Principles of Digital Image Processing Fundamental Techniques
This easy-to-follow textbook provides 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 critical analysis and concrete implemen
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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 (denoted by { }).
(a1 , a2 , . . . an )
A vector ; i. e., a fixed-size, ordered 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., an ordered collection of elements of variable length. Elements can be added to the sequence (inserted) or deleted from the sequence. A sequence may be empty (denoted by [ ]). In programming, sequences are usually implemented with dynamic data structures, such as linked lists. Java’s Collections framework (see also 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.
∗
Linear convolution operator (Sec. 5.3.1).
⊕
Morphological dilation operator (Sec. 7.2.3).
Morphological erosion operator (Sec. 7.2.4).
∂
Partial derivative operator (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), ond derivative, etc.
∂2 f ∂ 2 x (x, y)
is the sec-
∇
Gradient. ∇f is the vector of partial derivatives of a multidimensional function f (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(x, y)
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 ) (Secs. 6.3, B.1.6).
card{. . .}
Cardinality (size) of a set, card A ≡ |A| (Sec. 3.1).
h(i)
Histogram of an image at pixel value (or bin) i (Sec. 3.1).
H(i)
Cumulative histogram of an image at pixel value (or bin) i (Sec. 3.6).
I(u, v)
Intensity or color value of the image I at (integer) position (u, v).
K
Number of possible pixel values.
M, N
Number of columns (width) and rows (height) of an image (0 ≤ u < M , 0 ≤ v < N ).
mod
Modulus operator: (a mod b) is the remainder of the integer division a/b (Sec. B.1.2).
A.3 Algorithmic Complexity and O Notation
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p(i)
Probability density function (Sec. 4.6.1).
P(i)
Probability distribution function or cumulative probability density (Sec. 4.6.1).
round(x)
Rounding function: rounds x to the nearest
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