José M. Medina and José A. Díaz, University of Granada, Spain
The measurement and characterization of lightness fluctuations, their scaling properties and their potential causes could help design low-noise
industrial painting processes and help engineer innovative color and texture effects.
The appearance of an automobile’s color is an essential quality criterion. Modern automotive exterior finishes often incorporate flake-shaped or effect pigments in the base coat to obtain high-quality metallic effects.1,2 Metal flakes (e.g., aluminum) are an important group of effect pigments with typical size of a few microns.
They act as tiny mirrors to produce specular reflection and are mainly responsible for lightness changes as a function of viewing angle.2
Extensive research has studied how metal flakes are deposited in automotive coatings. In research on macro-appearance, the issue of trial-to-trial color fluctuations of car parts is a complex matter that depends on flake orientation, among other factors. These random color fluctuations have a direct impact on color-matching between the car body and assembled add-on parts – called color harmony1 (Figure 1a) – and increase costs and time related to car manufacturing.
Figure 1. (a) The color-matching effect between the hood and bumper of an automobile – both the hood and bumper were coated by different manufacturers with a metallic-blue paint containing aluminum flakes. (b) Schematic representation of a multi-angle spectrophotometer.
Reflectance and colorimetric specifications
Industry specification standards for metal flakes have established a few angles of detection defined by the specular reflection, called aspecular angle γ (Figure 1b).2 Paint manufacturers, add-on parts suppliers and car production plants have established color tolerances or threshold values relative to a reference or “master” panel to accept or reject the painted pieces. However color tolerances are difficult to define near specular reflection2 and don’t take into account trial-to-trial correlations of color coordinates during the painting process.3 A pattern-matching method has been proposed for spectroscopic quality control before analysis of color tolerances.4
In a different analysis, research on the micro-appearance of metallic coatings has revealed distinct spatial nonuniformities such as sparkles, which are bright spots observed over a darker background under direct sunlight (Figure 2a).2 They are characteristic of flake-shaped pigments with large particle diameter (Figure 2b) and depend on flake orientation.2
Figure 2. (a) Color photograph of a metallic-blue paint under direct sunlight. Sparkles are brighter spots. (b) Reflection optical micrograph (20×) of a metallic-blue paint under bright-field illumination. The optical micrograph shows lenticular, or ‘silver dollar,’ aluminum flakes of different sizes covered with blue absorption pigment particles.
Spatially varying reflectance properties of sparkles is important to achieve better color and texture effects. However standard spectrometers are limited, with an average size of a few millimeters, so individual flakes can’t be easily discerned.
The color variability of add-on parts can be measured by analyzing the colorimetric coordinates of a predetermined number of identical painted pieces in exact temporal order. Therefore the trial number can index raw sequences of color coordinates. For instance, trial-to-trial lightness fluctuations from aluminum flakes can be visualized in the CIE 1976 perceptual color space (CIELAB) by using the lightness coordinate L*. The CIELAB space is recommended for pigmented thin films.1,2 Figure 3a shows lightness variations ΔL* for a series of identical planar wings coated with a conventional metallic-green paint.
Lightness fluctuations are more irregular near the specular reflection at γ = 15°. However, they gradually smooth and are farther from the specular at γ = 110°.
Figure 3. (a) Trial-to-trial CIELAB lightness variations ΔL* of a metallic-green paint containing aluminum flakes as a function of the aspecular angle γ. ΔL* variations are referred to the corresponding master panel. Horizontal gray dashed lines indicate the color tolerances, except at γ = 15° (not available). (b) Log-log plot of the power spectrum estimation of time series. (c) Simulation of fractional Gaussian noise for different values of the Hurst exponent H. (c) Log-log plot of the power spectra of fractional Gaussian noise. To enhance visualization in (a) and (c), time series were scaled in the vertical axis by different normalization factors (a.u. = arbitrary units).
Correlations are visually noticeable by the presence of wavelike patterns across time series.3 A plausible approach to examining color fluctuations consists of correlated noises.3 Each time series can be treated as a discrete-time stochastic process that is susceptible to power spectral analysis in the Fourier domain.
Figure 3b exemplifies the power spectrum of each time series in Figure 3a. The frequency f is measured in cycles per trial. The experimental results indicate the existence of an important class of power law noises that are ubiquitous in many complex systems.3 The scaling exponent or slope indicates weak correlations that resemble a flat power spectrum (white noise) at γ = 15°. However, the slope approaches unity far from the specular, suggesting stronger correlations referred to as 1/f noise.3 An interpretation of these results could be based on fractional Gaussian noise (FGN). FGN comes from fractional Brownian motion, and the Hurst exponent, H, characterizes it. The Hurst exponent, ranging between zero and unity, measures the strength of correlations. Values for H that are higher than 0.5 indicate persistent correlations; a positive deviation is more likely to be followed by another positive deviation (Figure 3c). Antipersistent correlations denote those values for H that are lower than 0.5, and the stochastic process becomes more irregular. In FGN, the power spectrum decays as f−(2H−1) (Figure 3d) and replicates the experimental results in Figures 3a and 3b.
Multispectral surface images
A multispectral imaging system usually collects a set of images with high spatial resolution across the electromagnetic spectrum, producing a multidimensional representation of an imaged surface. Multispectral methods provide more spatial-spectral details and more colors not discernible by the human eye than standard camera-based systems and spectrometers. Multispectral imaging in automotive coatings can be performed in the visible range at the micron scale (Figure 4a).
