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\Huge
\begin{center}
Chapter 9 Texture
\end{center}

\LARGE
9.1 Introduction \\
formal approach or precise definition of texture does not exist \\
texture discrimination techniques are for the most part ad hoc  \\
principal characteristic of texture: repetition of basic pattern \\
texture: nonlocal property, characteristic of region larger than its size \\
=====/usr/local/vision/images/t\_bark.im===== \\
=====/usr/local/vision/images/t\_clouds.im===== \\
=====/usr/local/vision/images/t\_fur\_r.im===== \\
=====/usr/local/vision/images/t\_pebbles.im===== \\
=====/usr/local/vision/images/t\_straw.im===== \\
=====/usr/local/vision/images/t\_water\_r.im===== \\
visual cliff by William Vandivert, 1960 \\
=====V. S. Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 6.1===== \\
(a) water (b) beach pebbles (c) raffia weave (d) brick wall \\
=====V. S. Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 6.2===== \\
texture: repeating patterns of local variations in image intensity which are \\
\hspace*{1cm} too fine to be distinguished as separate objects \\
\hspace*{1cm} at the observed resolution \\
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

\vspace{1cm}
\noindent 
texture analysis issues 
\begin{enumerate}
\item pattern recognition: \\
      given textured region, determine the class the region belongs to
\item generative model: \\
      given textured region, determine a description or model for it 
\item texture segmentation: \\
      given image with many textured areas, determine boundaries
\end{enumerate}
(a) multispectral image (b) printout of machine classification results \\
=====Gonzalez, {\em Digital Image Processing}, Fig. 9.13=====

\vspace{1cm}
\noindent 
texture: fineness, coarseness, contrast, directionality, roughness, regularity\\
statistical texture-feature approaches use
\begin{itemize}
\item autocorrelation function
\item spectral power density function
\item edgeness per unit area
\item spatial gray level co-occurrence probabilities
\item gray level run-length distributions
\item relative extrema distributions
\item mathematical morphology
\end{itemize}

\vspace{1cm}
\noindent 
Given 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. \\
image texture analysis: verification and estimation \\
verification: verify given image texture sample consistent with model \\
estimation: estimate values of model parameters based on observed sample \\
examples of model-based techniques
\begin{itemize}
\item autoregressive, moving-average, time-series models (extended to 2D)
\item Markov random fields
\item mosaic models
\end{itemize}

\vspace{1cm}
\noindent 
texel: texture element: basic textural unit of some textural primitives 
qualitatively evaluated image texture properties
\begin{itemize}
\item fineness
\item coarseness
\item smoothness
\item granulation
\item randomness
\item lineation
\item mottled
\item irregular
\item hummocky
\end{itemize}
For any textural surface there exists a scale at which, when the surface 
is examined, it appears smooth and textureless 
(e.g. see from infinite distance). \\
Then as resolution increases, the surface appears as a fine texture and then
a coarse one, and for multiple-scale textural surfaces the cycle of smooth,
fine, and coarse may repeat. \\
texture: cannot be analyzed without frame of reference on scale, resolution \\
texture: scale-dependent phenomenon \\
=====Fig. 9.1=====

\vspace{1cm}
\noindent 
to characterize texture 
\begin{itemize}
\item characterize gray level primitive properties
\item characterize spatial relationships between them
\end{itemize}
=====Oldie 33:20=====

\vspace{1cm}
9.2 Gray Level Co-Occurrence \\
first-order gray-level statistics: statistics of single pixels, \\
\hspace*{1cm} e.g. histogram, mean, median, variance \\
second-order gray-level statistics: the combined statistics of the gray levels\\
\hspace*{1cm} of pairs of pixels in which each two pixels in a pair have a \\
\hspace*{1cm} fixed relative position, e.g. co-occurrence \\
gray level spatial dependence: characterize texture by co-occurrence \\
% coarse texture: distribution changes only slightly with distance \\
% fine texture: distribution changes rapidly with distance \\
The gray level co-occurrence can be specified in a matrix of relative 
frequencies $P_{ij}$ with which two neighboring pixels separated by distance
$d$ occur on the image, one with gray level $i$ and the other with gray 
level $j$. \\
co-occurrence matrix: symmetric, function of angle and distance between pixels\\
=====Fig. 9.2===== 

