Spline wavelet

Short description: Wavelet constructed using a spline function

Animation showing the compactly supported cardinal B-spline wavelets of orders 1, 2, 3, 4 and 5.

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]

## Contents

* 1 Cardinal B-splines
* 2 Properties of the cardinal B-splines
* 2.1 Elementary properties
* 2.2 Two-scale relation
* 2.3 Riesz property
* 3 Cardinal B-splines of small orders
* 3.1 Constant B-spline
* 3.2 Linear B-spline
* 3.3 Quadratic B-spline
* 3.4 Cubic B-spline
* 3.5 Bi-quadratic B-spline
* 3.6 Quintic B-spline
* 4 Multi-resolution analysis generated by cardinal B-splines
* 5 Wavelets from cardinal B-splines
* 6 Wavelets relative to cardinal B-splines using fundamental interpolatory splines
* 6.1 Fundamental interpolatory spline
* 6.1.1 Definitions
* 6.1.2 Procedure to find the fundamental cardinal interpolatory spline
* 6.1.3 Example
* 6.2 Wavelet using fundamental interpolatory spline
* 6.3 Example
* 6.4 Two-scale relation
* 7 Compactly supported B-spline wavelets
* 7.1 Definition
* 7.2 Properties
* 7.3 Two-scale relation
* 7.4 Decomposition relation
* 8 Compactly supported B-spline wavelets of small orders
* 8.1 Compactly supported B-spline wavelet of order 1
* 8.2 Compactly supported B-spline wavelet of order 2
* 8.3 Compactly supported B-spline wavelet of order 3
* 8.4 Compactly supported B-spline wavelet of order 4
* 8.5 Compactly supported B-spline wavelet of order 5
* 8.6 Images of compactly supported B-spline wavelets
* 9 Battle-Lemarie wavelets
* 9.1 Definition
* 10 References
* 11 Further reading

## Cardinal B-splines

Let 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.

N1(x)={10≤x<10otherwise
Nm(x)=∫01Nm−1(x−t)dt, for m>1.

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

1. The support of Nm(x) is the closed interval [0,m].
2. The function Nm(x) is non-negative, that is, Nm(x)>0 for 0<x<m.
3. ∑k=−∞∞Nm(x−k)=1 for all x.
4. The cardinal B-splines of orders m and m-1 are related by the identity: Nm(x)=xm−1Nm−1(x)+m−xm−1Nm−1(x−1).
5. The function Nm(x) is symmetrical about x=m2, that is, Nm(m2−x)=Nm(m2+x).
6. The derivative of Nm(x) is given by Nm′(x)=Nm−1(x)−Nm−1(x−1).
7. ∫−∞∞Nm(x)dx=1

### Two-scale relation

The cardinal B-spline of order m satisfies the following two-scale relation:

Nm(x)=∑k=0m2−m+1(mk)Nm(2x−k).

### Riesz property

The cardinal B-spline of order m satisfies the following property, known as the Riesz property: There exists two positive real numbers A and B such that for any square summable two-sided sequence {ck}k=−∞∞ and for any x,

A‖{ck}‖2≤‖∑k=−∞∞ckNm(x−k)‖2≤B‖{ck}‖2

where ‖⋅‖ is the norm in the ℓ2-space.

## Cardinal B-splines of small orders

The cardinal B-splines are defined recursively starting from the B-spline of order 1, namely N1(x), 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-spline

The B-spline of order 1, namely N1(x), is the constant B-spline. It is defined by

N1(x)={10≤x<10otherwise

The two-scale relation for this B-spline is

N1(x)=N1(2x)+N1(2x−1)
Constant B-spline
N1(x) |  |

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### Linear B-spline

The B-spline of order 2, namely N2(x), is the linear B-spline. It is given by

N2(x)={x0≤x<1−x+21≤x<20otherwise

The two-scale relation for this wavelet is

N2(x)=12N2(2x)+N2(2x−1)+12N2(2x−2)
Linear B-spline
N2(x) |  |

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### Quadratic B-spline

The B-spline of order 3, namely N3(x), is the quadratic B-spline. It is given by

N3(x)={12x20≤x<1−x2+3x−321≤x<212x2−3x+922≤x<30otherwise

The two-scale relation for this wavelet is

N3(x)=14N3(2x)+34N3(2x−1)+34N3(2x−2)+14N3(2x−3)
Quadratic B-spline
N3(x) |  |

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### Cubic B-spline

The cubic B-spline is the cardinal B-spline of order 4, denoted by N4(x). It is given by the following expressions:

