iff “if and only if”“there exists” “there does not exist” “for each” Sets Theoretic Notation and Operations X, Y, Z Bold, uppercase characters are used to represent point sets.. ÇS x Ima
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Handbook of Computer Vision Algorithms in Image Algebra
by Gerhard X Ritter; Joseph N Wilson
CRC Press, CRC Press LLC
ISBN: 0849326362 Pub Date: 05/01/96 Search this book:
Preface Acknowledgments
Chapter 1—Image Algebra
1.1 Introduction 1.2 Point Sets 1.3 Value Sets 1.4 Images 1.5 Templates 1.6 Recursive Templates 1.7 Neighborhoods
1.8 The p-Product
1.9 References
Chapter 2—Image Enhancement Techniques
2.1 Introduction 2.2 Averaging of Multiple Images 2.3 Local Averaging
2.4 Variable Local Averaging 2.5 Iterative Conditional Local Averaging 2.6 Max-Min Sharpening Transform 2.7 Smoothing Binary Images by Association 2.8 Median Filter
2.9 Unsharp Masking
Title
-invariant
Trang 22.10 Local Area Contrast Enhancement
2.11 Histogram Equalization
2.12 Histogram Modification
2.13 Lowpass Filtering
2.14 Highpass Filtering
2.15 References
Chapter 3—Edge Detection and Boundary Finding Techniques
3.1 Introduction
3.2 Binary Image Boundaries
3.3 Edge Enhancement by Discrete Differencing
3.4 Roberts Edge Detector
3.5 Prewitt Edge Detector
3.6 Sobel Edge Detector
3.7 Wallis Logarithmic Edge Detection
3.8 Frei-Chen Edge and Line Detection
3.9 Kirsch Edge Detector
3.10 Directional Edge Detection
3.11 Product of the Difference of Averages
3.12 Crack Edge Detection
3.13 Local Edge Detection in Three-Dimensional Images
3.14 Hierarchical Edge Detection
3.15 Edge Detection Using K-Forms
3.16 Hueckel Edge Operator
3.17 Divide-and-Conquer Boundary Detection
3.18 Edge Following as Dynamic Programming
3.19 References
Chapter 4—Thresholding Techniques
4.1 Introduction
4.2 Global Thresholding
4.3 Semithresholding
4.4 Multilevel Thresholding
4.5 Variable Thresholding
4.6 Threshold Selection Using Mean and Standard Deviation
4.7 Threshold Selection by Maximizing Between-Class Variance
4.8 Threshold Selection Using a Simple Image Statistic
4.9 References
Chapter 5—Thining and Skeletonizing
5.1 Introduction
5.2 Pavlidis Thinning Algorithm
5.3 Medial Axis Transform (MAT)
Trang 35.4 Distance Transforms
5.5 Zhang-Suen Skeletonizing
5.6 Zhang-Suen Transform — Modified to Preserve Homotopy 5.7 Thinning Edge Magnitude Images
5.8 References
Chapter 6—Connected Component Algorithms
6.1 Introduction
6.2 Component Labeling for Binary Images
6.3 Labeling Components with Sequential Labels
6.4 Counting Connected Components by Shrinking
6.5 Pruning of Connected Components
6.6 Hole Filling
6.7 References
Chapter 7—Morphological Transforms and Techniques
7.1 Introduction
7.2 Basic Morphological Operations: Boolean Dilations and Erosions 7.3 Opening and Closing
7.4 Salt and Pepper Noise Removal
7.5 The Hit-and-Miss Transform
7.6 Gray Value Dilations, Erosions, Openings, and Closings
7.7 The Rolling Ball Algorithm
7.8 References
Chapter 8—Linear Image Transforms
8.1 Introduction
8.2 Fourier Transform
8.3 Centering the Fourier Transform
8.4 Fast Fourier Transform
8.5 Discrete Cosine Transform
8.6 Walsh Transform
8.7 The Haar Wavelet Transform
8.8 Daubechies Wavelet Transforms
8.9 References
Chapter 9—Pattern Matching and Shape Detection
9.1 Introduction
9.2 Pattern Matching Using Correlation
9.3 Pattern Matching in the Frequency Domain
9.4 Rotation Invariant Pattern Matching
9.5 Rotation and Scale Invariant Pattern Matching
Trang 4Table of Contents
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Trang 5iff “if and only if”
“there exists”
“there does not exist”
“for each”
Sets Theoretic Notation and Operations
X, Y, Z Bold, uppercase characters are used to represent point sets
x, y, z Bold, lowercase characters are used to represent points, i.e., elements of point sets
The set = {0, 1, 2, 3, }
The set of integers, positive integers, and negative integers
The set = {0, 1, , n - 1}.
