Approximation of the Distance from a Point to an Algebraic Manifold Alexei Yu. Uteshev and Marina V. Goncharova Faculty of Applied Mathematics, St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia Keywords: Algebraic Manifold, Distance Approximation, Discriminant, Level Set. 2 Abstract: The problem of geometric distance d evaluation from a point X0 to an algebraic curve in R or manifold 3 G(X) = 0 in R is treated in the form of comparison of exact value with two its successive approximations d(1) and d(2). The geometric distance is evaluated from the univariate distance equation possessing the zero 2 set coinciding with that of critical values of the function d (X0), while d(1)(X0) and d(2)(X0) are obtained 2 via expansion of d (X0) into the power series of the algebraic distance G(X0). We estimate the quality of approximation comparing the relative positions of the level sets of d(X), d(1)(X) and d(2)(X). 1 INTRODUCTION Approximation (2) is known as Sampson’s dis- tance (Sampson, 1982). We aim at comparison of We treat the problem of Euclidean distance evaluation the qualities of approximations (2) and (3). We deal from a point X0 to manifold defined implicitly by the mostly with the case of algebraic manifolds (1), i.e. equation G(X) 2 R[X]. For this case, the tolerances of the ap- G(X) = 0 (1) proximations can be evaluated via comparison with n the true (geometric) distance value determined from in R ;n 2 f2;3g. Here G(X) is twice differentiable real valued function and it is assumed that the equa- the so-called distance equation (Uteshev and Gon- n charova, 2017; Uteshev and Goncharova, 2018). In tion (1) defines a nonempty set in R . This prob- lem arises in image processing, multi-object move- Section 2, we outline the background of this approach ment simulation, and in the scattered data approxima- for the case of a quadric manifold, while in Section 3 tion problems. Being the problem of nonlinear con- we extend it to the case of manifold of an arbitrary strained optimization, it can be solved with the aid order. In Section 4 we discuss an applicability of the of traditional Newton-like iteration methods. How- proposed approximations to the case of non algebraic ever for the applications connected with the parame- curve (1). ter synthesis such as, for instance, the manifold selec- tion best fitting to the given data set (Ahn et al., 2002; Aigner and Jutler, 2009; Cheng and Chiu, 2014), an 2 QUADRICS analytical representation is needed for the distance as a function of parameters of the problem (point coor- For the particular case of quadric polynomials G(X): dinates and coefficients of G(X) if the latter is a poly- G(X) := X>AX + 2B>X − 1 = 0; (4) nomial). > n×n n We concern here with the following approxima- where A = A 2 R and fX;Bg ⊂ R are the col- tions for the distance d umn vectors, formula (2) is represented as d = jGj=k∇Gk; (2) 1 jG(X0)j (1) d(1) = · : (5) 2 p(AX + B)>(AX + B) ∇G> · H (G) · ∇G 0 0 d = d 1 + G (3) (2) (1) 2k∇Gk4 The counterpart for the formula (3) can be originated from the following approximation Here ∇G stands for the gradient column vector, > — s 1 (AX + B)>A(AX + B) for transposition, H (G) is the Hessian of G(X), k · k d = d + 0 0 G(X ) e(2) (1) 1 > 2 0 ; is the Euclidean norm, and the right-hand sides in (2) 2 [(AX0 + B) (AX0 + B)] and (3) are calculated at X = X0. (6) 715 Uteshev, A. and Goncharova, M. Approximation of the Distance from a Point to an Algebraic Manifold. DOI: 10.5220/0007483007150720 In Proceedings of the 8th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2019), pages 715-720 ISBN: 978-989-758-351-3 Copyright c 2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved ICPRAM 2019 - 8th International Conference on Pattern Recognition Applications and Methods suggested in (Uteshev and Goncharova, 2018). For a0 6= 0 , b0 6= 0; then this determinant equals this particular case, both approximations d and de (1) (2) a0 a1 a2 a3 a4 0 0 can be deduced via the following consideration. First 0 a0 a1 a2 a3 a4 0 compute the distance equation, i.e. an algebraic 0 0 a0 