The thesis is organised as follows: state of the art textural representations and textural databases are reviewed in the next chapter. The proposed textural representation is de-scribed in Chapter 3. Chapter 4 concerns with illumination invariance and it contains derivation of novel illumination invariants based on the proposed textural representa-tion. In Chapter 5 rotation invariance is incorporated into the textural representarepresenta-tion.
Experimental results of the proposed methods are presented in Chapter 6 and appli-cations follow in Chapter 7. Finally, the thesis is concluded and further directions of development are outlined. Appendices include additional derivations, experiments and examples from texture databases.
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Chapter 2
State of the Art
Informally, a texture can be described as an image that consists of primitives (micro structures) placed under some placement rules, which may be randomised somehow.
This texture primitive may be considered to be an object, and vice versa many objects may form a texture, it all depends on the resolution scale. Crucial properties of all textures are homogeneity and translation invariance. The homogeneity is understood quite vaguely and it means that any subwindow of a single texture posses some common characteristics. The translation invariance implies that these texture characteristics do not depend on texture translation. To name a few examples, an appearance of many materials or regular patterns is perceived as a texture.
Although the notion of texture is tied to human perception, there is no mathemat-ically rigorous definition that would be widely accepted. In our work we assume that texture is a kind of random field and the texture image is the realisation of random field.
The following review of textural representations begins with known findings of human perception, continues with representations used in computers, and then these represen-tations are considered according to invariant properties they provide. Finally, existing texture databases and comparisons are listed.
2.1 Human perception of textures
Julesz (1962) published one of the first works on visual texture discrimination, and he devoted next thirty years (Julesz, 1991) to work on human perception of textures, which was highly influential for construction of texture discrimination algorithms.
In order to explain the psychophysical findings, some image statistics have to be clarified (Julesz, 1962),
”The nth-order statistic (or nth-order joint probability distribution) of an image can be obtained by randomly throwing n-gons of all possible shapes on the image and observing the probabilities that their n vertices fall on certain colour combinations.”
Chapter 2. State of the Art
The n-gons are geometrical objects: points (1-gon), line segments (2-gons, or dipoles), triangles (3-gons), etc.
Firstly, Julesz (1962) experimented with a spontaneous visual discrimination of tex-tural images, which were generated by the Markov process as a realisation of a random field. He posed a conjecture that textures cannot be spontaneously discriminated if they have the same first-order and second-order statistics and if they differ only in their third or higher order statistics. However, this conjecture was later disproved when several counterexamples were published (Julesz et al., 1978; Yellot, 1993). Consequently, such images cannot be discriminated by texture recognition algorithms that rely only on first or second order statistics (e.g. histograms or co-occurrence matrices). Our textural fea-tures (Section 3.1) use higher order statistics, although their interaction range is locally limited, so we expect their ability to recognise even textures with identical second-order statistics.
Yellot (1993) also proved that the third-order statistics of any monochromatic image of finite size uniquely determine this image up to translation. Although Julesz et al.
(1978); Julesz (1991) presented examples of distinguishable textures with same second-order and third-second-order statistics, Yellot (1993) argued that the actual sample third-second-order statistics were not identical. It is worth to stress that the theorem of Yellot (1993) does not claim that images with close statistics up to the third order look similar.
In later work, Julesz (1991) tended to characterise textures by small texture elements (textons) instead of global statistics. Similar paradigm was adopted by micropattern and texton based texture representations (Sections 2.2.4, 2.2.5). Julesz (1991) also demon-strated that texture discrimination is not symmetric: a small piece of one texture can be distinguished from another texture background, but if the textures are swapped the discriminability is weaker. Finally, the human texture discriminability is not linear in the sense that if an image with two highly discriminable textures is added to a homo-geneous texture, the textures in the resulting image may be nondiscriminable, because the texture elements became too complex (Julesz, 1991).
Rao and Lohse (1996) performed a psychophysical experiment with 56 textures, where the subjects were asked to group the textures and to describe the characteristics of created groups. Rao and Lohse (1996) concluded that texture can be described in three orthonormal dimensions:
repetitive/regular/non-random vs. non-repetitive/irregular/random granular/coarse/low-complexity vs. non-granular/fine/high-complexity
low contrast/directional vs. high contrast/non-directional.
Rao and Lohse (1996) argued that the joint axis of contrast and directionality is a new complex texture dimension, similarly as is the perception of colour hue (which can be decomposed into red–green and yellow–blue opponent components). However, we doubt about that and we would decompose this axis into two different properties.
Natural materials are recognised not only from the texture, but also from their re-flectance properties as lightness and gloss. Fleming et al. (2003) showed that humans are 8