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