- Email: [email protected]
Integral representation of local and global diffeomorphisms
Nonlinear Analysis: Real World Applications 4 (2003) 203 – 221 www.elsevier.com/locate/na
Integral representation of local and global di$eomorphisms Lubomir Dechevsky∗ Centre of Applied Mathematics, Lulea University of Technology, S-97187 Lulea, Sweden Received 3 November 1999; accepted 7 August 2001
Abstract The identi.cation of a smooth invertible map between two closed domains in Rn is of great importance in neuroimaging and other .elds where the domains of images must be nonlinearly transformed so as to align important features. The theory for locally di$eomorphic maps is classic, but much less is known about global di$eomorphisms between closed simply connected sets. These bijectively map interiors onto interiors and boundaries onto boundaries. We use this property and the integral of the exponential map to construct representations of global di$eomorphisms for star-shaped domains. As a simple example, details are provided for maps in two dimensions. Applications are outlined for three types of inverse problems related to image registration, di$eomorphic splines, and dynamical systems. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Locally invertible continuously di$erentiable surjection; Smooth bijection; Taylor formula with integral remainder for operators; Exponential map; Banach space; Small perturbation
1. Introduction Recent results on the interface of computer graphics with medical imaging require the selection of an optimal di$eomorphism within a large class of admissible smooth mappings which are invertible between two .xed subsets of Rn (n = 1; 2; 3). This requires the development of representation theory for the di$eomorphisms acting on a “reference” subset of Rn . In this connection, the only relevant general result known to us belongs to Hadamard [5] (see also [12]), and it treats the simple case without boundary when the reference subset is Rn itself. Here we propose a di$erent approach to that of Hadamard [5] for the study of di$eomorphisms where the reference subset is ∗
Corresponding author. E-mail addresses: [email protected], [email protected] (L. Dechevsky).
1468-1218/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 4 6 8 - 1 2 1 8 ( 0 2 ) 0 0 0 0 5 - 6
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L. Dechevsky / Nonlinear Analysis: Real World Applications 4 (2003) 203 – 221
a star-shaped domain in Rn or Cn . As an intermediate stage, we obtain representations for the smooth locally invertible surjections, or local di$eomorphisms between subsets of Rn or Cn . We were motivated to study these local and global di$eomorphisms between subsets of Rn by problems originating in functional data analysis. This type of statistical analysis di$ers from conventional multivariate analysis in that the observations are curves or surfaces rather than vectors. An important preliminary step is often the registration or alignment of essential features of the curve or surface by suitable “smooth” bijections of the argument(s). The problem of transforming the arguments of curves so as to align various salient features has a very large literature in many di$erent .elds. The problem is referred by Silverman [20], Ramsay and Li [17] and Kneip et al. [10] as curve registration; the engineering literature uses the term time warping [19,25], and the process of registering curves for the purpose of computing average curves is called structural averaging by Kneip and Gasser [8,9]. Marker registration is the process of aligning curves by identifying the timing of certain salient features in the curves, of which the zero of acceleration during the pubertal growth spurt and optimal temperature timings are examples. However, marker registration of curves or surfaces presents some problems, especially if markers are hard to identify. Silverman [20] developed a technique for curve registration that does not require explication of markers by optimizing a global .tting criterion with respect to a parametric family of transformations or time warpings. Ramsay and Li [17] applied an arbitrarily Hexible yet computationally convenient smooth monotone transformation family developed by Ramsay [16], and thus gave a nonparametric curve registration approach. Local linear techniques for this problem were developed in [10]. Recently, however, the problem of registering surfaces and volumes has become especially important in medical imaging of the brain [2]. This higher dimensional problem is much more diIcult than the one-dimensional curve registration problem because monotone and bijective are no longer synonymous. We have paid careful attention to the unidimensional case as presented in [12,15– 17]. However, as the range of a real-valued functional is a fully ordered set, while the range of a vector-valued mapping is only partially ordered, some of the key results on real-valued functions are no longer valid for vector-valued mappings (see e.g. [13,18]) and it became necessary to develop di$erent methods here. One possible approach would be to both characterize monotone “smooth” additive transformations between subsets of Rn and to suggest a computationally convenient method for their estimation with a view to their use in the problem of surface registration. However, the additivity assumption can be quite restrictive so we prefer here to study representations of locally invertible surjections as a .rst step in the registration problem. For the mappings to be represented, continuous di$erentiability will be assumed in the sense of [18, 9.11, 9.15], [22, Chap. I, Sections 4.3–5]. This ensures that the Gateau and Frechet derivatives coincide (see [22, Theorem 2.1]). In particular, this coincidence holds in the .nite dimensional case F : ⊂ Rn → Rl or F : ⊂ Cn → Cl if all partial derivatives (@F =@x )(x0 ), = 1; : : : ; n, = 1; : : : ; l, exist and are continuous at x0 ∈
