M. Unser and A. Aldroubi, "Polynomial splines and wavelets---A signal processing perspective," in Wavelets - A tutorial in theory and applications, C.K. Chui, Ed., San Diego: Academic Press, pp. 91-122, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....supported orthogonal wavelets [3] UnfEI9kSfiRE6 , the orthonormality of shifIfi waveletfetfiEVGV has not allowed us to construct wavelets easily. By relaxing such conditions, Unser and Adroubi have constructed biorthogonal wavelets using a two scale relationof B splinefspl tions with odd order [4]. However, the useof these specialfalfiV77I such as splineflinefiE6 does not allow us to havef99V6fi to design wavelet filters. The design of optimal wavelet filtersft data compression has been studied in the recent past. Kirac et al. proposed the design theoryof optimal energy compaction filter ....

M. Unser and A. Aldroubi, "Polynomial splines and wavelets---A signal processing perspective," in Wavelet---A utorial in heory and Applications, ed. C.K. Chui, pp.91-- 122, Academic Press, New York, 1992.

....une fonction d echelle OE(x) correspondant a un filtre passe bas. c(0; k) f(x) OE(x Gamma k) 9) 9 Les coefficients permettent de reconstruire un signal f 0 (x) par l expression: f 0 (x) X c(0; k) OE(x Gamma k) 10) o u OE(x) d esigne la fonction d echelle duale de OE(x) [34]. Nous avons: OE( OE( D( 11) o u: D( X n j OE( n)j 2 (12) D( est une fonction 1 p eriodique qui ne doit pas s annuler sur [0; 1] Le passage de f(x) a f 0 (x) est une projection dans L 2 (R) f 0 (x) appartient au sous espace V 0 de L 2 (R) engendr e par les ....

....dans l espace direct, il importe de bien d elimiter la bande de fr equence a chaque echelle, en ayant une fonction d echelle aussi compacte que possible en fr equence. Ces crit eres nous ont conduit a utiliser de mani ere syst ematique les interpolations B splines centr ees d ordre 2l Gamma 1 [34]. Nous avons: OE 2l Gamma1 ( sin ) 2l (32) Dans le cas le plus simple de l interpolation lin eaire, nous avons l = 1, soit: OE 1 (x) 1 Gamma j x j si x 2 [ Gamma1; 1] OE 1 (x) 0 si x 62 [ Gamma1; 1] Sur la figure 1, nous avons trac e la fonction d echelle correspondante. Dans ce ....

M. Unser, A. Aldroubi, "Polynomial Splines and Wavelets - A Signal Processing Perspective ", Wavelets: A Tutorial in Theory and Applications pp.91-122 ed. C.K.Chui Academic Press New York, 1992.

....5.5. Fast B spline based running wavelet transform algorithm. A similar approach is developed in an adaptation of the fast B spline based running wavelet transform to the Morlet wavelet [111] The # n (t) B spline of order n is defined as the (n 1) fold convolution by a square window [112, 114, 115, 116]. Because the functions are splines, the following relationships exist: # n t a = # X k= # b n a (k)# n (t k) 48) and Z b n a (z) z (n 1) a 1) 2 a n a 1 X k = 0 z k n 1 . 49) FAST QUASI CONTINUOUS WAVELET TRANSFORM 1777 n k z S k=0 a 1 S ....

....WAVELET TRANSFORM 1777 n k z S k=0 a 1 S k z a 1 z k=0 1 n (n 1) a 1) 2 ( z C(z ) a a g (a,b) f k=0 a 1 S k b a (z) n 1 moving sums . f(t) FIG. 7. The fast B spline based running wavelet transform algorithm. The B spline functions easily generate discrete time WTs [112, 115, 116, 117]. The technique relies on the development of the generating analysis wavelet in the B spline representation, g(t) # X k = # c(k)# n (t k) 50) The coe#cients c(k) are obtained by least squares approximations. Then, the WT is obtained as WT f (a, k, g) 1 # a [c] # a # ....

M. UNSER AND A. ALDROUBI, Polynomial splines and wavelets---A signal processing perspective, in Wavelet Analysis and Its Applications: A Tutorial in Wavelet Theory and Applications, C. Chui, ed., Academic Press, New York, 1992.

....width parameter, and P(x) is the Sinc interpolation kernel. The Gaussian window in our DAF wavelets efficiently smoothes out the Gibbs oscillations, which plague most conventional wavelet bases. The following equation shows the connection between the B spline function and the Gaussian window [4]: 1 6 exp ) 1 ( 6 ) 2 N x N x N p b (8) for large N. As in Fig. 6, if we choose the window width 12 ) 1 ( N h s (9) the Gaussian window can be regarded as a fine estimation of the B spline function. The cascade smooth construction of DAFwavelets using lifting ....

M. Unser, A. Aldroubi, "Polynomial splines and wavelets-a signal processing perspective," Wavelets-A Tutorial in Theory and Applications, C. K. Chui (ed.), pp.91-122, Academic Press, 1992.

