Introduction to Computer Vision
Image Texture Analysis
Lecture 12
1
A few examples
• Morphological processing for background
illumination estimation
• Optical character recognition
Roger S. Gaborski
2
Image with nonlinear illumination
Original Image
Thresholded with graythresh
3
Obtain Estimate of Background
background = imopen(I,strel('disk',15)); %GRAYSCALE
figure, imshow(background, [])
figure, surf(double(background(1:8:end,1:8:end))),zlim([0 1]);
Roger S. Gaborski
4
%subtract background estimate from original image
I2 = I - background;
figure, imshow(I2), title('Image with background removed')
level = graythresh(I2);
bw = im2bw(I2,level);
figure, imshow(bw),title('threshold')
Roger S. Gaborski
5
Comparison
Original Threshold
Background Removal - Threshold
Roger S. Gaborski
6
Optical Character Recognition
• After segmenting a character we still need to
recognize the character.
• How do we determine if a matrix of pixels
represents an ‘A’, ‘B’, etc?
Roger S. Gaborski
7
Roger S. Gaborski
8
Roger S. Gaborski
9
Approach
• Select line of text
• Segment each letter
• Recognize each letter as ‘A’, ‘B’, ‘C’, etc.
Roger S. Gaborski
10
Select line 3:
Samples of segment of individual letters in line 3:
Roger S. Gaborski
11
• We need labeled samples of each potential
letter to compare to unknown
• Take the product of the unknown character
and each labeled character and determine
with labeled character is the closest match
Roger S. Gaborski
12
%Load Database of characters (samples of known characters)
load charDB08182009.mat
whos char08182009
Name
Size
Bytes Class
Attributes
char08182009 26x1050
218400 double
EACH ROW IS VECTORIZED CHARACTER BITMAP
Roger S. Gaborski
13
BasicOCR.m
CODE SOMETHING LIKE THIS:
cc = ['A' 'B' 'C' 'D' 'E' 'F' 'G' 'H' 'I' 'J' 'K' 'L' 'M' 'N' 'O' ...
'P' 'Q' 'R' 'S' 'T' 'U' 'V' 'W' 'X' 'Y' 'Z'];
First, convert matrix of text character to a row vector
for j=1:26
score(j)= sum(t .* char08182009R(j,:));
end
ind(i)=find(score= =max(score));
fprintf('Recognized Text %s, \n', cc(ind))
OUTPUT: Recognized Text HANSPETERBISCHOF,
Roger S. Gaborski
14
How can I segment this image?
Assumption:
uniformity of
intensities in local
image region
Roger S. Gaborski
University of Bonn
15
What is Texture?
Roger S. Gaborski
University of Bonn
16
Roger S. Gaborski
17
• Edge Detection
• Histogram
• Threshold - graythresh
Roger S. Gaborski
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Roger S. Gaborski
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Roger S. Gaborski
20
Roger S. Gaborski
21
lev = graythresh(I)
lev =
0.5647
>> figure, imshow(I<lev)
Roger S. Gaborski
22
What is Texture
• No formal definition
– There is significant variation in intensity levels
between nearby pixels
– Variations of intensities form certain repetitive
patterns (homogeneous at some spatial scale)
– The local image statistics are constant, slowly varying
• human visual system: textures are perceived as
homogeneous regions, even though textures do not
have uniform intensity
Roger S. Gaborski
23
Texture
• Apparent homogeneous regions:
Sand on a beach
A brick wall
– In both cases the HVS will interpret areas of sand or
bricks as a ‘region’ in an image
– But, close inspection will reveal strong variations in pixel
intensity
Roger S. Gaborski
24
Texture
• Is the property of a ‘group of pixels’/area; a single
pixel does not have texture
• Is scale dependent
– at different scales texture will take on different properties
• Large number of (if not countless) primitive objects
– If the objects are few, then a group of countable objects
are perceived instead of texture
• Involves the spatial distribution of intensities
– 2D histograms
– Co-occurrence matrixes
Roger S. Gaborski
25
Scale Dependency
• Scale is important – consider sand
• Close up
– “small rocks, sharp edges”
– “rough looking surface”
– “smoother”
• Far Away
– “one object
–  brown/tan color”
Roger S. Gaborski
26
Terms (Properties) Used
to Describe Texture
• Coarseness
• Roughness
• Direction
• Frequency
• Uniformity
• Density
How would describe dog fur, cat fur, grass, wood grain,
pebbles, cloth, steel??
