An Effective Color Scale for SimultaneousColor and Gray-Scale Publications

IEEE SIGNAL PROCESSING MAGAZINE [82] JANUARY 2006

[dsp TIPS&TRICKS]James McNames

1053-5888/06/$20.00©2006IEEE

The use of images to representproperties of signals is ubiq-uitous in signal processingapplications. In most cases, acolor scale is used to repre-

sent a single dependent variable as afunction of two independent variables.Figure 1 shows a typical example of aspectrogram that represents how the sig-nal power is distributed across time and

frequency. The image is pseudocoloredwith a common color scale that roughlyresembles a rainbow. This articledescribes a new color scale that has sev-eral advantages over existing color scalessuch as the one shown in Figure 1.

Many conference proceedings andjournal publications are now distributedin both printed and electronic forms.Gray scale is required to unambiguouslyrepresent images in printed publica-tions. However, color is increasinglyavailable for electronic publications at

no additional cost. Unfortunately, mostcolor scales are distorted when convertedto gray scale. This diminishes the abilityof the image to convey critical informa-tion and can cause ambiguities. To facil-itate publications in both formssimultaneously, this article proposes acolor scale that appears as a monotonicgray scale in printed form and signifi-cantly enhances the image resolutionwhen viewed in color.

Luminance (or gray scale) is crucialto discern spatially high-frequency fea-tures and specific forms in images, suchas peaks, ridges, and valleys [1], [2].However, the addition of color canimprove the accuracy with which we candiscern specific values from images [3].

Although some commercial tools areavailable to tune a color scale to enhancethe features of interest in a specificimage [2], in most applications and pub-lications, a general-purpose color scale isused. An algorithm for finding an opti-

mal color scale was proposed in [4], butthis color scale was inferior to a lin-earized gray scale in a lesion detectiontask for brain images. The color scaleproposed here is easier to implement,has fewer design constraints, and mayhave better image resolution. A nine-level color scale with monotonic lumi-nance was described in [5], but it is notclear that this color scale has any advan-tages over other current color scaleswith monotonic luminance.

Selecting a single color scale for pub-lications is difficult, partly because mostdevices (e.g., monitors, color printers)have a different gamut of colors that theyare capable of displaying. This is also dif-ficult because the appearance of a colordepends on the colors in the neighboringregions of the image. Thus, the bestcolor scale depends on the image proper-ties, the features of interest, and thegamut of colors available for the displaymedium.

SELECTING A COLOR SCALEFour design principles guide the selec-tion of an effective color scale. First, tomaximally enhance the image resolu-tion when displayed in color, the colorscale should cover as much of therange of available colors as possible,

“DSP Tips and Tricks” introduces prac-

tical tips and tricks of design and

implementation of signal processing

algorithms so that you may be able

to incorporate them into your

designs. We welcome readers who

enjoy reading this column to submit

their contributions. Contact Column

Editors Rick Lyons (r.lyons@ieee.org)

or Amy Bell (abell@vt.edu).

[FIG1] Spectrogram of an arterial blood pressure signal.

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IEEE SIGNAL PROCESSING MAGAZINE [83] JANUARY 2006

subject to the constraints that theluminance increases monotonically(for gray scale publications).

Second, neighboring colors through-out the scale should be as distinct as pos-sible. This is motivated in part by thelinear separation theory, which statesthat objects can be more easily identifiedwhen their color cannot be formed by a

linear combination of the neighboringcolors [6]. For example, it is more diffi-cult to identify an orange object sur-rounded by red and yellow than an objectsurrounded by red or yellow colors alone.

Third, the perceptual differencebetween two colors should be approxi-mately proportional to the differencebetween their positions along the color

scale [7]. This is difficult to achieve inpractice because different devices rendercolors differently and because most common computer packages, such asMATLAB, do not permit the specificationof colors in perceptually uniform colorspaces. At the very least, similar colorsshould not appear at points that are farapart on the color scale.

[FIG2] Projections of the color spiral onto (a), (d), and (g) the red-green, (b), (e), and (h) red-blue, and (c), (f), and (i) green-blue planes inthe RGB color space. The window functions for the initial green and blue values are shown by the green and blue boundaries,respectively. (a), (b), and (c): The initial windowed complex sinusoid. (d), (e), and (f): Projections after the first rotation around the blueaxis. (g), (h), and (i): Projections after a second rotation around the green axis. The projection onto the unit cube is shown by the grayregion. After the second rotation, all three projections show that the color spiral is confined to the unit cube.

