An authentic material's surface reflectance function is a complex function of over sixteen physical variables, which are both unfeasible to measure as well as to mathematically model. The best simplified measurable material texture representation and approximation of this general surface reflectance function is the seven-dimensional Bidirectional Texture Function (BTF). BTF can be simultaneously measured and modeled using state-of-the-art measurement devices and computers and the most advanced mathematical models of visual data. However, such an enormous amount of visual BTF data, measured on the single material sample, inevitably requires state-of-the-art storage, compression, modeling, visualization, and quality verification. Storage technology is still the weak part of computer technology, which lags behind recent data sensing technologies, thus, even for virtual reality correct materials modeling, it is infeasible to use BTF measurements directly. Hence, for visual texture synthesis or analysis applications, efficient mathematical BTF models cannot be avoided. The probabilistic BTF models allow unlimited seamless material texture enlargement, texture restoration, tremendous unbeatable appearance data compression (up to 1:1000 000), and even editing or creating new material appearance data. Simultaneously, they require neither storing actual measurements nor any pixel-wise parametric representation. Unfortunately, there is no single universal BTF model applicable for physically correct modeling of visual properties of all possible BTF textures. Every presented model is better suited for some subspace of possible BTF textures, either natural or artificial. In this contribution, we intend to survey existing mathematical BTF models which allow physically correct modeling and enlargement measured texture under any illumination and viewing conditions while simultaneously offer huge compression ratio relative to natural surface materials optical measurements. Exceptional 3D Markovian or mixture models, which can be either solved analytically or iteratively and quickly synthesized, are presented. Illumination invariants can be derived from some of its recursive statistics and exploited in content-based image retrieval, supervised or unsupervised image recognition. Although our primary goal is physically correct texture synthesis of any unlimited size, the presented models are equally helpful for various texture analytical applications. Their modeling efficiency is demonstrated in several analytical and modeling image applications, in particular, on a (un)supervised image segmentation, bidirectional texture function (BTF) synthesis and compression, and adaptive multi-spectral and multi-channel image and video restoration.