Roger S. Gaborski
27
“The object has a fine grain and a
smooth surface”
• Can we define these terms precisely in
order to develop a computer vision
recognition algorithm?
Roger S. Gaborski
28
Features
• Tone – based on pixel intensity in the texture
primitive
• Structure – spatial relationships between primitives
• A pixel can be characterized by its Tonal/Structural
properties of the group of pixels it belongs to
Roger S. Gaborski
29
• Tonal:
–
–
–
–
Average intensity
Maximum intensity
Minimum intensity
Size, shape
• Spatial Relationship of Primitives:
– Random
– Pair-wise dependent
Roger S. Gaborski
30
Artificial Texture
























Roger S. Gaborski
31
Artificial Texture
























Segmenting into regions based on texture
Roger S. Gaborski
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Color Can Play an Important role in Texture
























Roger S. Gaborski
33
Color Can Play an Important Role in Texture
























Roger S. Gaborski
34
Statistical and Structural Texture
Consider a brick wall:
• Statistical Pattern – close up pattern in bricks
• Structural (Syntactic) Pattern – brick pattern
 on previous slides can be represented by a grammar,
such as, ababab )
Roger S. Gaborski
35
Most current research focuses on statistical texture
Edge density is a simple texture measure
- edges per unit distance
Segment object based on edge density
HOW DO WE ESTIMATE
EDGE DENSITY??
Roger S. Gaborski
36
Segment object based
on edge density
Move a window across the image
and count the number of edges in
the window
ISSUE – window size?
How large should the window be?
What are the tradeoffs?
How does window size affect
accuracy of segmentation?
Roger S. Gaborski
37
Segment object based
on edge density
Move a window across the image
and count the number of edges in
the window
ISSUE – window size?
How large should the window be?
Large enough to get a good estimate
Of edge density
What are the tradeoffs?
Larger windows result in larger overlap
between textures
How does window size affect
Accuracy of segmentation?
Smaller windows result in better region
segmentation accuracy, but poorer
Estimate of edge density
Roger S. Gaborski
38
Average Edge Density Algorithm
•
•
•
•
Smooth image to remove noise
Detect edges by thresholding image
Count edges in n x n window
Assign count to edge window
• Feature Vector  [gray level value, edge density]
• Segment image using feature vector
Roger S. Gaborski
39
Run Length Coding Statistics
• Runs of ‘similar’ gray level pixels
• Measure runs in the directions 0,45,90,135
0
0
2
3
1
2
1
0
2
3
1
3
3
3
1
0
Y( L, LEV, d)
Where L is the number of runs of
length L
LEV is for gray level value and
d is for direction d
Image
Roger S. Gaborski
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Image
0
0
2
3
1
2
1
0
2
3
1
3
3
3
1
0
45 degrees
0 degrees
Run Length, L
Run Length, L
2
3
4
1
Gray Level, LEV
Gray Level, LEV
1
0
1
2
3
Roger S. Gaborski
2
3
4
0
1
2
3
41
Image
0
0
2
3
1
2
1
0
2
3
1
3
3
3
1
0
45 degrees
0 degrees
Run Length, L
1
2
3
4
0
4
0
0
0
1
1
0
1
0
2
3
0
0
0
3
3
1
0
0
Gray Level, LEV
Gray Level, LEV
Run Length, L
Roger S. Gaborski
1
2
3
4
0
4
0
0
0
1
4
0
0
0
2
0
0
1
0
3
3
1
0
0
42
Run Length Coding
• For gray level images with 8 bits 256 shades of
gray  256 rows
• 1024x1024  1024 columns
• Reduce size of matrix by quantizing:
– Instead of 256 shades of gray, quantize each 8 levels into
one resulting in 256/8 = 32 rows
– Quantize runs into ranges; run 1-8  first column, 9-16 the
second…. Results in 128 columns
Roger S. Gaborski
43
Gray Level Co-occurrence Matrix, P[i,j]
• Specify displacement vector d = (dx, dy)
• Count all pairs of pixels separated by d having gray
level values i and j. Formally:
P(i, j) = |{(x1, y1), (x2, y2): I(x1, y1) = i, I(x2, 21) = j}|
Roger S. Gaborski
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Gray Level Co-occurrence Matrix
• Consider simple image with
gray level values 0,1,2
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
x
• Let d = (1,1)
x
y
One pixel right
One pixel down
y
Roger S. Gaborski
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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
Count all pairs of pixels in which the
first pixel has value i and the second
value j displaced by d.