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IEEE SIGNAL PROCESSING MAGAZINE [84] JANUARY 2006

Fourth, the color scale should beintuitive. Humans perceive color inthree dimensions because we havethree types of receptors that sense lightwith sensitivities near blue, yellow-green, and yellow-red frequencies.However, we perceive and process colorimages in terms of luminance (onedimension) and color (two dimen-sions). Although there are many possi-ble representations of color, peoplemost naturally perceive color in termsof its saturation and hue. Saturation is

a measure of how pure or vivid thecolor is, as compared to a gray of thesame luminance. Hue is the actualshade or tint of the color and usuallycorresponds to a specific wavelength oflight in the visible spectrum (i.e., therainbow). The luminance, saturation,and hue should vary smoothly alongthe color scale. People can also per-ceive color in terms of opponent colorchannels that consist of blue-to-yellowand red-to-green variations [1], [8].Thus, the color scale will have greaterperceptual color resolution if it con-

tains a blue-to-yellow transition and ared-to-green transition.

PROPOSED COLOR SCALEIt is generally accepted that the bestcolor scales have monotonic lumi-nance and cycle through a variety ofhues [1]. In most three-dimensional(3-D) color spaces, these scales appearas spirals. There is little in the litera-ture to guide the mathematical con-struction and implementation of thesecolor scales. A particularly simple and

efficient spiral can be formed out of thefamiliar complex sinusoid that is thebasis of linear systems analysis andspectral signal analysis.

Suppose that we specify colors interms of red, green, and blue (RGB) val-ues that range from zero (off) to one(maximum intensity). The gamut ofpossible colors is confined to the unitcube within this color space. Define t asan independent index to the color scalethat monotonically increases from zeroto a maximum value of

√3, the

Euclidean distance from black (RGB =

000) to white (RGB = 111) in the unitcolor cube. Define the initial RGB val-ues as follows:

r0(t) = t

g0(t) = w(t)cos

(p

2π√3

(t −

√3

2

))

b0(t) = w(t)sin

(p

2π√3

(t −

√3

2

)),

where p is a user-specified parameterthat controls the number of cycles thatoccur over the range of the color scale,and w(t) is a window function that con-trols how rapidly the spiral expands andcontracts. Figure 2(a), (b), and (c) showsthe projections of this initial spiral onthe red-green, blue-red, and blue-greenplanes of the color cube.

The second set of RGB values isformed by rotating the spiral counter-clockwise around the blue axis by theangle given by θt = sin−1(1/

√3)

(approximately 35.2◦). This is most easilycomputed by converting r0(t) and g0(t)to polar coordinates, increasing theangle by θ1, and converting them back torectangular coordinates. This rotationdoes not alter the initial blue values.After this rotation, the green values areconfined to the allowable range of 0–1.Figure 2(d), (e), and (f) shows the projec-tions of the resulting spiral onto each ofthe three color planes.

The final set of RGB values is formedby performing a second rotation counter-clockwise around the green axis by 45◦.This rotation does not alter the greenvalues. After this rotation, all three pro-jections are confined to the unit colorcube. Figure 2(g), (h), and (i) shows theprojections of the resulting spiral ontoeach of the three color planes. Figure 3shows a 3-D plot of the spiral.

The code that implements this colorscale in MATLAB is given below. The usermust define the number of colors withnc and the number of periods (i.e., thefrequency) of the complex sinusoid withnp. A MATLAB function implementationof this color scale is available athttp://bsp.pdx.edu.

[FIG3] Trajectory of the spiral color scale through the RGB unit color cube.

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wn = sqrt(3/8)*[0;triang(nc-

2);0]; % Triangular

window function

a12 = asin(1/sqrt(3));

% First rotation angle

a23 = pi/4;

% Second rotation angle

t = linspace(sqrt(3),0,nc).’;

% Independent variable

r0 = t; % Initial red values

g0 = wn.*cos(((t-

sqrt(3)/2)*np*2*pi/sqrt(3)

)); % Initial green values

b0 = wn.*sin(((t-sqrt(3)/2)

*np*2*pi/sqrt(3)));

% Initial blue values

[ag,rd] = cart2pol(r0,g0);

