Unser, M., A. Aldroubi, and M. Eden: 1993, `A Family of Polynomial Spline Wavelet Transforms'. Signal Processing 30, 141-162.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[3] All methods in common use, work the same way: the image signal bandwidth is reduced by filtering with a finite impulse response filter and the filtered signal is then down sampled. Contenders for the filter include: Laplacian of Gaussians [2] Gabor filters [4] wavelets [5] or B splines [6]. Discretized Gaussians have the particular advantage that they preserve scalespace causality [7] However, the method described here does not rely on linear filtering at all: mathematical morphology provides the theoretical framework space. Mathematical morphology is the analysis of signals, ....

M. Unser, A. Aldroubi, and M. Eden. A family of polynomial spline wavelet transforms. IEEE Trans. on Sig. Proc., 30(2):141--162, Jan 1993.

....of different wavelet systems. I. INTRODUCTION A. Previous Works M ANY families of wavelet systems have been constructed in the last decade, for instance, orthonormal Daubechies wavelets [1] orthonormal coiflets [2] biorthogonal spline wavelets (BSW s) 3] semiorthogonal spline wavelets [4] [5], and general biorthogonal Coifman wavelets (GBCW s) 6] 7] Most families of wavelet bases are indexed by the number of vanishing moments of the wavelet or, equivalently, the number of zeros at of the transfer function of the associated lowpass filter. For instance, if the synthesis wavelet and ....

M. Unser, A. Aldroubi, and M. Eden, "A family of polynomial spline wavelet transforms," Signal Process., vol. 30, pp. 141--162, Jan. 1993.

....4.5.2 Classification experiments A database of 1920 image regions of 30 texture classes was constructed by subdividing each 512x512 image into 64 non overlapping 64x64 subimages. Each image region was decomposed into a wavelet basis of depth 4 using a biorthogonal spline wavelet of order 2 [46]. Four different classification experiments were conducted as described in section 4.4, using the following feature sets: 1. 12 energy signatures. 2. 24 histogram signatures. 3. 96 cooccurrence features (quantization interval Deltau i = 1 8i, ffi = 1, matrices for = 0; 45; 90; 135 ffi were ....

M. Unser, A. Aldroubi, and M. Eden. A family of polynomial spline wavelet transforms. Signal Processing, 30:141--162, 1993.

....c k ( sup f2A k j( f)j k k 2 ; where A k = f f j kf (k) k 2 1 g; k 2 N: It is known that for different the constants c k ( may differ significantly in size. Precise relations for the orthogonal Daubechies wavelets (cf. 5] and the semiorthogonal spline wavelets (cf. [3, 9]) are proved in [7] These results show that asymptotically, i.e. for increasing order, the spline constants are considerably smaller. In contrast to these asymptotical results, in the present paper we are interested in precise constants for practical wavelets with small support lengths. We ....

....Numerical results The computational procedure desribed in the preceeding section can easily be implemented on a computer. We have computed the constants c k ( up to rounding error, for the practically important Daubechies orthonormal 6 wavelets [5] and the semiorthogonal spline wavelets from [3, 9] for k = 1; 2; 3 and support length up to 11. The spline wavelets are defined by (see [3, Theorem 1] 9] S m (x) 2m Gamma2 X =0 ( Gamma1) 2 m Gamma1 N 2m ( 1)N (m) 2m (2x Gamma ) x 2 R; 11) where Nm is the mth order B spline (of degree m Gamma 1) defined by Nm (x) 1 (m ....

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M. Unser, A. Aldroubi & M. Eden, A family of polynomial spline wavelet transforms, Signal Processing 30 (1993), 141-162.

....1.1.3 Orthonormal Daubechies wavelets and spline wavelets Among the orthonormal wavelets, the Daubechies wavelets D m (see, e.g. 5] have the shortest support length, 2m Gamma 1, with m vanishing moments. The semiorthogonal (s. o. spline wavelets S m are defined by (see [3, Theorem 1] [15]) S m (x) 2m Gamma2 X =0 ( Gamma1) 2 m Gamma1 N 2m ( 1)N (m) 2m (2x Gamma ) x 2 R; 10) where Nm is the mth order B spline (of degree m Gamma 1) defined by Nm (x) 1 (m Gamma 1) m X k=0 ( Gamma1) k i m k j (x Gamma k) m Gamma1 ; and this normalisation is such ....