\vspace{1cm}
\noindent 
probability of horizontal, $d$ pixels apart pixels
\[
P(i,j,d, 0^\circ)=\#\{[(k,l),(m,n)]|k-m=0,|l-n|=d, I(k,l)=i, I(m,n)=j\}
\]
probability of $45^\circ$, $d$ pixels apart pixels
\[
P(i,j,d, 45^\circ)=\#\{[(k,l),(m,n)]|[(k-m=d,l-n=-d) {\rm \ or \ } (k-m=-d,
\]
\[
         l-n=d)], I(k,l)=i, I(m,n)=j\}
\]
probability of vertical, $d$ pixels apart pixels
\[
P(i,j,d, 90^\circ)=\#\{[(k,l),(m,n)]||k-m|=d,l-n=0, I(k,l)=i, I(m,n)=j\}
\]
probability of $135^\circ$, $d$ pixels apart pixels
\[
P(i,j,d,135^\circ)=\#\{[(k,l),(m,n)]|[(k-m=d,l-n=d) {\rm \ or \ } (k-m=-d,
\]
\[
         l-n=-d)], I(k,l)=i, I(m,n)=j\}
\]
matrix symmetric: $P(i,j,d,a)=p(j,i,d,a)$  \\
=====Fig. 9.3===== \\
entropy highest: when all entries in $P_{ij}$ are equal \\
entropy highest: image where no preferred gray-level pairs exist \\
features calculated from the co-occurrence matrix \\
=====Fig. 9.4===== \\
gray-level co-occurrence matrix: widely used in remote sensing applications \\
\hspace*{1cm} for land-use classification

\vspace{1cm}
9.2.1 Generalized Gray Level Spatial Dependence Models for Texture \\
simple generalization: consider more than two pixels at a time

\vspace{1cm}
9.3 Strong Texture Measures and Generalized Co-Occurrence \\
generalized co-occurrence: consider co-occurrence between texture primitives \\
texture primitive: connected set of pixels characterized by attribute list\\
simplest texture primitive: pixel with gray level attribute \\
more complicated primitive: connected set of pixels homogeneous in level \\
primitive: characterized by size, elongation, orientation, average intensity 

\vspace{1cm}
9.3.1 Spatial Relationships \\
primitives constructed: list of primitives, center coordinates, attributes \\
generalized co-occurrence matrix $P$
\[
P(t_1, t_2) = \frac{\#\{(q_1,q_2)\in S | f(q_1)=t_1 {\rm \ and \ } f(q_2)=t_2\}}{\#S}
\]
$S \subseteq Q \times Q$: binary relation satisfying spatial relationship \\
$Q$: set of all primitives on the image \\
$T$: set of primitive properties \\
$t_1, t_2$: properties which primitives have \\
$f$: function assigning to each primitive in $Q$ a property of $T$