% Convert RG to polar

[r1,g1] = pol2cart

(ag+a12,rd); % Rotate

& convert back

b1 = b0;

[ag,rd] = cart2pol(r1,b1);

% Convert RB to polar

[r2,b2] = pol2cart

(ag+a23,rd); % Rotate

& convert back

g2 = g1;

r = max(min(r2,1),0);

% Ensure finite precision

g = max(min(g2,1),0);

% effects don’t exceed

b = max(min(b2,1),0);

% unit cube boundaries

mp = [r g b];

% Final colormap matrix

colormap(mp); % Apply to

the current figure

The gray scale equivalent of a colorin this space is the average of the RGBintensities. Geometrically, this is theorthogonal projection from the 3-Dcolor space onto the one-dimensionalchord that spans from black (RGB =000) to white (RGB = 111). This pro-jection of the proposed color spiralonto the gray scale axis is equivalent toprojecting g0(t) and b0(t) onto the redaxis prior to the two rotations,g0(t) = b0(t) = 0. In this case, the grayscale equivalent is just a line of length

√3 that initially lies along the red axis

and is rotated twice to fit in the unitcolor cube. This is shown as the graychord in Figure 3.

To ensure that the final color spiralfits entirely within the unit color cube,the window function must not be anygreater than a triangular window func-tion with a maximum amplitude of

√3/8. Although a triangular window

could be used, the slope of this functionis discontinuous at the midpoint of thecolor scale t = √

3/2. In some applica-tions, this may create an artificialboundary in the pseudocolored image.The hyperbolic function is a good alter-native that circumvents this problem.All of the examples in this article used a

[FIG4] Examples of (a) nine color scales and (b) their corresponding luminance, whichwould be produced in gray scale reproductions. Row (9) depicts the new color scale.

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(9)

[FIG5] (a) Red, (b) green, (c) blue, and (d) luminance values of each of the nine colorscales in Figure 4. Column (d) shows if the luminance is monotonic or not and, ifmonotonic, if the luminance is linear.

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IEEE SIGNAL PROCESSING MAGAZINE [86] JANUARY 2006

hyperbolic window with a maximumvalue of maxt w(t) = 0.95

√3/8, This is

an analytic function that prevents theappearance of artificial boundaries butclosely approximates the triangular win-

dow at values that are distant from themidpoint of the color scale.

The parameter p is the frequency ofthe complex sinusoid, and it controls thenumber of periods that occur over range

of the color scale. If p is too large, thensimilar colors may be assigned to regionsthat are far apart on the color scale. If pis too small, then only a small portion ofthe possible colors will be used, and little

[FIG6] Spectrogram of an arterial blood pressure signal created using each of the nine color scales. (a) The spectrograms in full color. (b)The spectrograms as they would appear in a gray-scale reproduction.

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additional resolution will be gained bythe application of color. In practice, val-ues in the range 1.5 ≤ p ≤ 3.0 offer areasonable compromise. All of the exam-ples in this article used p = 2.

COLOR SCALE PROPERTIESThe proposed spiral color scale has manynice properties and follows the fourdesign principles discussedearlier. The spiral isdesigned to smoothly cyclethrough a variety of hueswith maximum saturation,subject to the constraintthat the luminancedecreases linearly. Whenonly the luminance is viewed, the colorscale is identical to a linear gray scale. If aperceptual gray scale is preferred, it canbe obtained by scaling the initial scaleindex variable t accordingly [1]. The spi-ral is designed such that the brightesthue of green is obtained at the midpointof the scale. The dominant color preced-ing green is blue and the dominant colorthat follows is red. In this way, the spiralroughly resembles the familiar and intu-itive rainbow color scale.

EXAMPLESThe advantages of the proposed colorscale properties are illustrated in thefollowing examples. Figure 4 showsseven popular color scales that areincluded in MATLAB, the color scaleproposed in [5] (row 6), and the newcolor scale (row 9). Figure 4(a) and (b)shows how each color scale wouldappear in color and gray scale reproduc-tions, respectively.

The columns of Figure 5 show thecorresponding red, green, blue, andoverall luminance of the color scales.Notice that two color scales that roughlyresemble the natural “rainbow” spec-trum (rows 7 and 8) do not have amonotonic luminance (right column).The color scales in rows 3–6 have amonotonic but nonlinear luminance.This causes a higher contrast in someregions of the color scale. Only the colorscales in rows 1–2 and the new colorscale have linear luminance, whichensures that gray-scale contrast is uni-

form over the full range of the scale. Ofall the color scales with monotonicluminance, only the new color scale pos-sesses both a linear luminance and awide range of distinct colors to maxi-mize the color resolution.

