Statistical Color Models (SCM)

1. Statistical Color Models (SCM) Kyungnam Kim

2. Contents • Introduction • Trivariate Gaussian model • Chromaticity models • Fixed planar chromaticity models • Zhu and Yuille’s SCM • Oriented planar model • Healey’s model • Bingham model • Intensity models • Other types of model

3. Introduction • Dichromatic and unichromatic reflection models • Dichromatic: • Surface reflected light: Varying with the geometry of the scene, i.e., relative viewing, illumination and surface normal directions. The angular distribution of this geometric factor tends to be strongly peaked around the specular direction. • Body reflected light: Arises from subsurface reflection where light enters the body of the material and gets scattered and partially absorbed by subsurface pigment before eventually being reflected back out of the surface. s(): surface reflectance SPD(spectral power density), i(): illuminant SPD, ms(g) and mb(g): coefficients, g: a vector defining scene geometry • Unichromatic: • Outside the specular peak, the scene radiance (L) is dominated by the body reflected light. • This model in fact is often sufficient since specular highlights are localized, sparse (can be considered noise).

4. Introduction • Sensor model: A light sensor samples the SPD of the light, L(), entering the camera to produce its response. This process is well modeled by a spectral integration. The sensor response x is defined: • If the reflection is unichromatic, x is constrained to the positive segment of the line passing through the origin parallel to xb. The direction of this line depends only on the shape of the sensor response functions, surface reflectance and illumination SPDs. • Lines radiating from the origin of the sensor response space may be considered to be chromatic equivalence classes. • The chromatic properties or chromaticity of a surface under a fixed illumination is thus characterized by the direction of the ray joining the origin to x. The distance from the origin to characterizes the intensity of the data.

5. Introduction • (1) Sensor noise, (2) variations in the SPD of the incident light, and (3) variations in the reflectance SPD will perturb x (intensity and chromaticity). • These perturbations give rise, given surface patch, illuminant and sensor, to statistical variations in the color of the corresponding image region.  needs statistical modeling of color.

6. Trivariate Gaussian model • A common statistical model is provided by the multivariate Gaussian distribution. • The parameters of the distribution, the mean and covariance, are straightforward to estimate from a sample distribution of data by maximum likelihood estimations. • The fixed likelihood boundary has the ellipsoidal shape characteristics of the multivariate Gaussian distribution.

7. Chromaticity models • Assuming unichromatic model of reflection, the intensity and chromaticity of the data are independently distributed. • For an approximately single object, we expect the chromaticity distribution to have a single mode. which is peaked about a mean direction. • Chromaticity based SCMs generally consist of a Gaussian model of normalized RGB data (chromaticity). These models have conical contours in the RGB color cube with elliptical cross-section.

8. Fixed Planar Chromaticity Models • Represent chromaticity by the point of intersection of the corresponding line through the origin of the RGB cube with a fixed plane, ex., unit plane. • Normalized color coordinates are obtained by taking the L1 norm of the RGB vector: • Model chromaticity distribution with 2D Gaussians in the normalized color space (NCS).

9. Zhu and Yuille’s SCM • Ensure a consistently tight model by choosing a plane whose normal is aligned with the mean direction of the distribution (xf). • Its contours in the RGB cube are cylindrical rather than conical. • This strategy has the effect of down-weighting low intensity data. • Chromatic component of low intensity data is more sensitive to noise (poor resolution in the dark corner of the RGB cube).

11. Oriented Planar Model • Unlike Zhu and Yuille’s model, chromaticity data is obtained by mapping each RGB vector to the point of intersection of the ray from the origin to that RGB point with the plane perpendicular to the mean of the sample distribution. • The distribution of points in that plane is modeled by a 2-D Gaussian.

12. Healey’s Model • A more natural and consistent representation for the direction of a ray from the origin is by the L2 unit vector in that direction (by a point on the unit sphere). • Based on L2 normalization of RGB data. • First a trivariate Gaussian model is constructed. The intensity component of the SCM is then made uniform by L2 normalizing the parameters of the Gaussian model. The mean of the model is found by normalizing the mean vector of the trivariate Gaussian model and the covariance matrix is found by normalizing the trivariate covariance matrix by the squared magnitude of the mean.

13. Intensity Models • Different definitions of the intensity, |x|, that are consistent with the individual models need to be adopted. • For the spherical model – Healey model • For the Gaussian NCS model based on L1 norm • For the oriented planar, and Zhu and Yuille models • Uniform intensity model: highly non-rigid body. no knowledge about the distribution of intensities. • Gaussian intensity model: rigid object. The combination of a Gaussian intensity model with any directional chromaticity model.

14. Other Types of Model • Many common color spaces in which the chromaticity and intensity components are explicitly separated. • HSV space is a non-linear transformation of the original RGB space. • Fitting a Gaussian distribution in the chromatic subspace (H and S) is generally a poor choice since the non-linearity of the transformation can greatly complicate the shape of the chromaticity distribution. • Hue is an angular measurement which contains a jump discontinuity at 0 degree (pure red) and is undefined for all shades of grey (R=G=B). • Non-parametric SCMs is more accurate when sufficient sample date is available.