a1 a2 a3 a4 equation F (z) = 0;F (z) 2 [z] whose roots coin- R 0 0 0 b0 b1 b2 b3 cide with the critical values of the squared distance 0 0 b0 b1 b2 b3 0 function from X0 to (4), and, generically, the small- 0 b0 b1 b2 b3 0 0 est positive root of this equation equals d2. For the b0 b1 b2 b3 0 0 0 case of an ellipse G(x;y) := x2=a2 + y2=b2 − 1 = 0 and X0 = (x0;y0), this equation takes the form while for the general case it is composed similarly from degg rows of coefficients of f (x) and deg f rows 2 4 F (z;x0;y0) := L z (7) of coefficients of g(x). The resultant is a polynomial function of the coefficients of f (x) and g(x), and its vanishment yields the necessary and sufficient con- −2L L(a2 + b2 + x2 + y2) + a2y2 − b2x2 z3 0 0 0 0 dition for the existence of a common zero for these polynomials. 0 4 2 2 4 4 2 2 4 2 2 2 2 For the particular case g(x) ≡ f (x), the expression + 6L[a y0 + a y0 − b x0 − b x0 + L(a b + x0y0)] 0 Dx( f (x)) := Rx( f ; f )=a0 +[L2 − (a2x2 + b2y2)]2 z2 − 2a2b2 a2b2MG2 defines (up to a sign) the discriminant of the polyno- 0 0 0 mial f (x). Its vanishment yields the necessary and sufficient condition for the existence of a multiple 2 2 2 2 2 4 4 zero for f (x). − (a + b )M + 3a b M − 6a b S4 G0 Theorem 1. Let G(X0) 6= 0. Distance from X0 to the quadric (4) equals the square root from the minimal 2 2 2 4 4 2 2 2 2 +2a b M S4 z + a b G0 M + 4a b G0 = 0: positive zero of the distance equation (z) := (F(µ;z)) = 0; (8) 2 2 F Dµ Here L := a − b , G0 := G(x0;y0), where 2 2 2 2 2 4 2 4 M := x0 + y0 − a − b ; S4 := x0=a + y0=b : A B −I X F(µ;z) := det + µ 0 B> − X> z − X>X Represent the squared distance value as the formal se- 1 0 0 0 ries provided that this zero is not a multiple one. Here 2 2 3 2 n×n is the identity matrix. d (x0;y0) = `2G0 + `3G0 + ::: I R In (Uteshev and Goncharova, 2018) error esti- in powers of the algebraic distance G0 (which can be treated as a small parameter in a vicinity of the mations for the approximations (5) and (6) are de- considered ellipse) and substitute it into the distance duced in terms of maximal deviations of the mani- equation. Equate to zero the coefficients of G2 and folds d(1) = const and de(2) = const from the quadric 0 (4). G3. This results in the approximations (5) and (6). 0 2 2 2 2 Further expansion of the radical in (6) in power se- Example 1. For the ellipse x =18 + y =5 = 1, ries of G0, yields an analogue of approximation (3). approximate equidistant curves d(1)(x;y) = 3 and Similar distance equation of the degree 6 in z can be d(2)(x;y) = 3 in comparison with the true equidistant written down for the point-to-quadric problem in R3. d(x;y) = 3 are presented in Fig. 1 and 2 respectively. n The general expression (valid for a quadric in R for arbitrary n) can be represented via a special function of the coefficients of a polynomial known as the dis- 3 GENERAL ALGEBRAIC criminant. We first remind a more general notion. For MANIFOLDS the univariate polynomials f f (x);g(x)g ⊂ R[x] their For the general case of algebraic manifold, the con- resultant Rx( f ;g) can be defined in the form of Sylvester’s determinant (Uspensky, 1948). For in- struction of distance equation is also possible, at least, stance, if in principle. For the planar case, this construction is based on the following results (Uteshev and Gon- 4 3 f (x) := a0x + ··· + a4; g(x) := b0x + ··· + b3; charova, 2017): 716 Approximation of the Distance from a Point to an Algebraic Manifold Theorem 2. Let G(0;0) 6= 0 and G(x;y) be an even the minimal absolute value of real zeros of the equa- polynomial in y. Expand G in powers of y2 and de- tion G(x;0) = 0 or the square root from the minimal 2 note Ge(x;y ) ≡ G(x;y). Equation G(x;y) = 0 does positive zero of the equation F1(z) = 0 provided that not define a real curve if this zero is not a multiple one. (a) equation G(x;0) = 0 does not have real zeros Remark 1. The conditions (a) and (b) of Theo- and rems 2 and 3 (as well as their counterparts from the (b) equation undermentioned Theorem 5) can be verified without utilization of any numerical method.