An Algorithm for the Selection of High Contrast Color Sets

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1 An Algorithm for the Selection of High Contrast Color Sets Paola Campadelli, Roberto Posenato, Raimondo Schettini Dipartimento di Scienze dell Informazione Università degli Studi di Milano via Comelico 39/40, I-2035 Milano (Italy) Dipartimento Scientifico e Tecnologico Università degli Studi di Verona strada Le Grazie, I-3734 Verona (Italy) posenato@sci.univr.it Istituto di Tecnologie Informatiche Multimediali, CNR Via Ampere 56, I-203 Milano (Italy) centaura@mail.itim.mi.cnr.it Abstract Nominal color coding is the aesthetic and functional use of color to convey qualitative information in graphical environments. The specification of high contrast color sets is a fundamental step in this process. We formulate the color coding problem here as a combinatorial optimization problem on graphs and present an algorithm which performs well and does not require that the function used to code the similarity between colors be a distance function. Keywords: high contrast color sets, visual coding, optimization. Correspondence should be sent to: Paola Campadelli, Dipartimento di Scienze dell Informazione via Comelico 39/40, I-2035 Milano (Italy). Introduction Developments in the technology of color devices have brought a rapid increase in the number of applications in which color is used to convey information. In fact, since color is pre-attentively analyzed by the human visual system, it can be particularly effective in the identification and classification of various information in graphical environments. However the broad range of variables involved in perception, as well as the interaction and trade-offs between them, makes it extremely difficult to select easily distinguishable color sets from the range of those available when more than six or seven colors are required [4], [3], [9] and [8]. We address here the algorithmic aspect of the selection of high contrast color sets, and present an effective algorithm which has an innovative feature that could also be useful when one has to take into account visual features such as the shape, size and texture of the graphical items to be coded [2]. The algorithm does not require that the function adopted to code the similarity between two colors (or graphical codes Supported by the MURST 60% Progetto Disegno e analisi di algoritmi

2 obtained by combining color with other visual features) be a distance function; in other words, it works whatever the measure of perceptual similarity selected. Ergonomical constraints can also be easily taken into account. Several heuristic procedures have been proposed to define high contrast sets of colors. Kelly [] has conceived a list of 22 maximally contrasting surface colors, such that each color of the list is maximally different from the one immediately preceding it. In 982 Carter and Carter [4] formulated the first algorithm to compute easily discriminable sets of colors. Several authors [6], [7], [8], [5], [6] have devised algorithms which can also fulfil a number of ergonomical requirements. In 995 [3] we have presented an abstract formulation of the problem of selecting high contrast color sets, defining it as a combinatorial optimization problem on graphs. We have then transformed the problem into the one of minimizing an appropriate neural network energy function, and solved it approximately by symmetric networks of non-linear graded response neurons. The requirement of tuning some network parameters as a function of the problem instance and the fact that the network converges to a valid solution in about 90% of the trials have led us to explore a different approach which is presented in this paper. In section 2 we formally define the problem of selecting high contrast color sets, while in section 3 the algorithm is described. In section 4 we report and comment some experimental data regarding the selection of high contrast color sets within the 267 colors of the ISCC-NBS color naming system [2], the 502 different colors of the X Windows System palette (rgb.txt) [20] and the 2745 samples of the Munsell atlas. 2 Problem definition According to Carter and Carter [4], selecting a subset of highly contrasting colors from a given range of colors means choosing a subset such that in it the minimal distance among all possible couples of colors is maximal. More formally, we let N = {c,..., c n } be the given set of n colors, and K any subset of N of cardinality k << n. Denoting by C the set of all possible subsets of N with exactly k elements, by d ij the distance between color c i and color c j, and by d min the minimal distance among couples of colors belonging to an arbitrary subset K, the problem can be formulated as follows: max {d min} = max { min {d ij}} () C C {c i,c j} K We have, in a previous study [3], transformed this problem into a combinatorial optimization problem on graphs in order to be free to consider any color space and arbitrary dissimilarity functions among colors (not necessarily a distance in a given color space) and thus take into account some ergonomical or application requirements. With this abstract formulation of the problem, we can also precisely state its computational complexity. We let G = (V, E, W ) a complete, undirected weighted graph; V is the set of nodes, E the set of edges, and W : E R + a function which assigns to each edge a positive weight. We interpret each element of V as representing a color of the given set N and assign a weight to each edge {i, j} E throughout the function W ({i, j}) = d ij where d ij, is the chosen dissimilarity measure between the colors whose corresponding nodes in G are connected by the edge {i, j}. Of course, W ({i, i}) = 0 for all i ( i n). For the sake of simplicity, from now on we shall write W ij instead of W ({i, j}). The task of selecting a subset K of k highly contrasting colors from N, the set of n colors, can be considered equivalent to that of choosing, among all subgraphs 2

