Nonlinear Stationary Subdivision

Transcription

1 Nonlinear Stationary Subdivision Michael S. Floater SINTEF P. O. Box 4 Blindern, 034 Oslo, Norway michael.floater@math.sintef.no Charles A. Micchelli IBM Corporation T.J. Watson Research Center Department of Mathematical Sciences P. O. Box 8 Yorktown Heights, N. Y. U.S.A cam@watson.ibm.com Abstract In this paper we study a concrete interpolatory subdivision scheme based on rational interpolation and show that it preserves convexity.. Introduction There are two ideas which interest us in this paper. The first is to explore concrete possibilities of nonlinear stationary subdivision strategies. The linear case is treated in some detail in [7] where its connection to modeling of curves as well to wavelet construction is highlighted. The second idea that is featured here is that of convexity preserving interpolatory subdivision, as studied, for example, in [6]. One of the most celebrated concrete interpolatory subdivision schemes was introduced in []. This scheme is based upon local polynomial interpolation and is intimately connected to orthonormal wavelet construction. In [8] local exponential interpolation with real frequencies was considered and also shown to lead to a multiparameter family of wavelets of minimal support. Further developments of these ideas appear in [5] and [9]. The paper [9] reveals their applicability to wavelet construction in Sobolev spaces

2 Floater and Micchelli while [5] focuses upon the connection to the construction of conjugate filters with prescribed zeros. In this paper we propose to generate nonlinear interpolatory subdivision by rational interpolation. We study one such example and prove that when the data is convex the interpolant to the data generated by the subdivision scheme is likewise convex. A counterexample to higher order convexity preservation is also presented.. Nonlinear Subdivision In this section we formulate a notion of nonlinear stationary subdivision. All the methods we consider are stationary and homogeneous. This terminology, borrowed from the theory of Markoff chains, means that we always iterate the same operator homogeniety and this operator commutes with shift by an integer stationarity. Stationarity of subdivision must be formulated with care. To be precise, we consider the linear space X of all bi-infinite real sequences x = (x j : j ZZ). On this space acts the forward shift operator T : X X defined by the equation (T x) j = x j+, j ZZ. (.) Let F : X X be any mapping from X into itself. We say F is stationary provided that it commutes with the shift operator T, in other words T F = F T. Every such mapping is determined by one scalar-valued function f : X IR by the formula (F x) i = f(t i x), i ZZ. For instance, if a = (a i : i ZZ) is a bi-infinite vector of compact support, that is a j = 0, j < l or j > m for some integers l and m, and f is given by f(x) = j ZZ a j x j, then the function F (x) becomes the convolution of a with x, that is, (F x) i = j ZZ a j x i j, i ZZ. (.) This is the only linear and stationary mapping.

3 Nonlinear Stationary Subdivision 3 A subdivision operator S h : X X is determined by two scalar-valued mappings h := (h 0, h ) : X IR and is defined by the formula h 0 (T j x), i = j; (S h x) i := (.3) h (T j x), i = j +. If we introduce the bi-infinite sequence (y k (x) : k ZZ/) defined by y k (x) := (S h x) k, k ZZ/ (.4) then it easily follows that y k+ (x) = y k (T x), that is, S h is stationary when the vector in the range of S h is indexed over the fine lattice ZZ/. When each of the functions h 0 and h are linear, in particular, h l (x) = a k+l x k, k ZZ for l {0, }, x = (x k : k ZZ) X and a = (a i : i ZZ) is a prescribed bi-infinite vector of finite support then S h has the familiar form (L a x) i = k ZZ a i k x k. (.5) Similar to the linear case we say that S h converges with respect to some subspace Y X if for every x Y there is a function f x which is continuous on IR such that lim r sup{ (Sr hx) j f x (j/ r ) : j ZZ} = 0 (.6) and for some x Y we have that f x 0. We will now give an example of a nonlinear subdivision scheme generated by local rational interpolation. To this end, let us consider the following problem. Given points x 0, x,..., x n+ we wish to find a rational function R of the form P/Q where P is a polynomial of degree at most n and Q has degree at most one such that R(t j ) = x j, j = 0,,..., n + (.7) where t 0 < t < < t n+ are prescribed. There is a special circumstance which should be considered separately. To this end, for i = 0 or we let w i = [x i, x i+,..., x i+n ], be the divided difference of x i, x i+,..., x i+n at t i, t i+,..., t i+n. Recall that w := w w 0 t n+ t 0 = [x 0, x,..., x n+ ]