P(1,0)
1
0
P(2,1)
2
1
Etc.
Roger S. Gaborski
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Co-occurrence Matrix, P[i,j]
j
2
1
2
0
1
0
1
2
0
0
2
2
0
2
1
1
2
0
1
2
2
0
1
2
2
0
1
1
2
1
2
2
0
1
0
1
2
2
3
2
i
P(i, j)
There are 16 pairs, so normalize by 16
Roger S. Gaborski
47
Uniform Texture
d=(1,1)
x
y
Let Black = 1, White = 0
P[i,j]
P(0,0)=
P(0,1)=
P(1,0)=
P(1,1) =
Roger S. Gaborski
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Uniform Texture
d=(1,1)
x
y
Let Black = 1, White = 0
P[i,j]
P(0,0)= 24
P(0,1)= 0
P(1,0)= 0
P(1,1) = 25
Roger S. Gaborski
49
Uniform Texture
d=(1,0)
x
y
Let Black = 1, White = 0
P[i,j]
P(0,0)= ?
P(0,1)= ?
P(1,0)= ?
P(1,1) = ?
Roger S. Gaborski
50
Uniform Texture
x
d=(1,0)
y
Let Black = 1, White = 0
P[i,j]
P(0,0)= 0
P(0,1)= 28
P(1,0)= 28
P(1,1) = 0
Roger S. Gaborski
51
Randomly Distributed Texture
What if the Black and white pixels where randomly distributed?
What will matrix P look like??
1
0
1
0
1
1
0
0
1
0
1
1
1
1
0
0
1
1
0
1
0
0
1
0
0
0
0
1
0
0
0
1
0
1
0
0
1
1
0
1
1
0
1
0
1
1
1
0
0
0
0
1
0
1
0
1
0
1
1
1
0
1
1
1
No preferred set of gray level
pairs, matrix P will have
approximately a uniform
population
Roger S. Gaborski
52
Co-occurrence Features
• Gray Level Co-occurrence Matrices(GLCM)
– Typically GLCM are calculated at four different
angles: 0, 45,90 and 135 degrees
– For each angles different distances can be used,
d=1,2,3, etc.
– Size of GLCM of a 8-bit image: 256x256 (28).
Quantizing the image will result in smaller matrices.
A 6-bit image will result in 64x64 matrices
– 14 features can be calculated from each GLCM. The
features are used for texture calculations
Roger S. Gaborski
53
Co-occurrence Features
• P(ga,gb,d,t):
–
–
–
–
ga  gray level pixel ‘a’
gb  gray level pixel ‘b’
d  distance d
t  angle t (0, 45,90,135)
In many applications the transition ga to gb and gb to ga are
both counted. This results in symmetric GLCMs:
For P(0,0,1,0)
0
0
results in an entry of 2 for the ‘0 0’ entry
Roger S. Gaborski
54
Co-occurrence Features
• The data in the GLCM are used to derive the
features, not the original image data
Contrast   Pi , j  (i  j )2
i, j
• How do we interpret the contrast equation?
Roger S. Gaborski
55
Co-occurrence Features
• The data in the GLCM are used to derive the features,
not the original image data: Measures the local
variations in the gray-level co-occurrence matrix.
Contrast   Pi , j  (i  j )2
i, j
• How do we interpret the contrast equation?
The term (i-j)2: weighing factor (a squared term)
– values along the diagonal (i=j) are multiplied by zero. These values
represent adjacent image pixels that do not have a gray level
difference.
– entries further away from the diagonal represent pixels that have a
greater gray level difference, that is more contrast, and are
multiplied by a larger weighing factor.