M. Unser, A. Aldroubi & M. Eden, A family of polynomial spline wavelet transforms, Signal Processing 30 (1993), 141-162. 24

....supported wavelets in 1987 [18] Later many generalizations to the biorthogonal or semiorthogonal (pre wavelet) case were introduced. Biorthogonality allows the construction of symmetric wavelets and thus linear phase filters. Examples are: the construction of semiorthogonal spline wavelets [1, 10, 12, 13, 50], fully biorthogonal compactly supported wavelets [14, 57] and recursive filter banks [28] Recently a new angle to study these constructions was provided by the so called lifting scheme [46] The basic idea behind lifting is that it provides a simple relationship between all multiresolution ....

M. Unser, A. Aldroubi, and M. Eden. A family of polynomial spline wavelet transforms. Signal Process., 30:141--162, 1993.

....k 2 : 4 1.1.2 Orthonormal Daubechies wavelets and spline wavelets Among the orthonormal wavelets, the Daubechies wavelets D m (see, e.g. 6] have the shortest support length, 2m 1, with m vanishing moments. The semiorthogonal (s.o. spline wavelets S m are de ned by (see [4, Theorem 1] [16]) S m (x) 2m 2 X =0 ( 1) 2 m 1 N 2m ( 1)N (m) 2m (2x ) x 2 R; 10) where Nm is the mth order B spline (of degree m 1) de ned by Nm (x) 1 (m 1) m X k=0 ( 1) k m k (x k) m 1 ; and this normalisation is such that X 2Z Nm (x ) 1: Here, for k 1, y) ....

M. Unser, A. Aldroubi & M. Eden, A family of polynomial spline wavelet transforms, Signal Processing 30 (1993), 141-162. 25

....way. 16 2. 10 Denoising Example In this denoising example, the behavior of the slantlet basis is compared to other wavelet bases having two vanishing moments, the D 2 basis, the biorthogonal 2,2 and 2,4 bases (page 273 of [13] and the piecewise linear semi orthogonal (spline) bases (page 147 of [32]) For the non orthogonal bases, the DWT was carried out with symmetric extensions. A hard threshold was applied uniformly to each scale. We chose the signal to be the Houston skyline function by H. Guo, for it is piecewise linear and has numerous discontinuities. Figure 6 illustrates the ....

M. Unser, A. Aldroubi, and M. Eden. A family of polynomial spline wavelet transforms. Signal Processing, 30(2):141--162, January 1993.

....terms of which a signal may be decomposed into a weighted sum of scaled and translated wavelets. Many of the wavelets which are employed today are defined by means of recursion formulae and do not have a simple closed form expression. The Spline wavelets introduced by C. K. Chui [4] and M. Unser [8] are well suited to analytic manipulations, possessing many desirable properties in addition to having a closed form representation. These properties 2. Spline Wavelets and Multi Resolution Analysis 2 include rapid computation of the wavelet representation and preservation of certain kinds of ....

M. Unser, A. Aldroubi, and M. Eden. A family of polynomial spline wavelet transforms. Signal Processing, 30:141--162, 1993.

....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, A. ALDROUBI, AND M. EDEN, A family of polynomial spline wavelet transforms, Signal Process., 30 (1993), pp. 141--162.

....is a 2 D description of the signal with respect to time b and scale a. The scale a is inversely proportional to the central frequency of the rescaled wavelet (x) x a) which is typically a bandpass function; b represents the time location at which we analyze the Corrcsponding author. Tcl. 41 21 693 5142; fax: 41 21 69 3701. E mail addresses: mTate.munozepfi.ch (A. Mufioz) raphael.ertle compaq.com (R. Ertlb) michael.unser epfi.ch (M. Unser) signal. The larger the scale a, the wider the analyzing function , and hence smaller the corresponding analyzed frequency. The output value is ....