....many applications, yielding a broader class of transforms and more flexible filter bank designs. For instance, several authors have recently developed biorthogonal spline filter banks that generate multiresolution (MR) bases with symmetry and excellent approximation power and regularity properties [3, 4, 5, 6]. In addition, spline filters have rational coefficients and may be implemented using simple hardware (adders and shift registers) without the degradations associated with coefficient quantization. In image coding, the use of biorthogonal spline filter banks has led to some of the best results ....

M. Unser and A. Aldroubi, "Polynomial Splines and Wavelets -- A Signal Processing Perspective," pp. 91---122 in [4].

....basis functions. Basis functions that are local in both frequency and spatial domain can be constructed. These functions can then be shifted and dilated in the spatial domain. This corresponds to shifting the centers and varying the spans of the Gaussians in our GRBF network. Unser and Aldroubi [16] have presented a framework for representing signals using spline basis functions and wavelets. They also discuss a fast implementation of this technique using filter banks. This technique can be extended to tune Gaussian GRBF networks. The reason that this technique holds promise is that a ....

M. Unser, A. Aldroubi, and M. Eden, "Polynomial Splines and Wavelets - A Signal Processing Perspective," Wavelets - A Tutorial in Theory and Applications, C. K. Chui editor, pp. 91-122, Academic Press, 1992.

....splines is widespread in numerical analysis and many other fields because they have simple structure, good localization properties, and computing stability. See [7] Based on this fact, recently splines are often used to construct wavelet bases. See [1] 2] 3] 6] 11] 12] 14] [17]. In applications, wavelets on a bounded interval are also very useful. Chui and Quak first construct spline wavelet bases of L 2 (I) See [4] A general discussion of the construction of wavelets on interval [0; 1] can be found in paper [13] written by Micchelli and Xu. The aim of this paper ....

Unser., M. and Aldroubi, A., Polynomial splines and wavelets--A signal processing prospective, in "Wavelets: A tutorial in theory and applications." C. K. Chui Ed. Academic Press, 1992.

.... algorithm allowing the initial image to be rebuilt is obvious: the last smoothed array F (I ; k; l) is added to all the I differences W (i; k; l) I is the number of scales) We choose the following scaling function: OE(x; y) B 3 (x)B 3 (y) 15) where B 3 is the B spline function of degree 3 [24]. This function is very similar to a Gaussian one. The iterative relation (13) can be computed separately in row and column with the 1D mask h(n) f 1 16 ; 1 4 ; 3 8 ; 1 4 ; 1 16 g: 1.4 The pyramidal algorithm. The approximation is decimated at each scale: F (i; k; l) 1 4 i f(x; y) ....

M Unser and A. Aldroubi. Polynomial splines and wavelets - a signal processing perspective. In Wavelets: a tutorial in theory and applications, pages 91--122. C.K. Chui, Academic Press, New York, 1992.

....by using relations (8) S 2 j 1f(m) n=b X n=a ff n d j m Gamma2 j n and W 2 j 1f(m) n=d X n=c fi n d j m Gamma2 j n (9) x4. Discrete Dyadic Spline Wavelet Transform It is well known that the polynomial splines spaces are well fitted to multiscale wavelet transform [2] [5]. A spline function of order a is a piecewise polynomial function of degree a Gamma 1 which is C a Gamma2 for a 2. Its p th derivative may be used to detect singularities of f (p Gamma1) if 2p = a Gamma 2. If the wavelet (x) is the p th derivative of a spline of order 2p 2, to ....

Unser M. and Aldroubi A., Polynomial splines and Wavelets: A signal Processing Perspective, in Wavelets : A tutorial, C.K.Chui (eds.), Academic Press, (1991). Christine Potier and Christine Vercken Telecom Paris 46 rue Barrault 74634, Paris Cedex13, FRANCE Christine.Potier@ inf.enst.fr Christine.Vercken@ inf.enst.fr

....Every spline can be represented as a linear combination of B splines, which are smooth functions with compact support. The B spline of order 2 is plotted in Fig. 8. Several polynomial spline wavelets are described in the literature. Examples are the B spline wavelets of compact support [Chu92a, UA92] However, these wavelets are constructed by a finite linear combination of B splines. Therefore, we used the B splines themselves as tf atoms in our experiments. B splines can be defined in a recursive way: Bm (t) Bm Gamma1 B 0 ) t) 22) 2 1.5 1 0.5 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 ....

M. Unser and A. Aldroubi. Polynomial splines and wavelets -- a signal processing perspective. In Chui [Chu92b], pages 91--122.

.... by the equations N 1 (x) 0,1) x) Nm (x) Nm Gamma1 N 1 ) x) Z 1 0 Nm Gamma1 (x Gamma t) dt; m = 2; 3; Delta Delta Delta The above cardinal spline wavelet has been studied by several other authors as well (see, for example, Auscher [1] Micchelli [24] and Unser Aldroubi [27]) The advantage of the above cardinal spline wavelet is that the wavelet spaces are kept orthogonal and the wavelets are still symmetric or anti symmetric. The dual wavelet is still an m th order spline function. In a previous paper [3] we extended Chui and Wang s work and proved that the ....