Roger S. Gaborski
56
Co-occurrence Features
• Dissimilarity:
dissimilarity   Pi , j  | i  j |
i, j
– Dissimilarity is similar to contrast, except the
weights increase linearly
Roger S. Gaborski
57
Co-occurrence Features
• Inverse Difference Moment
Pi , j
IDM  
2
1

(
i

j
)
i, j
– IDM has smaller numbers for images with high contrast,
larger numbers for images low contrast
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Co-occurrence Features
• Angular Second Moment(ASM) measures orderliness: how
regular or orderly the pixel values are in the window
ASM   Pi ,2j
i, j
• Energy is the square root of ASM
E
2
P
 i, j
i, j
• Entropy:
Entropy   Pi ,2j  ( ln Pi , j )
i, j
where ln(0)=0
Roger S. Gaborski
59
Matlab Texture Filter Functions
Function
Description
rangefilt
Calculates the local range of an image.
stdfilt
Calculates the local standard deviation of an image.
entropyfilt
Calculates the local entropy of a grayscale image. Entropy is a
statistical measure of randomness
Roger S. Gaborski
60
rangefilt
A=
1
4
8
6
1
3
3
7
2
8
5
4
3
7
9
5
2
5
2
6
2
6
4
2
7
Symmetrical Padding
1
1
4
8
6
1
1
1
1
4
8
6
1
1
3
3
3
7
2
8
8
5
5
4
3
7
9
9
5
5
2
5
2
6
6
2
2
6
4
2
7
7
2
2
6
4
2
7
7
max = 4, min = 1, range = 3
Roger S. Gaborski
61
rangefilt Results (3x3)
A=
1
4
8
6
1
3
3
7
2
8
5
4
3
7
9
5
2
5
2
6
2
6
4
2
7
>> R = rangefilt(A)
R=
3
7
6
7
7
4
7
6
8
8
3
5
5
7
7
4
4
5
7
7
4
4
4
5
5
Roger S. Gaborski
62
rangefilt Results (5x5)
A=
1
4
8
6
1
3
3
7
2
8
5
4
3
7
9
5
2
5
2
6
2
6
4
2
7
>> R = rangefilt(A, ones(5))
R=
7
7
8
8
8
7
7
8
8
8
7
7
8
8
8
5
5
7
7
7
4
5
7
7
7
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63
Original image
Roger S. Gaborski
64
Imfilt = rangefilt(Im);
figure, imshow(Imfilt, []), title('Image by rangefilt')
Roger S. Gaborski
65
Imfilt = stdfilt(Im);
figure, imshow(Imfilt, []), title('Image by stdfilt')
Roger S. Gaborski
66
Imfilt = entropyfilt(Im);
figure, imshow(Imfilt, []), title('Image by entropyfilt')
Roger S. Gaborski
67
Matlab function: graycomatrix
• Computes GLCM of an image
– glcm = graycomatrix(I) analyzes pairs of
horizontally adjacent pixels in a scaled version of I.
If I is a binary image, it is scaled to 2 levels. If I is
an intensity image, it is scaled to 8 levels.
– [glcm, SI] = graycomatrix(...) returns the scaled
image used to calculate GLCM. The values in SI are
between 1 and 'NumLevels'.
Roger S. Gaborski
68
Parameters
• ‘Offset’ determines number of co-occurrences
matrices generated
• offsets is a q x 2matrix
– Each row in matrix has form [row_offset,
col_offset]
– row_off specifies number of rows between pixel
of interest and its neighbors
– col_off specifies number of columns between
pixel of interest and its neighbors
Roger S. Gaborski
69
Offset
•
•
•
•
•
•
[0,1] specifies neighbor one column to the left
Angle
Offset
0
[0 D]
45
[-D D]
90
[-D 0]
135
[-D –D]
Roger S. Gaborski
70
Orientation of offset
• The figure illustrates the array: offset = [0 1; -1
1; -1 0; -1 -1]
90, [-1,0]
135, [-1,-1]
Roger S. Gaborski
45, [ -1,1]
0
,
[
0
,
1
]
71
Intensity Image
– mat2gray Convert matrix to intensity image.
I = mat2gray(A,[AMIN AMAX]) converts the matrix A
to the intensity image I.