N4(x)={16x30≤x<1−12x3+2x2−2x+231≤x<212x3−4x2+10x−2232≤x<3−16x3+2x2−8x+3233≤x<40otherwise

The two-scale relation for the cubic B-spline is

N4(x)=18N4(2x)+12N4(2x−1)+34N4(2x−2)+12N4(2x−3)+18N4(2x−4)
Cubic B-spline
N4(x) |  |

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### Bi-quadratic B-spline

The bi-quadratic B-spline is the cardinal B-spline of order 5 denoted by N5(x). It is given by

N5(x)={124x40≤x<1−16x4+56x3−54x2+56x−5241≤x<214x4−52x3+354x2−252x+155242≤x<3−16x4+52x3−554x2+652x−655243≤x<4124x4−56x3+254x2−1256x+625244≤x<50otherwise

The two-scale relation is

N5(x)=116N5(2x)+516N5(2x−1)+1016N5(2x−2)+1016N5(2x−3)+516N5(2x−4)+116N5(2x−5)

### Quintic B-spline

The quintic B-spline is the cardinal B-spline of order 6 denoted by N6(x). It is given by

N6(x)={1120x50≤x<1−124x5+14x4−12x3+12x2−14x+1201≤x<2112x5−x4+92x3−192x2+394x−79202≤x<3−112x5+32x4−212x3+712x2−2314x+731203≤x<4124x5−x4+192x3−892x2+4094x−1829204≤x<5−1120x5+14x4−3x3+18x2−54x+32455≤x<60otherwise

## Multi-resolution analysis generated by cardinal B-splines

The cardinal B-spline Nm(x) of order m generates a multi-resolution analysis. In fact, from the elementary properties of these functions enunciated above, it follows that the function Nm(x) is square integrable and is an element of the space L2(R) of square integrable functions. To set up the multi-resolution analysis the following notations used.

* For any integers k,j, define the function Nm,kj(x)=Nm(2kx−j).
* For each integer k, define the subspace Vk of L2(R) as the closure of the linear span of the set {Nm,kj(x):j=⋯,−2,−1,0,1,2,⋯}.

That these define a multi-resolution analysis follows from the following:

1. The spaces Vk satisfy the property: ⋯⊂V−2⊂V−1⊂V0⊂V1⊂V2⊂⋯.
2. The closure in L2(R) of the union of all the subspaces Vk is the whole space L2(R).
3. The intersection of all the subspaces Vk is the singleton set containing only the zero function.
4. For each integer k the set {Nm,kj(x):j=⋯,−2,−1,0,1,2,⋯} is an unconditional basis for Vk. (A sequence {xn} in a Banach space X is an unconditional basis for the space X if every permutation of the sequence {xn} is also a basis for the same space X.[6])

## Wavelets from cardinal B-splines

Let m be a fixed positive integer and Nm(x) be the cardinal B-spline of order m. A function ψm(x) in L2(R) is a basic wavelet relative to the cardinal B-spline function Nm(x) if the closure in L2(R) of the linear span of the set {ψm(x−j):j=⋯,−2,−1,0,1,2,⋯} (this closure is denoted by W0) is the orthogonal complement of V0 in V1. The subscript m in ψm(x) is used to indicate that ψm(x) is a basic wavelet relative the cardinal B-spline of order m. There is no unique basic wavelet ψm(x) relative to the cardinal B-spline Nm(x). Some of these are discussed in the following sections.

## Wavelets relative to cardinal B-splines using fundamental interpolatory splines

### Fundamental interpolatory spline

#### Definitions

Let m be a fixed positive integer and let Nm(x) be the cardinal B-spline of order m. Given a sequence {fj:j=⋯,−2,−1,0,1,2,⋯} of real numbers, the problem of finding a sequence {cm,k:k=⋯,−2,−1,0,1,2,⋯} of real numbers such that

∑k=−∞∞cm,kNm(j+m2−k)=fj for all j,

is known as the cardinal spline interpolation problem. The special case of this problem where the sequence {fj} is the sequence δ0j, where δij is the Kronecker delta function δij defined by

δij={1, if i=j0, if i≠j,

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 Lm(x) and is given by

Lm(x)=∑k=−∞∞cm,kNm(x+m2−k)

where the sequence {cm,k} is now the solution of the following system of equations:

∑k=−∞∞cm,kNm(j+m2−k)=δ0j

#### Procedure to find the fundamental cardinal interpolatory spline

The fundamental cardinal interpolatory spline Lm(x) can be determined using Z-transforms. Using the following notations

A(z)=∑k=−∞∞δk0zk=1,

Bm(z)=∑k=−∞∞Nm(k+m2)zk,

Cm(z)=∑k=−∞∞cm,kzk,

it can be seen from the equations defining the sequence cm,k that

Bm(z)Cm(z)=A(z)

from which we get

Cm(z)=1Bm(z).

