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634 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 23, NO. 6, JUNE 2001Consider a closed region V, bounded by a surface S.Green's Theorem states that, for any vector field F,ZZ…r:F† dV ˆ F:dS;…1†Vwhere dS is the unit vector pointing out of the surface S.Assuming the volume has a constant cross-section boundedby the curve C and that the variation of the field along thez-direction is zero, we can restrict the theorem to twodimensions as,Z @F xS @x ‡ @F Iydxdy ˆ …F y dx F x dy†: …2†@yCThe first integral is evaluated over the area S enclosed bythe curve and the second one along the curve C in theclockwise direction. The computation of the momentsinvolves the evaluation of the integral R S xm :y n :dxdy on thesurface bounded by the curve. This, by (2), is equivalent toII m;n ˆCSx m y n‡1n ‡ 1 dx;with F ˆ e y … xm y n‡1n‡1 †; e y denotes the unit vector along they direction. Note that the choice of F is not unique. Wechoose the vector field F that makes the computationsimple. Another possible choice that has the same computationalcomplexity is F ˆe x … xm‡1 y nm‡1 †.2.2 Parametric Representation of a CurveA curve in the x Ð y plane can be represented in terms of anarbitrary parameter t as r…t† ˆ…x…t†;y…t††. If the curve isclosed, as discussed in the paper, the functions x…t† and y…t†are periodic.When the curve C is represented as above, r…t† can beapproximated efficiently as linear combinations of somebasis functions, which makes the representation compactand easy to handle.In this paper, we mainly focus on the representation ofthe function vector r…t† in a scaling function basis asr…t† ˆ X1kˆ1b k '…t k†:Here, b k denotes the sequence of vector coefficients givenby b k ˆ…c k ;d k †. If the period, M, is an integer, we haveb k ˆ b k‡M . This reduces the infinite summations towherer…t† ˆXM1kˆ0' p …t† ˆ X1kˆ1b k ' p …t k†;'…t k:M†:In the context of wavelets, ' is called the scaling function; itsatisfies the two-scale difference equation'…t† ˆXh…k†'…2t k†;…7†k…3†…4†…5†…6†Fig. 1. Example: A polygon and its parametric representation.where h…k† is the mask of the corresponding refinementfilter [14]. The scaling function representation enables us tohave local control of the contour, which is desirable in manyapplications. It also permits a multiresolution representationof the curve [9], [17]. Moreover, the scaling functionrepresentation is affine-invariant; an affine transformationof the curve is achieved simply by transforming thecoefficient vector b k ;kˆ 0; 1; ...;M 1. This is because ofthe linearity of the representation and the partition of unitycondition:X 1kˆ1'…t k† ˆ1;which is satisfied by all valid scaling functions in wavelettheory. Among the scaling functions, a case of specialinterest is ' ˆ n , where n is the causal B-spline of degree n[18] defined by its Fourier transform^ s …!† ˆ 1 s‡1 ej!: …9†j!This yields spline curves which are frequently used incomputer graphics [19] and computer vision [6], [20], [21].We now consider a simple example to illustrate this kindof curve representation. Given in Fig. 1 is a polygon and itsparametric representation. The dotted lines show the linear…8†
JACOB ET AL.: AN EXACT METHOD FOR COMPUTING THE AREA MOMENTS OF WAVELET AND SPLINE CURVES 635B-spline basis functions, 1 …t k† (tent function), multipliedby the corresponding coefficients.The description of C in the scaling function basis isequivalent to a periodized wavelet representation [9]. Thisimplies that, if we have a wavelet description of the curve,the scaling function coefficients at any scale can be obtainedfrom the wavelet coefficients using the fast reconstructionequation described in [22]. Hence, the theory is sufficientlygeneral to include the wavelet curve descriptors as well.The representation of the curves in a sinc basis also fallsin this class, as sinc is a valid scaling function. Thedescription of the curve in the sinc basis as (5) is notefficient, as sinc has an infinite mask unlike most of thewidely used scaling functions. It is well known (c.f. [23])that the sinc interpolation of a periodic signal can beformulated into a numerically stable and efficient expressionasr…t† ˆ XLkˆLb k expj2kt ; …10†Mwhere 2L ‡ 1 ˆ M, assuming M to be odd. A similarexpression is obtained for even M as well. Here, b k is thediscrete Fourier transform of the vector sequence r…k†. Notethat (10) provides the Fourier series description of thecurve, which is frequently used for the representation ofclosed curves [24], [25].2.3 Differentiation of Scaling FunctionsWe will use the property that the kth derivative of a scalingfunction ' can be expressed as [26]' …k† …x† ˆ k ' fkg …x†;…11†where ' fkg …x† denotes the scaling function whose mask isgiven by H fkg 2 k…z† ˆ1 ‡ z 1 H…z†;H…z† is the mask of '. denotes the backward differenceoperator, defined as …x† ˆ…x†…x 1†.The relation (11) follows from the fact that any mth orderscaling function can be written as^'…!† ˆ 1 mej!^…!†;j!|‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚}^ m1 …!†where is a refinable distribution which doesnot satisfythemHpartition ofunity.The mask of ' is H…z† ˆ 1‡z12…z†.m1‡zNote that12 is the mask of m1 and H the mask of .Differentiating ' with respect to x, k number of times(k m) yields' …k† …x† ! F … j! † k ^'…!† ˆ 1 e j! k 1 e j! mk^…!†j!|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}^' fkg …!†!F 1 k ' fkg …x†: …12†Thus, the mask of ' fkg …x† isH fkg …z† ˆ 1 ‡ mk z12 kH …z† ˆ21 ‡ z 1 H…z†:3 COMPUTATION OF THE MOMENTS OF AN AREABOUNDED BY A PARAMETRIZED CURVETo facilitate the understanding of our method, we first givea detailed derivation of the formula for the area of theregion bounded by the curve. We then extend ourformulation to the general case.3.1 Computation of the AreaFor the parametric representation of the curve, the area ofthe region is given byZ MI 0;0 ˆ y…t† dx…t† dt:…13†0 dtWhen the curve is described in a scaling function basis as in(5), we havewhereXM1 Z MI 0;0 ˆ d i c ji;jˆ00' p …t i†' 0 p …t j†dt; …14†' 0 p …t† ˆd' p…t†:dtSubstituting for ' 0 p…t† from (6), we getXM1 Z 1I 0;0 ˆ d i c j ' p …t i†' 0 …t j†dt;i;jˆ0which is equivalent to1…15†XM1 Z 1I 0;0 ˆ d i c j ' p …t i ‡ j†' 0 …t†dti;jˆ0 1|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}g p 0 …ij† : …16†Again, substituting for ' p from (6), we get the kernel g p 0…l† asthe M periodized version ofg 0 …l† ˆZ 11' 0 …t†'…t l†dt…17†as g p 0…l† ˆP1kˆ1 g 0…l ‡ k:M†. With the simplification (11),the above equation becomesandf 0 …x† ˆg 0 …l† ˆf 0 …l†;Z 11' f1g …t†'…t x†dt:…18†…19†Note that, if '…t† ˆ'… t†, then f 0 …x† can be written as the convolution ' f1g ' … ‡ x†. We prefer to represent thekernel g p in terms of f due to its nice properties, discussedlater.
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