## Ten Good Reasons For Using Spline Wavelets (1997)

Venue: | Proc. SPIE vol. 3169, Wavelet Applications in Signal and Image Processing V |

Citations: | 20 - 5 self |

### BibTeX

@INPROCEEDINGS{Unser97tengood,

author = {Michael Unser},

title = {Ten Good Reasons For Using Spline Wavelets},

booktitle = {Proc. SPIE vol. 3169, Wavelet Applications in Signal and Image Processing V},

year = {1997},

pages = {422--431}

}

### OpenURL

### Abstract

The purpose of this note is to highlight some of the unique properties of spline wavelets. These wavelets can be classified in four categories: othogonal (Battle-Lemari), semi-orthogonal (e.g., B-spline), shift-orthogonal, and biorthogonal (Cohen-DaubechiesFeauveau) . Unlike most other wavelet bases, splines have explicit formulae in both the time and frequency domain, which greatly facilitates their manipulation. They allow for a progressive transition between the two extreme cases of a multiresolution: Haar's piecewise constant representation (spline of degree zero) versus Shannon's bandlimited model (which corresponds to a spline of infinite order). Spline wavelets are extremely regular and usually symmetric or anti-symmetric. They can be designed to have compact support and to achieve optimal time-frequency localization (B-spline wavelets). The underlying scaling functions are the B-splines, which are the shortest and most regular scaling functions of order L. Finally, splines have the best approximation properties among all known wavelets of a given order L. In other words, they are the best for approximating smooth functions.

### Citations

2385 | A theory for multi-resolution signal decomposition: The wavelet representation
- Mallat
- 1989
(Show Context)
Citation Context ... Z 2 2 = - , (13) where h(k) is the corresponding (lowpass) reconstruction filter. However, in the spline case, there will always exist a sequence p(k) such that j j ( ) ( ) ( ) x p k x k n k Z = - . =-=(14)-=- Such specific B-spline characterizations for various kinds of spline scaling functions (orthogonal, dual, or interpolating) can be found elsewhere 32 . Note that the sequence p(k) defines an invertib... |

1728 |
Lectures on Wavelets
- Daubechies, Ten
- 1992
(Show Context)
Citation Context ...nd synthesis waveletss( ) y x and y( ) x are then contructed by taking linear combinations of these scaling functionss( / ) ( )s( ) y j x g k x k k 2 2 = - (8) y j ( / ) ( ) ( ) x g k x k k 2 2 = - . =-=(9)-=- They form a biorthogonal set in the sense thats= - - y y d i k j l i j k l , , , ,s, (10) with the short form convention y y i k i x k , ( ) = - - - 2 2 2 . This allows us to obtain the wavelet expan... |

1587 | Biorthogonal bases of compactly supported wavelets
- Cohen, Daubechies, et al.
- 1992
(Show Context)
Citation Context ...linessj n k Z x k ( ) - { }sconstitute a Riesz basis of V n 0 in the sense that there exist two constants A n > 0 and B nssuch that "s-sc l A c c k x k B c n l n k Z L n l 2 2 2 2 2 2 2 , ( ) ( )=-= j . (7)-=- The lower inequality implies that the B-splines are linearly independent (i.e., s x 0 0 ( ) =s= c k ( ) 0 ). The upper inequality guarantees that V L n 0 2s. Hence, any polynomial spline has a unique... |

1093 |
A Practical Guide to Splines
- Boor
- 1978
(Show Context)
Citation Context ...s of these scaling functionss( / ) ( )s( ) y j x g k x k k 2 2 = - (8) y j ( / ) ( ) ( ) x g k x k k 2 2 = - . (9) They form a biorthogonal set in the sense thats= - - y y d i k j l i j k l , , , ,s, =-=(10) with-=- the short form convention y y i k i x k , ( ) = - - - 2 2 2 . This allows us to obtain the wavelet expansion of any L 2 -function as "s=sf L f f k i k k Z i Z 2, , ,sy y . (11) Note that the und... |