The returned matrix I contains values in the range
0.0 (black) to 1.0
Roger S. Gaborski
72
graycomatrix Example
From textbook, p 649
>> f = [ 1 1 7 5 3 2;
5 1 6 1 2 5;
8 8 6 8 1 2;
4 3 4 5 5 1;
8 7 8 7 6 2;
7 8 6 2 6 2]
f=
1
5
8
4
8
7
1
1
8
3
7
8
7
6
6
4
8
6
5
1
8
5
7
2
3
2
1
5
6
6
2
5
2
1
2
2
Need to convert to an Intensity image
[0,1]
Roger S. Gaborski
73
>> fm = mat2gray(f)
fm =
0
0.5714
1.0000
0.4286
1.0000
0.8571
0
0.8571 0.5714 0.2857 0.1429
0 0.7143
0 0.1429 0.5714
1.0000 0.7143 1.0000
0 0.1429
0.2857 0.4286 0.5714 0.5714
0
0.8571 1.0000 0.8571 0.7143 0.1429
1.0000 0.7143 0.1429 0.7143 0.1429
Roger S. Gaborski
74
Quantize to 8 Levels
IS =
1
5
8
4
8
7
1
1
8
3
7
8
7
6
6
4
8
6
5
1
8
5
7
2
3
2
1
5
6
6
2
5
2
1
2
2
Roger S. Gaborski
75
>> offsets = [0 1]; 
>> [GS, IS] =
graycomatrix(fm,'NumLevels', 8, 'Offset', offsets)
GS =
1
0
0
0
2
1
0
1
2
0
1
0
0
3
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
1
0
1
0
1
1
0
0
0
0
1
2
1
0
0
0
0
0
0
2
0
0
0
0
0
1
2
1
Roger S. Gaborski
See NEXT PAGE
76
GS =
1
0
0
0
2
1
0
1
2
0
1
0
0
3
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
1
0
1
0
1
1
0
0
0
0
1
2
1
1
8
3
7
8
7
6
6
4
8
6
5
1
8
5
7
2
3
2
1
5
6
6
2
5
2
1
2
2
1
0
0
0
0
0
0
2
0
0
0
0
0
1
2
1
IS =
1
5
8
4
8
7
Roger S. Gaborski
77
'GrayLimits'
Two-element vector, [low high], that specifies how the grayscale
values in I are linearly scaled into gray levels. Grayscale values
less than or equal to low are scaled to 1. Grayscale values greater
than or equal to high are scaled to NumLevels. If graylimits is set
to [], graycomatrix uses the minimum and maximum grayscale
values in the image as limits, [min(I(:)) max(I(:))].
>> [GS, IS] = graycomatrix(f,'NumLevels', 8, 'Offset', offsets, 'G',[])
Roger S. Gaborski
78
>> [GS, IS] = graycomatrix(f,'NumLevels', 8, 'Offset', offsets, 'G',[])
>> I = rand(5)
I=
0.0085 0.8452 0.2026 0.1901 0.6818
0.6311 0.1183 0.1947 0.1580 0.5397
0.2303 0.8539 0.6766 0.8251 0.9968
0.4624 0.7807 0.7231 0.5540 0.1104
0.3995 0.4229 0.7560 0.3559 0.6204
Roger S. Gaborski
79
>> [GS, IS] = graycomatrix(f,'NumLevels', 8, 'Offset', offsets, 'G',[])
GS =
1
0
0
0
2
1
0
1
2
0
1
0
0
3
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
1
0
1
0
1
1
0
0
0
0
1
2
1
1
8
3
7
8
7
6
6
4
8
6
5
1
8
5
7
2
3
2
1
5
6
6
2
5
2
1
2
2
1
0
0
0
0
0
0
2
0
0
0
0
0
1
2
1
IS =
1
5
8
4
8
7
Roger S. Gaborski
80
>> [GS, IS] = graycomatrix(f,'NumLevels', 4, 'Offset', offsets, 'G',[])
GS =
3
1
6
1
0
2
1
0
3
1
1
4
1
0
1
5
1
1
4
2
4
4
4
3
3
2
4
3
3
1
4
3
4
1
IS =
1
3
4
2
4
4
2
1
1
3
3
3
1 ORIGINAL IMAGE QUANTIZED
3 TO 4 LEVELS
1
1
1
1
Roger S. Gaborski
81
Texture feature
Energy
P
2
i, j
i, j
formula
Provides the sum of squared elements
in the GLCM. (square root of ASM)
Entropy
Measure uncertainty of the
P  ( ln Pi , j )

image(variations)
i, j
Contrast
2
P

(
i

j
)
 i, j
2
i, j
i, j
Homogeneity
Pi , j
 1 || i  j ||
i, j
Measures the local variations in the
gray-level co-occurrence matrix.
