Course 9
Texture
Course 9 Texture
Definition: Texture is repeating patterns of local variations
in image intensity, which is too fine to be distinguished.
Texture evokes the image of large number of structural
primitive of (statistically) identical shape and size placed
(statistically) uniformly.
To perceive a homogeneous texture,
What features should be homogeneous?
What features are unimportant?
The answer may lie in what signifies the constant of
changing scene geometry.
The texture image includes two
homogeneous regions
The answer may lie in what signifies the constant of
changing scene geometry.
The texture image includes two
homogeneous regions
1. Gray-level co-occurrence Matrix
• Characterized to capture spatial dependence of image graylevel value, which contribute to the perception of texture.
• Gray-level co-occurrence matrix P is defined by first

specifying a displacement vector d  (dx, dy ) , and counting

all pairs of pixels separated by d having gray level values i
and j.
Specifically, for a given image of size M  N and with L gray
levels.

For a defined displacement vector d  (dx, dy )
a) Systemically scan image from top to the row of
M  dx ,from left-most to the column of N  dy
For
f ( x, y )  i and
f ( x  dx, y  dy
, )  j i, j [0, L);
count the number of occurrence, say beingl .
b) Set matrix element P[i, j ]  l
c) Repeat operations b) and c) until all L  L
combinations of i and j are completed.
d) Normalize matrix P by
1
 p[i, j ]
i
j
.

For example: d  (1,1)

Note: for the same image with different d displacement
vector , it will yield different gray-level co-occurrence matrix,
which characterizes the texture homogeneity of different
special distribution and orientations.
From co-occurrence matrix, some useful measurements can
be derived.

Energy :
  P 2 i, j 
i
•
Contrast:
j
2
(
i

j
)
P i, j 

i
j
 Homogeneity:
P i, j 

i j 1 i  j
 Entropy :
   P i, j  log P i, j 
i
j
2. Structural analysis of texture
Assumptions:
－ Texture is ordered.
－ Texture primitives are large enough.
－ Texture primitives can be separated.
After primitive regions are identified, homogeneity
properties can be measured.
－ Centroid distances of different directions.
－ Size.
－ Elongation.
－ Orientation.
Then, co-occurrence based measurements of the
primitives are used to characterize the texture.
3. Model-based Texture Analysis
Concept:
－ Establish an analytical model of the given
textured image.
－ Then, analysis the model.
Difficulty:
too many parameters in the model to be
determined.
Example: Gauss-Markove random field model:
f i, j  
 f i  k , j  l  hk , l   ni, j 
[ k ,l ]
where: h[i, j ] is weight , n[i, j ] is an additive noise.
In this model, any pixel is modeled as a linear combination of
gray level of its neighbors to pixel [i, j ] . The parameters are
the weight h[i, j ] , which can be estimated by least-square
from the given textured image.
4. Shape From Texture
－ Recover 3D information from 2D image clues.
－ Image clues: variations of size, shape, and density of
texture primitives.
－ Yield: surface shape and orientation.
An simple example for analysis
Assumptions:
1) 3D surface is slanted with angel .
2) Till angel being zero, i.e. points along horizontal line on
the surface have the same depth from the camera.
3) Texture primitive is disk.
4) Perspective projection imaging model.
Observations from image:
1) 3D disks appear to be ellipses in image plane.
2) The size of ellipses decrease as a function of y , the ycoordinate of image plane, causing “density gradients”.
3) The ratio of minor to major diameters of ellipses does
not remain constant along y -axis.
Def. aspect ratio =
minor diameter
major diameter
At image center:
let the diameter of 3D disk
be d, then
d
d major (0,0)  f
z
d
d min or (0,0)  f cos 
z
Thus, aspect ration is
d minor
 cos 
d major
At point of (0, y)
y
tan  
f
So,
s
sz
tan  
s tan 
z
1 tan θ tan α
Thus,
d
d (1  tan  tan  )
d major (0, y ' )  f  f
s
z
AC  d cos 
(AC parallel to image plane)
BC  d sin  tan 
(Assume
1   2  
AB  AC  BC  d cos   d sin  tan 
 d cos  (1  tan  tan  )
AB
d minor (0, y ' )  f
f
s
d cos  (1  tan  tan  )
z
1  tan  tan 
d
2
 f cos  (1  tan  tan  )
z
Thus, aspect ration = cos  (1  tan  tan  )
)
y
Note : tan   is known from image plane,
f
the 3D surface orientation
from aspect ratio.
 can be computed
5. Surface Orientation from statistic Texture
Assumptions:
1) 3D texture primitives are small line segments, called
needles.
2) Needles are distributed uniformly on 3D surface, their
directions are independent.
3) Surface is approximately planar.
4) Orthographic image projection
Given: image of N needles with needle angle  i from x-axis.
Find: surface orientation ( ,  )
Method (omitting detail deriving):
For a give image needle
direction, define an auxiliary
vector (cos 2i , sin 2i )
The average of the vector is:
1 N
C   cos 2 i
N i1
1 N
S   sin 2 i
N i1
From orthographic projection
1  cos 
C  cos 2
1  cos 
1  cos 
S  sin 2
1  cos 
Thus, one can solve for:
1 Q
  cos
1 Q
1

    (mod 2 )
2
(i.e.,  [0,  ))
where
Q  C2  S 2
1
1 S
ψ  tan
2
C