449 |
Multiresolution approximations and wavelet orthonormal bases of L2
- Mallat
- 1989
(Show Context)
Citation Context ...ine of degree n --- unless h(k) is precisely the binomial filter (5). This function is usually specified indirectly as the solution of the two-scale relation j j ( / ) ( ) ( ) x h k x k k Z 2 2 = - , =-=(13)-=- where h(k) is the corresponding (lowpass) reconstruction filter. However, in the spline case, there will always exist a sequence p(k) such that j j ( ) ( ) ( ) x p k x k n k Z = - . (14) Such specifi... |

213 |
Wavelets and dilation equations: A brief introduction
- Strang
- 1989
(Show Context)
Citation Context ...cit characterization of all spline wavelets. By contrast, most other scaling functions are only defined through an infinite product in the frequency domain 9, 23, 36s( ) ( ) j w w =s= + 1 2 2 1 H e . =-=(21)-=- Applying this latter result to the B-spline case yields the identity 1 1 2 1 2 1 1 -s= +s- + - + = + e j e j n n i w w w . (22) 3.2 Simple manipulation Splines are piecewise polynomial, which greatly... |

197 |
Zur Theorie der orthogonalen Funktionensysteme
- Haar
- 1910
(Show Context)
Citation Context ...i j k l , , , ,s, (10) with the short form convention y y i k i x k , ( ) = - - - 2 2 2 . This allows us to obtain the wavelet expansion of any L 2 -function as "s=sf L f f k i k k Z i Z 2, , ,sy=-= y . (11)-=- Note that the underlying basis functions are usually specified indirectly in terms of the four sequences h k ( ), ( ) h k , g k ( ) and ( ) g k , which are the filters for the fast wavelet transform ... |

164 |
Contributions to the problem of approximation of equidistant data by analytic functions, Parts A
- Schoenberg
(Show Context)
Citation Context ... w j w w w n n n e j = ( ) = -s+ - + 0 1 1 1 , (16) wheres( ) j w n denotes the Fourier transform of j n x ( ) . This may also be rewritten ass( ) sinc ( / ) ( ) j w w w n j n n e = p - + + 1 2 1 2 , =-=(17)-=- which involves the (n+1)th power of the sinc function. Next, we expand (16) using the binomial expansions( ) ( ) ( ) j w w w n k n k j k n n k e j = +s- = + - +s1 1 0 1 1 . (18) The crucial step is t... |

152 |
A Fourier analysis of the finite element variational method. In: Constructive aspects of Functional Analysis. Edizione Cremonese
- Strang, Fix
- 1973
(Show Context)
Citation Context ...uct in the frequency domain 9, 23, 36s( ) ( ) j w w =s= + 1 2 2 1 H e . (21) Applying this latter result to the B-spline case yields the identity 1 1 2 1 2 1 1 -s= +s- + - + = + e j e j n n i w w w . =-=(22)-=- 3.2 Simple manipulation Splines are piecewise polynomial, which greatly simplifies their manipulation 10, 16 . In particular, it is straightforward to obtain spline derivatives and integrals. For ins... |

121 |
The theory of splines and their applications
- Ahlberg, Nilson, et al.
- 1967
(Show Context)
Citation Context ...lines which can be constructed from the (n+1)-fold convolution of the indicator function in the unit interval (causal B-spline of degree 0) j j j n n times x x ( ) ( ) ( ) = * * + 0 0 1 L 1 2 4 3 4 , =-=(2)-=- where j 0 1 0 1 0 ( ) , x x = otherwise. (3) The B-spline of degree n satisfies the two-scale relation 31 j j n n n k Z x h k x k ( / ) ( ) ( ) 2 2 = - , (4) where h k n ( ) is the binomial filter of... |

108 | On compactly supported spline wavelets and a duality principle
- Chui, Wang
- 1992
(Show Context)
Citation Context ...the integers could be represented as a linear combination of shifted B-splines 17 . Thus, our basic spline space V n 0 can also be specified as V s x c k x k c l n n k Z 0 0 2 = = -s( ) ( ) ( ) | j , =-=(6)-=- where the weights c(k) are the so-called B-spline coefficients of the spline function s x 0 ( ) . In addition, it can be shown that the Bsplinessj n k Z x k ( ) - { }sconstitute a Riesz basis of V n ... |