4 4 Floater and Micchelli is the leading coefficient of the polynomial of degree at most n+ (one more than we require for P above) which satisfies the interpolation conditions (.7). Hence, when w = w 0 this polynomial has at most degree n and is the solution R to our interpolation problem. There remains the case that w w 0. We only discuss the case that w w 0 > 0. In this case our rational interpolant is given by the formula R(t) = w (t n+ t)p (t) + w 0 (t t 0 )p + (t) w (t n+ t) + w 0 (t t 0 ) (.8) where in this formula p and p + are polynomials of degree at most n which interpolate the n+ data x 0, x,..., x n at t 0, t,..., t n and x, x,..., x n+ at t, t,..., t n+, respectively. It is important to rewrite this function in another form. To this end we let p be the unique polynomial of degree at most n which solves the interpolation problem Proposition.. where p(t j ) = x j, j =,,..., n. w 0 w R(t) = p(t) + (t t ) (t t n ) w 0 λ 0 (t) + w λ (t) λ (t) = t n+ t t n+ t 0, λ 0 (t) = t t 0 t n+ t 0 are the barycentric coordinates of t relative to t 0 and t n+. Proof: Using the Newton form of polynomial interpolation we have that and p + (t) = p(t) + w (t t ) (t t n ) p (t) = p(t) + w 0 (t t ) (t t n ). Substituting these two formulas into equation (.8) and simplifying the resulting expression proves the formula. To motivate the use of this result for subdivision we restrict ourselves to the case that n =. We choose t i = i, i = 0,,, 3, interpolate the data x, x 0, x, x and evaluate the rational interpolant at t = /. This gives us the formula R(/) = x 0 + x 8 H(x x 0 + x, x 0 x + x ) (.9)

5 Nonlinear Stationary Subdivision 5 where H(a, b) is the harmonic mean defined for a, b IR + by the formula ab a+b, (a, b) (0, 0); H(a, b) = (.0) 0, (a, b) = (0, 0). To ensure the validity of formula (.9) we must demand that both x x 0 + x and x 0 x + x are positive. Note the important fact that H(a, b) min (a, b), a, b R +. In formula (.9) we think of the nonlinear term as a perturbation of the linear term. This suggests to us to multiply the nonlinear term by a relaxation parameter and also use this formula to generate the following nonlinear subdivision scheme. Let E : IR IR IR be a given continuous function. For any bi-infinite vector x = (x i : i ZZ) we set for l ZZ x l = x l x l+ + x l+. Choose w IR and define the bi-infinite vector y = (y i : i ZZ) by the formulas y i = x i, i ZZ y i+ = x i + x i+ λe( x i, x i ), i ZZ. (.) This defines a nonlinear subdivision scheme. The special case above would correspond to λ = /8 and E(a, b) := H( a, b ). (.) Hence, when the vector x is convex, that is x i 0, i ZZ, and λ = /8, (.) reduces to (.9). In this case the scheme reproduces rational polynomials whose numerators are of degree at most two and whose denominators have degree at most one. We also remark that when E(a, b) = (a + b), a, b IR and λ = /8 the linear subdivision scheme (.) is obtained by interpolating the data x i, x i, x i+, x i+ at the points i, i, i+, i+, respectively

6 6 Floater and Micchelli by a cubic polynomial and evaluating the resulting polynomial at the point i + /. This is the special case of the interpolatory subdivision scheme introduced in [] Returning to (.) we shall rewrite it in another form. To this end, we introduce the sequence m = (m j : j ZZ) defined by setting m 0 =, m = m = / and m j = 0, j {, 0, }. Then the subdivision scheme (L m x) i = j ZZ m i j x j (.3) corresponds to the linear term in (.) and L m converges with limit f x (t) = j ZZ x j M(t j), t IR (.4) where M(t) = max( t, 0), t IR. The function f x is the piecewise linear function with breakpoints at integers such that f x (j) = x j, j ZZ. Also, we set for x X e(x) = (0, E(x x 0 + x, x 0 x + x )). (.5) This mapping e : X IR determines a nonlinear subdivision scheme and thus our scheme (.) has the form F m (x) = L m (x) λs e (x), x X. When E is nonnegative and λ 0 then formula (.) shows for i ZZ that F m (x) L m (x). For λ and E further constrained the subdivision scheme (.) has the useful property of being convexity preserving. This fact is proved next. We let C = {x : x X, x i 0, i ZZ}. Proposition.. Suppose there is a constant µ > 0 such that for a, b IR + Then for 0 λ 4µ, F m (C ) C. 0 E(a, b) µ min(a, b). (.6)