Figure 4. (a) Representation of a calibrated multispectral camera in a goniometric setup. (b) Schematics of multispectral imaging of a metallic-blue coating. (c) Representation in the sRGB color space of an imaged area (400 × 400 pixels) at the illumination θ = 15°. Open squares labeled as 1 and 2 exemplify two sensor pixels (size 14.4 µm per pixel) at different spatial positions. These two pixels capture an aluminum flake covered with blue pigment particles and the blue background (i.e., no sparkles), respectively. (d) Spectral reflectance factor of selected pixels. Their lightness difference ΔL* is displayed in the upper-right corner.
Angle-dependent lightness variations can be captured by changing the illumination angle θ from near to far in the surface normal (Figure 4b). For instance, multispectral images can be combined by producing a 3-D (x, y, λ) data cube or “image cube.” Each image cube contains the two spatial dimensions (x, y) with high resolution and the spectral dimension at a discrete sequence of wavelengths λ. User-defined regions of interest can be selected for classification of sparkles in the standard (sRGB) color space (Figure 4c) – this is recommended for visualization in displays and online. Figure 4d exemplifies the estimation of the reflectance factor of a single panel coated with a metallic-blue paint at θ = 15°. The reflectance factor is referred to a white reference reflectance standard. Sparkles from aluminum flakes could have reflectance values higher than unity, which are compatible with the law of energy conservation.2 In general, sparkles have higher reflectance and lightness L* values than the background (Figure 4d).5
Color maps from aluminum flakes, shown in Figure 4c, can be transformed pixel-by-pixel to the CIELAB color space. These color maps do not exhibit conventional Euclidean forms, but rather irregular point patterns.5,6 The shapes of these point patterns resemble fractal-like geometric forms and provide useful information about the colorimetric organization of sparkles.5 Fractals are objects or processes where every piece at a specific scale resembles the entire structure; that is, they are self-similar. One example is the Koch snowflake (Figure 5a).
Figure 5. (a) The Koch snowflake and its Hausdorff dimension. (b) Schematics of box-counting analysis of the Koch snowflake. (c) Ln-ln plot of the number of boxes as a function of the box size. The slope of the red straight line gives the box-counting dimension DB of the Koch snowflake. (d)-(e) Projection in the CIELAB L*b* plane of the color coordinates of an imaged area (200 × 200 pixels) of a metallic-blue and a metallic-yellow paint, respectively, at different illumination angles. The size of symbols was increased to enhance visualization. (f) Representation of the box-counting dimension DB at the illumination angle θ of 15°, 45° and 75°.
In the Koch snowflake, each segment is divided into four identical, self-similar parts and reduced by a factor of three. The length of the Koch snowflake depends on the scale of measurement and can be characterized by the Hausdorff dimension DH (Figure 5a). An estimation of DH can be obtained by box-counting analysis, wherein the number of superimposed grid boxes that cover the entire structure is related to their box size by a power law (Figure 5b). The box-counting dimension DB is a non-integer value or the exponent of power law scaling (Figure 5c). The higher DB, the more rugged the structure will be, and vice versa. The box-counting approach can be used to investigate fractal organization of sparkles relative to the background in the CIELAB color space.5 For instance, Figures 5d and 5e represent the 2-D L*b* planes for each pixel of the imaged scene of a metallic-blue and a metallic-yellow paint, respectively.
The coordinate b* is the orthogonal blue-yellow axis in the CIELAB space2 encoding the color gamut that is expanded by aluminum flakes and covered with conventional light-absorption pigments.5 L* values higher than 100 are usual in metallic coatings near the specular reflection2 (Figures 5d, 5e). In the examples analyzed, DB values from 2-D box counting indicate rugged color maps at the illumination angle θ of 15° and 45°. However, CIELAB color maps are smoother at θ = 75° (Figures 5d, 5e). This angle-dependent effect on DB is due to the presence of sparkles from aluminum flakes and could be displayed in a 3-D space for further classification analysis (Figure 5f).5
Meet the authors
Dr. José M. Medina is a visiting researcher at the University of Granada in Spain; email: email@example.com. Dr. José A. Díaz is an associate professor at the University of Granada in Spain; email: firstname.lastname@example.org.
1. H.J. Streitberger et al (2008). Automotive Paints and Coatings. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.
2. G.A. Klein (2010). Industrial Color Physics. Springer Science+Business Media LLC, New York.
3. J.M. Medina et al (2012). Low-frequency correlations (1/fα type) in paint application of metallic colors. Opt Express, Vol. 20, Issue 16, pp. 17560-17565 (doi: 10.1364/OE.20.017560).
4. J.M. Medina et al (2015). Classification of batch processes in automotive metallic coatings using principal component analysis similarity factors from reflectance spectra (Submitted).
5. J.M. Medina et al (2014). Fractal dimension of sparkles in automotive metallic coatings by multispectral imaging measurements. ACS Appl Mater Interfaces, Vol. 6, Issue 14, pp. 11439-11447 (doi: 10.1021/am502001m).
6. J.M. Medina et al (2011). Scattering characterization of nanopigments in metallic coatings using hyperspectral optical imaging. Appl Optics, Vol. 50, Issue 31, pp. G47-G55 (doi: 10.1364/AO.50.000G47).