3 of G having k nodes, the subgraph Ĝ such that the minimal weight is maximal. More precisely: MDC (Maximal Dissimilarity among colors): Instance: a complete, undirected, weighted graph G = (V, E, W ), a positive integer k < V. Solution: Ĝ = ( V, Ê, Ŵ ), a subgraph of G having exactly k nodes; Ŵ is the restriction of the function W on the set Ê E. Cost function: H(Ĝ) = min {i,j} Ê {Ŵij} Goal: Max The decision version of this combinatorial optimization problem is NP complete; in fact it can easily be proved that it is in NP, and a simple polynomial time reduction can be designed from the well known NP complete problem CLIQUE [0]. Since the decision version is NP complete, the corresponding optimization version can not be solved exactly by polynomial time algorithms unless P = NP. Developing an idea presented by De Corte [7], instead of addressing this problem directly we take a different, but related one for which we have designed an algorithm that gives good approximate solutions. We call this different problem MSID. We are able to show the precise relation between the two problems; in particular, we can estimate a lower bound to the error that can be made by solving MSID instead of MDC. MSID (Minimum of the sum of the inverses of the dissimilarity raised to α): Instance: a complete, undirected, weighted graph G = (V, E, W ), a positive integer k < V. Solution: Ĝ = ( V, Ê, Ŵ ), a subgraph of G having exactly k nodes; Ŵ is the restriction of the function W on the set Ê E. Cost function: F α (Ĝ) = {i,j} Ê Goal: Min (Ŵ ij) α It can be proved, with arguments very similar to those presented above, that the decision version of this problem is also NP complete. Solving MSID exactly means finding the minimum of the sum of the inverses of the dissimilarities raised to α; this may produce pairs of colors which are not dissimilar enough. However, the following theorems show the relation between α and the error ɛ that can be incurred by solving PROBLEM 2 instead of MDC. De Corte [7] has already noted that both formulations coincide when α. Given G = (V, E, W ) we define H α (G) = min {i,j} E { ij ) α }. The relation between F α (G) and H α (G) is given by Theorem 2. H α(g) F α(g) H α(g) V 2 Proof : The left inequality follows directly from definitions of H α (G) and F α (G); the right inequality is derived as follows F α (G) = H α (G) {i,j} E ij ) α H α(g) (2) NP is the class of problems solvable by non deterministic Turing machines in polynomial time; P is the class of problems solvable by deterministic Turing machines in polynomial time. 3