Figure 6 shows the spectrogram ofFigure 1 for each of the nine color scales(a) and their corresponding gray scales

(b). The dominant features of this spec-trogram are the cardiac components(2.16 and 4.32 Hz), the cardiac sidebandsdue to pulse pressure variation (2.16 ±0.45 Hz and 4.32 ± 0.45 Hz), the respira-tory component (0.45 Hz), and the sec-ond respiratory harmonic (0.90 Hz).Each color scale is able to represent theamplitudes of these four componentsand the background intensity with differ-ent resolutions. The color scale in row 2does not have any better resolution thanthe linear gray scale in row 1. The colorscales in rows 3−6 show the low-ampli-tude harmonics and cardiac sidebandsless clearly than the linear gray scale.One can most clearly discern the ampli-tudes of each of the components fromthe color scale in row 7, but the gray-scale version of this figure is more diffi-cult to analyze due to the ambiguitiescaused by the nonmonotonic luminance.The amplitudes of each of the four domi-nant features can be discerned clearlyfrom the new spiral color scale (bottomrow) without compromising the resolu-tion of the gray scale version.

SUMMARYPseudocolored images are used in manysignal processing applications to repre-sent signal properties as a function oftwo independent variables, such as timeand frequency. The color scale describedin this article can be used to unambigu-ously represent these images in printedgray-scale publications as well as fullcolor electronic publications. This color

scale also allows for correct interpreta-tion by people with common types ofcolor blindness, which accounts forapproximately 10% of the male popula-tion and 1% of the female population.

While it may be possible to obtain abetter resolution with a color scale thatis designed for a particular image onhand, the proposed color scale works

well in many applicationswhere the user does nothave the time or resourcesto tailor the color scale to aspecific image. The pro-posed color scale is alsoeasy to implement and hasan intuitive interpretation

in that it cycles through a sequence ofhues that roughly resembles the famil-iar rainbow.

AUTHORJames McNames has been with theElectrical and Computer EngineeringDepartment at Portland State Universitysince 1999, where he is an associate pro-fessor and director of the BiomedicalSignal Processing Laboratory. His pri-mary research interest is statistical sig-nal processing with applications tobiomedical engineering and semiconduc-tor manufacturing. He has publishedmore than 90 peer-reviewed journal andconference papers.

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[2] B.E. Rogowitz and L.A. Treinish, “Data visualiza-tion: The end of the rainbow,” IEEE Spectr., vol. 35, no. 12, pp. 52–59, Dec. 1998.

[3] P. Rheingans, “Task-based color scale design,” in Proc.SPIE-Int. Soc. Opt. Eng., 2000, vol. 3905, pp. 33–43.

[4] H. Levkowitz and G.T. Herman, “Color scales forimage data,” IEEE Comput. Graph. Appl., vol. 12, no.1, pp. 72–80, Jan. 1992.

[5] C. Rappaport, “A color map for effective black-and-white rendering of color-scale images,” IEEEAntennas Propagat. Mag., vol. 44, no. 3, pp. 94–96,Jun. 2002.

[6] C.G. Healey and J.T. Enns, “Large datasets at aglance: Combining textures and colors in scientificvisualization,” IEEE Trans. Visual. Comput.Graphics, vol. 5, no. 2, pp. 145–167, Apr.–Jun. 1999.

[7] A.D. Kalvin, B.E. Rogowitz, A. Pelah, and A.Cohen, “Building perceptual color maps for visualiz-ing interval data,” in Proc. SPIE-Int. Soc. Opt. Eng.,2000, vol. 3959, pp. 323–335.

[8] P.K. Robertson, “Visualizing color gamuts: Auser interface for the effective use of perceptual colorspaces in data displays,” IEEE Comput. Graph. Appl.,vol. 8, no. 5, pp. 50–64, Sept. 1988.

THIS ARTICLE PROPOSES A COLORSCALE THAT APPEARS AS A

MONOTONIC GRAY SCALE IN PRINTEDFORM AND ENHANCES THE IMAGE

RESOLUTION WHEN VIEWED IN COLOR.

[SP]