1. Computer Vision Chapter 9
Texture
Presented by 王夏果 and 傅楸善教授
Cell phone:
Digital Camera and Computer Vision Laboratory
Department of Computer Science and Information Engineering
National Taiwan University, Taipei, Taiwan, R.O.C.

2. Introduction
What does texture mean? Formal approach or precise definition of texture does not exist!
Texture discrimination techniques are for the part ad hoc.
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3. Definition of Texture
Non-local property, characteristic of region larger than its size
Repeating patterns of local variations in image intensity which are too fine to be distinguished as separated objects at the observed resolution
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4. Definition of Texture (cont.)
For humans, texture is the abstraction of certain statistical homogeneities from a portion of the visual field that contains a quantity of information grossly in excess of the observer’s perceptual capacity
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11. Texture Analysis Issues
Pattern recognition: given texture region, determine the class the region belongs to
Generative model: given textured region, determine a description or model for it
Texture segmentation: given image with many textured areas, determine boundaries
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13. Statistical Texture-Feature Approaches
Autocorrelation function
Spectral power density function
Edgeness per unit area
Spatial gray level co-occurrence probabilities
Graylevel run-length distributions
Relative extrema distributions
Mathematical morphology
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14. Image Texture Analysis
Give a generative model and the values of its parameters, one can synthesize homogeneous image texture samples associated with the model and the given value of its parameters.
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15. Image Texture Analysis (cont.)
Verification: verify given image textures sample consistent with model
Estimation: estimate values of model parameters based on observed sample examples of model-based techniques
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16. Some Model-Based Techniques
Autoregressive, moving-average, time-series models (extended to 2D)
Markov random fields
Mosaic models
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17. Texel
Texture element, basic textural unit of some textural primitives qualitatively evaluated image texture properties
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18. Some Texture Features Fineness Coarseness Contrast Directionality
Roughness
Regularity
Smoothness
Granulation
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19. Some Texture Features (cont.)
Randomness
Lineation
Mottled
Irregular
Hummocky
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20. Take a Break
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21. Texture and Scale
For any textural surface, there exists a scale at which, when the surface is examined, it appears smooth and textureless. (see from infinite distance)
As resolution increases, the surfaces appears as a fine texture and then a coarse one, and for multiple-scale textural surface the cycle of smooth, fine, and coarse may repeat.
一定是這種順序嗎?
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22. Texture and Scale (cont.)
Thus, texture cannot be analyzed without frame of reference on scale or resolution.
Texture is a scale-dependent phenomenon.
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24. Characterizing Texture
Characterize gray level primitive properties
Characterize spatial relationships between them
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25. First-Order Gray-Level Statistics
Statistics of single pixels
E.g. Histogram, mean, median, variance
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26. Second-Order Gray-Level Statistics
The combined statistics of gray levels of pairs of pixels in which each two pixels in a pair have a fixed relative position
E.g. co-occurrence
Gray level spatial dependence: characterize texture by co-occurrence
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27. Co-Occurrence Matrix
The gray level co-occurrence can be specified in a matrix of relative frequencies Pij with which two neighboring pixels separated by distance d occur on the image, one with gray level i and the other with gray level j
Symmetric matrix
Function of angle and distance between pixels
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28. 2)
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29. Co-Occurrence Matrix (cont.)
Probability of horizontal, d pixels apart pixels
P(i, j, d, 0°) =
#{[(k, l), (m, n)] | k-m = 0, |l-n| = d, I(k, l) = i, I(m,n) = j}
Probability of 45°, d pixels apart pixels
P(i, j, d, 45°) =
#{[(k, l), (m, n)] | (k-m = d, l-n = -d) or (k-m = -d, l-n = d),
I(k, l) = i, I(m,n) = j}
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30. Co-Occurrence Matrix (cont.)
Probability of 90°, d pixels apart pixels
P(i, j, d, 90°) =
#{[(k, l), (m, n)] | |k-m| = d, l-n = 0, I(k, l) = i, I(m,n) = j}
Probability of 135°, d pixels apart pixels
P(i, j, d, 135°) =
#{[(k, l), (m, n)] | (k-m = d, l-n = d) or (k-m = -d, l-n = -d),
I(k, l) = i, I(m,n) = j}
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32. Co-Occurrence Matrix (cont.)
Matrix symmetric: P(i, j, d, a) = P(j, i, d, a)
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33. Take a Break
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35. Matrix with Highest Entropy
When all entries in Pij are equal
Image where no preferred gray-level pairs exist features calculated from the co-occurrence matrix
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36. Generalized Gray Level Spatial Dependence Models for Texture
Simple generalization: consider more than two pixels at a time
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37. Generalized Co-Occurrence
Strong texture measures take into account the co-occurrence between texture primitives.