4 = = H α (G) H α (G) {i,j} E {i,j} E ij ) α ( min ij) α ) {i,j} E ( ) α H(G) H α (G) V 2. W ij This last inequality derives from the observation that H(G) W ij for all {i, j} and that the edges in G are at most V 2. Now, given two arbitrary complete, undirected, weighted graphs, G = (V, E, W ) and G 2 = (V 2, E 2, W 2 ), such that V = V 2 = k, we can state that Theorem 2.2 If H(G ) < H(G 2 ) then there is an α such that F α (G ) > F α (G 2 ) Proof : From theorem 2. we can write the following relations: H α (G 2 ) F α(g 2 ) H α (G 2 ) k2 (3) k 2 H α(g ) F α (G ) H α(g ) (4) and multiplying side by side, considering only the right inequality, obtain ( ) α F α (G 2 ) F α (G ) H(G ) k 2 (5) H(G 2 ) and Since by hypothesis H(G ) < H(G 2 ), there is an ɛ > 0 such that H(G ) H(G 2 ) = ɛ (6) F α (G 2 ) F α (G ) ( ɛ)α k 2 (7) We want to find an α such that ( ɛ) α k 2 <. Taking the logarithm and approximating lg( ɛ) by its Taylor first order expansion, we obtain α > 2 lg k ɛ Theorem 2.2 implies that if we knew the ɛ > 0 that satisfies for each G r, G s pair of complete undirected weighted graphs with k nodes the relation (8) H(G r ) H(G s ) ɛ (9) we could set α so as to make MSID equivalent to MDC. Since we usually do not know ɛ, the relation α > 2 lg k ɛ gives us an idea of the error that we must accept in minimizing the sum of the inverses of the dissimilarity raised to a fixed α instead of maximizing the minimal dissimilarity. This result is useful, as it is easier to design efficient algorithms for minimizing the sum. We have already used continuous Hopfield networks to do so [3] since this problem formulation can be very simply translated into a neural network algorithm. The local search algorithm presented in the following section outperforms the Hopfield network. 4

5 3 Algorithm The problem of selecting high contrast color sets can be expressed as one of minimizing a quadratic function subject to integer constrains: we denote by x i ( i n) n binary variables, each corresponding to a graph node. If we select node i for the set of k maximally contrasting colors, then x i assumes value, otherwise it assumes value 0. Finding the desired subgraph Ĝ having k nodes is the equivalent of minimizing the function that with abuse of notation we still denote as F α, F α (x,..., x n ) = n i=,j>i ij) x α i x j under the condition that only k of the n binary variables have value. More formally { n min i=,j>i ij) x α i x j ( n i= x (0) i) = k We recall the bijective correspondence among subsets with k elements of a given set with n elements and the vectors in the hypercube {0, } n with exactly k elements equal to. In particular, denoting with V k an arbitrary subset with k elements of the set V with n elements, the corresponding vector x(v k ) = (x,..., x n ) {0, } n is defined: x i = iff i V k. Given x(v k ) we say that vector x can be reached from x in a step if it can be obtained by selecting two elements, x i = and x j = 0, of x and exchanging their values, that is setting x i = 0 and x j =. A neighborhood of x is given by all the vertices of the hypercube {0, } n obtainable from x vertex by setting one of the n k elements which have value 0 at and setting one of the k elements which have value at 0. Without loss of generality let us consider a vector x = (x,..., x n ) with k elements with value and the vector x obtained from x by interchanging x and x 2. If F α (x) = F α (x, x 2,..., x n ), then F α (x ) = F α ( x, x 2,..., x n ) and F α = F α (x, x 2,..., x n ) F α ( x, x 2,..., x n ) () = n n i=3 j>i ij ) α x ix j + 2 ) α x x 2 + j>2 j ) α x x j + i>2 i2 ) α x ix 2 n n ij ) α x ix j + 2 ) α ( x )( x 2 ) i=3 j>i + j>2 j ) α ( x )x j + i>2 i2 ) α ( x 2)x j = (x + x ) j ) α x j + (x 2 + x 2 ) j>2 j>2 2j ) α x j Denoting (x + x ) with t, it is easily seen that (x 2 + x 2 ) = t and F α = t j>2 j ) α x j t j>2 2j ) α x j (2) = t j>2 j ) α x j j>2 2j ) α x j Since the goal is the minimization of F α, we must find the conditions which make 5

