What is Texture?
Even when the concept of texture is intuitive, no single unifying definition has been given by the image analysis community. Most of the numerous definitions that are present in the literature have some common elements that emerge from the etymology of the word: texture comes from the Latin textura, the past participle of the verb texere, to weave (Webster, 2004). This implies that a texture will exhibit a certain structure created by common elements, repeated in a regular way, as in the threads that form a fabric. Three ingredients of texture were identified in (Hawkins, 1970):
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some local ‘order’ is repeated over a region which is large in comparison to the order’s size,
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the order consists of the non-random arrangement of elementary parts, and,
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the parts are roughly uniform entities having approximately the same dimensions everywhere within the textured region.
Yet these ingredients could still be found in the very different contexts, not just in imagery, such as food, painting, haptics or music. Wikipedia cites more than 10 contexts of tactile texture alone (http://en.wikipedia.org/wiki/Texture). In this work, we will limit ourselves to visual non-tactile textures, sometimes referred as visual texture or simply image texture, which is defined in (Tuceryan & Jain, 1998) as: a function of the spatial variation in pixel intensities. Although brief, this definition highlights a key characteristic of texture, that is, the spatial variation. Texture is therefore inherently scale dependent (Bouman & Liu, 1991; Hsu et al., 1992; Sonka et al., 1998). The texture of a brick wall will change completely if we get close enough to observe the texture of a single brick. Furthermore, the texture of an element (pixel or voxel) is implicitly related to its neighbors. It is not possible to describe the texture of a single element, as it will always depend on the neighbors to create the texture. This fact can be exploited through different methodologies that analyze neighboring elements, for example: Fourier domain methods, which extract frequency components according to the frequency of elements; a Markovian process in which the attention is restricted to the variations of a small neighborhood; a Fractal approach where the texture is seen as a series of self-similar shapes; a Wavelet analysis where the similarity to a prototypical pattern (the Mother wavelet) at different scales or shifts can describe distinctive characteristics; or a co-occurrence matrix where occurrence of the grey levels of neighboring elements is recorded for subsequent analysis.
Some authors have preferred to indicate properties of texture instead of attempting to provide a definition. For instance in (Gonzalez & Woods, 1992) texture analysis is related to its Statistical (smooth, coarse, grainy,...), Structural (arrangement of feature primitives sometimes called textons) and Spectral (global periodicity based on the Fourier spectrum) properties. Some of these properties are visually meaningful and are helpful to describe textures. In fact, studies have analyzed texture from a psycho-visual perspective (Ravishankar-Rao & Lohse, 1993; Tamura et al., 1978) and have identified the properties such as: Granular, marble-like, lace-like, random, random non-granular and somewhat repetitive, directional locally oriented, repetitive, coarse, contrast, directional, line-like, regular or rough.
It is important to note that these properties are different from the features or measurements that can be extracted from the textured regions (although confusingly, some works refer to these properties as features of the data). When a methodology for texture analysis is applied, sub-band filtering for instance, a measurement is extracted from the original data and can be used to distinguish one texture from another one (Hand, 1981). A measurement space (it can also be called the feature space or pattern representation (Kittler, 1986)) is constructed when several measurements are obtained. Some of these measurements are selected to build the reduced set called a feature space. The process of choosing the most relevant measurements is known as feature selection, while the combination of certain measurements to create a new one is called feature extraction (Kittler, 1986).
Throughout this work we will refer to Volumetric texture as the texture that can be found in volumetric data ((Blot & Zwiggelaar, 2002) used the term solid texture). All the properties and ingredients that were previously mentioned about texture, or more specifically, visual, or 2D texture, can be applied to volumetric texture. Figure 1 shows three examples of volumetric data in which textured regions can be seen and analyzed.
Figure 1. Three examples of volumetric data from where textures can be analyzed: (a) A sample of muscle from MRI, (b) A sample of bone from MRI, (c) The vasculature of a growing tumor from multiphoton microscopy. Unlike two dimensional textures, the characteristics of volumetric texture cannot always be observed from their 2D projection