....the signal with respect to time b and scale a. The scale a is inversely proportional to the central frequency of the rescaled wavelet (x) x a) which is typically a bandpass function; b represents the time location at which we analyze the Corrcsponding author. Tcl. 41 21 693 5142; fax: 41 21 69 3701. E mail addresses: mTate.munozepfi.ch (A. Mufioz) raphael.ertle compaq.com (R. Ertlb) michael.unser epfi.ch (M. Unser) signal. The larger the scale a, the wider the analyzing function , and hence smaller the corresponding analyzed frequency. The output value is maximized when the ....

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M. Unser, A. AlPh'oubi, M. Eden, A family of polynomial spline wavelet transforms, SignaI Process. Mag. 30 (1993) 141 162.

....for i 7 generates a multiresolution of L2( in the sense specified by Mallat [24] Hence, it is possible to construct a whole variety of corresponding wavelet bases using any of the standard design techniques. Specific examples of linear spline wavelets that are orthogonal [24] semi orthogonal [25], biorthogonal [26] or even shift orthogonal [27] have been de scribed in the literature. C. Nonuniform Linear Splines The power of the present formulation really becomes apparent if we move one step further and consider a given nonuniform sequence of knots. xk z . with k E 7. We ....

M. Unser, A. Aldroubi, and M. Eden, "A family of polynomial spline wavelet transforms," Signal Process., vol. 30, pp. 141 162, 1993.

....several other subclasses of spline wavelets available; they differ in the type of projection used and in their orthogonality properties. Corresponding to an orthogonal projection (and to the L 2 pyramid above) is the class of semi orthogonal wavelets which are orthogonal with respect to dilation [98]. These wavelets span the same space as the Battle Lemari splines, but are not constrained to be orthogonal. This gives flexibility and makes it possible to design wavelets with many interesting properties [5] and almost any desirable shape [1] Of particular interest are the B spline wavelets ....

M. Unser, A. Aldroubi and M. Eden, "A family of polynomial spline wavelet transforms," Signal Processing, vol. 30, no. 2, pp. 141-162, 1993.

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M. Unser, A. Aldroubi and M. Eden, "A family of polynomial spline wavelet transforms", Signal Processing, Vol. 30, No. 2, pp. 141-162, January 1993.

No context found.

M. Unser, A. Aldroubi and M. Eden, "A family of polynomial spline wavelet transforms", Signal Processing, Vol. 30, No. 2, pp. 141-162, January 1993.

....form. All other wavelet bases are defined indirectly through an infinite recursion (or an infinite product in Fourier domain) To date, four sub families of spline wavelets have been characterized explicitly: the orthogonal Battle Lemari wavelets [5, 8] the semiorthogonal spline wavelets [6, 21, 22], the biorthogonal splines [7] and the shift orthogonal spline wavelets [24] The first two types span the same spline multiresolution subspaces and are based on an orthogonal projection; the more general semi orthogonal splines are orthogonal with respect to dilation but not necessarily with ....

....have essentially two strategies: either to perform an infinite summation in the Fourier domain, or to compute the B spline samples b a ( 2 1 k explicitly. In the standard polynomial case, the most efficient approach is clearly the second one because the B splines are compactly supported [22]. Here, in the fractional case, there is no clear advantage of one method over the other. In our implementation, we have chosen the former for we have a relatively convenient Fourier domain expression for ( b w a (cf. 13) and (14) Using the general approach in [2] we express the ....

M. Unser, A. Aldroubi and M. Eden, "A family of polynomial spline wavelet transforms," Signal Processing, vol. 30, no. 2, pp. 141162, 1993.

....to evaluate the spline kernel. In other words, the original method had a strong penalty for large reduction factors. When the reduction factor is an integer, there exist alternative filtering decimation techniques which are equivalent to the present algorithm (least squares spline approximation) [31]; these are also very efficient computationally, but they require the design of a separate prefilter for each scale factor . When the scale parameter is a power of 2, the method is equivalent to a wavelet decomposition [15] because splines satisfy a two scale difference equation. B. ....