6 F α > 0, that is: t > 0 and j>2 j ) α x j > j>2 2j ) α x j (3) or t < 0 and j>2 j ) α x j < j>2 2j ) α x j (4) Supposing without loss of generality that x =, then F α > 0 only if: j>2 j ) α x j > j>2 2j ) α x j (5) If this condition is satisfied, the interchange between the values of x and x 2 makes F α decrease. After these preliminary observations, we can now sketch the algorithm. Algorithm A Input: A complete undirected weighted graph G = (V, E, W ), an integer k and a real α. Step : Choose randomly a subset V k V with k elements and build the corresponding vector x(v k ) Step 2: For each couple x i, x j x(v k ) such that x i = and x j = 0, Estimate A = il ) α x l (6) and B = l i,l j l j,l i If A > B then x i := 0, x j := and restart the cycle. Output: The chosen subset V k V. jl ) α x l (7) It can easily be seen that this algorithm works in a time polynomial with respect to the dimension of the input. In particular (k )(n k) iterations are required at most, and each iteration costs O(n). In the worst case (k n/2) the time complexity is therefore O(n 3 ). 4 Experimental results The performance of the algorithm 2 has been evaluated on three different color sets: the 267 colors of the ISCC-NBS color naming system [2], the 502 different colors of the X Windows System palette (rgb.txt) [20] and the 2745 samples of the Munsell atlas as indicated in [9]. The first two sets have been chosen since each color is described not only by its coordinates but also by a name and color names can be useful in most applications. The Munsell atlas offers a wider gamut of colors and allows, by the ISCC-NBS system, to associate a name to the selected colors []. The ISCC-NBS color naming system was proposed in 955 by the Inter-Society Color Council and the National Bureau of Standards to provide a standard color naming system that could be used in non-specialized applications. It partitions 2 the code is written in C++ and is available free of charge by request. 6

7 the Munsell color system into 267 blocks, each named in terms of its hue (red, orange, yellow, green, etc.), saturation (grayish, moderate, strong and vivid) and lightness (very dark, dark, medium, light, and very light). Each block contains approximately 40, 000 discriminable color stimuli, and is represented, by name and by the Munsell coordinates of the centroid of the color block. These centroids are usually determined by graphical interpolation and in general do not correspond to the center of mass of the blocks, which have an irregular shape and vary greatly in size. The rgb.txt palette, which is of common use in the UNIX c environment, is composed of a 502 colors labelled by linguistic tags. As dissimilarity function we have chosen the Euclidean distance Eu,v in the CIELUV space, since this choice allows us to compare our results with those found in the literature [] [5] [5]. We realize that the CIELUV color space is designed to measure small color differences between uniform color samples on a neutral background, and that, although it has been used to compute medium and high color differences in nominal coding, there is some doubt as to its accuracy in this task (see, for example, [7].) However we are not defining an actual application-independent function here, but evaluating the performance of the algorithm, which is independent of the dissimilarity function chosen. We have implemented an interpolation technique to obtain from the Munsell triple the corresponding CIE(x,y,Y) coordinates, which we have then transformed into CIELUV coordinates. The mapping between the RGB values and the corresponding CIELUV ones has been done for a Barco Calibrator display using its own colorimetric profile. The experiments have been conducted in this manner: for each set of colors (ISCC-NBS system, rgb.txt palette and Munsell atlas) we have selected subsets with cardinality k varying from 4 to 50. For having the problem MSID equivalent to the problem MDC, the higher the parameter α the better is (see Section 2). We set α = 90 in order to avoid approximation errors using standard data types to represent real numbers (double of ANSI/IEEE Std ). Since the algorithm has a random initialization, we present the best results obtained on 0 different random initial conditions. Results obtained on the ISCC-NBS set are presented in Table ; the cardinality of the selected subset is denoted by k, d min, d av, d max denote the minimal, the average, and the maximal distance among all pairs of colors selected by the algorithm, while time gives the number of seconds required for running the algorithm on a Sun Ultra Creator 3D (UltraSPARC 67MHz processor with 52MB of RAM). K d min d av d max time (s) Table : The results of A on the ISCC-NBS palette Table 2 and Table 3 show the results obtained for the same values of k with initial sets the rgb.txt palette and the Munsell atlas respecitvely. To give an idea of the selected colors, the outputs of the algorithm with k = 8 and the three initial palettes is reported in Table 4, 5, and 6. Table 7 shows the 7