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38. Texture Primitive
Connected set of pixels characterized by attribute set
Simplest primitive: pixel with gray level attribute
More complicated primitive: connected set of pixels homogeneous in level, characterized by size, elongation, orientation, and average gray level
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39. Spatial Relationship
We have a list of primitives, their center coordinate, and their attributes after the primitives have been constructed.
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40. Spatial Relationship (cont.)
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41. Autocorrelation Function
Texture relates to the spatial size of the gray level primitives on an image
Gray level primitives of larger size are indicative of coarser texture
Gray level primitives of smaller size are indicative of finer texture
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42. Autocorrelation Function (cont.)
Autocorrelation function describes the size of gray level primitives
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43. Autocorrelation Function (cont.)
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44. Autocorrelation Function (cont.)
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45. Autocorrelation Function (cont.)
If the gray level on image is relatively large: texture is coarse, autocorrelation drops off slowly with distance
If the gray level on image is relatively small: texture is fine, autocorrelation drops off quickly with distance
Periodic
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46. Take a Break
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47. Digital Transform Methods and Texture
In the digital transform method of texture analysis, the digital image is typically divided into a set of non-overlapping small square subimages
The vectors is reexpressed in a new coordinate system
Fourier transform uses the complex sinusoid basic set, Handamard transfer uses the Walsh function basic set, …..
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48. Texture Energy The image is first convolved with a variety of kernels
Then each convolved image is processed with a nonlinear operator to determine the total textural energy in each pixel’s neighborhood
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49. Texture Edgeness
Autocorrelation function and digital transform both reference texture to spatial frequency
Texture Edgeness: conceive texture in terms of edgeness per unit area
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50. Texture Edgeness (cont.)
Use small neighborhood to detect microedge
Use large neighborhood to detect macroedge
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52. Vector Dispersion
Divide the texture into mutually exclusive neighborhoods
A sloped plane fit to the gray levels is performed for each neighborhood
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54. Relative Extrema Density
Count the number of extrema per unit area for a texture measure
One dimension, along a horizontal scan
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55. Relative Extrema Density (cont.)
Relative minimum:
g(i) ≦ g(i+1) and g(i) ≦ g(i-1)
Relative maximum:
g(i) ≧ g(i+1) and g(i) ≧ g(i-1)
Pixels in a constant run: both minimum and maximum
Center a square window around each pixel, and count the number of extrema pixels
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56. Mathematical Morphology
Granularity of a binary image F:
#F: number of elements in F
: disk structuring element of diameter d
G(d) measures the proportion of pixel participating in grains of size smaller than d
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57. Mathematical Morphology (cont.)
Scale-k volume of the blanket around a gray level intensity surface I:
⊕k: k-fold dilation
Θk: k-fold erosion
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58. Autoregression Models
Doing linear estimates of a pixel’s gray level with the gray levels in the neighborhood
For coarse texture, coefficient will be similar
For fine texture, coefficient will vary widely
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59. Autoregression Models
Next gray value aN+1 : linear combination of synthesized data and noise value
ak : given starting sequence
bk : randomly generated noise image
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61. Autoregression Models (cont.)
Two-dimensional autoregression model:
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63. Autoregression Models (cont.)
Easy to use the estimator in a node that synthesized textures from any initially given linear estimator
Sufficient to capture everything in a texture
But the textures it can characterize are likely to consist mostly of microtextures
Microtexture: gray level primitives are small, spatial interaction between primitives is local
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64. Take a Break
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65. Discrete Markov Random Fields
Assumption: the texture field is stochastic and stationary and satisfies a conditional independence assumption
When the distributions are Gaussian, each pixel’s value is a combination of the value in its neighborhood plus a noise term
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66. Discrete Markov Random Fields (cont.)
h: coefficients, computed from texture image with least-square method
u: joint set of possible correlated Gaussian random variables
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67. Random Mosaic Models Constructing steps:
1. Provide a mean of tessellating a plane into
cells
2. Assign a property value to each cell
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68. Structural Approaches to Texture Models
Pure structural model: primitives in regular repetitive spatial arrangements
To describe the texture, describe the primitives and the placement rules
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69. Texture Segmentation
Each region has homogeneous texture, and each pair of adjacent regions is differently textured
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70. Synthetic Texture Image Generation
Fractals: shapes that exhibit recursive self-similarity
Every fractal can be recursively subdivided into smaller non-overlapping shapes, each of which is a scale-down version of the whole, either in a deterministic sense or in a statistical sense
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71. Shape from Texture
Use image texture gradients to estimate surface orientation of the observed 3D object
Assumption: no depth changes and no texture changes in observed texture area, and no subtextures
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75. Shape from Texture (cont.)
Unknown plane where texture observed
Ax + By + Cz + D = 0
where
From perspective projection, 3D point
(x, y, z) with projection (u, v)
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76. Shape from Texture (cont.)
Solving z, …., use similar triangle geometry
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79. Summary Texture: in terms of primitives and spatial relationships
Qualitatively, shape from texture can work
Quantitatively, the techniques are generally not dependable
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