E.G.M. Petrakis Texture 1

Texture

• Repeative patterns of local variations of intensity on a surface– texture pattern: texel

• Texels: similar shape, intensity distribution and probably orientation or size– geometric shapes, lines, dots, points

• Different applications, different texture

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Texture Resolution

• The resolution at which the image is observed determines the scale at which the texture is perceived

• The texture changes or vanishes depending on the distance from which an image is observed– e.g., when a tiled floor is observed from a large

distance the texture is formed by the placement of tiles– when the same image is observed from a closer

distance only a few tiles are within the field of view and the texture is formed by the placement of dots, lines etc. composing each tile

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examplesof texture

from Ballardand Brown ‘84

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examples of texels(a) circles (b) circles and (c) ellipses line segments

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examples of aerial image textures

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Texture Analysis

• Segmentation: determine the boundaries of textured regions– region and boundary-based methods

• Classification: identify a textured region – find the most similar from multiple classes of texture– extract and classify texels

• Shape recovery from texture: use variations in size and orientation of texels to estimate surface shape and orientation

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Texture Classification

• Structural: the texels are large enough (can be distinguished from the background) to be individually segmented and described – e.g., grammars, tessellation models etc.

• Statistical: describe the gray level distribution of textured areas – apply when the texels exhibit variations which

can be described statistically

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Tessellation Methods

• The texels tessellate the place in an ordered way– in a regular tessellation the polygons surrounding a

vertex have the same number of sides

– semi-regular tessellations have two or more kinds of polygons

– these tessellations are described by listing in order the number of sides of the polygons surrounding each vertex

• e.g., hexagonal tessellations: (6,6,6)

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semi-regulartessellations

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Grammatical Methods

• A grammar describes how to generate patterns by applying rewriting rules to a small number of symbols– a grammar can generate complex textural patterns

– stochastic grammars: real world variations can be incorporated into a grammar by attaching probabilities to different rules

– no unique grammar for a given texture

– variants: shape, tree and array grammars

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Shape Grammar

• Hexagonal texture: the grammar is a 4-tuple– G = <Vt, Vm, R, S>

– Vt ={ }: finite set of “terminal” shapes

– Vm: finite set of shapes such that Vt/Vm = 0 (non-terminal shape elements or markers)

– R: set of rules for producing patterns of Vt S

– Vt S: elements of Vt used a multiple number of times in any location, orientation and scale

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Example of Texture Grammar

• Apply the rules in reverse order until a symbol in S is produced

• A failure means that the texture cannot be recognized from G

textures to be recognized

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Statistical Methods

• The texture is described by the spatial distribution of intensities – the texels cannot be recognized individually

• Texture recognition as a pattern classification: – compute a vector V = (v1,v2, …, vn)

– classify the vector into one of M classes

– select the features of V

– classify a vector according to its minimum distance from a class vector

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effective features ineffective features

texture classification as as pattern recognition problem

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Co-Coherence Matrix

• P[i,j] : counts pairs of pixels separated by distance d = (dx,dy) having gray levels i, j in [0,2]

2 1 2 0 1

0 2 1 1 2

0 1 2 2 0

1 2 2 0 1

2 0 1 0 1

i

j

0 2 2

1 1 2

2 3 2

1

16

P[i,j]d = (1,1)

0 1 2

0

1

2

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Co-Coherence Matrix (cont.)

• There are 16 pairs of pixels which satisfy the spatial separation d in direction 45o

• Count all pairs of pixels in which the first pixel has value i and its matching pair displaced by d, θ has value j

• Enter this count in the (i,j) position of P– e.g., if there are 3 pairs [2,1] then P[2,1] = 3– P is not symmetric– normalize P by the total number of pairs– P: probability mass function

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Co-Coherence Matrix (cont.)

• P captures the spatial distribution of gray levels for the specified d,θ– repeat the same for all i, j

• Textured regions exhibit a non-random distribution of values in P

• Entropy: feature which measures randomness

– takes high values for uniform P (no preferred gray-level, no texture)

i j

jiPjiPEntropy ],[log],[

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More Texture Feaures

i j

i j

i j

|ji|

]j,i[PyHomogeneit

]j,i[P)ji(Contrast

]j,i[PEnergy

1

2

2

• Measure P for several d,θ• Existence of texture:

– maximization of one of these measures or

– minimization of entropy

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Auto-Correlation

• Compute A of image f of size N x N for various k,l

• A measures the periodicity of texture• Textured images: A exhibits periodic behavior

with a period equal to the spacing between pixels– coarse texture: A drops slowly– fine texture: A drops rapidly

101 1

21

1

1

11

2

Nl,k,

]j,i[f

]lj,ki[f]j,i[f]l,k[A N

i

N

jN

)kN(

i

N

j)lN)(kN(

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Fourier Method

• When texture exhibits periodicity or orientation– detect peaks in the power spectrum

– partition the Fourier space into bins of r or θ

– texture features are defined on the spectrum |F|2

v

u u

v

rθ

angualbins

radialbins

Partitioning the Fourier Spectrum

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Radial Features

• Radial feature vector: – V = (Vr0r0,Vr0r1,…, Vrkrl), 0<= k,l <= m– Vrkrl is the spectral content of a ring [rk,rl]

• Exploit the sensitivity of the power spectrum to the size of the texture

1,0

],[|

2222

nvu

rvur

dudvvuFrVr

lk

lk

[r1,r2] define oneof the radial bins

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Angular Features

• Angular feature vector: – V = (Vθ0θ0,Vθ0θ1,…, Vθkθl), 0<= k,l <= m

– Vθkθl is the spectral content in a piece [θk,θl]

• Exploit the sensitivity of the power spectrum to the directionality of the texture

1,0

][tan

],[|

1

nvu

dudvvuFV

luv

k

lk

[θk,θl] define one of the sectors

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Comments on Fourier Method

• Radial features are correlated with texture coarseness– smooth texture has high Vr1r2 for small radii

– coarse texture has higher Vr1r2 for larger radii

• Angular features are correlated with the directionality of the texture– if the texture has many lines or edges in a given

direction θ then |F|2 tends to give high values between θ and θ + π/2