8 K d min d av d max time (s) Table 2: The results of A on the rgb.txt palette K d min d av d max time (s) Table 3: The results of A on the Munsell atlas selection of 22 colors from the Munsell atlas. Comparing Table 4 and 6 it can be seen that five out of the eight colors have the same name but different coordinates, while Tables 7 shows that two couples of colors labeled with the same names have been selected. To avoid this fact one can simply build the initial graph in such a way that all the couples of colors in the same ISCC-NBS block, within the Munsell atlas, are assigned distances very close to zero. L* u* v* Hue Value Chroma Color name Light Pink Vivid Orange Vivid Yellow Green Vivid Green Very Dark Green Vivid Greenish Blue Vivid Purple Vivid Purplish Red Table 4: The 8 colors chosen by A from the ISCC-NBS palette Another simple observation is that the couple of black and white is missing from all the tables. Black and white would probably have been selected by most people working in an interactive mode, but since the algorithm simply minimizes the objective function, sets selected on the basis of the CIELUV distance may not look optimal from a perceptual point of view. The algorithm, however, allows one or more colors to be fixed and completing the set with other maximally contrasting colors. As an example we made the algorithm to choose 22 colors from the Munsell atlas having fixed black and white. The result is shown in Table 8 and it can be 8

9 L* u* v* Red Green Blue Color name black blue green yellow sienna red magenta PaleTurquoise Table 5: The 8 colors chosen by A from the rgb.txt palette L* u* v* Hue Value Chroma Color name Vivid Orange Vivid Greenish Yellow Vivid Yellowish Green Vivid Green Vivid Greenish Blue Vivid Purple Light Purlish Pink Vivid Purplish Red Table 6: The 8 colors chosen by A from the Munsell atlas L* u* v* Hue Value Chroma Color name Vivid Reddish Orange Strong Reddish Orange Vivid Orange Yellow Dark Brown Light Olive Brown Vivid Greenish Yellow Vivid Yellow Green Very Light Yellowish Green Deep Yellowish Green Vivid Yellowish Green Vivid Bluish Green Vivid Bluish Green Vivid Greenish Blue Very Light Blue Vivid Blue Vivid Purlish Blue Vivid Violet Vivid Purple Vivid Reddish Purple Vivid Purplish Red Light Purlish Pink Vivid Purplish Red Table 7: The 22 colors chosen by A from the Munsell atlas 9

10 seen that the minimal distance among couple of colors is reduced. K d min d av d max time (s) Table 8: The results of A on Munsell atlas with White and Black fixed Our results have been sistematically compared with those obtainable by the well known Carter and Carter s algorithm [5] applied to the Munsell atlas (Table 9). Of course the best of 0 different trials has been considered for comparison. It can be noted that the minimal, average, and maximal distances obtained by our algorithm are significantly higher than those obtained by the Carter and Carter s, however the running time of the latter is significantly lower. K d min d av d max time (s) Table 9: The results of Carter and Carter s algorithm on the Munsell atlas Table 0 presents the 22 colors selected by Carter and Carter s algorithm. Also in this case two couples of colors labeled with the same names have been selected and black and white are missing. As regards as the comparison with the 22 colors selected by Kelly [] from the ISCC-NBS set we must say that the minimal distances found by our algorithm is at least double, but we observe that Kelly selection has been done heuristically by the author and it has already been pointed out [] that the selected list does not provide maximum discriminability for any given number of colors. The results we have obtained are very satisfactory and we can say that our approach is particularly well suited to work with finite set of colors and makes it very simple to provide for constraints while constructing the graph: the algorithm does not need to take them into account. In some nominal coding tasks, such as maps of land cover, the assignment of colors to the various classes may apply certain conventions: for example certain types of vegetation may be coded by colors chosen from the gamut of greens, while different types of bare soil may be coded by yellows and browns. The algortithm permits this type of color selection. The user can run the algorithm to select a given number of colors in the first range of hues (the greens, for example). Once selected the greens a new initial palette is built adding the yellow-brown range; the algorithm is then run again keeping the greens fixed. The output will be therefore the already selected greens and the required number of yellow-brown colors that will be higly contrasting both with respect to the greens and one with respect to each other. To conclude we observe that on sets with cardinality much greater than that of the Munsell atlas the algorithm may be too time consuming to be used in an interactive environments. We have therefore designed a modified version of A to deal with this aspect: 0

