T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I---Interpolators and projectors," IEEE Trans. Signal Processing, vol. 47, pp. 2783--2795, Oct. 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....m kk m kPk m mCP rp rprp jppjp = 4.19) If we may take rj= then ( S jj and thus j is refinable. In such a case, we obtain from (4.19) a formula for the constant C j 2 21 C . 4.20) Formula (4. 20) for the refinable case is exactly equation (26) in [BU3]. Lemma 4.17 Let 0 m and assume that j L fr # are such that , j suppsuppfrWwhere W is some bounded domain. Then we have 32 00 22 CkkC rf rpfp . 4.21) Proof By Lemma 2.4, for any wT , 22 j wkwk rpfp . 4.22) In particular we have (4.22) for 0 w = ....

....( mr j H 1. We use the cascade result Theorem 3.9. 2. See the proof of Corollary 4.15. 3. For each 1 j we use Corollary 4.15. We then apply Theorem 4.2. 4. The convergence CC rf follows from Lemma 4.17. The estimate (4.23) is obtained using Lemma 4. 18 and some techniques from [BU3]. Example 4.20 Let 04 OM r = 4 N f = and let r be the sequence constructed in Theorem 4.19 . Then one can compute using (4.19) 44 41 1.4631.07 . This means that the first generator 1 r constructed by the cascade process is not as good as the initial optimal 04 : OM r = but ....

T. Blu and M. Unser, " Quantitative Fourier Analysis of approximation techniques, part IIWavelets ", IEEE Trans. Signal Processing, vol. 47, no. 10, pp. 2796-2806, 1999.

....(5) where Phi( and Phi( denote the Fourier transforms of (x) and (x) respectively. In this section, we will briefly compare the L2 error bounds of interpolation and LS approximation. The detailed proof is not included in this paper, but some relevant information can be found in [6] [7]. 2.1. L2 Error of Interpolation The L2 error bound of the interpolator is given as following. If is a quasi interpolant of order L with sufficient decay, then: 8s 2 W L 2 ; ks Gamma Ihsk2 C ;L Delta h L Delta ks (L) k2 (6) C ;L = 1 L sup x2[0;1] X k2Z jx Gamma kj L j (x ....

T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I -- interpolators and projectors," IEEE Trans. on Signal Processing, Vol. 47, No. 10, pp. 2783 - 2795, October 1999.

.... spaces that are not band limited is a suitable and realistic model for many applications, e.g. for taking into account real acquisition and reconstruction devices, for obtaining smoother frequency cut o#s than is the case with band limited functions, or for the numerical implementation [8, 11, 17, 20, 72, 87, 88, 91, 94, 98, 99]. These requirements can often be met by choosing appropriate functions # i , 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 ....

T. Blu and M. Unser. Quantitative fourier analysis of approximation techniques: Part i-interpolators and projectors. IEEE Trans. Signal Process., 47(10):2783--2795, 1999.

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T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I---Interpolators and projectors," IEEE Trans. Signal Processing, vol. 47, pp. 2783--2795, Oct. 1999.

....rendering by computing splats via least squares approximation. The method decomposes the volume data into a wavelet pyramid representation in the B spline domain. The splats of the basis functions are projected onto a multiresolution grid. The approximation error on the grid is derived, applying [2], as a function of the sampling step h. Choosing at each step the appropriate wavelet space and spatial resolution produces the smallest possible filters and reduces computation. Our approach ensures a maximal image quality when rendering at low resolution. keywords: volume rendering, wavelet ....

T.Blu, M.Unser, "Quantitative Fourier Analysis of Approximation Techniques", submitted tod56tr IEEE Transaction on Signal Processing, June 5, 1998.

.... = the kernels (and therefore the errors) tend to be smaller for splines of higher degree. 16 Recently, it has become possible to determine the approximation error much more precisely by simply integrating the whole spectrum of the function to approximate against a frequency kernel E ( [15]. The justification for this procedure is the error formula sx W ( 2 , sPs ETs d Ts T nrr 0 1 2 12 3 , 25) where 3 is bounded by some known constant [13] The second term in (25) is a correction that may take positive or negative values. It is zero for bandlimited ....

....over all possible shifts of the input function; this is a reasonable thing to do since the sampling phase is usually arbitrary. Thus, the first term on the right hand side in (25) provides a very accurate prediction of the error which can be the basis for a quantitative Fourier domain evaluation [15]. The error kernel for a least squares spline approximation of degree n is EHH ( 1 (26) where H ( and H ( are the spline filters defined by (16) and (22) respectively. The main point is that the study of these kernels gives us a very direct way to assess the performance ....

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T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I---interpolators and projectors," IEEE Transactions on Signal Processing, in press.