7 Nonlinear Stationary Subdivision 7 Proof. First, we observe for all j ZZ that ( F m (x)) j = (F m (x)) j (F m (x)) j+ + (F m (x)) j+ and then we verify that ( F m (x)) j = x j { x j + x j+ λe( x j, x j )} + x j+ = λe( x j, x j ) 0 = (F m (x)) j (F m (x)) j + (F m (x)) j+ = x j + x j x j + x j + x j+ λ{e( x j, x j ) + E( x j, x j )} = x j λ{e( x j, x j ) + E( x j, x j )} x j λµ x j = ( 4λµ) x j. Remark.3. When in addition E has the property that E(a, b) > 0 whenever a and b are positive then strictly convex data, x i > 0, i ZZ is mapped into such by the the nonlinear scheme (.). From Proposition. follows the next result. Theorem.4. Let x 0 C, define x r := F r m(x 0 ), r ZZ + and suppose the hypothesis of Proposition. holds. Then the sequence of polygonal lines f r (t) := j ZZ x r j M( r t j), t IR for r ZZ + satisfies the following properties. (i) For all t IR and r ZZ + f r (t) f r+ (t). (ii) For all r ZZ +, f r is convex on IR. (iii) lim r f r (t) = f x (t), t IR, f x is continuous on IR and f x (j) = x j, j ZZ.

8 8 Floater and Micchelli Proof : Since M is the refinable function associated with the linear subdivision scheme L m we have for t IR and r ZZ + that f r+ (t) f r (t) = j ZZ (F r+ m (x 0 ) L m F r m(x 0 )) j M( r+ t j). (.7) But and so (i) follows. F r+ m (x 0 ) = F m (F r m(x 0 )) L m (F r m(x 0 )) (.8) The function f r is convex if and only if x r C and that is assured by Proposition. which takes care of (ii). Finally, considering (iii), for i ZZ let l i : IR IR be the unique linear interpolant satisfying l i (i) = x i and l i (i + ) = x i+ and let b : IR IR be the piecewise linear function defined for t [j, j + ], j ZZ, as b(t) = max(l j (t), l j+ (t)). Due to standard properties of convex functions, all convex interpolants g : IR IR, g(j) = x j, are bounded below by b and therefore the sequence of functions f r is bounded below by b. Hence the limit in (iii) exists and must be continuous since it is convex. Remark.5. For any p > 0 we define E p (a, b) = φ ( φ( a ) + φ( b ) where φ(t) := t p, t IR +. Then for any a, b IR ) 0 E p (a, b) /p min ( a, b ) = /p a b ( a p + b p ) /p (.9) so that this family of means satisfies the hypothesis of Proposition. with µ = /p and so the corresponding scheme produces a convex interpolant to convex data {x j : j ZZ} for 0 λ (+/p). When E(a, b) = (a + b), a, b IR (.0) the hypothesis of Proposition. is not satisfied. This linear subdivision scheme is precisely the one studied in [] in which w = λ/ plays the role of tension parameter. This scheme does not preserve convexity even when it converges. The case of the nonlinear means above for p = corresponding to the harmonic mean was independently considered in [3]. We fell upon it through the process of rational interpolation. This fact seems to have

9 Nonlinear Stationary Subdivision 9 not been noticed in [3] as well as the monotonicity of the polygonal lines embodied in (i) of Theorem.4. Since then we received [4] where similar issues are studied further. We remark that the proof of Proposition. shows that for arbitrary λ 0, property (i) still holds. Hence the subdivision scheme converges for every x X in the sense that lim f r (t) r exists for each t IR but we cannot rule out that the limit function may be at some point and also not continuous. For a more restrictive range of λ we prove convergence in terms of our definition (.6). To this end we define the subspace of X Y = {x : x X, x < }, where x := sup{ x i : i ZZ} for a bi-infinite sequence x = {x i : i ZZ}. Theorem.6. Suppose there exists a constant ρ > 0 such that for all a, b IR E(a, b) ρ max ( a, b ). (.) Then for λ < (4ρ), the subdivision scheme (.) converges with respect to Y and the limit function f x interpolates x j at t = j, j ZZ, that is, f x (j) = x j, j ZZ. Remark.7. When φ is a monotonic function on IR + (either increasing or decreasing) then ( ) φ( a ) + φ( b ) E(a, b) = φ has the property that E(a, b) max { a, b }. In fact, if a b and φ increases then φ also does so that { } { } φ( a ) + φ( b ) φ( b ) φ φ = b. Likewise, if a b and φ decreases then so does φ and so since φ( a ) φ( b ), φ( a ) + φ( b ) φ( b )