The set = {1, 2, , n}.
The set = {-n+1, , -1, 0, 1, , n - 1}.
The set of real numbers, positive real numbers, negative real numbers, and positive real numbers including 0
The set of complex numbers
An arbitrary set of values
The set unioned with {}
The set unioned with {}
The set unioned with {-,}
The empty set (the set that has no elements).
2X The power set of X (the set of all subsets of X).
“is an element of”
“is not an element of”
Union
X * Y = {z : z X or z Y}
Let be a family of sets indexed by an indexing set › = {x : x X»
for at least one » ›}
= X1 * X2 * * X n
= {x : x X i for some i }
X ) Y = {z : z X and z Y}
Let be a family of sets indexed by an indexing set › = {x : x X»
for all » ›}
= X ) X ) ) X
Trang 6= {x : x X i for all i }
X × Y {(x, y) : x X, y Y}
= {(x1,x2, ,x n ) : x i X i}
= {(x1,x2,x3, ) : x i X i}
The Cartesian product of n copies of , i.e.,
Let X and Y be subsets of some universal set U, X \ Y = {x X : x Y}.
X2 = U \ X, where U is the universal set that contains X.
Point and Point Set Operations
1 + y1, , x n + y n)
1 - y1, , x n - y n)
1y1, , x n y n)
1/y1, , x n /y n)
1 ¦ y1, , x n ¦ y n)
x ¥ y If x, y , then x ¥ y = (x1 ¥ y1, , x n ¥ y n)
1³y1, , x n ³y n)
1, , k³x n)
1y1 + x2y2 + ··· + x n y n
2y3 - x3y2, x3y1 - x1y3, x1y2 - x2y1)
If x and y then = (x1, , x n , y1, , y m)
1, , -x n)
1, , x n)
1, , x n)
[x] If x , then [x] = ([x1], , [x n])
pi(x) If x = (x1, x2, , x n) , then pi (x) = x i
1 + x2 + ··· + x n
1x2 ··· x n
1 ¦ x2 ¦ ··· ¦ x n
¥x If x , then ¥x = x ¥ x ¥ ··· ¥ x
Trang 7If x , then ||x||2 =
1 = |x1| + |x2| + ··· + |x n|
||x|| If x , then ||x|| = |x
1| ¦ |x2| ¦ ··· ¦ |x n|
X\Y If X, Y , then X\Y = {z : z X and z Y}
If X , then = {z : z and z X}
sup(X)
If X , then sup(X) = the supremum of X If X = {x1, x2, , xn }, then sup(X) = x1
¦ x2 ¦ ¦ xn
X For a point set X with total order , x0 = X Ô x x0, x X \ {x0}
inf(X)
If X , then inf(X) = the infimum of X If X = {x1, x2, , xn }, , then sup(X) = x1
¥ x2 ¥ ¥ xn
X For a point set X with total order , x0 = X Ô x0 x, x X \ {x0}
choice(X)
If X then, choice(X) X (randomly chosen element)
Morphology
In following table A, B, D, and E denote subsets of
A* The reflection of A across the origin 0 = (0, 0, 0) .
A2 The complement of A; i.e., A2 = {x : x A}.
A × B Minkowski addition is defined as A × B = {a + b : a A, b B} (Section 7.2)
A B The opening of A by B is denoted A B and is defined by A B = (A/B) × B.
(Section 7.3)
A " B The closing of A by B is denoted A " B and is defined by A " B = (A × B)/B (Section
7.3)
A C Let C = (D, E) be an ordered pair of structuring elements The hit-and-miss transform
of the set A is given by A C = {p : D p 4 A and E p 4 A2} (Section 7.5)
Functions and Scalar Operations
Trang 8f : X ’ Y f is a function from X into Y.
domain(f) The domain of the function f : X ’ Y is the set X.
range(f) The range of the function f : X ’ Y is the set {f (x) : x X}.
f-1 The inverse of the function f.