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(see [18, 9.16], [23, Append. IV]). In this paper, m times continuously di$erentiable mappings, m ¿ 1, will also be called “smooth” mappings or “C m -mappings”, for short. Continuous mappings will also be referred to as “C 0 -mappings”. In our construction of di$eomorphism representations we will make use of: the Taylor expansion with integral remainder of Frechet-di$erentiable mappings; the fact that C∞ is dense in Cm ; m ∈ N (see, e.g., [1]); the exponent mapping eT ; T ∈ L(X ) (L(X )) being the space of linear bounded operators on the Banach space X where usually X will be Rn or Cn ; and the following Theorem A (Cf., e.g., [6]). Let be a simply connected open set in Rn or Cn , n ∈ N, M is continuous on the closure M and is a C 1 -mapping and assume that F : M → F() on . Let N0 ={x ∈ : det(F (x))=0}, where F denotes the Frechet derivative of F on . Assume that N0 = ∅ or N0 is an isolated subset of and that the restriction F|@ is bijective between the boundary @ and F(@). Then, F is bijective between M M and F(). For ease of presentation, we shall develop in detail the theory of C 1 -di$eomorphisms, and then we shall revisit certain aspects of this theory in the more general context of C m -di$eomorphisms. The organization of the paper is as follows. In Section 2 we derive an integral representation for the elements of C 1 and discuss the exponent mapping. Section 3 contains the main results about the integral representation of local di$eomorphisms. We also derive integral representation theorems for global di$eomorphisms on open and closed star-shaped domains in Cn and Rn in terms of boundary mappings. The essence of these results is that we reduce the dimension of the representation problem by passing from the domain to its boundary. In Section 3.3 we show how this can be used to completely solve by induction the problem of representation of the global di$eomorphisms on the closed unit ball in Rn . In this way we construct explicitly, as a useful example, the representation in the case n = 2. We also brieHy discuss the case of C m -di$eomorphisms, m ¿ 1. In Section 4 we discuss some applications and we suggest solving ill-posed problems in brain imaging by Tikhonov regularization. We also consider the concept of “di$eomorphic splines”, and an application to the study of dynamical systems. Some concluding remarks can be found in Section 5. 2. Preliminary results 2.1. Integral representation of smooth mappings The integral-remainder Taylor expansion is valid for star-shaped sets. The set ⊂ X , X = Rn or X = Cn , is called star-shaped with respect to x0 ∈ (“with center x0 ” for short) if for every x ∈ and every s ∈ [0; 1] the convex combination (1 − s)x0 + sx is also in . A necessary and suIcient condition for this is, if for every x ∈ there exist ∈ and s ∈ [0; 1] with x = (1 − s)x0 + s, or, equivalently, for every x ∈ Rn such that there exist ∈ and s ∈ [0; 1] with x = (1 − s)x0 + s, it follows that x ∈ . A
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convex set is star-shaped with respect to every x0 ∈ . In particular, when n = 1, the notions of a convex and a star-shaped set coincide. A non-void simply connected open set 1 ⊂ X will be called an open domain; its closure M 1 in the topology of X will be called a closed domain. A set ⊂ X will ◦ be called a domain if its closure M is a closed domain. Then its open interior is an ◦
◦
open domain; @ := M \ is the boundary of ; clearly ⊂ ⊂ M holds. Let ⊂ X be a domain and Y = Rl or Y = Cl ; l ∈ N. We de.ne the space 0 C = C 0 (; Y ) of all C 0 -mappings F : → Y such that
F C 0 (; Y ) = sup F(x) Y ¡ ∞ x∈
and the space C 1 = C 1 (; Y ) of all C 1 -mappings F : → Y for which
F C 1 () = F C 0 (; Y ) + sup F (x) L(X; Y ) ¡ ∞:
(1)
x∈
Here L(X; Y ) is the space of all bounded linear operators between X and Y , L(X ) := L(X; X ): (Since in .nite-dimensional space all norms are equivalent, we shall be assuming in this case that the norms · X and · Y are the usual Hilbert norms.) We prove the following integral representation of the elements of C 1 . ◦
Theorem 1. Let be a bounded star-shaped domain with centre x0 ∈. Then, F ∈ C 1 (; Y ) if and only if there exists a unique pair (f0 ; ), f0 ∈ Y , ∈ C 0 ◦
(; L(X; Y )), such that 1 F(x) = f0 + (x0 + t(x − x0 ))(x − x0 ) dt; 0
x ∈ ;
(2)
holds, with f0 = F(x0 ); ≡ F
◦
on ;
(3)
where the integral in (2) is the Riemann integral for vector-valued functions (see, e.g., [4, Chap. III, Section 5], [22, Chap. I, Sections 4.3–5]). Proof. The “only if ” part follows immediately from the Taylor integral-remainder expansion of the elements of C 1 . Namely, if F ∈ C 1 (; Y ), then 1 F(x) = F(x0 ) + F (x0 + t(x − x0 ))(x − x0 ) dt: (4) 0
To prove the “if ” part, we .rst assume that is smooth enough, namely, ∈ C 1 (; L(X; Y )). Now we apply Frechet di$erentiation to both sides of (2). Since [0; 1] is compact, the integrand and its Frechet derivative are uniformly continuous and Frechet di$erentiation commutes with integration in t. Here ∈ C 0 (; L(X 2 ; Y )), where L(X 2 ; Y ) is the space of bounded bilinear operators from the Cartesian product
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X 2 to Y , endowed with the uniform operator norm. We obtain 1 F (x) = [t (x0 + t(x − x0 ))(x − x0 ) + (x0 + t(x − x0 ))] dt 0
=
1
t d(x0 + t(x − x0 )) +
0
= [t(x0 + t(x − +
1
0
= (x);
x0 ))]|10
−
1
0 1
0
(x0 + t(x − x0 )) dt
(x0 + t(x − x0 )) dt
(x0 + t(x − x0 )) dt x ∈ :
The equality f0 = F(x0 ) coincides with (2) for x = x0 . This proves (3) and the “if ” part of the theorem in the partial case when is suIciently smooth. To prove the general case, we shall .rst show that the mapping 1 S = Sx : ∈ L(X; Y ) → (x0 + t(x − x0 ))(x − x0 ) dt ∈ Y 0
is continuous, uniformly in x ∈ . Indeed, by the triangle inequality, 1
Sx (1 ) − Sx (2 ) Y = [1 (x0 + t(x − x0 )) − 2 (x0 + t(x − x0 ))] 0 (x − x0 ) dt 6
0
Y
1
[1 (x0 + t(x − x0 )) − 2 (x0 + t(x − x0 ))]
(x − x0 ) Y dt 6 diam() 1 − 2 C 0 (; L(X; Y )) : From here, and from the obvious continuity of the identities IX and IL(X; Y ) in the topologies of X and L(X; Y ), respectively, the “if ” part follows for general ∈ C 0 (; L(X; Y )) by the fact that C 1 (; L(X; Y )) is dense in C 0 (; L(X; Y )). 2.2. The exponent mapping on L(X ) Assume .rst that X is Banach space over C. Analytic functions : L(X ) → L(X ) can be de.ned via Dunford’s representation [4, Chapter VII]: 1 f(T ) = f()(IX − T )−1 d; T ∈ L(X ); (5) 2 i @U where U ⊂ C is open (not necessarily connected), such that the spectrum (T ) of T is compactly contained in U, the boundary @U being a Jordan recti.able curve. Here (IX − T )−1 is the resolvent of T .