10 0 Floater and Micchelli from which we get again Hence (.) converges for λ < 4. E(a, b) b. When E(a, b) = (a + b), (.) is the scheme in []. In this case the equivalent result that the scheme converges for w < 8 is proved there. Alternatively, this linear scheme can be written in the standard form (.5) where a j = 0, j > 3 and a 3 = a 3 = w, a = a = 0, a = a = + w, a 0 =. In this case, the symbol of the scheme is given by a(z) = j ZZ a j z j = + (w + )(z + z ) w(z 3 + z 3 ). When z = e iθ and x = cos θ we have that a(z) = ( + x)( + 8wx( x)). A direct computation confirms that when < w 6, a(eiθ ) 0 for θ π with equality if and only if x =. Hence it follows that the scheme converges for this range, [8]. Moreover, the refinable function f corresponding to the vector x j = 0, j ZZ\{0}, x 0 = which is supported on ( 3, 3) is the autocorrelation of a refinable function φ w which yields an orthonormal wavelet of finite support (for an explanation of wavelet construction see Chapter of [7]). When w = 6 this wavelet is one of the family constructed by Daubechies, [8]. The special case w = 6 corresponds to local cubic interpolation and is a special case of the schemes appearing in []. Proof: According to formula (.7) we have that f r+ f r λ ρ x r where x r = F r m(x) and f := sup{ f(t) : t IR}. Also, returning to the proof of Proposition. we see for j ZZ and r IN that ( x r ) j λ ρ x r (.) and ( x r ) j+ ( + λ ρ) x r. (.3) Combining the inequalities (.) and (.3) we obtain for r ZZ + the inequality f r+ f r λ ρ γ r x 0 (.4)

11 Nonlinear Stationary Subdivision where γ := + λ ρ. Since by hypothesis, γ <, this proves lim r f r = f x uniformly on IR. Next, we shall show f x is Hölder continuous and afterwards that it is the limit of the subdivision scheme in the sense of (.6). For the first claim we note that since f r ( t ) is a piecewise linear function which interpolates r x r j at t = j, j ZZ it follows for r ZZ +, t, s IR that f r (t) f r (s) r x r t s. From the formula (.) and our hypothesis it follows for r IN that Consequently, we have that x r xr + λ ρ x r. x r ( + λ ρ) xr ( + λ ρ)r x 0 and so for r ZZ +, t, s IR we have that f r (t) f r (s) (γ) r x 0 t s. (.5) Combining this inequality with (.4) we conclude with the help of Lemma. of [7], p.8, that f x is Hölder continuous with exponent µ = log γ which is positive since γ <. Thus there is a constant C > 0 such that f x (t) f x (s) C t s µ, for all t, s IR. To finish the proof we note that the function g r (t) = j ZZ f x ( j r )M(r t j), t IR has the property that Hence, we have the bound g r (t) f x (t) µr C, t IR. sup{ x r j f x ( j ) : j ZZ} r sup{ j ZZ(x r j f x ( j r ))M(r t j) : t IR} f r f x + µr C, r ZZ,

12 Floater and Micchelli and sending r proves the result. One may be optimistic that interpolatory subdivision based on rational interpolation with one pole as appears in Proposition. with n 3 would preserve higher order convexity. Unfortunately, this conjecture fails as the next observation indicates. Specifically, instead of the nonlinear scheme of the form (.) we consider, for a given function F : IR IR IR which is symmetric, i.e. F (s, t) = F (t, s), s, t IR, and a constant µ IR, the nonlinear subdivision scheme y i = x i, i ZZ, y i+ = A(x i, x i+ ) 8 A( x i, x i ) + µf ( 4 x i, 4 x i ), where A(s, t) = (t + s), t, s IR. i ZZ. (.6) For the choice µ = 3/8 and F (s, t) = A(s, t) this becomes the linear interpolatory subdivision scheme in [] corresponding to quintic interpolation. For the choice µ = 3/8 and F (s, t) = H(s, t), with H given in (.0), the scheme corresponds to rational interpolation with rational polynomials P/Q where P is quartic and Q linear. When µ = 0, the scheme reduces to that of [] corresponding to cubic interpolation, or the scheme (.) with λ = /8. For ease of notation we set α i = 4 x i and F i = F (α i, α i ), i ZZ in the computations we perform next. Let us now investigate the higher order convexity preservation of this scheme. To this end, we recall that 4 y l = y l 4y l+ + 6y l+ 4y l+3 + y l+4, and therefore we obtain l ZZ 4 y i = 4 α i 4µ(F i + F i+ ), i ZZ (.7) and 4 y i+ = 8 A(α i, α i ) + µ(f i + 6F i+ + F i+ ), i ZZ (.8) We note for the case µ = 3/8 and F (t, t) = t, t IR that whenever x i = p(i), i ZZ, p π 4, and α i = k, we have that 4 y i = k, i ZZ 6

