Principles of Digital Image Processing Core Algorithms
This easy-to-follow textbook is the second of three volumes which provide a modern, algorithmic introduction to digital image processing, designed to be used both by learners desiring a firm foundation on which to build, and practitioners in search of cri
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Mathematical Notation
A.1 Symbols The following symbols are used in the main text primarily with the denotations given below. While some symbols may be used for purposes other than the ones listed, the meaning should always be clear in the particular context. {a, b, c, d, . . .}
A set ; i. e., an unordered collection of distinct elements. A particular element x can be contained in a set at most once. A set may also be empty ({ }).
(a1 , a2 , . . . an )
A vector ; i. e., a fixed-size collection of elements of the same type. (a1 , a2 , . . . an )T denotes the transposed (i. e., column) vector. In programming, vectors are usually implemented as one-dimensional arrays, with elements being referred to by position (index).
[c1 , c2 , . . . cm ]
A sequence or list; i. e., a collection of elements of variable length. Elements can be added to the sequence (inserted) or deleted from the sequence. A sequence may be empty ([ ]). In programming, sequences are usually implemented with dynamic data structures, such as linked lists. Java’s Collections framework (see also Vol. 1 [14, Appendix B.2.7]) provides numerous ready-to-use implementations.
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A. Mathematical Notation
α1 , α2 , . . . αk
A tuple; i. e., an ordered list of elements, each possibly of a different type. Tuples are typically implemented as objects (in Java or C++) or structures (in C) with elements being referred to by name.
¬
Logical “not” operator.
∧
Logical “and” operator.
∗
Linear convolution operator.
Linear correlation operator (Sec. 11.1.1).
⊕
Morphological dilation operator (see Vol. 1 [14, Sec. 7.2.3]).
Morphological erosion operator (see Vol. 1 [14, Sec. 7.2.4]).
∂
Partial derivative operator (see Vol. 1 [14, Sec. 6.2.1]). For example, ∂f ∂x (x, y) denotes the first derivative of the function f (x, y) along the x variable at position (x, y), is the second derivative, etc.
∂2 f ∂ 2 x (x, y)
∇
Gradient. ∇f is the vector of partial derivatives of a multidimensional function f (see Vol. 1 [14, Sec. 6.2.1]).
x
“Floor” of x, the largest integer z ∈ Z smaller than x ∈ R (i. e., z = x ≤ x). For example, 3.141 = 3, −1.2 = −2.
a
Pixel value (usually 0 ≤ a < K).
Arctan(y, x)
Inverse tangent function, similar to arctan xy = tan−1 xy but with two arguments and returning angles in the range [−π, +π] (i. e., covering all four quadrants). It corresponds to the Java method Math.atan2(y,x ).
card{. . .}
Cardinality (size) of a set, card A ≡ |A|.
DFT
Discrete Fourier transform (Sec. 7.3).
F
Continuous Fourier transform (Sec. 7.1.4).
g(x), g(x, y)
One- and two-dimensional continuous functions (x, y ∈ R).
g(u), g(u, v)
One- and two-dimensional discrete functions (u, v ∈ Z).
G(m), G(m, n)
One- and two-dimensional discrete Fourier spectra (m, n ∈ Z).
A.2 Set Operators
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h(i)
Histogram of an image at pixel value (or bin) i (see Vol. 1 [14, Sec. 3.1]).
H(i)
Cumulative histogram of an image at pixel value (or bin) i (see Vol. 1 [14, Sec. 3.6]).
I(u, v)
Intensity value of the image I at (integer) position
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