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The mapping exp z = ez , z ∈ C, is an entire analytic function; therefore, it is analytic in a neighbourhood of (T ) for any T ∈ L(X ) and (5) can be made meaningful for any T ∈ L(X ) by a suitable choice of the contour @U. The analytic expansion ∞ Tk eT = (6) k! k=0
has radius of convergence ∞. Of course, (eT )−1 = e−T ; however, eT1 +T2 = eT1 eT2 is not true in general, unless T1 and T2 commute. The mapping ez is a 2 i-periodic surjection from C to C \ {0}, and a bijection from D0 = {z ∈ C : − ¡ Im z 6 } to C \ {0}. The following lemma will be essential for the representation of local and global di$eomorphisms. ˜ ), where L(X ˜ ) denotes the Lemma 1. (a) T → eT is a surjection from L(X ) to L(X open subset of L(X ) consisting of all invertible elements of L(X ) (see [4, Chapter VII, Section 6.1; 18, 9.8]). ˜ ). (b) T → eT is a bijection from {T ∈ L(X ); (T ) ⊂ D0 } to L(X ˜ ), and that, Proof. In part (a) we have to prove that, for every T ∈ L(X ); eT is in L(X ˜ ) there exists a (not necessarily unique) T ∈ L(X ), such that S = eT . for every S ⊂ L(X The .rst of these two assertions follows from the identity eT e−T = e−T eT = IX . The second assertion follows from part (b). To prove part (b) we note that the inverse ˜ ), then 0 ∈ (S), function ln z of ez is a bijection between C \ {0} and D0 . If S ∈ L(X and, since (S) is a compact in C, there exists a neighbourhood U containing (S) such that 0 ∈ U. Then, ln S = (1=2 i) @U ln · (IX − S)−1 d, and T = ln S is such that eT = S, which implies the surjectivity part of (b) and completes the proof of (a). To prove the injectivity part of (b), assume that Tj ∈ L(X ); j = 1; 2, with (T1 ) ∪ (T2 ) ⊂ D0 , and that T1 = T2 in L(X ). If ( (T1 ) ∪ (T2 )) \ ( (T1 ) ∩ (T2 )) = ∅, then, since for any analytic function on (T1 ) ∪ (T2 ) the identity (f(T )) = f( (T )) holds, and since ez is analytic and bijective from D0 to C \ {0}, we obtain ( (eT1 ) ∪ (eT2 )) \ ( (eT1 ) ∩ (eT2 )) = ∅, hence, eT1 = eT2 in L(X ). It remains to consider the more diIcult case when the sets (T1 ) and (T2 ) coincide, but T1 and T2 have di$erent invariant subspaces corresponding to the elements of their common spectrum. Assume again that T1 = T2 in L(X ), i.e., T1 − T2 L(X ) = 0; denote Sj = eTj ; j = 1; 2, and suppose that S1 − S2 L(X ) = 0. Then, by Hilbert’s identity for resolvents and the triangle inequality, 1 −1 −1
T1 − T2 L(X ) = ln [(IX − S1 ) − (IX − S2 ) ] d 2 @U L(X ) 1 −1 −1 = ln [(IX − S1 ) (S1 − S2 )(IX − S2 ) ] d 2 @U L(X ) 2
6
1
ln(·) H1 (@U) max (IX − Sj )−1 L(X ) S1 − S2 L(X ) : ∈@U 2 j=1
(7)
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Here (T1 ) = (T2 ) ⊂ U ⊂ D0 , and U can be chosen bounded, because (T1 ) is compact. Hence, @U can be chosen recti.able and of .nite length. So chosen, @U is also a compact in C. The space Hp (@); 1 6 p 6 ∞, is the Hardy space on @U and its norm is denoted by · Hp (@U) . Then, ln(·) H1 (@U) ¡ ∞ holds, because ln(·) ∈ H∞ (@U) and the length of @U is .nite. Now, max∈@U (IX − Sj )−1 L(X ) ¡ ∞ holds, because
(IX −Sj )−1 L(X ) is continuous in and @U is compact. Hence, by (7), T1 −T2 L(X ) = 0, which is a contradiction. This proves the injectivity part of (b) and completes the proof of the lemma. Now let us consider the case when X is a Banach space over the reals. In this ˜ ) is not connected, but is the union of two case there is one essential di$erence: L(X ˜ disjoint connected open sets L+ (X ) and L˜− (X ) such that ±IX ∈ L˜± (X ). Thus, the ˜ ) exponent mapping represents only the elements of L˜+ (X ); to represent the whole L(X when X is a real Banach space, an additional orientation parameter & = ±1 must be introduced. 3. Main results In this section X = Rn or Cn and ⊂ X is star-shaped bounded domain with centre ◦
x0 ∈. 3.1. Integral representation of local C 1 -di9eomorphisms We consider those F ∈ C 1 (; X ) which are local di$eomorphisms, that is, they are ◦
locally invertible from to subsets of X , i.e., F (x)−1 exists in L(X ) for every x ∈. Global di$eomorphisms from to subsets of X are included, of course, but the class of local di$eomorphisms is essentially larger [18, Chap. 9, Problem 12]. Corollary 1. F ∈ C 1 (; X ) is a local C 1 -di9eomorphism from to a subset of X if and only if there exists a couple (f0 ; G); f0 ∈ X; G ∈ C 0 (; L(X )), such that 1 eG(x0 +t(x−x0 )) (x − x0 ) dt; x ∈ (8) F(x) = f0 + 0
and F(x0 ) = f0 ; F ≡ eG n
on :
(9)
G
When X = R , in (8), (9) e must be factored by the orientation parameter & = ±1. If (G(x)) ⊂ D0 for every x ∈ , then representation (8) and (9) is 1–1. Proof. The proof follows from part (a) of Theorem 1 and Lemma 1. The integral representation (8) and (9) has been previously proposed by Ramsay [15–17] for representing global di$eomorphisms in the case X = R.