This can be used to obtain concrete expressions for cm,k.

#### Example

As a concrete example, the case L4(x) may be investigated. The definition of Bm(z) implies that

B4(x)=∑k=−∞∞N4(2+k)zk

The only nonzero values of N4(k+2) are given by k=−1,0,1 and the corresponding values are

N4(1)=16,N4(2)=46,N4(3)=16.

Thus B4(z) reduces to

B4(z)=16z−1+46z0+16z1=1+4z+z26z

This yields the following expression for C4(z).

C4(z)=6z1+4z+z2

Splitting this expression into partial fractions and expanding each term in powers of z in an annular region the values of c4,k can be computed. These values are then substituted in the expression for L4(x) to yield

L4(x)=∑k=−∞∞(−1)k3(2−3)|k|N4(x+2−k)

### Wavelet using fundamental interpolatory spline

For a positive integer m, the function ψm(x) defined by

ψI,m(x)=dmdxmL2m(2x−1)

is a basic wavelet relative to the cardinal B-spline of order Nm(x). The subscript I in ψI,m is used to indicate that it is based in the interpolatory spline formula. This basic wavelet is not compactly supported.

### Example

The wavelet of order 2 using interpolatory spline is given by

ψI,2(x)=d2dx2L4(2x−1)

The expression for L4(x) now yields the following formula:

ψI,2(x)=d2dx2∑k=−∞∞(−1)k3(2−3)|k|N4(2x+1−k)

Now, using the expression for the derivative of Nm(x) in terms of Nm−1(x) the function ψ2(x) can be put in the following form:

ψI,2(x)=∑k=−∞∞(−1)k43(2−3)|k|((N2(2x+k−1)−2N2(2x+k−2)+N2(2x+k−3))

The following piecewise linear function is the approximation to ψ2(x) obtained by taking the sum of the terms corresponding to k=−3,…,3 in the infinite series expression for ψ2(x).

ψI,2(x)≈{0.07142668x+0.17856670−2.5<x≤−2−0.48084803x−0.92598272−2<x≤−1.52.0088293x+2.8085333−1.5<x≤−1−7.5684795x−6.7687755−1<x≤−0.528.245949x+11.138439−0.5<x≤0−57.415316x+11.1384390<x≤0.557.415316x−46.2768780.5<x≤1−28.245949x+39.3843881<x≤1.57.5684795x−14.3372551.5<x≤2−2.0088293x+4.81736252<x≤2.50.48084803x−1.40683082.5<x≤3−0.07142668x+0.249993383<x≤3.50otherwise

### Two-scale relation

The two-scale relation for the wavelet function ψm(x) is given by

ψI,m(x)=∑−∞∞qnNm(2x−n) where qn=∑j=0m(−1)j(mj)cm+n−j−1.

## Compactly supported B-spline wavelets

The 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 Nm(x) of order m discovered by Chui and Wong and denoted by ψC,m(x), has as its support the interval [0,2m−1]. These wavelets are essentially unique in a certain sense explained below.

### Definition

The compactly supported B-spline wavelet of order m is given by

ψC,m(x)=12m−1∑j=02m−2(−1)jN2m(j+1)dmdxmN2m(2x−j)

This is an m-th order spline. As a special case, the compactly supported B-spline wavelet of order 1 is

ψC,1(x)=N2(1)ddxN2(2x)={10≤x<12−112≤x<10otherwise

which is the well-known Haar wavelet.

### Properties

1. The support of ψC,m(x) is the closed interval [0,2m−1].
2. The wavelet ψC,m(x) is the unique wavelet with minimum support in the following sense: If η(x)∈W0 generates W0 and has support not exceeding 2m−1 in length then η(x)=c0ψC,m(x−n0) for some nonzero constant c0 and for some integer n0.[8]
3. ψC,m(x) is symmetric for even m and antisymmetric for odd m.

### Two-scale relation

ψm(x) satisfies the two-scale relation:

ψC,m(x)=∑n=03m−2qnNm(2x−n) where qn=(−1)n2m−1∑j=0m(mj)N2m(n−j+1).

### Decomposition relation

The decomposition relation for the compactly supported B-spline wavelet has the following form:

Nm(2x−l)=∑k=−∞∞[am,l−2kNm(x−k)+bm,l−2kψC,m(x−k)]

where the coefficients am,j and bm,j are given by

am,j=−(−1)j2∑l=−∞∞q−j+2m−2l+1c2m,l,

bm,j=(−1)j2∑l=−∞∞p−j+2m−2l+1c2m,l.

