for selecting a set P j of three primaries from the K possibilities in P 3 Display Gamut Volume and P = G XYZ +β is the parallelepiped obtained by displacing 3.1 Gamut Volume in Additive Color Spaces j P j j XYZ Additive color spaces are three dimensional and are embedded in the gamut G , of the three primary system with primaries P j, P j R3. Therefore, they inherit the results of three dimensional ge- by the displacement vector β . The displacement vector β can be j j ometry, which can be applied in the manipulation and analysis of expressed as a linear combination of the primaries P where the co- display gamuts. For a system with the matrix of primaries P in efficients in the linear combination are 0 or 1 and the coefficients CIEXYZ, the gamut volume V (G XYZ) is the volume of the paral- corresponding to the primaries in P are necessarily zero. For P j lelepiped enclosed by primaries, and therefore the gamut volume space reasons, we omit a formal proof of this result and the algo- can be defined as [8, pp. 468], rithm for obtaining this representation for a given set of primaries, illustrating the result instead by examples, the first of which we V(G XYZ)=|det (P)|, (6) present pictorially. In Fig. 1, the gamuts for a three primary and P a multiprimary systems are shown. For the latter case, four par- where, det(·) and |·| are the determinant the magnitude opera- K allelepipeds partition the entire gamut, corresponding to 3 pos- tors, respectively. Note that det (P) represents the Jacobian (the sible different choices of the primaries, with K = 4. A numerical determinant of the matrix of derivatives) of the linear transfor- example illustrating this decomposition is presented in Table 1, mationdefinedbyP, from the space of primaries to the gamut for a four primary system from [7] specified by the 4 × 3 primary in the additive space.