M. Unser and A. Aldroubi, Polynomial splines and wavelets -- A signal processing Perspective, Wavelets -- A Tutorial in Theory and Applications (C. K. Chui, ed.), Academic Press, Boston, 1992, pp. 91-121.

....for the wavelet, scaling function, and dual functions are available. A disadvantage is that the dual functions do not have compact support, but have exponential decay instead. The same wavelets, but in a different setting, were also derived by Akram Aldroubi, Murray Eden and Michael Unser in [129, 131]. They also showed that for N going to infinity, the spline wavelets converge to Gabor functions [130] iii) Other semiorthogonal wavelets can be found in [89, 109, 110, 113] A characterization of all semiorthogonal wavelets is given in [1, 2] The properties of some of the orthogonal, ....

.... x ; is an interpolating scaling function. It is band limited, but it has very slow decay. ii) An interpolating scaling function, whose translates also generate V 0 , can be found by letting b interpol ( b ( X l (l)e Gammai l ; provided that the denominator does not vanish [1, 2, 129, 138]. Even if is compactly supported, interpol is in general not compactly supported. The cardinal spline interpolants of even order are constructed this way [118] iii) An interpolating scaling function can also be constructed from a pair of biorthogonal scaling functions as interpol (x) Z ....

M. Unser and A. Aldroubi, Polynomial splines and wavelets --- a signal processing perspective, in [21], pp. 543--601.

....resolution level (m Gamma 1) Fm Gamma1 (x) 1 X k= Gamma1 ff m;k OE m;k (x) 1 X k= Gamma1 ffi m;k m;k (x) 9) 2. 3 Wavelets in Higher Dimensions The one dimensional cubic B spline scaling functions and wavelets at any scale m and location k have the following general expressions [8]: OE m;k (x) 0:690988 p 2 Gammam expf Gamma1:5(2 Gammam x Gamma k) 2 g for (m; k)2Z 2 (10) m;k (x) 0:251477 p 2 Gammam cosf2:570935f2(2 Gammam x Gamma k) Gamma 1gg Theta expf Gamma0:222759f2(2 Gammam x Gamma k) Gamma 1g 2 g for (m; k)2Z 2 (11) ....

M. Unser and A. Aldroubi, "Polynomial Splines and Wavelets- A Signal Processing Perspective,"Wavelets: A Tutorial in Theory and Applications, Editor C. K. Chui, Academic Press Inc., San Diego, California.

.... 1 and Pf belong to the subspace W that is the complement of S with respect to S 1 ; i.e. SSW iii 222 1 =7 with SW ii 0 8= This is where the famous wavelet 9( x enters the scene: it generates the basis functions of the residual spaces W =span 9( xk 2 [51, 91]. There are many applications (e.g. coding) where it is more concise to express the residues Pf 1 Pf W using wavelets rather than the basis functions of V i 1 as has been is done in Fig. 10. An example of wavelet transform is shown in Fig. 11; this decomposition works well for ....

M. Unser and A. Aldroubi, "Polynomial splines and wavelets---A signal processing perspective," in Wavelets - A tutorial in theory and applications, C.K. Chui, Ed., San Diego: Academic Press, pp. 91-122, 1992.

...., e.g. function # i with some desirable shape corresponding to a particular impulse response of a device, a compactly supported function, or a function # with a smooth cut o# frequency # . Early results on sampling in shift invariant spaces concentrated on the problem of uniform sampling [6, 8, 9, 10, 27, 50, 89, 92, 91, 96, 99], or interlaced uniform sampling [94] Similar to the case of Shannon s sampling theorem, the problem of uniform sampling in shift invariant spaces requires only the Poisson summation formula and a few facts about Riesz bases [6, 8] The connection between interpolation in spline spaces, filtering ....

....to reduce the errors in reconstructions called the aliasing errors. It has been shown that the three steps of traditional sampling procedure, namely pre filtering, sampling, and post filtering for reconstruction, are equivalent to finding the best L 2 approximation of a function in L 2 # B# [8, 89]. This procedure and its implementation generalize to sampling in general shift invariant spaces [6, 8, 9, 89, 92] In fact, the reconstruction from the samples of a function should be considered as an approximation in the shift invariant space generated by the impulse response of the sampling ....

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M. Unser and A. Aldroubi. Polynomial splines and wavelets--a signal processing perspective. In [24], pages 543--601. 1992.

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M. Unser and A. Aldroubi. Polynomial splines and wavelets --- a signal processing perspective. In 4 , pages 543--601.

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Aldroubi, A. and Unser, M. (1992) Polynomial Splines and Wavelets -- a signal processing perspective. in Wavelets: a tutorial in theory and applications ed C.K. Chui. Academic Press, San Diego.

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