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638 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 23, NO. 6, JUNE 2001The area bounded by the curve (cf. (15)) is computed asI 0;0 ˆ XM1c p XN2kkˆ0 lˆN‡1d p k‡l g 0…l†;…45†where g 0 is given by (17). The sequences c p k and dp k areM-periodized versions of the coefficients c k and d k withrespect to the period M. This is simply because convolvinga nonperiodized sequence with a periodized kernel isequivalent to convolving a periodized sequence with anonperiodized kernel. We have also reduced the range ofsummation of the inner sum to N ‡ 1 to N 2, which istypically much less than the range 0 to M 1. Similarly, forthe higher order moments all the summations, except theouter one, are in the range N ‡ 1 to N 2.From (45), we see that the computation of the areainvolves just a filtering operation by g…l† ˆg T …l†, followedby an inner product. This can be written as,I 0;0 ˆhc p ;g T 0 dp i;…46†wherePh:; :i stands for the inner product hc; di ˆM1kˆ0c…k†d…k†. With a similar notation, the computationof the other moments are given asI m;n ˆ hcp ;g T m‡n …cp‰mŠ d p‰n‡1Š †in ‡ 1…47†ˆ hdp ;g T m‡n …cp‰m‡1Š d p‰nŠ †i: …48†m ‡ 1As the …m ‡ n ‡ 1†-D sequence is separable, the filteringoperation is much simpler than the usual …m ‡ n ‡1†-dimensional filtering.The complexity in the computation of the moment I m;n isM:…2N 2† …m‡n‡2† , without taking the symmetries intoaccount. Thus, for basis functions with small support andreasonable m and n, the complexity is quite managable.5 COMPUTATION OF THE KERNELIn this section, we propose two schemes for computing thekernel. An exact space domain scheme and an approximateone in the Fourier domain.5.1 Exact MethodIn this scheme, we compute the kernels in space domainmaking use of the properties of kernels discussed before.We start with the computation of f 0 and later extend it tothe general case. Making use of the finite support property,the two-scale relation (34) can be rewritten in the matrixform as,A 0 :f 0 ˆ f 0 ;…49†where A 0 is the square matrix with coefficients ‰A 0 Š k;l ˆH 0 …2k l† and f 0 is the vector whose elements are f 0 …n†. Asthe support of f 0 is ‰N ‡ 1;N 2Š, the indices of A 0 runfrom N ‡ 1 to N 2.It can be seen from (49) that f 0 is an eigen-vector of thematrix A 0 , with eigen-value 1. Solving for f 0 is equivalent tosolving for a vector which falls in the nullspace of …A 0 I†,where I is the identity matrix. Since f 0 6ˆ 0, A 0 must havethe eigen-value 1, which is in general single. This providesf 0 up to a constant which is further set by the normalizationidentityXf 0 …k† ˆ1;…50†kwhich can be seen from (19). This is because the function '…x†has at least an approximation order of one [14], which impliesPk '…x ‡ k† ˆ1. One of the equations in …A 0 I†:f 0 ˆ 0 canbe substituted for by the (50) to yield the system of equationsgiven byB:f 0 ˆ y;…51†B is the matrix obtained by substituting one of the rows of…A 0 I† with the row vector ‰1; 1; ...; 1Š and y is given by‰0; 0; 0...; 0; 0; 1Š T c.f [30]. Now, B is a full rank matrix and,hence, the eigen-vector f 0 can be solved by matrix inversion.To represent the two-scale relations of the higher orderkernels in the matrix form, we introduce a one-to-onefunction : ‰N ‡ 1;N 2Š m 7!‰0; …2N 2† m 1Š. Usingthis function, (34) can be rewritten asf m … 1 …k†† ˆ…2N2†Xm‡1 1lˆ0 H m 2 1 …k† 1 …l† fm 1 …l† ;which is a linear system of equations. This can be written inthe matrix form asA m f m ˆ f m ;…52†where ‰A m Š i;j ˆ H m … 2 1 …i† 1 …j†† and f m …i† ˆf … 1 …i††.This equation is of the same form as (49) and can be solvedinPthe same way, with the normalization constrainti f m…i† ˆ1.Let us now compare our computational solution with themethod developed for computing g m in the context ofwavelet-Galerkin approach [27]. For a scaling function ofsupport N, the kernel g m is zero outside the intervalI 0 ˆ‰N ‡ 1;N 1Š...…53†‰N ‡ 1;N 1Š‰N ‡ 1;N 1Š:as compared to f m whose support is given by (31). Thus, thedirect computation of g m involves a linear system with…2N 1† m variables as compared to …2N 2† m for f m in ourcase. For 3-dimensional kernels involving cubic splines, weachieve a 40 percent reduction in the number of equations.As the computational complexity in inverting a linearsystem is proportional to the third power of the number ofequations, this implies a performance improvement ofaround 5 times. The approach becomes even more rewardingfor higher order kernels. Moreover, the normalizationconstraint (50) that we use to make the system full rank ismuch more straightforward than the corresponding relationfor the derivative functions.Note that this simplification is covered by Dahmen andMichelli's general theory for integrals of multidimensionalscaling functions [28]. This is because the mask of anymth order 1 D scaling function can be always factored asproposed in [28, Corollary 3.3]. In the case of wavelet-Galerikin integrals, the performance improvement can evenmore substantial depending on the number of derivatives.