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From the point of view of the theory of Lie groups and Lie algebrae, Eq. (8) provides a transparent connection between the elements F of a global group of local di$eomorphisms and the elements G of the Lie algebra of this group. If the group is local in the sense of classical Lie theory [11,14] (that is, if only those F in a suIciently small neighbourhood of the restriction of IX onto are considered, so that G is close enough to OL(X ) on for the equality eG = IX + G to hold with suIcient accuracy), then, for the case X = Cn , (8) is linearized to 1 F(x) = x + (F(x0 ) − x0 ) + G(x0 + t(x − x0 ))(x − x0 ) dt; x ∈ ; (10) 0
with the additional orientation factor & = ±1 in the case of X = Rn . 3.2. Integral representation of local and global C 1 -di9eomorphisms on star-shaped domains Our aim in this subsection is to derive an integral representation for the global di$eomorphisms from an open (closed) star-shaped domain ⊂ X onto another open (closed), not necessarily star-shaped, domain ) ⊂ X . To achieve this, we .rst note that every global di$eomorphism F : → F() = ) can be extended by continuity to a ◦ ◦ M which maps surjectively M onto ), M onto ) and @ onto @). M de.ned on , mapping F, In fact, the last property is possessed, more generally, by any open homeomorphism (a mapping is called open, if it maps open sets onto open sets). Local di$eomorphisms F are certainly continuous; by the inverse function theorem (see [3], [18, 9.17–18], [23, Append. V]) they are also open mappings. This suIces to ensure that FM is a ◦ ◦ surjection from M onto )M and from onto ). This does not guarantee, however, that the local di$eomorphism FM is a surjection from @ onto @) (a counterexample is given in [5]). Thus, global di$eomorphisms from M onto )M fall in a special class ◦ ◦ M onto ), and @ of local di$eomorphisms: those which map surjectively M onto ), onto @). Our approach will, therefore, be to .rst represent this speci.c class of local di$eomorphisms F in terms of F and the trace F|@ in order to characterize all global ◦
di$eomorphisms on . Further narrowing of the trace class will give a representation M of the global di$eomorphisms on . The de.nition of a star-shaped set and of a domain and its boundary imply easily that ◦ if is a star-shaped domain with centre x0 ∈, then, for every x ∈ M \ {x0 }, there exist a unique x1 ∈ @ and a unique s ∈ [0; 1] with x = (1 − s)x0 + x1 . This correspondence de.nes a surjection * : M \ {x0 } → @ which is constant on every ray beginning at x0 : *(x0 + s1 (x − x0 )) = *(x0 + s2 (x − x0 )), where s1 ¿ 0; s2 ¿ 0; s1 = s2 , are such that M j = 0; 1. In particular, it suIces to consider only the value s = 1 x0 + sj (x − x0 ) ∈ ; which corresponds to *(x); x ∈ M \ {x0 }. For example, if is the ball with centre x0 and radius + ¿ 0, then *(x) = x0 + +(x − x0 )= x − x0 X . Besides *(x), we also consider the continuous surjective boundary mapping g : @ → @), and the composition - =g◦* M 0} which, in view of the properties of * and g, is a continuous surjection from \{x onto @).
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◦
Lemma 2. Let F : → X be a local di9eomorphism from onto a domain F() ⊂ ◦ ◦ X , and assume that FM maps M onto F(), onto F() and @ onto @F(). Let ◦
◦
◦
◦
) ⊂ X be a domain, such that ) ∩ F() = ∅ and f0 ∈) ∩ F(), where f0 = F(x0 ); x0 is a centre of . Let g be any continuous surjection of @ onto @). Then, for ˜ ) (T ∈ L˜+ (X ) in the case of real X ), such every x1 ∈ @ there exists T = Tx1 ∈ L(X that g(x1 ) − f0 = Tx1 (F(x1 ) − f0 ). Moreover, the dependence of Tx1 on x1 ∈ @ is continuous at F(x1 ) − f0 , that is, for any x1 ∈ @ and for any sequence x1; ∈ @ with
x1 − x1; X → 0; → +∞, it follows that (Tx1; − Tx1 )(F(x1 ) − f0 ) X → 0, where
F(x1 ) − f0 X = 0. ◦
◦
Proof. F(x1 )−f0 = 0 and g(x1 )−f0 = 0, because f0 ∈)∩F(), while F(x1 ) ∈ @F(), by assumption about F, and g(x1 ) ∈ @), by de.nition of g. This easily implies the ˜ ). Moreover, if X is a real Hilbert space, then Tx1 can be chosen existence of Tx1 ∈ L(X to be x1 Sx1 , where Sx1 is rotation on the positive angle between F(x1 )−f0 and g(x1 )− f0 , and x1 = g(x1 )−f0 X = F(x1 )−f0 X ¿ 0. Therefore, Tx1 ∈ L˜+ (X ). Hence, in both cases—whether X is complex or real—there exists H : @ → L(X ), such that Tx1 = eH (x1 ) . Moreover, for real Hilbert space X there exists an anti-symmetric operator-valued function H : @ → L(X ); H (x1 ) = −H (x1 ) , and there exists : @ → (0; ∞), such that Tx1 = (x1 )eH (x1 ) , x1 ∈ @. To prove continuity of Tx1 , we note .rst that max (x1 ) =
x1 ∈@
g(x1 ) − f0 X diam()) 6 ¡ ∞;
F(x1 ) − f0 X dist(f0 ; @F())
˜ ) in such a way that maxx1 ∈@ Tx1 L(X ) ¡ ∞. Now therefore, Tx1 can be chosen in L(X continuity of Tx1 at F(x1 ) − f0 follows from the bound
(Tx1; − Tx1 )(F(x1 ) − f0 ) X = g(x1; ) − g(x1 ) − Tx1; (F(x1; ) − F(x1 )) X 6 g(x1; ) − g(x1 ) X + max T L(X ) F(x1; ) ∈@