108 |
On trigonometric spline interpolation
- Schoenberg
- 1964
(Show Context)
Citation Context ... n n + where ( ) max( , ) x x n n + = 0 . By interpreting the complex exponentials as shift factors, we can get back to the time domain j n k k n n x n n k x k ( ) ! ( ) = - +s- ( ) = + +s1 1 1 0 1 . =-=(19)-=- This formula shows that j n x ( ) is piecewise polynomial of degree n. It is also clear from (18) that j n x ( ) can be differentiated n times before one starts uncovering discontinuities at the inte... |

89 |
Fast B-spline transforms of continuous image representation and interpolation
- Unser, Aldroubi, et al.
- 1991
(Show Context)
Citation Context ...the first integral relation 2 , which states that for any function f(x) whose mth derivative is square integrable, we have f dx s dx f s dx m m m m ( ) ( ) ( ) ( ) ( ) = ( ) + - ( ) - + - + - + 2 2 2 =-=(28)-=- where s x ( ) is the spline interpolant of degree n=2m-1 such that s k f k ( ) ( ) = . In particular, if we apply this decomposition to the problem of the interpolation of a given data sequence f(k),... |

83 |
Splines and Variational Methods
- Prenter
- 1975
(Show Context)
Citation Context ...frequency domain. To derive such formulae, we start by expressing the convolution property (2) by a product in the Fourier domain, which yieldss( )s( ) j w j w w w n n n e j = ( ) = -s+ - + 0 1 1 1 , =-=(16)-=- wheres( ) j w n denotes the Fourier transform of j n x ( ) . This may also be rewritten ass( ) sinc ( / ) ( ) j w w w n j n n e = p - + + 1 2 1 2 , (17) which involves the (n+1)th power of the sinc f... |

61 |
A family of polynomial spline wavelet transforms
- Unser, Aldroubi, et al.
- 1993
(Show Context)
Citation Context ...let of degree n, then we have the following approximate formula (cosine-modulated Gaussian) y s s n n w w x b n f x x n ( ) @ p + p - ( ) - - + + 4 2 1 2 2 1 2 1 2 1 1 0 2 2 ( ) cos ( ) exp ( ) ( ) , =-=(31)-=- with b=0.697066, f 0 =0.409177 and s w 2 =0.561145. The quality of this Gabor approximation improves rapidly with increasing n; for n=3, the approximation error is less than 3%. The implication is th... |

60 |
A block spin construction of ondelettes, Part I: Lemarié functions
- Battle
- 1987
(Show Context)
Citation Context ... two-scale relation 31 j j n n n k Z x h k x k ( / ) ( ) ( ) 2 2 = - , (4) where h k n ( ) is the binomial filter of order n+1 whose transfer function is h k H z z n z n n ( ) ( )s=s+s- + 2 1 2 1 1 . =-=(5)-=- In 1946, Schoenberg proved that any polynomial spline of degree n with knots at the integers could be represented as a linear combination of shifted B-splines 17 . Thus, our basic spline space V n 0 ... |

54 |
Cardinal interpolation and spline functions
- Schoenberg
- 1969
(Show Context)
Citation Context ...e = p - + + 1 2 1 2 , (17) which involves the (n+1)th power of the sinc function. Next, we expand (16) using the binomial expansions( ) ( ) ( ) j w w w n k n k j k n n k e j = +s- = + - +s1 1 0 1 1 . =-=(18)-=- The crucial step is then to identify ( ) ) j n w - +1 as the Fourier transform of the (n+1)-fold integral of the Dirac delta; i.e., the function ( ) / ! x n n + where ( ) max( , ) x x n n + = 0 . By ... |

50 |
On asymptotic convergence of B-spline wavelets to Gabor functions
- Unser, Aldroubi, et al.
- 1992
(Show Context)
Citation Context ... the approximation error decreases with the Lth power of the scale a i = 2 (cf. Strang 21, 22 ). Specifically, we can derive the following asymptotic relation 26 lim ( ) a a L L f P f C a f - =s0 j , =-=(29)-=- where P f a denotes the projection (orthogonal or oblique) of f onto the multiresolution space at scale a. This error formula becomes valid as soon as the sampling step a is sufficiently small with r... |

