55:148 Digital Image Processing
55:247 Image Analysis and Understanding

Chapter 13,
Texture: Statistical texture description

• The concept of texture is intuitively obvious but has no precise definition.
• Texture is something consisting of mutually related elements.
• Texture primitives (or texture elements, texels) are building blocks of a texture.
• Texture description is scale dependent.

• Humans describe texture as fine, coarse, grained, smooth, etc.
• These description are imprecise and non-quantitative.

• Texture can be described by its tone and structure.
• Tone ... based on pixel intensity properties.
• Structure ... describes spatial relationships of primitives.
• Texture can be described by the number and types of primitives and by their spatial relationships.

• The following figure shows that neither identical primitives nor identical structure is sufficient for texture description if considered alone.

• Weak and strong textures ...
• Weak textures
• have small spatial interactions between primitives, can be described by frequencies of primitive types appearing in some neighborhood;
• statistical approaches are adequate for weak texture description.
• Strong textures
• spatial interactions among primitives are somewhat regular;
• frequency of occurrence of primitive pairs in some spatial relationship can be used for description;
• frequently accompanied by exact definition of texture primitives.

• Definition of a constant texture:
• An image region has a constant texture if a set of its local properties in that region is constant, slowly changing, or approximately periodic.

• Two basic texture description approaches:
• statistical
• syntactic

Methods based on spatial frequencies

• coarse textures ... larger primitives ... lower spatial frequencies
• fine textures ... smaller primitives ... higher spatial frequencies

• pixel = texture primitive
• linear spatial properties evaluated by correlation coefficients
• coarse texture ... autocorrelation function value decreases slowly with distance
• fine texture ... autocorrelation function value decreases rapidly with distance

• average values of energy of wedges or rings of Fourier spectrum can be used as features

Co-occurrence matrices

• based on repeated occurrence of some gray-level configuration in the texture
• this configuration varies rapidly in fine textures, more slowly in coarse textures
• occurrence of gray-level configuration may be described by matrices of relative frequencies, called co-occurrence matrices

• these matrices are symmetric but asymmetric definition is also possible

• Example image:

• corresponding co-occurrence matrices:

• Texture classification can be based on criteria (features) derived from the co-occurrence matrices.
• Energy

• Entropy

• Maximum probability

• Contrast (typically, kappa=2, lambda=1)

• Inverse difference moment

• Correlation

• where means and standard deviations are defined as

• Co-occurrence matrices description uses 2^nd order image statistics.
• Advantage ... consideration of spatial properties
• Limitation: does not consider primitive shape
• Therefore, not recommended for textures with large primitives

• features based on distance-related gradient
• micro-edges can be detected using small-distance operators
• macro-edges need large-size edge detectors
• g(d) is a distance-dependent texture description function:

• g(d) is similar to negative autocorrelation function
• dimensionality of texture description is specified by the number of considered distances d

• Texture features can be derived from 1^st and 2^nd order statistics of edge distributions
• 1^st order
• coarseness ~~edge density
• contrast ~~ large edge magnitudes = high contrast
• randomness ~~ entropy of edge magnitude histogram
• 2^nd order
• directivity ~~ entropy of edge direction histogram
• linearity ~~ co-occurrence of edge pairs with same direction (a,b in figure below)
• periodicity ~~ co-occurrence of edge pairs with same direction (a,c in figure below)
• size ~~ co-occurrence of edge pairs with opposite direction (a,d in figure below)

• large number of neighboring pixels o the same gray level - coarse texture
• small number ... fine texture
• lengths of texture primitives in different directions can be used for texture description
• primitives - length, direction, gray level

• B(a,r) ... number of primitives of all directions having length r and gray level a
• M, N ... image dimensions
• L ... number f image gray levels
• N_r ... max length of primitive
• K ... total number of primitives

• Short primitive emphasis

• Long primitive emphasis

• Gray-level uniformity

• Primitive length uniformity

• Primitive percentage

• texture description uses
• average gray level
• edges
• spots
• ripples
• waves

• Features are derived from 3 vectors:
• L₃ = (1,2,1) ... averaging
• E₃ = (-1,0,1) .. first difference - edges
• S₃ = (-1,2,-1) ... second difference - spots
• After convolution of the 3 vectors with themselves and each other, five vectors result:

• By mutual multiplying these 5 vectors, 5x5 Laws' masks result, e.g.:

• Convolution texture with Laws' masks and calculation of energy statistics gives description features.

Fractal texture description

Other statistical methods of texture description

• Mathematical morphology
• Texture transform
• Autoregression texture model
• Peak and valley method
• Markov chain approach
• Gabor transforms, wavelets