The natural world is rich in texture: the surface of any visible object is textured at certain scale. A wealth of textures are observed on both artificial and natural objects such as those on wood, plants, materials and skin. In a general sense, the word texture refers to surface characteristics and appearance of an object given by the size, shape, density, arrangement, proportion of its elementary parts . A texture is usually described as smooth or rough, soft or hard, coarse of fine, matt or glossy, and etc.
Textures might be divided into two categories, namely, tactile and visual textures. Tactile textures refer to the immediate tangible feel of a surface. Visual textures refer to the visual impression that textures produce to human observer, which are related to local spatial variations of simple stimuli like colour, orientation and intensity in an image. This thesis focuses only on visual textures, so the term `texture' thereafter is exclusively referred to `visual texture' unless mentioned otherwise.
Textures are usually given by examples in digitised images. Figures 4.1 and 4.2 show a few natural and man-made textures, respectively, which could be met in daily life.
Although texture is an important research area in computer vision, there is no precise definition of the notion texture. The main reason is that natural textures often display different yet contradicting properties, such as regularity versus randomness, uniformity versus distortion, which can hardly be described in a unified manner. Many researchers have been trying to define textures from a certain perspective of their nature. Haralick considers a texture as an ``organised area phenomenon'' which can be decomposed into `primitives' having specific spatial distributions . This definition, also known as structural approach, comes directly from human visual experience of textures. For instance, each texture in Figs 4.1 and 4.2 is composed of particular texture elements, e.g., objects (windows), shapes (jigsaw pieces), or simply colour patterns. Meanwhile, these primitives are organised in a particular spatial structure indicating certain underlying placement rules. Alternatively, as Cross and Jain suggested, a texture is ``a stochastic, possibly periodic, two-dimensional image field'' . This definition describes a texture by a stochastic process that generates the texture, which is also known as stochastic approach. These different definitions usually lead to different computational approaches to texture analysis.
Nevertheless, an apparent consensus that spatial homogeneity is the most important property of a texture has been reached. From the statistical point of view, homogeneity means statistical stationarity, i.e. that certain signal statistics of each texture region have the same values. This property relates directly to self-similarity: the patterns at different magnifications, although not identical, are represented by the same signal statistics .
Textures also exhibit local non-homogeneity, i.e. departures from strict homogeneity to some extent in a local image region. For example, in the image `leaves' in Fig 4.1, every single leaf is slightly different from another (local non-homogeneity), but as a whole they display approximate spatial uniformity and consistency (global homogeneity).
Due to the diversity and complexity of natural textures, it is useful to separate them into categories. For instance, textures can be classified into regular and stochastic ones by their degree of randomness. A regular texture is formed by regular tiling of easily identifiable small size elements organised into strong periodic patterns. A stochastic texture exhibits less noticeable elements and display rather random patterns. For examples, textures in Figs 4.1 and 4.4 are mostly stochastic, and those in Figs 4.2 and 4.3 are regular. Most of real world textures, however, are mixtures of the above-mentioned categories.
By spatial homogeneity, textures can be classified into homogeneous, weakly-homogeneous, and inhomogeneous patterns. Specifically, homogeneous texture contains ideal repetitive structures, and such uniformity produces idealised patterns. Weak homogeneity involves local spatial variation in texture elements or their spatial arrangement, which leads to more or less violates the precise repetitiveness (See Fig 4.5). An `inhomogeneous texture' mostly refers to an image where repetition and spatial self-similarity are absent. Since spatial homogeneity is considered below as an essential property of a texture, an inhomogeneous image is not treated in this thesis as a `texture'.