.... order property implies that one has the following asymptotic form of the approximation error (cf. 89] 26 sPsC Ts TL = as T 0 (37) where Ps T , is the projection of s onto the space VxTk T kZ = span ( and where the constant C L , can be determined explicitly [89, 14]. This is essentially the same equation as (24) with an equality instead of an upper bound; the asymptotic leading constant C L , is therefore necessarily smaller than C L in (24) Among all known wavelet families, splines appear to have the best approximation property in the sense that the ....

....can be quite significant. For instance, Sweldens observed that splines at half the resolution could provide a better approximation than Daubechies wavelets [83] Recently, the exact subsampling factor such that the asymptotic errors in both cases are identical has been determined analytically [14] : it converges to as the order L gets sufficiently large 5.4 Maximum regularity and shortest support It is well known from wavelet theory that the B splines are the shortest scaling functions of order L [81, 23] They are also the most regular ones if one takes the size of the refinement ....

T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part II---wavelets," IEEE Transactions on Signal Processing, in press.

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T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I--interpolators and projectors", IEEE Transactions on Signal Processing, Vol. 47, No. 10, pp. 2783-2795, October 1999.

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T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part II--wavelets", IEEE Transactions on Signal Processing, Vol. 47, No. 10, pp. 2796-2806, October 1999.

....quality depends on the sampling step , the type of algorithm used (e.g. interpolation versus projection) and, most importantly, on the choice of the generating function . This can be quantified rather precisely, thanks to the availability of sharp mean square error estimates in the setting [3], 4] Bounds are also available for the approximation error (worst case scenario) 5] Manuscript received April 5, 2001; revised January 8, 2002. This work was supported by the Swiss National Science Foundation under Grant 2100 053 540. The associate editor coordinating the review of this paper ....

....In particular, we will derive a general predictive error formula that depends on the Fourier coefficients of . Interestingly, the formula bears a strong resemblance to the error expression of signals in . However, the recipe is different although the ingredients are more or less the same as in [3]; the average least squares error is obtained as a discrete sum of the Fourier series coefficients, as opposed to a continuous integral in [3] We also study the behavior of the approximation as the sampling step goes to zero. II. PRELIMINARIES A. Notations We denote the Fourier transform of a ....

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T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I---Interpolators and projectors," IEEE Trans. Signal Processing, vol. 47, pp. 2783--2795, Oct. 1999.

....calculated for different spline degrees. We observe that we have lower values of the entropy (maximum SNRep) with p s close to 1 when using spline basis of the same degree. The minimum 2 error is reached for the higher order splines. This finding is consistent with the standard theory of splines [25], 26] as the degree Fig. 8. a) Biomcdical imagcs, b) corrcsponding histogram, and (c) histogram of thc dctail imagcs (odgimd minus the approximatcd vcrsion at scale 2 using cubic splines as basis functions) for different values ofp. Note the high peak at zero for p close to 1. increases, the ....

T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I--Interpolators and projectors; Part II--Wavelets," IEEE Trans. Signal Processing, vol. 47, pp. 2783-2806, Oct. 1999.

....T (f) and T in the limit when T 0. The exact formula s j (L) 2n)j (4) will be needed in the next section, since this is precisely the quantity that we will minimize. B. Quantitative study The approximation error has been studied in a more quantitative way in [6] 7] 8] [9], showing that the quality of the approximation is fully characterized by a Fourier kernel which simplifies here to E( 1 1 Gamma j ( j : 5) In particular, if f satisfies Shannon s sampling conditions, i.e. if the support of f( is contained within [ Gamma T ; T ] then ....

.... equation also gives the average error over all possible shifts for an arbirary function f , not necessarily bandlimited (cf. 6] Thm.2) We recently showed the relevance of this approximation kernel as regards the computation of asymptotic expansions, upper bounds and shiftvariance estimates [8] [9]. III. Minimum support functions First, we give the following result which provides an explicit characterization of the shortest generating function of order L [11] Theorem 1: Minimizing for the support of the function (x) under the L th order constraint yields piecewise polynomials that ....

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T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: part II---wavelets," IEEE Trans. Sig. Proc., 1998, submitted.

.... T (f) and T in the limit when T 0. The exact formula s j (L) 2n)j (4) will be needed in the next section, since this is precisely the quantity that we will minimize. B. Quantitative study The approximation error has been studied in a more quantitative way in [6] 7] [8], 9] showing that the quality of the approximation is fully characterized by a Fourier kernel which simplifies here to E( 1 1 Gamma j ( j : 5) In particular, if f satisfies Shannon s sampling conditions, i.e. if the support of f( is contained within [ Gamma T ; T ] ....