Here the sequence c2m,l is the sequence of coefficients in the fundamental interpolatoty cardinal spline wavelet of order m.

## Compactly supported B-spline wavelets of small orders

### Compactly supported B-spline wavelet of order 1

The two-scale relation for the compactly supported B-spline wavelet of order 1 is

ψC,1(x)=N1(2x)−N1(2x−1)

The closed form expression for compactly supported B-spline wavelet of order 1 is

ψC,1(x)={10≤x<12−112≤x<10otherwise

### Compactly supported B-spline wavelet of order 2

The two-scale relation for the compactly supported B-spline wavelet of order 2 is

ψC,2(x)=112(N2(2x)−6N2(2x−1)+10N2(2x−2)−6N2(2x−3)+N2(2x−4))

The closed form expression for compactly supported B-spline wavelet of order 2 is

ψC,2(x)={16x0≤x<12−76x+2312≤x<183x−1961≤x<32−83x+29632≤x<276x−1762≤x<52−16x+1252≤x<30otherwise

### Compactly supported B-spline wavelet of order 3

The two-scale relation for the compactly supported B-spline wavelet of order 3 is

ψC,3(x)=1480[(N3(2x)−29N3(2x−1)+147N3(2x−2)−303N3(2x−3)+

303N3(2x−4)−147N3(2x−5)+29N3(2x−6)−N3(2x−7)]

The closed form expression for compactly supported B-spline wavelet of order 3 is

ψC,3(x)={1240x20≤x<12−31240x2+215x−13012≤x<1103120x2−221120x+2292401≤x<32−313120x2+1027120x−164324032≤x<2225x2−77940x+339162≤x<52−225x2+98140x−5411652≤x<3313120x2−70140x+2341803≤x<72−103120x2+809120x−316924072≤x<431240x2−139120x+6232404≤x<92−1240x2+124x−54892≤x<50otherwise

### Compactly supported B-spline wavelet of order 4

The two-scale relation for the compactly supported B-spline wavelet of order 4 is

ψC,4(x)=140320[N4(2x)−124N4(2x−1)+1677N4(2x−2)−7904N4(2x−3)+18482N4(2x−4)−

24264N4(2x−5)+18482N4(2x−6)−7904N4(2x−7)+1677N4(2x−8)−124N4(2x−9)+N4(2x−10)]

The closed form expression for compactly supported B-spline wavelet of order 4 is

ψC,4(x)={130240x30≤x<12−12730240x3+2315x2−1315x+1189012≤x<119280x3−47224x2+214710080x−10314401≤x<32−11092520x3+465224x2−3241310080x+165591008032≤x<252613360x3−334633360x2+420432016x−145193100802≤x<52−3503310080x3+935773360x2−1485172016x+216269336052≤x<34832945x3−27691560x2+113923720x−281451683≤x<72−4832945x3+583931008x2−52223240x+2048227756072≤x<43503310080x3−758271680x2+9811015040x−2341498404≤x<92−52613360x3+385091680x2−1124871008x+3034716892≤x<511092520x3−240773360x2+783112016x−14131120165≤x<112−19280x3+13611120x2−146172016x+4151288112≤x<612730240x3−55672x2+535910080x−11603100806≤x<132−130240x3+11440x2−71440x+494320132≤x<70otherwise

### Compactly supported B-spline wavelet of order 5

The two-scale relation for the compactly supported B-spline wavelet of order 5 is

ψC,5(x)=15806080[N5(2x)−507N5(2x−1)+17128N5(2x−2)−166304N5(2x−3)+748465N5(2x−4)

−1900115N5(2x−5)+2973560N5(2x−6)−2973560N5(2x−7)+1900115N5(2x−8)
−748465N5(2x−9)+166304N5(2x−10)−17128N5(2x−11)+507N5(2x−12)−N5(2x−13)]

The closed form expression for compactly supported B-spline wavelet of order 5 is