Y X The set of all functions from X into Y, i.e., if f YX, then f : X ’ Y.
f| A Given a function f : X ’ Y and a subset A 4 X, the restriction of f to A, f| A : A ’ Y, is
defined by f| A (a) = f(a) for a A.
f| g
Given: f : A ’ Y and g : B ’ Y, the extension of f to g is defined by
g f Given two functions f : X ’ Y and g : Y ’ Z, the composition g f : X ’ Z is defined by
(g f)(x) = g(f (x)), for every x X.
f + g Let f and g be real or complex-valued functions, then (f + g)(x) = f(x) + g(x).
f · g Let f and g be real or complex-valued functions, then (f · g)(x) = f(x) · g(x).
k · f Let f be a real or complex-valued function, and k be a real or complex number, then f
, (k · f)(x) = k · (f (x)).
|f| |f|(x) = |f(x)|, where f is a real (or complex)-valued function, and |f(x)| denotes the
absolute value (or magnitude) of f(x).
1X The identity function 1 X : X ’ X is given by 1 X (x) = x.
The projection function p j onto the jth coordinate is defined by p j (x1, ,x j , ,x n ) = x j
x For x the ceiling function x returns the smallest integer that is greater than or
equal to x.
x For x the floor function x returns the largest integer that is less than or equal to x.
[x] For x the round function returns the nearest integer to x If there are two such
integers it yields the integer with greater magnitude
x mod y For x, y , x mod y = r if there exists k, r with r < y such that x = yk + r.
ÇS (x)
Images and Image Operations
a, b, c Bold, lowercase characters are used to represent images Image variables will usually
be chosen from the beginning of the alphabet
a The image a is an -valued image on X The set is called the value set of a and X
the spatial domain of a.
1 Let be a set with unit 1 Then 1 denotes an image, all of whose pixel values are 1.
0 Let be a set with zero 0 Then 0 denotes an image, all of whose pixel values are 0.
a| Z
The domain restriction of a to a subset Z of X is defined by a| Z = a ) (Z × )
a||S
The range restriction of a to the subset S 4 is defined by a||S = a ) (X × S).
The double-bar notation is used to focus attention on the fact that the restriction is
applied to the second coordinate of a 4 X ×
Trang 9If a , Z 4 X, and S 4 , then the restriction of a to Z and S is defined as a| (Z,S) =
a ) (Z × S).
a| b
Let X and Y be subsets of the same topological space The extension of a to b
(a|b), (a1|a2| ···, |an) Row concatenation of images a and b, respectively the row concatenation of images
a1, a2, , an
Column concatenation of images a and b.
f(a)
If a and f : ’ Y, then the image f(a) YX is given by f a, i.e., f(a) = {(x, c(x)) : c(x) = f(a(x)), x X}.
a f If f : Y ’ X and a , the induced image a f is defined by a f = {(y,
a(f(y))) : y Y}.
a ³ b
If ³ is a binary operation on , then an induced operation on can be defined Let
a, b ; the induced operation is given by a ³ b = {(x, c(x)) : c(x) = a(x) ³ b(x), x
X}.
k ³ a
Let k , a , and ³ be a binary operation on An induced scalar operation on
images is defined by k ³ a = {(x, c(x)) : c(x) = k ³ a(x),x X}.
a b
Let a, b ; a b = {(x, c(x)) : c(x) = a(x) b(x) , x X}.
logb a
Let a, b logba = {(x, c(x)) : c(x) = logb(x)a(x), x X}.
a* Pointwise complex conjugate of image a, a* (x) = (a(x))*.
The following four items are specific examples of the global reduce operation Each assumes a and X = {x1, x2, , xn}
= a(x1) + a(x2) + ··· + a(xn)
= a(x1) · a(x2) ··· a(xn)
= a(x1) ¦ a(x2) ¦ ··· ¦ a(xn)
= a(x1) ¥ a(x2) ¥ ··· ¥ a(xn)
a " b
Dot product, a " b = £(a · b) = (a(x) · b(x)).
Templates and Template Operations
s, t, u Bold, lowercase characters are used to represent templates Usually characters from the
middle of the alphabet are used as template variables
t A template is an image whose pixel values are images In particular, an -valued
template from Y to X is a function t : Y ’ Thus, t and t is an -valued image on Y.
Trang 10t y
Let t For each y Y, t y = t(y) The image t y is given by t y = {(x, t y (x)) : x X}.
S(ty)
If and t , then the support of t is denoted by S(ty) and is
defined by S(ty ) = {x X : t y (x) ` 0}.
S (ty)
If t , then S (ty ) = {x X : t y (x) ` }.
S-(t y)
If t , then S-(t y ) = {x X : t y (x) ` -}.