50 |
Ondelettes a localisation exponentielle
- Lemarié
- 1988
(Show Context)
Citation Context ... synthesis functions ( ψ( x ) and ϕ( x ) ) are polynomial splines of degree n. This means that the synthesis wavelet can also be represented by its B-spline expansion n ψ( x/ 2 ) = ∑ w( k) ϕ ( x−k) . =-=(12)-=- k∈Z It is important to observe that the underlying scaling function ϕ( x) V n ∈ 0 is not necessarily the B-spline of degree n — unless h(k) is precisely the binomial filter (5). This function is usua... |

42 |
Cardinal spline filters: Stability and convergence to the ideal sinc interpolator
- Aldroubi, Unser, et al.
- 1992
(Show Context)
Citation Context ...x ( ) ( ) ( ) = * * + 0 0 1 L 1 2 4 3 4 , (2) where j 0 1 0 1 0 ( ) , x x = otherwise. (3) The B-spline of degree n satisfies the two-scale relation 31 j j n n n k Z x h k x k ( / ) ( ) ( ) 2 2 = - , =-=(4)-=- where h k n ( ) is the binomial filter of order n+1 whose transfer function is h k H z z n z n n ( ) ( )s=s+s- + 2 1 2 1 1 . (5) In 1946, Schoenberg proved that any polynomial spline of degree n with... |

39 |
Approximation power of biorthogonal wavelet expansions
- Unser
- 1996
(Show Context)
Citation Context ...0 1 ( ) / = - = - . If we now apply the convolution properties of B-splines, we gets( ) ( )s( ) j w j w n m n n m H z = ( ) + 0 1 , or, equivalently, j j m m n n k Z x m m h k x k ( / ) ( ) ( ) = - , =-=(26)-=- where H z m z m n n k k m n ( ) = - = - + 1 0 1 1 . (27) Thus, (26) provides us with a two-scale relation that is valid for any integer m. In addition, the refinement filter h k m n ( ) can be interp... |

37 | The -polynomial spline pyramid - Unser, Aldroubi, et al. - 1993 |

36 |
Families of multiresolution and wavelet spaces with optimal properties
- Aldroubi, Unser
(Show Context)
Citation Context ...-fold convolution of the indicator function in the unit interval (causal B-spline of degree 0) j j j n n times x x ( ) ( ) ( ) = * * + 0 0 1 L 1 2 4 3 4 , (2) where j 0 1 0 1 0 ( ) , x x = otherwise. =-=(3)-=- The B-spline of degree n satisfies the two-scale relation 31 j j n n n k Z x h k x k ( / ) ( ) ( ) 2 2 = - , (4) where h k n ( ) is the binomial filter of order n+1 whose transfer function is h k H z... |

36 |
On the optimality of ideal filters for pyramid and wavelet signal approximation
- Unser
- 1993
(Show Context)
Citation Context ...o a power of two. We will derive this property by first considering a B-spline of degree 0 expanded by a factor of m, which can obviously be represented as j j 0 0 0 1 ( / ) ( ) x m x k k m = - = - . =-=(25)-=- We rewritte this equation in the Fourier domain ass( ) ( )s( ) j w j w 0 0 0 m H z m =swhere H z z m m k k m 0 0 1 ( ) / = - = - . If we now apply the convolution properties of B-splines, we gets( ) ... |

26 |
Polynomial Spline Signal Approximations: Filter Design and Asymptopic Equivalence with Shannon’s Sampling Theorem
- Unser, Aldroubi, et al.
- 1992
(Show Context)
Citation Context ...step a is sufficiently small with respect to the smoothness scale of f(x). The constant C j is the same for all spline wavelet transforms of a given order L, and is given by 26 C B L L j = 2 2 ( )! , =-=(30)-=- where B L 2 is Bernouilli's number of order 2L. This turns out, by far, to be the smallest constant among all known wavelet transforms of the same order L (cf. Table I in the above mentioned referenc... |

26 | Fast implementation of the continuous wavelet transform with integer scales - Unser, Aldroubi, et al. - 1994 |