− F(x1 ) X → 0: Since we shall be always applying Tx1 to the corresponding argument F(x1 ) − f0 , we can assume, without loss of generality, that Tx1 = eH (*(x)) where x1 = *(x), M 0 }; L(X )). H ∈ C 0 (@; L(X )), H ◦ * ∈ C 0 (\{x ◦
Lemma 3. Assume that ⊂ X is a bounded star-shaped domain with centre x0 ∈, and that ) ⊂ X is a bounded domain. Let *(x); g(x1 ) and -(x)=g(*(x)) be as de:ned M 0 }; x1 ∈ @. Then, F ∈ C 1 (; M X ) is a local C 1 -di9eomorphism from M above, x ∈ \{x ◦ ◦ onto )M which maps surjectively onto ) and @ onto @) and for which F (x0 ) = 0 ◦
holds, if and only if there exists a 4-tuple (f0 ; g; G; 0), in which f0 ∈); g : @ → @)
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L. Dechevsky / Nonlinear Analysis: Real World Applications 4 (2003) 203 – 221 ◦
◦
is a continuous surjection, G ∈ C 0 (; L(X )), 0 ∈ C 0 (\{x0 }; R), limx→x0 0(x) = +∞, M such that, for x ∈ , F(x) = f0 +
0
1
eH (*(x)) eG(x0 +t(x−x0 ))−0(x0 +t(x−x0 ))IX (x − x0 ) dt;
(11)
where f0 = F(x0 ); -(x) = g(*(x)) = F(*(x)), F (x) = eH (*(x)) eG(x)−0(x)IX ;
◦
x ∈ \ {x0 };
F (x0 ) = 0:
(12)
Here H (*(x)) is as de:ned above. Equivalently, making the boundary conditions explicit,
x − x0 X
x − x0 X f0 + F(x) = 1 − -(x)
*(x) − x0 X
* − x0 X 1
x − x0 X + ; t eH (*(x)) eG(x0 +t(*(x)−x0 ))−0(x0 +t(*(x)−x0 )) K
*(x) − x0 X 0 (x − x0 ) dt;
(13)
M with kernel K is de:ned by K(2; t) = 1=2 − 1, 0 6 t ¡ 2 6 1, and by where x ∈ , K(2; t) = −1; 0 ¡ 2 6 t 6 1. Proof. Recalling that *(x) = *(x0 + t(x − x0 )), and substituting this into (11) we see that (11) and (12) coincide with (8) and (9). Hence, the only thing to verify is that F(*(x)) = -(x). In fact, by de.nition of H (*(x)), eH (*(x))
1
0
eG(x0 +t(*(x)−x0 ))−0(x0 +t(*(x)−x0 ))IX (*(x) − x0 ) dt = -(x) − f0 :
(14)
On the other hand, computing (11) at *(x) and comparing the result with (14) yields F(*(x)) = -(x). Formula (13) follows from (11) and (14) and an appropriate change of variable in the integral of (11). In fact, (13) is the Lagrange expansion of F with simple knots at x0 and -(x), and with integral remainder. When X = Rn , the product of exponents in (11)–(14) must be factored by & = ±1. Now we are in a position to prove the main result of this subsection. Theorem 2. Under the conditions of Lemma 3, F, with F (x0 ) = 0, represented via (11) and (12), is a global C 1 -di9eomorphism between ◦
◦
(a) the open domains and ), if and only if the continuous surjection g : @ → @) is a bijection;
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M if and only if F is de:ned and continuous on @, (b) the closed domains M and ), condition (a) about g is satis:ed and, additionally, g is C 1 -smooth. Proof. Part (a) follows from Lemma 3 and Theorem A in Section 1. The assumptions in part (b) are equivalent to continuous di$erentiability of F : M → )M on @, hence, part (b) follows from part (a). 3.3. Representation of boundary mappings in Rn Theorem 2(b) reduces the problem of representation of global C 1 -di$eomorphisms from M onto )M by one dimension, the new problem being to represent the global C 1 -di$eomorphisms from @ onto @). When X = Rn , this reduction suIces to solve the n-dimensional problem, by n consecutive reductions. In the case n=1, because of the full ordering on the real line, the Lagrange mean-value theorem holds, and monotonicity is equivalent to bijectivity. Therefore, for n = 1 the class of global C 1 -di$eomorphisms coincides with the whole class of local di$eomorphisms between these intervals, and can be represented easily (see Proposition 1). Consider now the unit circle in R2 . By polar change of variables, the unit circle is mapped di$eomorphically onto an interval. By utilizing the already available representation of C 1 -di$eomorphisms on the interval, and then applying inverse polar change of variables, we achieve a representation of the global C 1 -di$eomorphisms on the unit circle. Hence, by Theorem 2(b), we also obtain a representation of the global C 1 -di$eomorphisms on the unit disk in R2 . Thus, the problem is solved for n = 2. To solve it for n = 3, we reiterate the procedure. By spherical change of variables, we map the unit sphere onto a rectangle in R2 ; by inverse polar change of variables this rectangle is mapped onto the unit disk in R2 ; the already available representation on the unit disk is utilized and, going back via polar and inverse spherical change of variables to the unit sphere, we represent the C 1 -di$eomorphisms on the unit sphere, hence, by Theorem 2(b) the problem can be solved also for n = 3. And so on, the main tools at each iteration being Theorem 2(b) and the hyperspherical change of variables (r; 1 ; : : : ; n−1 ) → (x1 ; : : : ; x n ): x1 = r cos 1 ; x = r
−1
4=1
x = r
−1
sin 4 cos ; = 2; : : : ; n − 1; sin 4 ;
4=1
r ¿ 0; ∈ [0; ], = 1; : : : ; n − 2; n−1 ∈ [0; 2 ), with Jacobian J = r n−1 n−2 n−1−
. For r = 1 this change of variables maps di$eomorphically the unit
=1 sin sphere in Rn onto the hyperrectangle [0; ]n−2 × [0; 2 ) ⊂ Rn−1 . Below we give the results of this construction for n = 2, but .rst let us look into the simple case n = 1.
