ERCIM News No.44 - January 2001 [contents]
by Michal Haindl and Vojtech Havlícek
The Textures Modelling Project carried out at CRCIM - UTIA and supported by the Czech Grant Agency, aims at developing mathematical models capable to simulate natural colour textures.
Virtual models require object surfaces covered with realistic nature-like colour textures to enhance realism in virtual scenes. Although such models can work also with digitised natural textures, synthetic textures are far more convenient, not only because of their huge data compression ratio (dozens model parameters irrespective of the required texture size) but can be designed to have certain desirable properties (eg, they can be made smoothly periodic).
Among several possible modelling approaches which are capable to learn a given natural texture sample (eg, fractals, random mosaic models, syntactic models, SARMA, etc.) the Markov random fields (MRF) are the most powerful and flexible ones, because choosing different neighbour sets and different types of probability distribution for the random variables a variety of distinct image types (sharp edges, patchy images, etc.) can be generated. Unfortunately MRF models require in general time-consuming Markov chain Monte Carlo synthesis methods and MRF analysis is complicated task even for simple MRFs.
Modelling multi-spectral images requires three-dimensional models. However if we are willing to sacrifice some spectral information a 3D model can be approximated with a set of simpler 2D models without compromising its visual realism. Random field based models quite successfully represent high frequencies present in natural textures though low frequencies are much more difficult for them. One possibility how to overcome this drawback is to use a multi-scale random field model. The resolution hierarchy provides then a transition between pixel-level features and global features and hence enables to model a large variety of possible textures.
Two fast multi-spectral multi-scale MRF texture models developed in the project synthesize single mono-spectral single-resolution factors of a given natural colour texture using either Gaussian MRF (GMRF) or causal wide-sense Markov (WMRF) sub-models. Hence a complex 3D MRF texture model is approximated with a set of simpler 2D MRF models. WSM model parameters as well as the optimal contextual support set of the model can be found analytically using Bayesian estimators (the WMRF statistics can be evaluated also recursively if required) and the model is easily synthesized from the model equation. Fast GMRF parameter estimation requires an approximate maximum likelihood estimator and similarly fast estimators exist only for a sub-optimal contextual support set. However, also the Fourier transformation based GMRF synthesis avoids the time consuming Markov chain Monte Carlo iterations. Finally the set of single synthetic mono-spectral single-resolution factors from either of the models is transformed into the resulting synthetic texture.
Some our synthetic textures reproduce given natural texture so well that both textures are visually indiscernible. The original colour tones are reproduced realistically in spite of the restricted spectral modelling power of the model. The multi-scale approach is more robust and allows far better results than the single-scale one if the synthesis model is inadequate (lower order model, non-stationary texture, etc.). The GMRF multi-scale model seems to be superior to the causal WMRF model for some textures, however the WMRF model synthesis is much faster than the GMRF model synthesis. Another advantage of the WMRF model is its consistent and efficient analytical estimators for all model parameters which is not the case for the GMRF model. Both proposed methods allow large compression ratio for transmission or storing texture information while they have still moderate computation complexity (see our Java applet demo) and avoid any time-consuming numerical optimization.
Texture demos: http://www.utia.cas.cz/user_data/haindl/virtuous/dema/demtexsyn.html
Michal Haindl - CRCIM (UTIA)
Tel: +420 2 6605 2350