M. Unser, A. Aldroubi, and M. Eden, "A family of polynomial spline wavelet transforms," IEEE Trans. Signal Processing, vol. 30, pp. 141--162, Feb. 1993.

....# Z generates a multiresolution of L 2 (R) in the sense specified by Mallat [24] Hence, it is possible to construct a whole variety of corresponding wavelet bases using any of the standard design techniques. Specific examples of linear spline wavelets that are orthogonal [24] semi orthogonal [25], biorthogonal [26] or even shift orthogonal [27] have been described in the literature. C. Non uniform linear splines The power of the present formulation really becomes apparent if we move one step further and consider a given non uniform sequence of knots x k x k 1 with k # Z. ....

M. Unser, A. Aldroubi, and M. Eden, "A family of polynomial spline wavelet transforms," Signal Process., vol. 30, pp. 141--162, 1993.

No context found.

M. Unser, A. Aldroubi and M. Eden, "A family of polynomial spline wavelet transforms", Signal Processing, Vol. 30, No. 2, pp. 141-162, January 1993.

....is that it cannot express linear functions (a#ne deformations) The only two remaining candidate basis are therefore B splines and wavelets. 78 5.3. 1 Splines versus wavelets To make a fair comparison between B spline and wavelet bases, we consider compactly supported cubic B spline wavelets [99] spanning the same space as cubic B splines. First, let us analyze the task of evaluating the deformation at a single point. There are four participating B splines alltogether while there are four participating B spline wavelet at each level. Given that the complexity of one generating function ....

M. Unser, A. Aldroubi, and M. Eden, "A family of polynomial spline wavelet transforms," Signal Processing, vol. 30, no. 2, Jan. 1993.

....employing tricubic spline interpolation (Section IV C) orthogonal cubic spline wavelets were used for the statistical analyses. D. Periodic Implementations A simplified form for the transfer function of the low pass refinement filter (Fig. 1) for an orthogonal spline of degree is (cf. [42]) 16) where is the Fourier transform of the discrete Bspline of degree . Recursive formulas for computing can be found in [43] For example, for piecewise linear splines and for piecewise cubic splines . An expedient way to obtain these filter coefficients is to sample at the frequencies , and ....

M. Unser, A. Aldroubi, and M. Eden, "A family of polynomial spline wavelet transforms," Signal Processing, vol. 30, no. 2, pp. 141--162, 1993.

.... on the smoothness of the underlying basis functions: Most wavelet families exhibit a regularity index that is roughly proportional to (typically, with ) It is therefore quite natural to index common families of wavelets (Daubechies [6] orthogonal splines [7] 8] semi orthogonal splines [9], 10] biorthonormal splines [11] coiflets, etc. by the order parameter . The other remarkable consequence of the order constraint is that the residual error of a scale truncated wavelet expansion will decrease like the th power of that scale [12] 15] Manuscript received August 5, 1998; ....

....filter banks. least squares approximation . That is, orthogonal projection, for which such that (7) Using this function also called dual on the analysis side yields the smallest approximation error. This corresponds to the case of orthogonal [6] 23] and semiorthogonal wavelets [9], 10] interpolation . That is, with the property that by choosing , where (8) This is the inverse of a digital FIR filter since . A particular case is the spline interpolator, which is investigated in [24] and [25] Computing such an interpolation provides a consistent way of initializing ....

M. Unser, A. Aldroubi, and M. Eden, "A family of polynomial spline wavelet transforms," Signal Process., vol. 30, pp. 141--162, 1993.

No context found.

Unser, M., A. Aldroubi, and M. Eden: 1993, `A Family of Polynomial Spline Wavelet Transforms'. Signal Processing 30, 141-162.

No context found.

M. Uaser, A. Aldroubi and M. Eden, "A family of polynomial spline wavelet transforms", Signal Processing, vol. 30, pp. 141-162, January 1993.

No context found.

M. Unser, A. Aldroubi and M. Eden, "A Family of Polynomial Spline Wavelet Transforms," Signal Processing, vol. 30, pp. 141--162, 1993.

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