What is texture?

Even though texture is an intuitive concept, a formal definition of texture has proven elusive. In 1973, Haralick, Shanmugam and Dinstein noted (p611):

texture has been extremely refractory to precise definition.

There is no universally accepted definition for texture.

an exact definition of texture either as a surface property or as an image property has never been adequately formulated.

Texture [eludes] a formal definition,

Some researchers describe texture in terms of the human visual system: that textures do not have uniform intensity, but are none-the-less perceived as homogeneous regions by a human observer. For example, Bovik, Clarke and Geisler (p55) write

an image texture may be defined as a local arrangement of image irradiances projected from a surface patch of perceptually homogeneous irradiances.

Texture regions give different interpretations at different distances and at different degrees of visual attention. At a standard distance with normal attention, it gives the notion of macroregularity that is characteristic of the particular texture. When viewed closely and attentively, homogeneous regions and edges, sometimes constituting texels, are noticeable.

However, a definition based on human acuity poses problems when used as the theoretical basis for a quantitative texture analysis algorithm. Faugeras and Pratt (p323) note

The basic pattern and repetition frequency of a texture sample could be perceptually invisible, although quantitatively present.

There have been two main computational approaches to the definition of texture: the stochastic approach and the structural approach.

The stochastic approach considers that the intensities are generated by a two-dimensional random field. This approach is described by Faugeras and Pratt (p323)

The stochastic formulation is based on a model in which a texture region is viewed as a sample of a two-dimensional stochastic process describable by its statistical parameters.

We consider a texture to be a stochastic, possibly periodic, two-dimensional image field.

The brightness level at a point in an image is highly dependent on the brightness levels of neighbouring points unless the image is simply random noise.

Texture is characterized not only by the gray value at a given pixel, but also by the gray value `pattern' in a neighborhood surrounding the pixel.

This spatial structure is more strongly emphasised in the structural approach to texture. In this approach, a texture is composed of a primitive pattern which is repeated throughout the texture. The relative positioning of the primitives in the pattern are determined by the placement rule. Faugeras and Pratt (p323) describe this approach:

In the determinsitic formulation texture is considered as a basic local pattern that is periodically or quasi-periodically repeated over some area.

[the placement rule viewpoint] considers a texture to be composed of primitives. These primitives may be of varying or deterministic shape, such as circles, hexagons or even dot patterns. Macrotextures have large primitives, whereas microtextures are composed of small primitives. These terms are relative to image resolution. The textured image is formed from the primitives by placement rules which specify how the primitives are oriented, both on the image field and with respect to each other. Examples of such textures include tilings of the plane, cellular structures such as tissue samples, and a picture of a brick wall.

Francos, Meiri and Porat describe a texture model which unifies the stochastic and structural approaches. They assume that a texture is a realization of a 2-D homogeneous random field. This assumption allows the texture to be decomposed into orthogonal components.

In this framework, we will consider a texture to be a locally-structured 2-D homogeneous random field. That is, the intensity at a pixel location is a probabilistic function of the intensities at nearby pixel locations. This probabilistic function is independent of the location in the image, giving the perceptual impression of homogeneity.

Some other issues need to be addressed in defining the scope of the term texture, and what is meant by distinguishable textures. These issues include the palette of intensities available at each pixel, and whether texture discriminablility is affected by scale, rotation and illumination.

We will consider only grey-scale textures ; that is, monochrome textures with an effectively continuous range of values. This specifically excludes black and white images with only two intensity levels, and also excludes multispectral images. This restriction to monochrome images is consistent with the literature, which deals almost exclusively with monochrome images.

We will consider two textures to be distinct even if one is a rotation or magnification of the other. Of course, isotropic and fractal textures are not distinguishable after rotation and magnification, respectively. However, in general, such texture pairs are distinguishable by both human perception and machine algorithms. The rotation and magnification transformations clearly form families of textures which are, in some way, closely related. However, there are many transformations which define such families ; for example, elongation and perspective effects. In the absence of strong reasons for considering texture to be unchanged by a transformation, it is simplest to assume that transformations create distinct textures.

By this argument, we would consider a transformation of the intensity of each pixel, independent of the intensities of neighbouring pixels, would create distinguishable textures. In particular, linear functions of the intensities, such as might be caused by varying the level of illumination of a scene, would create distinct textures. This is contrary to the majority of the literature, which ignores or even explicitly removes intensity histogram information.

We argue that this discrepancy is not, in practice, an important discrepancy. If two images can be distinguished on the basis of the histogram of their intensities, regardless of their spatial structure, then the two images can be distinguished by applying standard statistical techniques to the histogram of their intensities. These statistical techniques are much better developed than texture analysis techniques ; any classification problem which can be addressed by standard statistical techniques is not, in practice, of interest as a texture classification problem. Thus, we may assume that a texture analysis algorithm may ignore first-order intensity distribution information for all interesting texture problems.