.... equation also gives the average error over all possible shifts for an arbirary function f , not necessarily bandlimited (cf. 6] Thm.2) We recently showed the relevance of this approximation kernel as regards the computation of asymptotic expansions, upper bounds and shiftvariance estimates [8], 9] III. Minimum support functions First, we give the following result which provides an explicit characterization of the shortest generating function of order L [11] Theorem 1: Minimizing for the support of the function (x) under the L th order constraint yields piecewise polynomials ....

T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: part I---interpolators and projectors," IEEE Trans. Sig. Proc., 1998, submitted.

....can be implemented at almost no additional computational cost [19] Regarding the approximation quality, it is not ensured that closeness to the is truly beneficial. To be more specific, the problem lies in how to measure this closeness. We have been investigating these issues in some depth [20], 21] and have proposed satisfactory solutions that are summarized in the following paragraphs. Our Approach to Interpolation [22] 23] A fast implementation of (1) can be devised, provided that itself can be expressed as an arbitrary linear combination of shifted versions of a ....

....content at high frequencies, the approximation error has to be characterized not only by a (integer) number, but by a function: the Fourier approximation kernel . When the approximation method is the orthogonal projection onto , the expression of this kernel is (7) We have recently shown [20], 21] that (8) where if . More remarkable, this term cancels on the average over all possible shifts of the function; i.e. if we denote , then Finally, we also know that the correcting term in (8) vanishes when satifies Nyquist s band limitation constraint. The approximation efficiency of a ....

T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I---Interpolators and projectors," IEEE Trans. Signal Processing, vol. 47, pp. 2783--2795, Oct. 1999.

....increase of the degree. Specifically, if where the sum is finite IV. SPLINE RESIZING ALGORITHM Our reason for using B splines or some close relatives is that these are functions for which we know how to compute the required inner products. They also have excellent approximation properties [23], 24] Moreover, they have the shortest support for a given approximation order, which means that the computational complexity is minimized. A. Derivation of the Algorithm Thus, we choose our basis functions to be B splines. In that case, our algorithm has the following parameters: and . This ....

....the O MOMS give the best value in terms of SNR, 1 dB over cubic splines, while the linear splines (resp. piecewise constant) are 10 dB (resp. 25 dB) below the cubic ones. Thus, it appears that higher order correlates with improved shift invariance, in accordance with the theoretical findings in [23]. 1374 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 9, SEPTEMBER 2001 Fig. 15. Loss in performance by using oblique projection instead of least squares. a) Full scale range [0.2 : 1.4] and (b) reduced scale range [0.2 : 1.0] to magnify the difference at low scale factors. C. Oblique ....

T. Blu and M. Unser, "Quantitative fourier analysis of approximation techniques: Part I---Interpolators and projectors," IEEE Trans. Signal Processing, vol. 47, pp. 2783--2795, Oct. 1999.

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T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I---Interpolators and projectors," IEEE Trans. Signal Process., vol. 47, no. 10, pp. 2783--2795, October 1999.

....filtering. Note that, in our tests, this handling cost is 6 times higher for non interpolating splines (i.e. for degree 2) because the prefilter requires additional line and column manipulations. III. Approximation Error To evaluate the approximation error involved, we use a recent result [4] which expresses the L 2 error as a scalar product between the squared Fourier transform of the signal and a Fourier kernel which depends on only. For the case where the s k are the samples f(k) of a well behaved function f(x) 4] 5] we have kf(x) Gamma s(x)k L 2 z f = s Z ....

....the approximation error involved, we use a recent result [4] which expresses the L 2 error as a scalar product between the squared Fourier transform of the signal and a Fourier kernel which depends on only. For the case where the s k are the samples f(k) of a well behaved function f(x) [4], 5] we have kf(x) Gamma s(x)k L 2 z f = s Z j f( j 2 E( d 2 z jf ae f (6) where E( fi fi fi X n2Znf0g ( 2n) fi fi fi 2 X n2Znf0g j ( 2n)j 2 fi fi fi X n2Z ( 2n) fi fi fi 2 (7) and where ae f is a usually ....

T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I---interpolators and projectors," IEEE Trans. Signal Process., 1999, To appear.

....spline model. a) Good approximation properties The error of a cubic spline approximation decreases asymptotically as h 4 (measured by any L p or l p norm, p # 1, 2, # ) Quantitative analyses indicate that splines perform well in comparison with other wavelet like basis functions [33]. b) Speed Cubic splines have a short compact support of length 4. They are symmetric and piecewise cubic. To evaluate # 3 (x) at one particular point, only 5 arithmetic operations (additions or multiplications) and 3 comparisons are needed. In multiple dimensions, where we will use tensor ....

T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I---interpolators and projectors," IEEE Transactions on Signal Processing, 1999.

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T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part II---wavelets," IEEE Trans. Signal Process. , 1999. To appear.

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T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I---interpolators and projectors," IEEE Trans. Signal Process. , 1999. To appear.