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Proposition 1. Let X = R, = M = [x0 − +; x0 + +], ) = )M = [y0 − r; y0 + r], x0 ∈ R, M = )M is a global di9eomorphism from M y0 ∈ R, + ¿ 0, r ¿ 0. Then, F : M → F() onto )M if and only if there exists a couple (G; &), in which G ∈ C 0 ([x0 − +; x0 + +]; R), & ∈ {−1; 1}, such that x eG(t) dt x0 −+ F(x) = y0 + &r −1 + 2 x0 ++ (15) ; x ∈ [x0 − +; x0 + +]: eG(t) dt x0 −+ Proof. When x varies on the ray { ∈ R: ¿ x0 − +}, the operator eH (*) in (11)–(14) reduces to a real constant c ¿ 0, it is unique and can be computed explicitly, which yields (15). The following is the main result of this subsection. Consider the canonical isometry between R2 and C, de.ned by (1; 0) → 1, (0; 1) → i, so that to every (x1 ; x2 ) ∈ R2 corresponds x1 + i · x1 ∈ C, given via its polar coordinates rei . M Theorem 3. Let X =R2 be endowed with the usual Hilbert norm. Let = M =)= )= ◦ 2 1 M 2 {x ∈ R : x X 6 1}, let F ∈ C (; R ), be a surjection on , , and @, and assume that F (0) = 0. Then, F is a C 1 -di9eomorphism on M if and only if there exists an 8-tuple (r0 ; 0 ; ˜1 ; ˜2 ; 8˜ 1 ; 8˜ 2 ; *; &), with the following properties r0 ∈ [0; 1), 0 ∈ [0; 2 ), ˜ j ∈ C 0 (\{0}; R), 8˜ j ∈ C 0 (; [0; 2 )), j = 1; 2, * ∈ C 0 ([0; 2 ); R), & ∈ {−1; 1}; for j de:ned by j (rei ) = ˜j (x1 ; x2 ), where x1 + ix2 = rei , there holds limr→0+ j (rei ) = −∞, for any in [0; 2 ); 8j are de:ned by 8j (rei ) = 8˜ j (x), x = (x1 ; x2 ); A1 ( ) + iB1 ( ) = 0 for ∈ [0; 2 ) where A1 ; B1 are de:ned in (18) and (19) and below Ar ( ) + iBr ( ) ig( ) − r0 ei 0 ) (e A1 ( ) + iB1 ( )
F(rei ) = r0 ei 0 + r holds, where
g( ) = 1 + & −1 + 2 Ar ( ) =
1
0
{e1 (rte
− e2 (rte Br ( ) =
0
1
i
)
{e1 (rte
+ e2 (rte r ∈ [0; 1], ∈ [0; 2 ).
i
)
i
)
0 2 0
e*(t) dt e*(t) dt
;
)
(17)
cos 81 (rtei ) cos [82 (rtei ) + ]
sin 81 (rtei ) sin [82 (rtei ) + ]} dt; i
(16)
(18)
sin 81 (rtei ) cos [82 (rtei ) + ]
cos 81 (rtei ) sin [82 (rtei ) + ]} dt;
(19)
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215
Proof. Assume .rst that F is a C 1 -global di$eomorphism. Then, F (x), x=(x1 ; x2 ) ∈ , can be written as ˜ 0 cos 8˜ 1 (x) −sin 8˜ 1 (x) e1 (x) F (x) = & ˜ sin 8˜ 1 (x) cos 8˜ 1 (x) 0 e2 (x) cos 8˜ 2 (x) −sin 8˜ 2 (x) ; (20) sin 8˜ 2 (x) cos 8˜ 2 (x) where ˜1 ; ˜2 ; 8˜ 1 ; 8˜ 2 have the properties described in statement of the theorem. The ◦
condition F (0) = 0 determines the behaviour of 1 and 2 at 0. F(0) ∈ , since F ◦
maps onto itself. Under a polar change of variables x = rei in the identity F(x) = 1 F(0) + 0 F (tx)x dt, and after using (20) and some computations, we have 1 (21) F (tx)x dt = r(Ar ( ) + iBr ( )); 0
where the equality is understood as a correspondence by way of the isometry between R2 and C. We also have F(0) = r0 ei 0 for some r0 ¡ 1, 0 ∈ [0; 2 ). Now we see that also A1 ( ) + iB1 ( ) = 0 holds, uniformly in ∈ [0; 2 ), because x1 = ei ∈ @, A1 ( )+iB1 ( )=F(x1 )−F(0) and F(x1 ) ∈ @ because F maps @ onto itself. Moreover, F is a C 1 -di$eomorphism on @, which by Proposition 1 proves the existence of g and * with the implied properties and (17) holds. Therefore, eig( ) = r0 ei 0 + A1 ( ) + iB1 ( )
(22)
and in (22) we can divide by A1 ( ) + iB1 ( ). Substituting the result in (21), we obtain (16). This completes the proof of the necessary part. To prove suIciency, we note that eig( ) = r0 ei 0 for every ∈ [0; 2 ), since r0 ¡ 1. Therefore, (A1 ( ) + iB1 ( ))=(eig( ) − r0 ei ) exists. Hence, by inverse polar change of variables, (eig( ) − r0 ei )=(A1 ( )+iB1 ( )) corresponds to an invertible operator, which we write as eH (*(x)) . By the same inverse polar change, after computations, (16) becomes (11), where one can .nd G(x); 0(x), such that eG(x)−0(x)IX is the RHS of (20). Therefore, F as determined by the representation 8-tuple via (16), is a local C 1 ◦
di$eomorphism which is surjective on and on @. Proposition 1 and (17) imply that it is also a C 1 -smooth bijective mapping on @. By Theorem 2(b), F is a C 1 -di$eomorphism. The proof is complete. We note that analogous results to those in 3.2 and 3.3 hold when is star shaped with respect to x0 ∈ @. Then, the restriction F (x0 )=0 can be avoided. This restriction can also be avoided in the present context, if we consider di$eomorphisms on the “punctured” domain \ {x0 }. 3.4. C m -di9eomorphisms, m ∈ N Detailed consideration of this topic will not be possible here; however, we give an outline of some basic details.