S±(t y)
If t , then S±(t y ) = {x X : t y (x) ` ±}.
t(p) A parameterized -valued template from Y to X with parameters in P is a function of
the form t : P ’
t2
Image-Template Operations
In the table below, X is a finite subset of
a t
Let ( , ³, ) be a semiring and a , t , then the generic right product
t a
With the conditions above, except that now t , the generic left product of a
a t
product (or convolution) is defined as
t a
With the conditions above, except that t , the left linear product (or
a t
For a and t , the right additive maximum is defined by
t a
For a and t , the left additive maximum is defined by
Trang 11
a t
For a and t , the right additive minimum is defined by
t a
For a and t , the left additive minimum is defined by
a t
For a and t , the right multiplicative maximum is defined by
t a
For a and t , the left multiplicative maximum is defined by
a t
For a and t , the right multiplicative minimum is defined by
t a
For a and t , the left multiplicative minimum is defined by
Neighborhoods and Neighborhood Operations
M, N Italic uppercase characters are used to denote neighborhoods.
A neighborhood is an image whose pixel values are sets of points In particular, a
neighborhood from Y to X is a function N : Y ’ 2X
N(p) A parameterized neighborhood from Y to X with parameters in P is a function of the
form N : P ’
N2
Let N , the transpose N2 is defined as N2(x) = {y Y : x N (y)} that
is, x N(y) iff y N2( x).
N1 • N2
The dilation of N1 by N2 is defined by N(y) = (N1(y) + (p - y)).
Image-Neighborhood Operations
In the table below, X is a finite subset of
right reduction of a with N is defined as (a N)(x) =
Trang 12N a With the conditions above, except that now N , the generic left reduction of a
with t is defined as (N a)(x) = (a N2)(x).
average of its image argument (a N)(x) = a(a| N(x))
average of its image argument, (a N)(x) = m(a| N(x))
Matrix and Vector Operations
In the table below, A and B represent matrices.
The p-product of matrices A and B.
The dual p-product of matrices A and B, defined by
References
1 G Ritter, “Image algebra with applications.” Unpublished manuscript, available via anonymous ftp
from ftp://ftp.cis.ufl.edu/pub/src/ia/documents, 1994
Dedication
To our brothers, Friedrich Karl and Scott Winfield
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Handbook of Computer Vision Algorithms in Image Algebra
by Gerhard X Ritter; Joseph N Wilson
CRC Press, CRC Press LLC
ISBN: 0849326362 Pub Date: 05/01/96 Search this book:
Previous Table of Contents Next
Chapter 1 Image Algebra
1.1 Introduction
Since the field of image algebra is a recent development it will be instructive to provide some background information In the broad sense, image algebra is a mathematical theory concerned with the transformation and analysis of images Although much emphasis is focused on the analysis and transformation of digital images, the main goal is the establishment of a comprehensive and unifying theory of image transformations, image analysis, and image understanding in the discrete as well as the continuous domain [1]
The idea of establishing a unifying theory for the various concepts and operations encountered in image and signal processing is not new Over thirty years ago, Unger proposed that many algorithms for image
processing and image analysis could be implemented in parallel using cellular array computers [2] These
cellular array computers were inspired by the work of von Neumann in the 1950s [3, 4] Realization of von Neumann’s cellular array machines was made possible with the advent of VLSI technology NASA’s massively parallel processor or MPP and the CLIP series of computers developed by Duff and his colleagues represent the classic embodiment of von Neumann’s original automaton [5, 6, 7, 8, 9] A more general class
of cellular array computers are pyramids and Thinking Machines Corporation’s Connection Machines [10, 11, 12] In an abstract sense, the various versions of Connection Machines are universal cellular automatons with
an additional mechanism added for non-local communication
Many operations performed by these cellular array machines can be expressed in terms of simple elementary operations These elementary operations create a mathematical basis for the theoretical formalism capable of expressing a large number of algorithms for image processing and analysis In fact, a common thread among designers of parallel image processing architectures is the belief that large classes of image transformations can be described by a small set of standard rules that induce these architectures This belief led to the creation
of mathematical formalisms that were used to aid in the design of special-purpose parallel architectures Matheron and Serra’s Texture Analyzer [13] ERIM’s (Environmental Research Institute of Michigan) Cytocomputer [14, 15, 16], and Martin Marietta’s GAPP [17, 18, 19] are examples of this approach
The formalism associated with these cellular architectures is that of pixel neighborhood arithmetic and
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