An exact method for computing the area moments of ... - IEEE Xplore

636 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 23, NO. 6, JUNE 2001Z 1Substituting for ' 0 p …t† from (6), we get Both these properties together imply 2……m ‡ n ‡ 1†!†relations, which are used to accelerate the computationFor the example given in Fig. 1, we haveI m;n ˆ 1 XXc k c ‰mŠ ‰n‡1Ši d j ' p …t i 1 † ...g 0 …l† ˆ 0 1 n ‡ 1…l ‡ 2† ˆ… 2 k2R i2R m‡11…25†j2R†…l ‡ 2†g 0 : …0:5; 0; 0:5†; l 2f1; 0; 1g;' p …t i m †' p …t j 1 † ...' p …t j n‡1 †' 0 …t k†dt:where n is the causal B-spline function of degree n. The integral in the above equation is equivalent toRNow, for the polygon, c…k† : …1; 1; 6; 8; 7; 4† and d…k† :11 '0 …t†' p …t‡ki 1 †...' p …t‡ki m †' p …t‡kj 1 †...' p …t‡kj n‡1 †dt : …26†|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}…1; 6; 8; 5; 1; 0†. Hence, by (16), we haveg p m‡n …ik;jk†I 0;0 ˆ 1 h…6; 8; 5; 1; 0; 1†; …5; 7; 1; 4; 6; 3†iHence, the …m; n†th order moment is2ˆ 42 units:I m;n ˆ 1 XXc k c ‰mŠ i d ‰n‡1Š j g pn ‡ 1m‡n…i k; j k†: …27†k2R i2RHere, hx 1 ;x 2 i stands for the `2 inner product given byj2RPk x 1…k†x 2 …k†.tu3.2 General FormulaAs in the case of the area, the kernel g p is obtained by theHaving shown how to compute the area, we proceed on to M-periodization ofZthe general case. The formula for the computation of the1g m‡n …k† ˆ ' 0 …t†'…t k 1 †::'…t k m‡n‡1 †:dt;general moments are given by the following theorem:1…28†Theorem 1. Let C be a closed curve in the x-y plane represented where k 2 Z m‡n‡1 . Expressing ' 0 in terms of ' f1g , we getin the parametric form in a periodized scaling function basis asg m‡n …k† ˆf m‡n …k†f m‡n …k 1†; …29†(4). Then, the …m; n†th order area moment of the region S,wherebounded by the curve C, given byZZ1I m;n ˆ x m y n fdxdy for m; n 0 …20† m‡n …x† ˆ ' f1g …t†'…t x 1 †::'…t x m‡n‡1 †dt; …30†1Swhere x ˆ…xcan be computed as1 ;x 2 ...;x m‡n‡1 †2 R m‡n‡1 . The kernel f hasmany interesting properties, which are discussed next.I m;n ˆ 1 XXc k c ‰mŠ i d ‰n‡1Š j g pn ‡ 1m‡n…i k; j k†; …21† 3.3 Properties of the KernelÐfk2R i2R m‡1j2R nwhere R is the integer range ‰0...M 1Š. The kernel g p m‡n in(21) isZ 11. Finite Support. As the kernel is an integral ofproducts of the translates of finitely supportedfunctions, it has a finite support as well. If thescaling function is continuous and has a supportg p m‡n…k† ˆ ' 0 …t† ' p …t k 1 † ...' p …t k m‡n‡1 † dt: …22† ‰0;NŠ, then the kernel will be supported on the1integer points in the intervalHere, c ‰mŠ stands for the m-times tensor product 1 c I ˆ‰N ‡ 1;N 2Š...c ... c and i k denotes the sequence‰N ‡ 1;N 2Š‰N ‡ 1;N 2Š:…31†…i 1 k; i 2 k; ...i m‡1 k†:Proof. For a parametric curve, the evaluation of the …m; n†thorder moment given by (20) can be reduced to2. Symmetry. The fact that the kernel is obtained fromthe integration of similar translated scaling functionsintroduces a lot of symmetry. As (30) is symmetricI m;n ˆ 1 Z Mx m …t†y n‡1 …t† dx…t†with respect to the parameters k 1 ;k 2 ;::, interchangingthem will not affect the value of the kernel. Thisdt …23†n ‡ 1 0dtimpliesby (3). When the curve is described in a scaling functionf…k† ˆf… i …k††;…32†basis, we havewhere i indicates all possible …m ‡ n ‡ 1†! permutationoperators. In addition, if the scaling functionsI m;n ˆ 1 XXZ Mc k c ‰mŠ ‰n‡1Ši d j ' p …t i 1 † ...n ‡ 1k2R i2R m‡10are symmetric as in the case of splines, we have…24†j2R n' p …t i m †' p …t j 1 † ...' p …t j n‡1 †' 0 p …t k†dt:f…k† ˆf…k†:…33†1. c ‰0Š is defined as the neutral element c ‰0Š c ‰mŠ ˆ c ‰mŠ .of the kernel as well as themoments.