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Given X and Y are Banach spaces, the 4th Frechet derivative of F(X ), F (4) (x), takes its values in the space L(X 4 ; Y ) of 4-multilinear operators, acting from the Cartesian product X × · · · × X = X 4 to Y , and bounded with respect to the uniform operator 4
norm [22, Chap. I, Section 4]. The respective Taylor expansion, with F (0) := F, is [22, Theorem 4.7] m−1 1 F (k) (x0 )(x − x0 )k F(x) = k! k=0
+ for
1 (m − 1)!
1
0
(1 − t)m−1 F (m) (x0 + t(x − x0 ))(x − x0 )m dt;
F ∈ C m (; Y ) = F ∈ C 0 (; Y ) : F C m (; Y ) = F C 0 (; Y )
+
m
sup F 4 (x) L(X 4 ; Y ) ¡ ∞
4=1 x∈
:
The notation F (4) (x )(x − x0 )4 has the meaning that the 4-polylinear operator F (4) (x ) is applied to (x − x0 )4 := (x − x0 ; : : : ; x − x0 ) ∈ X 4 [22, Chapter 4, Section 4]. For local C m -di$eomorphisms the basic representation remains, of course, (8) and (9), but now the function G itself is also to be represented. For m = 2; 3; : : :, m−2 1 1 G(x) = H4 (x − x0 )4 + 4! (m − 2)! 4=0
0
1
(1 − t)m−2 H (x0 + t(x − x0 ))(x − x0 )m−1 dt;
(23)
where H4 ∈ L(X 4+1 ; X )=L(X 4 ; L(X )), 4 =0; : : : ; m−2, and H ∈ C m−1 (; L(X m ; X )) are the parameters of the representation, together with f0 =F(x0 ) ∈ X . In the case of global di$eomorphisms, it can be shown that if F is C m -smooth and @ is C m−1 -smooth, Lemma 2 about the adjusting mapping Tx1 = eH (*(x)) can be strengthened to prove that Tx1 is in certain sense C m−1 -smooth, too. This means that the function GH (x) de.ned by eGH (x) = eH (*(x)) eG(x) is also C m−1 -smooth, hence, for @ smooth enough and m ¿ 1, the study of global C m -di$eomorphisms can be based on the representation of local C m -di$eomorphisms, similarly to the case m = 1. Instead of substituting (23) for G in (8), in practice one would prefer to integrate by parts appropriately in (8). Because of the fast convergence of series (6) in the norm topology of L(X ), given G ∈ C m−1 (; X ), it is always possible to integrate m − 1 times by parts in (8). The derivatives (d k =dt k )[eG(x0 +t(x−x0 )) ] can be computed in terms of analytic series, by Frechet di$erentiation in (5) or (6). For example, let us compute A ∈ L(X 2 ; X ) : A(x − x0 )2 := (d=dt[eG(x0 +t(x−x0 )) (x − x0 )]|t=0 in terms of H0 = G(x0 ) and
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217
H1 = G (x0 ). Frechet di$erentiation in (5) yields, by invoking Hilbert’s identity, −1 −1 Ah − 1 e (IX − H0 ) (H1 h)(IX − H0 ) d = 0; (H0 ) ⊂ U 2 i @U
L(X )
for any h ∈ X . This computation can be done also directly in (6), with the help of the identity [21, p. 156], T1j − T2j =
j−1
=0
T1j− (T1 − T2 )T2 ;
T1 ∈ L(X );
and the result is k−1 ∞ 1 k− −1
H0 (H1 h)H0 Ah − k! k=0
=0
T2 ∈ L(X )
=0
L(X )
for any h ∈ X . 4. Applications Here we outline the applications of our results to three types of inverse problems. Detailed considerations will be given elsewhere. 4.1. Smooth registration of images As mentioned in the introduction, this problem is the main motivation of our research. We present an outline of a technique, based on our results concerning representation of di$eomorphisms, which can be helpful in resolving the multidimensional problem of registration without markers, that is, in the case when the markers are the connected parts of the stationary manifold, {(x; f(x)): x ∈ ; f (x) = 0}, of the “smooth image”, f : → R, where f is the gradient of f. By a slight abuse of terminology, we use the term “C 1 -smooth reference image” for any g : → R of interest, where ⊂ Rn , n = 1; 2; 3, is a closed domain, and where g has continuous partial derivatives on up to order m ¿ 1. The relevance of the di$eomorphism-based approach is justi.ed by the following. Let us consider a “C 1 -smooth reference image” g de.ned on , the unit disk of 2 R . At a .nite subset of distinct points {x }N =1 , x ∈ , we have a corresponding set of N noisy observations {Y }N =1 , where it is known that Y = f(x ) + & . Here f is a C 1 -smooth function on (to be registered with respect to the “C 1 -smooth reference image” g), and the errors & are independent, identically distributed random variables with mean zero. The problem is to .nd a C 1 -smooth image f and a C 1 -di$eomorphism F on such that f(x) = g(F(x)) holds uniformly in x ∈ . We must .nd an optimal compromise between .tting the data and the smoothness of the image. Using the representation for F given in Theorem 3, with &=+1, it is possible to solve the above inverse stochastic problem by the following method of Tikhonov regularization: .nd optimal admissible values for the parameters of the representation of the C 1 -di$eomorphism F
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L. Dechevsky / Nonlinear Analysis: Real World Applications 4 (2003) 203 – 221
such that for .xed t ¿ 0, 2 ¿ 0 and = ¿ 0 the minimum of the following functional: N
2
w (Y − g(F(x ))) + t
2
j 2L2 ()
+2
j=1
=1
2
8k 2L2 () + = * 2L2 ([0;2 ))
k=1