ψC,5(x)={18709120x40≤x<12−731244160x4+18505x3−111340x2+134020x−127216012≤x<195814354560x4−194172177280x3+130396768x2−196092177280x+654729030401≤x<32−1189314354560x4+3661192177280x3−186253483840x2+121121311040x−427181290304032≤x<27592394354560x4−31465612177280x3+64666011451520x2−132028732177280x+2681989787091202≤x<52−29804094354560x4+5183893725760x3−13426333483840x2+4265898960x−1263524341472052≤x<378735774354560x4−16524079725760x3+738536969120x2−1786867180640x+49766854329030403≤x<72−147143274354560x4+1085430912177280x3−56901557207360x2+14544586512177280x−5286189059870912072≤x<4156193402x4−33822017435456x3+1582892932256x2−597598433435456x+2774136491935364≤x<92−156193402x4+38150335435456x3−2015724732256x2+859841695435456x−644723452764892≤x<5147143274354560x4−446613762208x3+165651247290304x2−875490655435456x+461490401517418245≤x<112−78735774354560x4+30717383725760x3−179437319483840x2+1660672911520x−869722273414720112≤x<629804094354560x4−12698561725760x3+1621166996768x2−1913889126880x+328978799329030406≤x<132−7592394354560x4+105197412177280x3−10403603207360x2+71964499311040x−34816468378709120132≤x<71189314354560x4−17746392177280x3+63025969120x2−14096161311040x+24510850129030407≤x<152−95814354560x4+21863311040x3−407387483840x2+97588732177280x−259714992903040152≤x<8731244160x4−43432177280x3+5273207360x2−3137032177280x+38087312441608≤x<172−18709120x4+1241920x3−117920x2+38960x−2735840172≤x<90otherwise

### Images of compactly supported B-spline wavelets

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B-spline wavelet of order 1 | B-spline wavelet of order 2 |
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B-spline wavelet of order 3 | B-spline wavelet of order 4 | B-spline wavelet of order 5

## Battle-Lemarie wavelets

The 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, F(t), is denoted by F^(ω).

### Definition

Let m be a positive integer and let Nm(x) be the cardinal B-spline of order m. The Fourier transform of Nm(x) is N^m(ω). The scaling function ϕm(t) for the m-th order Battle-Lemarie wavelet is that function whose Fourier transform is

ϕ^m(ω)=N^m(ω)(∑k=−∞∞|N^m(ω+2πk)|2)1/2.

The m-th order Battle-Lemarie wavelet is the function ψBL,m(t) whose Fourier transform is

ψ^BL,m(ω)=−e−iω/2ϕ^m(ω+2π)‾ϕ^m(ω2)ϕ^m(ω2+π)‾

## References

1. ↑ Michael Unser (1997). "Ten good reasons for using spline wavelets". Proc. SPIE Vol. 3169, Wavelets Applications in Signal and Image Processing V. Wavelet Applications in Signal and Image Processing V 3169: 422–431. doi:10.1117/12.292801. Bibcode: 1997SPIE.3169..422U. http://bigwww.epfl.ch/publications/unser9702.pdf. Retrieved 21 December 2014.
2. ↑ Chui, Charles K, and Jian-zhong Wang (1991). "A cardinal spline approach to wavelets". Proceedings of the American Mathematical Society 113 (3): 785–793. doi:10.2307/2048616. http://www.ams.org/journals/proc/1991-113-03/S0002-9939-1991-1077784-X/S0002-9939-1991-1077784-X.pdf. Retrieved 22 January 2015.
3. ↑ 3.0 3.1 Charles K. Chui and Jian-Zhong Wang (April 1992). "On Compactly Supported Spline Wavelets and a Duality Principle". Transactions of the American Mathematical Society 330 (2): 903–915. doi:10.1090/s0002-9947-1992-1076613-3. http://www.shsu.edu/~mth_jxw/pdfflies/CWTRAN.pdf. Retrieved 21 December 2014.
4. ↑ Charles K Chui (1992). An Introduction to Wavelets. Academic Press. p. 177.
5. ↑ Ingrid Daubechies (1992). Ten Lectures on Wavelets. Philadelphia: Society for Industrial and Applied Mathematics. pp. 146–153. ISBN 9780898712742. https://archive.org/details/tenlecturesonwav0000daub.
6. ↑ Christopher Heil (2011). A Basis Theory Primer. Birkhauser. pp. 177–188. ISBN 9780817646868. https://archive.org/details/basistheoryprime00chei.
7. ↑ Charles K Chui (1992). An Introduction to Wavelets. Academic Press. p. 249.
8. ↑ Charles K Chui (1992). An Introduction to Wavelets. Academic Press. p. 184.

## Further reading

* Amir Z Averbuch and Valery A Zheludev (2007). "Wavelet transforms generated by splines". International Journal of Wavelets, Multiresolution and Information Processing 257 (5). http://www.cs.tau.ac.il/~zhel/PS/splitr3AA.pdf. Retrieved 21 December 2014.
* Amir Z. Averbuch, Pekka Neittaanmaki, and Valery A. Zheludev (2014). Spline and Spline Wavelet Methods with Applications to Signal and Image Processing Volume I. Springer. ISBN 978-94-017-8925-7.

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