JACOB ET AL.: AN EXACT METHOD FOR COMPUTING THE AREA MOMENTS OF WAVELET AND SPLINE CURVES 6395.2 Approximate Method for SplinesBecause the spline kernel has a closed-form expression inthe frequency domain, the kernel can be obtained by takingthe inverse DFT of the above mentioned Fourier transform(40) sampled at an appropriate rate; we make use of thefinite support property of the kernel. As sinc is a decayingfunction, the periodization of the Fourier transform may beapproximated with an appropriately truncated sum toachieve any desired accuracy. This is because we can havean upper bound for the error that is a decreasing function ofthe summation range. Moreover, the symmetries of thekernel discussed before may be used for the efficientcomputation of the box spline kernels as in [31].However, this technique, besides being approximate, canbe used only for scaling functions that have a closed formexpression in the frequency domain, i.e., splines in practice.This scheme may be useful to precompute the spline kernelsfor very high order moments, where the exact scheme canbe computationally expensive.6 COMPUTATION OF THE AREA MOMENTS USINGRIEMANN SUMSAn alternate approach to compute the moments is toapproximate the integral (3) by a Riemann sum:1MP1I m;n ˆ…n ‡ 1†P : X‰x int …l=P †Š m :‰y int …l=P †Š n‡1 :‰x 0 int …l=P †Š;lˆ0…54†where P is an appropriate oversampling factor. We show inthis section that this quadrature formula is exact when thecurves are described in a sinc basis. For other representations,it can be used for the approximate computation ofhigher order moments.6.1 Sinc Representation of the CurveA curve represented in a sinc basis also falls into theframework of Theorem 1 because sinc…x† is a valid scalingfunction. However, computing the moments as described inSection 4 is expensive as the mask of the sinc function is notfinitely supported. We remind the reader that the representationof a periodic signal in the sinc basis is equivalentto the Fourier representation as seen in (10).In this particular case, the moments can be computedexactly and more efficiently using (54), where the oversamplingfactor, P , is any integer greater than m‡n‡22.Proposition 1. The quadrature formula (54) is exact for the sincrepresentation provided that P d m‡n‡22e.The continuously defined functions x int …t† and y int …t† areobtained by interpolating the sample values of the curve atthe integers, using the periodized sinc function. Thecomputation is exact because we implicitly assume thatthe functions x…t† and y…t† are bandlimited functions, withbandwidth B ˆ 2.Proof. The integral (3) can be considered as an L 2 …0;M† innerproduct of two functions, which are d m‡n‡22e and b m‡n‡22cfold 2 products of the corresponding band-limited2. bxc and dxe denote the floor and the ceil operators, operating on afraction x to yield the lower and upper integers that bound x.functions. Hence, they are bandlimited by B 0 ˆBd m‡n‡22e and B 00 ˆ Bb m‡n‡22c, respectively. So, thesefunctions are exactly represented in the basis fsinc…Pxk†; 8k 2 Zg, where 2P B 0 . Because the sinc basis isorthogonal, the L 2 …0;M† inner product is equivalent to the`2 …0;MP1† inner product. Hence, it is sufficient tocompute the discrete summation instead of the integral.Finally, the sinc function is interpolating, so that thecoefficients of the basis functions are the resampledcurve values and, hence, the result (54).tuUsing the equivalence of the sinc and the Fourierrepresentations, we can compute the interpolated samplesefficiently with a MP point inverse FFT of the Fouriercoefficients c k and d k .We will compare the sinc moment estimator with thescaling-function-based moment estimator in the next section.One disadvantage of the Fourier(sinc) representationof curves is the loss of local control property that we werehaving with the finitely supported scaling functions.The complexity in the computation of the momentsin this scheme is MP…3 log…MP†‡…m ‡ n ‡ 2††. Here,3MP log…MP† is the cost of the inverse FFT of thesequences c k , d k , and k:c k , and …m ‡ n ‡ 2†MP correspondsto the multiplications.6.2 Spline Representation of the CurveThe quadrature formula (54) is also applicable to the splinerepresentation, provided that the functions x int …t† and y int …t†are obtained by interpolating the integer sample values,using the corresponding B-spline functions. This scheme isno longer exact, but it may be a viable alternative forcomputing the higher order moments. The necessarycondition for the computation to be reliable is that theFourier transform of the B-spline function is essentiallybandlimited to 2P, where P is the oversampling factor.The error in the moments computed with the approximatemethod is, thus, proportional to the residual energy of theB-spline function in the corresponding outband. As theFourier transform of the B-spline is a decaying function ofthe frequency, the error will be a decaying function of P aswell. Thus, any desirable accuracy may be achieved bychoosing P sufficiently large.The complexity of the spline quadrature formula is O…M…m ‡ n ‡ 2†…m ‡ n ‡ 2 ‡ 3N†P†, where M…m ‡ n ‡ 2†P isthe total number of resampled points. The evaluation of thespline representation requires N multiplications to obtainone resampled point from the corresponding B-splinerepresentation. Then, the computation of the discrete sumcosts m ‡ n ‡ 2 multiplications per resampled point. Interestingly,the approximate scheme will give better results forhigher order splines as these functions will becomebandlimited as the order tends to infinity [32].7 EXPERIMENTS AND RESULTSIn this section, we compare the new technique with theexisting ones: approximation using polygons and rasterizing.We first consider the exact scheme proposed inSection 4. We try to estimate the parameters of a known
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