\vspace{1cm}
9.4 Autocorrelation Function and Texture \\
texture: relates to spatial size of gray level primitives on image \\
gray level primitives of larger size: coarser texture \\
gray level primitives of smaller size: finer texture \\
autocorrelation function: describes the size of gray level primitives \\
autocorrelation function
\[
p(x,y)=\frac{\frac{1}{(L_x-|x|)(L_y-|y|)}\int_{-\infty}^\infty 
\int_{-\infty}^\infty I(u,v)I(u+x,v+y)dudv}{\frac{1}{L_xL_y}
\int_{-\infty}^\infty \int_{-\infty}^\infty I^2(u,v)dudv} \ \ \ \ \ 
|x|<L_x, |y| < L_y
\]
autocorrelation function $p[k,l]$ for $N\times N$ image:
\[
p[k,l]=\frac{\frac{1}{(N-k)(N-l)}\sum_{i=1}^{(N-k)}
\sum_{j=1}^{(N-l)} f[i,j]f[i+k,j+l]}{\frac{1}{N^2}
\sum_{i=1}^N \sum_{j=1}^N f^2[i,j]} \ \ \ \ \ 
0\leq k,l \leq N-1
\]
gray level primitives on image relatively large: texture coarse: \\
\hspace*{1cm} autocorrelation drops off slowly with distance \\
gray level primitives on image relatively small: texture fine: \\
\hspace*{1cm} autocorrelation drops off quickly with distance \\
autocorrelation function: measure of periodicity and scale of texture

\vspace{1cm}
9.5 Digital Transform Methods and Texture \\
In the digital transform method of texture analysis, the digital image is
typically divided into a set of nonoverlapping small square subimages \\
Fourier transform: uses the complex sinusoid basic set

\vspace{1cm}
9.6 Texture Energy \\
In the textural energy approach, 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.

\vspace{1cm}
9.7 Textural Edgeness \\
autocorrelation, digital transforms: reference texture to spatial frequency \\
texture: in terms of edgeness per unit area \\
small neighborhoods: to detect microedges \\
large neighborhoods: to detect macroedges \\
=====Fig. 9.6=====

\vspace{1cm}
9.8 Vector Dispersion \\
In the vector dispersion technique, the image texture is divided into
mutually exclusive neighborhoods, and a sloped plane fit to the gray levels
is performed for each neighborhood. \\
=====Fig. 9.7=====

\vspace{1cm}
9.9 Relative Extrema Density \\
a texture measure: number of extrema per unit area \\
in any row of pixels, a pixel $i$ is a relative minimum if
\[
g(i) \leq g(i+1) {\rm \ and \ } g(i) \leq g(i-1)
\]
pixel $i$ is relative maximum if
\[
g(i) \geq g(i+1) {\rm \ and \ } g(i) \geq g(i-1)
\]
constant gray level run of pixels: minimum and maximum simultaneously

\vspace{1cm}
9.10 Mathematical Morphology \\
granularity of the binary image $F$
\[
G(d)= 1-\frac{\#F \open H_d}{\#F} = \frac{\#F - \#F \open H_d}{\#F}
\]
$\#F$: number of elements in $F$ \\
$H_d$: disk structuring element of diameter $d$ \\
$G(d)$ measures the properties of grain pixels that cannot be contained in
some translated structuring element of size $d$ that is entirely contained
in the grain and contains the given pixel \\
$G(d)$ measures the proportion of pixels participating in grains of a size 
smaller than $d$

\vspace{1cm}
\noindent
scale-$k$ volume of the blanket around a gray level intensity surface $I$
\[
V(k) = \sum_{(r,c)} (I\oplus_k H)(r,c) - (I \ominus_k H)(r,c)
\]
$\oplus_k$: $k$-fold dilation with the structuring element \\
$\ominus_k$: $k$-fold erosion with the structuring element \\
=====joke=====

\vspace{1cm}
9.11 Autoregression Models \\
autoregression models for texture synthesis: \\
next gray level value $a_{N+1}$: linear combination of synthesized and 
noise values
\[
a_{N+1}=\sum_{k=0}^{K-1} \alpha_k a_{N-k} + \sum_{l=0}^{L-1} \beta_l b_{N-l}
\]
$a_k$: given starting sequence \\
$b_k$: randomly generated noise image \\
=====Fig. 9.8===== \\
two-dimensional autoregressive model: better performance on general textures
\[
a(i,j)=\sum_{(k,l)\in N(i,j)} \alpha(i-k,l-j)a(k,l)
+\sum_{(k,l)\in N(i,j)} \beta(i-k,l-j) b(k,l)
\]
=====Fig. 9.9===== \\
The power of the autoregression linear estimator approach is that it is easy
to use the estimator in a mode that synthesizes textures from any initially
given linear estimator. \\
In this sense the autoregressive approach is sufficient to capture everything
about a texture. \\
Its weakness is that the textures it can characterize are likely to consist
mostly of microtextures

