词条 | Spline wavelet | ||||||||||||||||||||||
释义 |
In the mathematical theory of wavelets, a spline wavelet is a wavelet constructed using a spline function.[1] 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 formula.[2] Though these wavelets are orthogonal, they do not have compact supports. There is a certain class of wavelets, unique in some sense, constructed using B-splines and having compact supports. Even though these wavelets are not orthogonal they have some special properties that have made them quite popular.[3] 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.[4] The Battle-Lemarie wavelets are also wavelets constructed using spline functions.[5] Cardinal B-splinesLet n be a fixed non-negative integer. Let Cn denote the set of all real-valued functions defined over the set of real numbers such that each function in the set as well its first n derivatives are continuous everywhere. A bi-infinite sequence . . . x−2, x−1, x0, x1, x2, . . . such that xr < xr+1 for all r and such that xr approaches ±∞ as r approaches ±∞ is said to define a set of knots. A spline of order n with a set of knots {xr} is a function S(x) in Cn such that, for each r, the restriction of S(x) to the interval [xr, xr+1) coincides with a polynomial with real coefficients of degree at most n in x. If the separation xr+1 - xr, 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 = {. . ., -2, -1, 0, 1, 2, . . .} 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 Nm(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-splinesElementary properties
Two-scale relationThe cardinal B-spline of order m satisfies the following two-scale relation: . Riesz propertyThe cardinal B-spline of order m satisfies the following property, known as the Riesz property: There exists two positive real numbers and such that for any square summable two-sided sequence and for any x, where is the norm in the ℓ2-space. Cardinal B-splines of small ordersThe cardinal B-splines are defined recursively starting from the B-spline of order 1, namely , which takes the value 1 in the interval [0, 1) and 0 elsewhere. Computer algebra systems may have to be employed to obtain concrete expressions for higher order cardinal B-splines. The concrete expressions for cardinal B-splines of all orders up to 6 are given below. The graphs of cardinal B-splines of orders up to 4 are also exhibited. In the images, the graphs of the terms contributing to the corresponding two-scale relations are also shown. The two dots in each image indicate the extremities of the interval supporting the B-spline. Constant B-splineThe B-spline of order 1, namely , is the constant B-spline. It is defined by The two-scale relation for this B-spline is
Linear B-splineThe B-spline of order 2, namely , is the linear B-spline. It is given by The two-scale relation for this wavelet is
Quadratic B-splineThe B-spline of order 3, namely , is the quadratic B-spline. It is given by The two-scale relation for this wavelet is
Cubic B-splineThe cubic B-spline is the cardinal B-spline of order 4, denoted by . It is given by the following expressions: The two-scale relation for the cubic B-spline is
Bi-quadratic B-splineThe bi-quadratic B-spline is the cardinal B-spline of order 5 denoted by . It is given by The two-scale relation is Quintic B-splineThe quintic B-spline is the cardinal B-spline of order 6 denoted by . It is given by Multi-resolution analysis generated by cardinal B-splinesThe cardinal B-spline of order m generates a multi-resolution analysis. In fact, from the elementary properties of these functions enunciated above, it follows that the function is square integrable and is an element of the space of square integrable functions. To set up the multi-resolution analysis the following notations used.
That these define a multi-resolution analysis follows from the following:
Wavelets from cardinal B-splinesLet m be a fixed positive integer and be the cardinal B-spline of order m. A function in is a basic wavelet relative to the cardinal B-spline function if the closure in of the linear span of the set (this closure is denoted by ) is the orthogonal complement of in . The subscript m in is used to indicate that is a basic wavelet relative the cardinal B-spline of order m. There is no unique basic wavelet relative to the cardinal B-spline . Some of these are discussed in the following sections. Wavelets relative to cardinal B-splines using fundamental interpolatory splinesFundamental interpolatory splineDefinitionsLet m be a fixed positive integer and let be the cardinal B-spline of order m. Given a sequence of real numbers, the problem of finding a sequence of real numbers such that for all , is known as the cardinal spline interpolation problem. The special case of this problem where the sequence is the sequence , where is the Kronecker delta function defined by , is the fundamental cardinal spline interpolation problem. The solution of the problem yields the fundamental cardinal interpolatory spline of order m. This spline is denoted by and is given by where the sequence is now the solution of the following system of equations: Procedure to find the fundamental cardinal interpolatory splineThe fundamental cardinal interpolatory spline can be determined using Z-transforms. Using the following notations it can be seen from the equations defining the sequence that from which we get . This can be used to obtain concrete expressions for . ExampleAs a concrete example, the case may be investigated. The definition of implies that The only nonzero values of are given by and the corresponding values are Thus reduces to This yields the