Measures the closeness of the
distribution of elements in the GLCM to
the GLCM diagonal.
Roger S. Gaborski
82
glcms = graycomatrix(Im, 'NumLevels', 256, 'G',[]))
stats = graycoprops(glcms, 'Contrast Correlation Homogeneity’);
figure, plot([stats.Correlation]);
title('Texture Correlation as a function of offset');
xlabel('Horizontal Offset'); ylabel('Correlation')
Roger S. Gaborski
83
Texture Measurement
Quantize 256
Gray Levels to 32
Data Window
31x31 or 15x15
GLCM0 GLCM45 GLCM90 GLCM135
Feature for Each Matrix
ENERGY
ENTROPY
CONTRAST
etc
Roger S. Gaborski
Generate
Feature
Matrix
For Each
Feature
84
image
Ideal
map
Roger S. Gaborski
85
Classmaps generated using the 3 best CO feature images
Roger S. Gaborski
86
31x31 produces the
Best results, but
large errors at
borders
Classmaps generated using the 7 best CO feature images
Roger S. Gaborski
87
Law’s Texture Energy Features
• Use texture energy for segmentation
• General idea: energy measured within textured
regions of an image will produce different values for
each texture providing a means for segmentation
• Two part process:
– Generate 2D kernels from 5 basis vectors
– Convolve images with kernels
Roger S. Gaborski
88
Law’s Kernel Generation
Level L5 = [ 1 4 6 4 1 ]
Spot S5 = [ -1 0 2 0 –1 ]
Ripple R5 = [ 1 –4 6 –4 1 ]
Wave W5 = [ -1 2 0 -2 1 ]
Edge E5 = [ -1 –2 0 2 1 ]
To generate kernels, multiply one vector by the transpose of itself or
another vector:
L5E5 = [ 1 4 6 4 1 ]’ * [ -1 –2 0 2 1 ]
-1
-2
0
2
1
-4
-8
0
8
4
-6
-12
0
12
6
-4
-8
0
8
4
-1
-2
0
2
1
• 25 possible 2D kernels are
possible, but only 24 are used
• L5L5 is sensitive to mean
brightness values and is not used
Roger S. Gaborski
89
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92
textureExample.m
•
•
•
•
•
Reads in image
Converts to double and grayscale
Create energy kernels
Convolve with image
Create data ‘cube’
Roger S. Gaborski
93
stone_building.jpg
Roger S. Gaborski
94
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Test 2
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99
Roger S. Gaborski
100
Scale
• How will scale affect energy measurements?
• Reduce image to one quarter size
imGraySm = imresize(imGray, 0.25, bicubic');
Roger S. Gaborski
101
Data ‘cube’
>> data = cat(3, im(:,:,1), im(:,:,2), im(:,:,3), imL5R5, imR5E5);
>> figure, imshow(data(:,:,1:3))
>> data_value=data(7,12,:)
data_value(:,:,1) = 142
data_value(:,:,2) = 166
data_value(:,:,3) = 194
data_value(:,:,4) = 22
data_value(:,:,5) = 10
Roger S. Gaborski
102
Fractal Dimension
• Hurst coefficient can be used to calculate the fractal
dimension of a surface
• The fractal dimension can be interpreted as a
measure of texture
Consider the 5 pixel wide neighborhood (13 pixels)
d
c
b
c
d
b
a
b
d
c
b
c
Pixel
Distance
Number
Class
from center
d
a
b
1
4
0
1
c
4
1.414
d
4
2
Roger S. Gaborski
103
Fractal Dimension Algorithm
•
•
•
•
Lay mask over original image
Examine pixels in each of the classes
Record the brightest and darkest for each class
The pixel brightness difference (range) for each pixel
class is used to generate the Hurst plot
• Use least squares fit to construct a ln distance vs ln
range plot
• The slope of this line is the Hurst coefficient for the
specific pixel
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104