\vspace{1cm}
9.12 Discrete Markov Random Fields \\
The Markov random field model for texture assumes that the texture field is
stochastic and stationary and satisfies a conditional independence assumption.\\
when the distributions are Gaussian, the discrete Gauss-Markov field can be 
written as an equation in which each pixel's value is a linear combination of
the values in its neighborhood plus a correlated noise term
\[
I(r,c)=\sum_{(i,j) \in N(0,0)} I(r-i, c-j)h(i,j)+u(r,c)
\]
function $h$: coefficients of linear combination \\
model parameters $h$: computed from texture image with least-squares method \\
$u$: joint set of possible correlated Gaussian random variables

\vspace{1cm}
9.13 Random Mosaic Models \\
random mosaic models: constructed in two steps. \\
first step: provides a means of tessellating a plane into cells \\
second step: assigns a property value to each cell 

\vspace{1cm}
9.14 Structural Approaches to Texture Models \\
pure structural models: primitives in regular repetitive spatial arrangements \\
to describe texture: describe primitives and placement rules \\
Escher-inspired pattern with two of several possible candidate texels \\
=====V. S. Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 6.5===== \\
(a) simple texture formed by repeated placement of disks (b) corrupted \\
=====R. Jain, {\em Machine Vision}, Fig. 7.3=====

\vspace{1cm}
9.15 Texture Segmentation \\
The constraint is that each region has a homogeneous texture, such as that
arising from a frontal view, and that each pair of adjacent regions is 
differently textured

\vspace{1cm}
9.16 Synthetic Texture Image Generation \\
fractals: shapes that exhibit recursive self-similarity \\
Every fractal can be recursively subdivided into smaller nonoverlapping
shapes, each of which is a scaled-down version of the whole, either in
a deterministic sense, or in a statistical sense \\
fractals: shapes that exhibit recursive self-similarity \\
=====V. S. Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 6.4===== 

\vspace{1cm}
9.17 Shape from Texture \\
image texture gradients on oblique photography can be used to estimate
surface orientation of the observed three-dimensional object \\
assumption: no depth change and no texture changes within the observed area \\
assumption: no subtextures \\
results of integrated approach to identification of image texels \\
=====V. S. Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 6.9===== \\
image textural variation as cue to three-dimensional shape \\
=====V. S. Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 6.6===== \\
shape of golf ball recovered from the projective distortion of circles \\
=====V. S. Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 6.11===== 

\vspace{1cm}
\noindent
unknown plane where textural surface observed
\[
Ax+By+Cz+D=0
\]
where $A^2+B^2+C^2=1$ \\
from perspective projection, 3D point $(x,y,z)$ with projection $(u,v)$ 
\[
u = f \frac{x}{z} \ \ \ \ \ \ v=f \frac{y}{z}
\]
$f$: distance from the image plane to center of projection
\[
x = \frac{uz}{f} \ \ \ \ \ \ y = \frac{vz}{f}
\]
solving for $z$
\[
z = \frac{-Df}{Au+Bv+Cf}
\]
keep going on ... \\
use similar triangle geometry \\
=====Fig. 9.11=====

\vspace{1cm}
\noindent
3D implications of discontinuities in image characteristics of spatial texture\\
=====V. S. Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 6.14===== 

\vspace{1cm}
9.18 Summary \\
texture: in terms of primitives and spatial relationships \\
qualitatively: shape from texture can work \\
quantitatively: techniques generally not dependable \\
=====joke=====

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