following expression for . Splitting this expression into partial fractions and expanding each term in powers of z in an annular region the values of can be computed. These values are then substituted in the expression for to yield Wavelet using fundamental interpolatory splineFor a positive integer m, the function defined by is a basic wavelet relative to the cardinal B-spline of order . The subscript I in is used to indicate that it is based in the interpolatory spline formula. This basic wavelet is not compactly supported. ExampleThe wavelet of order 2 using interpolatory spline is given by The expression for now yields the following formula: Now, using the expression for the derivative of in terms of the function can be put in the following form: The following piecewise linear function is the approximation to obtained by taking the sum of the terms corresponding to in the infinite series expression for . Two-scale relationThe two-scale relation for the wavelet function is given by where Compactly supported B-spline waveletsThe spline wavelets generated using the interpolatory wavelets are not compactly supported. Compactly supported B-spline wavelets were discovered by Charles K. Chui and Jian-zhong Wang and published in 1991.[3][7] The compactly supported B-spline wavelet relative to the cardinal B-spline of order m discovered by Chui and Wong and denoted by , has as its support the interval . These wavelets are essentially unique in a certain sense explained below. DefinitionThe compactly supported B-spline wavelet of order m is given by This is an m-th order spline. As a special case, the compactly supported B-spline wavelet of order 1 is which is the well-known Haar wavelet. Properties
Two-scale relationsatisfies the two-scale relation: where . Decomposition relationThe decomposition relation for the compactly supported B-spline wavelet has the following form: where the coefficients and are given by Here the sequence is the sequence of coefficients in the fundamental interpolatoty cardinal spline wavelet of order m. Compactly supported B-spline wavelets of small ordersCompactly supported B-spline wavelet of order 1The two-scale relation for the compactly supported B-spline wavelet of order 1 is The closed form expression for compactly supported B-spline wavelet of order 1 is Compactly supported B-spline wavelet of order 2The two-scale relation for the compactly supported B-spline wavelet of order 2 is The closed form expression for compactly supported B-spline wavelet of order 2 is Compactly supported B-spline wavelet of order 3The two-scale relation for the compactly supported B-spline wavelet of order 3 is The closed form expression for compactly supported B-spline wavelet of order 3 is Compactly supported B-spline wavelet of order 4The two-scale relation for the compactly supported B-spline wavelet of order 4 is The closed form expression for compactly supported B-spline wavelet of order 4 is Compactly supported B-spline wavelet of order 5The two-scale relation for the compactly supported B-spline wavelet of order 5 is The closed form expression for compactly supported B-spline wavelet of order 5 is Images of compactly supported B-spline wavelets
Battle-Lemarie waveletsThe Battle-Lemarie wavelets form a class of orthonormal wavelets constructed using the class of cardinal B-splines. The expressions for these wavelets are given in the frequency domain; that is, they are defined by specifying their Fourier transforms. The Fourier transform of a function of t, say, , is denoted by . DefinitionLet m be a positive integer and let be the cardinal B-spline of order m. The Fourier transform of is . The scaling function for the m-th order Battle-Lemarie wavelet is that function whose Fourier transform is The m-th order Battle-Lemarie wavelet is the function whose Fourier transform is References1. ^{{cite journal|last1=Michael Unser|title=Ten good reasons for using spline wavelets|journal=Proc. SPIE Vol. 3169, Wavelets Applications in Signal and Image Processing V|date=1997|pages=422–431|url=http://bigwww.epfl.ch/publications/unser9702.pdf|accessdate=21 December 2014}} 2. ^{{cite journal|last1=Chui, Charles K, and Jian-zhong Wang|title=A cardinal spline approach to wavelets|journal=Proceedings of the American Mathematical Society|date=1991|volume=113|issue=3|pages=785–793|url=http://www.ams.org/journals/proc/1991-113-03/S0002-9939-1991-1077784-X/S0002-9939-1991-1077784-X.pdf|accessdate=22 January 2015|doi=10.2307/2048616|jstor=2048616}} 3. ^1 {{cite journal|last1=Charles K. Chui and Jian-Zhong Wang|title=On Compactly Supported Spline Wavelets and a Duality Principle|journal=Transactions of the American Mathematical Society|date=April 1992|volume=330|issue=2|pages=903–915|url=http://www.shsu.edu/~mth_jxw/pdfflies/CWTRAN.pdf|accessdate=21 December 2014|doi=10.1090/s0002-9947-1992-1076613-3}} 4. ^{{cite book|last1=Charles K Chui|title=An Introduction to Wavelets|date=1992|publisher=Academic Press|page=177|ref=Chui}} 5. ^{{cite book|last1=Ingrid Daubechies|title=Ten Lectures on Wavelets|date=1992|publisher=Society for Industrial and Applied Mathematics|location=Philadelphia|pages=146–153}} 6. ^{{cite book|last1=Christopher Heil|title=A Basis Theory Primer|publisher=Birkhauser|pages=177–188}} 7. ^{{cite book|last1=Charles K Chui|title=An Introduction to Wavelets|date=1992|publisher=Academic Press|page=249|ref=Chui}} 8. ^{{cite book|last1=Charles K Chui|title=An Introduction to Wavelets|date=1992|publisher=Academic Press|page=184}} Further reading
3 : Wavelets|Continuous wavelets|Splines (mathematics) |
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