....model. a) Good approximation properties The error of a cubic spline approximation decreases asymptotically as h 4 (measured by any L p or l p norm, p # 1, 2, # ) Quantitative analyses indicate that splines perform well in comparison with other wavelet like basis functions [92]. b) Speed Cubic splines have a short compact support of length 4. They are symmetric and piecewise cubic. To evaluate # 3 (x) at one particular point, only 5 arithmetic operations (additions or multiplications) and 3 comparisons are needed. In multiple dimensions, where we will use tensor ....

Thierry Blu and Michael Unser, "Quantitative Fourier analysis of approximation techniques: Part I---interpolators and projectors," IEEE Transactions on Signal Processing, 1999.

....quality depends on the sampling step h, the type of algorithm used (e.g. interpolation vs. projection) and most importantly, on the choice of the generating function . This can be quanti ed rather precisely, thanks to the availability of sharp error estimates in the L 2 (R) setting [2] [3]. In this paper we are interested in the case where the input signal s(t) is periodic, which is an assumption that is commonly made in practice. When the period T is an integer multiple of the sampling step (T = Nh) it is straightforward to adapt most of the L 2 techniques to the periodic case ....

T. Blu and M. Unser, \Quantitative Fourier analysis of approximation techniques: Part II{Wavelets," IEEE Trans. Signal Processing, vol. 47, pp. 2796-2806, October 1999.

....quality depends on the sampling step h, the type of algorithm used (e.g. interpolation vs. projection) and most importantly, on the choice of the generating function . This can be quanti ed rather precisely, thanks to the availability of sharp error estimates in the L 2 (R) setting [2], 3] In this paper we are interested in the case where the input signal s(t) is periodic, which is an assumption that is commonly made in practice. When the period T is an integer multiple of the sampling step (T = Nh) it is straightforward to adapt most of the L 2 techniques to the periodic ....

.... 2n )j 2 (7) 1 j ( j 2 a ( z Emin ( a ( d ( 2 z E res ( 8) where a ( P 1 k=1 j ( 2n )j 2 and d ( a ( Note that this kernel is identical to the one obtained in the case of signals in L 2 (R) [2]. When = d , the kernel reduces to E min ( which depends only on . The analysis function that gives this minimum error approximation is d (the dual function of ) as in the case of signals in L 2 (R) 5] This case corresponds to the orthogonal projection. 0 5 10 15 20 1 0.5 0 ....

T. Blu and M. Unser, \Quantitative Fourier analysis of approximation techniques: Part I{Interpolators and projectors, " IEEE Trans. Signal Processing, vol. 47, pp. 2783{ 2795, October 1999.

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Blu T. and Unser M., "Quantitative Fourier analysis of approximation techniques: Part I-interpolators and projectors; part II-wavelets," IEEE Trans. Signal Processing, vol. 47, no. 10, pp. 2783--2806, October 1999.

....functions at the next finer scale (1) with the short form notation . This suggests that we can estimate the wavelet approximation error by studying the properties of the projector operator (2) simply because . This problem therefore clearly falls into the general framework of the companion paper [17], except that the present situation is more constrained: The functions and are biorthonormal and both satisfy a twoscale relation. From what is known in approximation theory, we would expect higher order wavelets to provide better approximations of piecewise smooth functions, at least in the ....

....studying the error behavior of the more general projection operator [15] This work resulted in an exact asymptotic error formula as well as a computational method for obtaining the leading constant in the wavelet case. Thanks to the general results that have been presented in our companion paper [17], we are now in the position to go further. The main problem with wavelets, however, is 1053 587X 99 10.00 1999 IEEE BLU AND UNSER: QUANTITATIVE FOURIER ANALYSIS OF APPROXIMATION TECHNIQUES: PART II 2797 that they generally have no closed form representation, which implies that we cannot simply ....

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T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I---Interpolators and projectors," IEEE Trans. Signal Processing, this issue, pp. 2783--2795.

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T.Blu, M.Unser, "Quantitative Fourier Analysis of Approximation Techniques: Part I---Interpolators and Projectors, Part II---Wavelets", to appear in IEEE Trans. Signal Process., 1999.

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T. Blu and M. Unser. Quantitative Fourier analysis of approximation techniques: Part I---Interpolators and projectors. IEEE Transactions on Signal Processing, 47(10):2783-- 2795, October 1999.

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T. Blu and M. Unser, " Quantitative Fourier Analysis of approximation techniques, part IInterpolators and projectors", IEEE Trans. Signal Processing, vol. 47, no. 10, pp. 2783-2795, 1999.

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Blu, T. and Unser, M. (1999b). Quantitative Fourier analysis of approximation techniques: Part II - wavelets. IEEE Transactions on Signal Processing , 47, 2796-2806.

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