19 |
A modified Franklin system and higher-order spline systems on Rn as unconditional basis for Hardy spaces, Conference in Harmonic Analysis in Honor of Antoni Zygmund
- Strömberg
- 1981
(Show Context)
Citation Context ...elebrated Daubechies wavelets 8. 3.4 Shortest scaling function of order L We recall that the refinement filter for an Lth order wavelet transform can be factorized as 9 −1 L Hz () = ( 1+ z ) ⋅Qz () , =-=(24)-=- where Qz () is the transfer function of a stable filter. The B-spline of degree n=L-1 corresponds to the shortest Lth order refinement filter with Q(z)=1. One can therefore conclude that the B-spline... |

14 |
Designing multiresolution analysis-type wavelets and their fast algorithms
- Abry, Aldroubi
- 1995
(Show Context)
Citation Context ...omial on each segment 2 2 1 0 0 i k k , ( ) + [ ) is also included in any of the finer subspaces V i n with i is0 . Thus, we have the following inclusion property L V V V n n i n 2 1 0 0s- L L L { }. =-=(1)-=- Furthermore, it is well known that one can approximate any L 2 -function by a spline as closely as one wishes by letting the knot spacing (or scale) go to zero ( is- ). This means that the above sequ... |

9 |
Notes on spline functions. III. On the convergence of the interpolating cardinal splines as their degee tends to infinity
- Schoenberg
- 1973
(Show Context)
Citation Context ...ther interesting observation is that (19) can also be interpreted as the (n+1)th forward difference of the one-sided power function ( ) x n + . In other words, we have j n n n x x ( ) = D ( ) + + 1 , =-=(20)-=- where D + n 1 denotes the (n+1) iteration of the forward difference operator D = - - f x f x f x ( ) ( ) ( ) 1 . The important point here is that these formulae together with (12) provide an explicit... |

9 | Shift-orthogonal wavelet bases using splines - Unser, Thévenaz, et al. - 1996 |

8 |
A practical guide to the implementation of the wavelet transform, in
- Unser
- 1996
(Show Context)
Citation Context ...rties of B-splines, we gets( ) ( )s( ) j w j w n m n n m H z = ( ) + 0 1 , or, equivalently, j j m m n n k Z x m m h k x k ( / ) ( ) ( ) = - , (26) where H z m z m n n k k m n ( ) = - = - + 1 0 1 1 . =-=(27)-=- Thus, (26) provides us with a two-scale relation that is valid for any integer m. In addition, the refinement filter h k m n ( ) can be interpreted as a cascade of (n+1) moving sum filters. Each of t... |

6 | The L 2 polynomial spline pyramid - Unser, Aldroubi, et al. - 1993 |

4 |
Ondelettes localisation exponentielles
- Lemari
- 1988
(Show Context)
Citation Context ...s functions ( y( ) x and j( ) x ) are polynomial splines of degree n. This means that the synthesis wavelet can also be represented by its B-spline expansion y j ( / ) ( ) ( ) x w k x k n k Z 2 = - . =-=(12)-=- It is important to observe that the underlying scaling function j( ) x V ns0 is not necessarily the B-spline of degree n --- unless h(k) is precisely the binomial filter (5). This function is usually... |

3 |
Cardinal spline interpolation operators on ` p data
- Marsden, Richards, et al.
- 1975
(Show Context)
Citation Context ..., where P e j ( ) w denotes the Fourier transform of p. If we combine (9) with (14), we obtain the B-spline coefficients of the wavelet y( ) x : w k p g k W z P z G z z ( ) ( )( ) ( ) ( ) ( ) = *s= . =-=(15)-=- These spline wavelets are quite attractive because they are extremely regular. In fact, any spline wavelets of degree n is (n+1) times differentiable almost everywhere. It has a Sobolev regularity in... |

2 |
A modified Franklin system and higher-order spline system of R n as unconditional bases for Hardy spaces
- Strmberg
- 1983
(Show Context)
Citation Context ...ted Daubechies wavelets 8 . 3.4 Shortest scaling function of order L We recall that the refinement filter for an Lth order wavelet transform can be factorized as 9 H z z Q z L ( ) ( ) ( ) = +s- 1 1 , =-=(24)-=- where Q z ( ) is the transfer function of a stable filter. The B-spline of degree n=L-1 corresponds to the shortest Lth order refinement filter with Q(z)=1. One can therefore conclude that the B-spli... |

1 | Construction of shift-orthogonal spline wavelets using splines - Unser, Thvenaz, et al. |