11 L* u* v* Hue Value Chroma Color name Vivid Red Vivid Reddish Orange Strong Yellowish Pink Vivid Orange Strong Orange Yellow Vivid Yellow Green Light Yellowish Green Vivid Yellowish Green Vivid Yellowish Green Vivid Green Vivid Bluish Green Brilliant Bluish Green Strong Greenish Blue Vivid Purlish Blue Vivid Purlish Blue Pale Violet Vivid Purple Strong Reddish Purple Vivid Reddish Purple Strong Purlish Pink Brilliant Purlish Pink Vivid Purplish Red Table 0: The 22 colors chosen by Carter and Carter s algorithm from the Munsell atlas

12 Algorithm A M Input: A complete undirected weighted graph G = (V, E, W ), an integer k, a subgraph G s = (V s, E s, W s ) of G with V s = n >> k, a control parameter r. Step : Apply algorithm A to the graph G s Step 2: For each node i ( i n) selected by algorithm A, search all nodes j G s such that W ij r Step 3: G s := G s {j j G s and W ij r} Step 4: If no node has been added then STOP, else GO TO Step. Output: The chosen subset V k V. Acknowledgments We thank Profesor Alberto Bertoni for many fruitful discussions and helpful suggestions. References [] Agoston, A. (987) Color theory and its applications in art and design, Springer-Verlag. [2] Bertin, J. (983) Semiology of graphics, University of Wisconsin Press, Madison. [3] Campadelli, P., Mora, P., and Schettini, R. (995) Color set selection for nominal coding by Hopfield networks. The Visual Computer, : [4] Carter, R. C., Carter, E. C. (98) Color and conspicuousness. Journal of Optical Society of America, 7: [5] Carter, R. C., Carter, E. C. (982) High-contrast sets of colors. Applied Optics [6] Carter, R. C., Carter, E. C. (988) Color coding for rapid location of small symbols. Color research and applications 3: [7] De Corte, W. (988) Ergonomically optimal CRT colours for non fixed ambient illumination conditions. Color Research and Application 3(5): [8] De Corte, W. (990) Recent developments in the computation of ergonomically optimal contrast sets of CRT colours. Displays, : [9] Della Ventura, A., and Schettini, R. (993) Computer aided color coding. Communication with Virtual Word (N.M. Thalmann, D. Thalmann eds), Springer- Verlag, Tokyo, [0] Garey, M. R., Johnson, D.S. (979) Computers and intractability. Freeman and Company. [] Kelly, K. L. (976) Twenty-two colors of maximum contrast. Color Engineering, 3: [2] Kelly, K. L, Judd, D.B. (976) Universal language and dictionary of names. NBS Special Publication 440. Washington D.C., U.S. Government Printing Office. 2

13 [3] Mac Donald, L.W. (990) Using colour effectively in displays for computerhuman interface. Displays, :29-4. [4] Olson, J.M. (987) Color and Computer in Cartography. In Color and the Computer, Academic Press [5] Silverstein, L.D., Lepkowski, J.S., Carter, R.C., Carter, E. C. (986) Modeling of display color parameters and algorithm color selection. Procedings of SPIE, Advances in Display Technology 6 624: [6] Van Laar, D., Flavell, A. (994) A name based algorithm for generating maximally discriminable color sets. J. of Photo. Science 42: [7] Stalmeier, P.F.M., de Weert, C.M.M. (99) Large Color differences and selective attention Journal of Optical Society of America (A) 8: [8] Travis, D. (99) Effective Color Displays, Theory and Practice. Academic Press. [9] Wyszecki, G., Stiles, W.S. (982) Color Science. John Wiley and Sons. [20] O Really and Associates, ( ) Definitive Guides to the X Windows System, Vol. : Xlib Programming Manual. O Really and Associates. 3