
In the
mathematical theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
of
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the n ...
s, a spline wavelet is a wavelet constructed using a
spline function.
There are different types of spline wavelets. The interpolatory spline wavelets introduced by C.K. Chui and J.Z. Wang are based on a certain
spline interpolation
In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one ...
formula. Though these wavelets are
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
, they do not have
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
support
Support may refer to:
Arts, entertainment, and media
* Supporting character
* Support (art), a solid surface upon which a painting is executed
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Su ...
s. There is a certain class of wavelets, unique in some sense, constructed using
B-spline
In numerical analysis, a B-spline (short for basis spline) is a type of Spline (mathematics), spline function designed to have minimal Support (mathematics), support (overlap) for a given Degree of a polynomial, degree, smoothness, and set of bre ...
s and having compact supports. Even though these wavelets are not orthogonal they have some special properties that have made them quite popular.
The terminology ''spline wavelet'' is sometimes used to refer to the wavelets in this class of spline wavelets. These special wavelets are also called B-spline wavelets and cardinal B-spline wavelets. The Battle-Lemarie wavelets are also wavelets constructed using spline functions.
Cardinal B-splines
Let ''n'' be a fixed non-negative
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
. Let ''C''
''n'' denote the set of all
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real ...
s defined over the set of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s such that each function in the set as well its first ''n''
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
everywhere. A
bi-infinite sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
. . . ''x''
−2, ''x''
−1, ''x''
0, ''x''
1, ''x''
2, . . . such that ''x''
''r'' < ''x''
''r''+1 for all ''r'' and such that ''x''
''r'' approaches ±∞ as r approaches ±∞ is said to define a set of knots. A ''spline'' of order ''n'' with a set of knots is a function ''S''(''x'') in ''C''
''n'' such that, for each ''r'', the restriction of ''S''(''x'') to the interval
r, ''x''''r''+1) coincides with a polynomial">'x''r, ''x''''r''+1) coincides with a polynomial with real coefficients of degree at most ''n'' in ''x''.
If the separation ''x''
''r''+1 - ''x''
''r'', where ''r'' is any integer, between the successive knots in the set of knots is a constant, the spline is called a ''cardinal spline''. The set of integers ''Z'' = is a standard choice for the set of knots of a cardinal spline. Unless otherwise specified, it is generally assumed that the set of knots is the set of integers.
A cardinal B-spline is a special type of cardinal spline. For any positive integer ''m'' the cardinal B-spline of order ''m'', denoted by ''N''
''m''(''x''), is defined recursively as follows.
:
:
, for
.
Concrete expressions for the cardinal B-splines of all orders up to 5 and their graphs are given later in this article.
Properties of the cardinal B-splines
Elementary properties
# The support (mathematics)">support
Support may refer to:
Arts, entertainment, and media
* Supporting character
* Support (art), a solid surface upon which a painting is executed
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Su ...