13 Nonlinear Stationary Subdivision 3 which means the scheme reproduces quartic polynomials. If the subdivision procedure (.6) preserves four-convexity, that is, whenever 4 x i 0, i ZZ it follows that 4 y i 0, i ZZ we conclude first from (.7) that for any sequence α i 0, i ZZ we have and then from (.8) that α i 6µ(F i + F i+ ), i ZZ (.9) α i + α i 6µ(F i + 6F i+ + F i+ ), i ZZ. (.30) For any a, b 0, letting α i = a, α i = α i = b in (.9) yields a 3µF (a, b) (.3) and letting α i = α i = a, α i = α i+ = b in (.30) yields a + b 8µF (a, b). (.3) Combining (.3) and (.3) we conclude for a, b 0 that an apparent contradiction. 3. Numerical Examples b 3a, The subdivision scheme (.) was applied to two data sets (i) and (ii), tabulated in Tables and respectively t i x i Table. Data set (i) t i x i Table. Data set (ii) Figures,, 3, 4, and 5 show the result of applying the scheme to data set (i) while Figures 6 and 7 depict the ouput when starting with data set (ii). In Figures and 6 we used the linear scheme in which E(a, b) is the arithmetic mean (.0) and λ = /8. Figures and 7 show the result of the

14 4 Floater and Micchelli rational convexity preserving scheme defined by setting E(a, b) to be the harmonic mean (.) and λ = /8. In Figure 3 the harmonic mean was used again, this time setting λ = /6, thereby increasing the tension. This scheme also preserves convexity. The harmonic mean (.) is equivalent to the function E where E p is defined in (.9). Figure 4 shows the result of applying instead E and this time λ = /(4 ) which is the upper limit of the range of λ for which Proposition. guarantees convexity preservation. Figure 5 on the hand shows the output using E / and λ = /6 which again is the limit of the range in Proposition Fig.. Linear, λ = /8, data (i). Fig.. Harmonic, λ = /8, data (i) Fig. 3. Harmonic, λ = /6, data (i). Fig. 4. E, λ = /(4 ), data (i).

15 Nonlinear Stationary Subdivision Fig. 5. E /, λ = /6, data (i). Fig. 6. Linear, λ = /8, data (ii) Fig. 7. Harmonic, λ = /8, data (ii).

16 6 Floater and Micchelli Acknowledgement Much of this work was done during the Spring of 996 when the authors were visiting the University of Zaragoza. We wish to thank our host, Mariano Gasca, for providing a friendly environment that led to this stimulating scientific exchange. We would also like to thank Narendra Govil for his generous and indispensable help in the preparation of this manuscript. References [] G. Deslauriers and S. Dubuc Symmetric iterative interpolation processes, Constr. Approx. 5 (989), [] N. Dyn, D. Levin and J. A. Gregory, A 4-point interpolatory subdivision scheme for curve design, Computer Aided Geometric Design, 4 (987), [3] F. Kuyt and R. van Damme, Smooth interpolation by a convexity preserving nonlinear subdivision algorithm, preprint 996. [4] F. Kuyt and R. van Damme, Convexity preserving interpolatory subdivision schemes, Memorandum No. 357, University of Twente, The Netherlands, 996. [5] W. Lawton and C. A. Micchelli, Construction of Conjugate Quadrature Filters with Specified Zeros, to appear in Numerical Algorithms. [6] A. Le Méhauté and F. I. Utreras, Convexity-preserving interpolatory subdivision, Computer Aided Geometric Design, (994), [7] C. A. Micchelli, Mathematical Aspects of Geometric Modeling, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol 65, SIAM, Philadelphia, 995. [8] C. A. Micchelli, Interpolatory subdivision schemes and wavelets, J. Approx. Theory 86 (996), 4 7. [9] C. A. Micchelli, On a family of filters arising in wavelet construction, Applied and Computational Harmonic Analysis 4 (997),