Polynomial Texture Mapping (Malzbender et al. 2001) refers to a set of computer graphics and image processing methods where each pixel in the image contains information about luminance as a function of the direction of incoming light. The light direction (lx,ly) is given as the x and y coordinates of the projection of the normalized light vector onto the image plane. The luminance function for each pixel is approximated by a biquadratic polynomial in lx and ly:
L = a0 lx2 + a1 ly2 + a2lxly + a3lx + a4ly + a5
The six coefficients a0–a5 are stored in addition to the unscaled RGB (red/green/blue) values of the pixel. Alternatively, such a function can be used for the R, G, and B color channels separately in order to capture possible change in color as a function of the direction of incoming light.
By making a series of photographs with camera and specimen in fixed positions, but with varying directions of incoming light, the coefficients of the luminance polynomial for each pixel can be computed by a least-squares method (regression) using Singular Value Decomposition. Given this data set, the digital image can be interactively manipulated by moving virtual light sources on the computer screen, giving the illusion of moving one or several lamps around the real specimen. The luminance polynomial can also be extrapolated beyond the original range of lighting angles, simulating the effect of light sources in "impossible" positions almost parallel to, or even below, the surface of the specimen.
The surface-normal vector for each pixel can be estimated by finding the direction of maximal luminance. A new image can then be generated using any of the synthetic light reflection models known from computer graphics, such as Phong lighting (Phong 1975). The parameters of the synthetic reflection model can be freely manipulated, giving control of surface parameters such as the quantity of diffuse reflection, specular reflection (shininess), and the specular exponent. Perception of surface shape can be strongly enhanced by specular enhancement (Malzbender et al. 2000). A more subtle enhancement technique is to increase the curvature of the reflectance function in the space of light direction, reducing the range of light directions giving high luminance. This is referred to as diffuse gain (Malzbender et al. 2001). The techniques of specular enhancement and diffuse gain are examples of reflectance transformation.
Finally, one can try to estimate the true 3D relief from surface normals. This can be done by converting the surface normals to partial derivatives (slopes) in the x and y directions, and finding a surface that minimizes least-squares deviations from the observed slopes. Using a variational approach, it can be shown that this optimization criterion leads to a Poisson equation which can be solved numerically with an iterative algorithm (Horn 1990). The resulting digital elevation map can be exaggerated vertically in order to amplify relief.