Efficient robust image interpolation and surface properties using polynomial texture mapping

Polynomial texture mapping (PTM) uses simple polynomial regression to interpolate and re-light image sets taken from a fixed camera but under different illumination directions. PTM is an extension of the classical photometric stereo (PST), replacing the simple Lambertian model employed by the latter with a polynomial one. The advantage and hence wide use of PTM is that it provides some effectiveness in interpolating appearance including more complex phenomena such as interreflections, specularities and shadowing. In addition, PTM provides estimates of surface properties, i.e., chromaticity, albedo and surface normals. The most accurate model to date utilizes multivariate Least Median of Squares (LMS) robust regression to generate a basic matte model, followed by radial basis function (RBF) interpolation to give accurate interpolants of appearance. However, robust multivariate modelling is slow. Here we show that the robust regression can find acceptably accurate inlier sets using a much less burdensome 1D LMS robust regression (or 'mode-finder'). We also show that one can produce good quality appearance interpolants, plus accurate surface properties using PTM before the additional RBF stage, provided one increases the dimensionality beyond 6D and still uses robust regression. Moreover, we model luminance and chromaticity separately, with dimensions 16 and 9 respectively. It is this separation of colour channels that allows us to maintain a relatively low dimensionality for the modelling. Another observation we show here is that in contrast to current thinking, using the original idea of polynomial terms in the lighting direction outperforms the use of hemispherical harmonics (HSH) for matte appearance modelling. For the RBF stage, we use Tikhonov regularization, which makes a substantial difference in performance. The radial functions used here are Gaussians; however, to date the Gaussian dispersion width and the value of the Tikhonov parameter have been fixed. Here we show that one can extend a theorem from graphics that generates a very fast error measure for an otherwise difficult leave-one-out error analysis. Using our extension of the theorem, we can optimize on both the Gaussian width and the Tikhonov parameter.