is achieved, with j ; 8k and * as in (17)–(19). Here L2 is a weighted Lebesgue N space of square integrable functions; the weights w ¿ 0 are such that =1 w = 1; mi for example, we could use wi = 1=(3V ) =1 Vi , where Vi are the areas of those triangles in the Voronoi triangulation of the convex hull of {xk }Nk=1 which have xi as a vertex, mi being the number of all such triangles, V being the area of the convex hull of {xk }Nk=1 . The choice of the optimal regularization parameters t = tN∗ , 2 = 2∗N , = = =N∗ is subject to statistical considerations (see, e.g., [24]) and tN∗ → 0, 2∗N → 0, =N∗ → 0 as N → ∞. 4.2. Di9eomorphic splines Consider a bounded closed domain D ⊂ Rn with C 1 -smooth boundary, and a .nite cover {Uk }lk=1 of D, where Uk ⊂ D is a closed, say, convex domain, k = 1; : : : ; l. Consider a corresponding set {Vk }lk=1 of closed domains Vk ⊂ Rn . Based on the idea of Theorem 2(b), a representation of global di$eomorphisms Fk : Uk → Vk , k = 1; : : : ; l, can be considered, so that varying the parameters of the representation of Fk , k=1; : : : ; l, leads to a representation of any possible atlas of C 1 -manifolds on D over the .xed ◦
◦
j
k
family of maps {(Uk ; Vk )}lk=1 . For every j; k, such that U ∩ U = ∅ the change Fjk ◦
◦
◦
◦
j
k
j
k
of variables between Fk (U ∩ U ) and Fj (U ∩ U ) is continuously di$erentiable and can be obtained explicitly from the parameters of the representations of Fk and Fj ; for example, the Jacobian is −1
det(Fjk ()) = &j &k eTr Gj (Fk ◦
◦
j
k
())−Tr Gk (Fk−1 ())
= 0;
where ∈ Fk (U ∩ U ), Fk = eGk , Fj = eGj , and Tr G is the trace of G. Let Sk be the set of nk representation parameters corresponding to Fk , k = 1; : : : ; l. l We call k=1 Sk a C 1 -di$eomorphic spline. We see the potential importance of this concept in the fact that for every -dimensional manifold M ⊂ D, 1 6 6 n, such that its maps are {(Uk ; Vk )}, Uk =Uk ∩M , Vk ⊂ R , with corresponding C 1 -di$eomorphisms k : Uk → Vk , there exists a (generally non-unique) C 1 -di$eomorphic spline with corresponding {Vk }lk=1 , Vk ⊂ Rn and C 1 -di$eomorphisms {Fk }lk=1 , Fk : Uk → Vk , such that Fk (Uk )=Vk and Fk |Uk =k . This induces a representation of an arbitrary -dimensional manifold M ⊂ D whose maps are generated by an appropriate di$eomorphic spline on D. This construction can be applied to study the following inverse problem which is a generalization of the problem considered in 4.1. Consider a bounded closed domain D ⊂ R3 and a C 1 -smooth reference surface M0 ⊂ D. Consider also a scattered set of noisy data {X4 }N4=1 , X4 ∈ D. Find a C 1 -smooth surface M ⊂ D which is C 1 -di$eomorphic to M0 and which is “optimally near” to the data {X4 } and “optimally
L. Dechevsky / Nonlinear Analysis: Real World Applications 4 (2003) 203 – 221
219
smooth”, with respect to a certain criterion. The proposed type of method for solving this problem is again via Tikhonov regularization based on a penalization criterion. 4.3. Systems of di9erential equations and Lie groups of di9eomorphisms Consider a dynamical system d y4 = B4 (x; ˜y); x ∈ R; y˜ = (y1 ; : : : ; yn ) ∈ Rn ; 1 6 4 6 n; dx allowing a fundamental set of solutions, i.e., having a superposition formula ˜ y 1 (x); : : : ; ˜y k (x); C1 ; : : : ; Cn ); ˜y(x) = F(˜
(24)
(25)
expressing the general solution ˜y(x) in terms of a .nite number k of particular solutions ˜y i (x) and n signi.cant constants C1 ; : : : ; Cn , specifying the initial conditions. A theorem by Lie [11] states that a necessary and suIcient condition for a fundamental set of solutions and a superposition formula (25) to exist is that the RHS of (24) should lie in a .nite-dimensional Lie algebra, i.e., r B4 (x; y) = X (x)Y4 (y); (26) Z =
n 4=1
=1
Y4 (y)
@ ; @y4
[Zi ; Zj ] =
r
ij Z ;
(27)
=1
where [·; ·] is the Lie bracket and ij , i; j; = 1; : : : ; r, are the structural constants of the r-dimensional Lie algebra. Our results about representation of local (and global) di$eomorphisms can be used in the design of dynamical systems (24) satisfying (25)–(27) whose solutions have additional special properties, by imposing appropriate constraints on the functions X (x) in (26). Let us consider one simple model example for the case n = 1. Lie has shown that this case is very simple: the most general single Eq. (24) which can be solved by superposition (25) is the Riccati equation y = A(x) + B(x)y + C(x)y2 :
(28)
The model problem is: .nd the constraints on A(x), B(x), C(x) in (28) so that the solution y(x) of (25) is de.ned on the interval [x0 ; x2 ], increases from x0 to x1 , decreases from x1 to x2 , takes values y(xj ) = yj , j = 0; 1; 2, and is C 1 -smooth on (x0 ; x2 ). Here −∞ ¡ x0 ¡ x1 ¡ x2 ¡ ∞ and y0 ; y1 ; y2 are a priori given, with max(y0 ; y2 ) ¡ y1 . To solve this problem, we employ representation (15) in Proposition 1. The resulting constraint on A(x), B(x), C(x) in (28) is that on (x0 ; x1 ) ∪ (x1 ; x2 ) there exists a continuous real-valued function g(x) which has a vertical asymptote at x1 : lim g(x)= lim g(x)= −∞, and for which x x [ − aeg(x) + A(x) + B(x)(y1 + aeg(t) ) x1
x1
+ C(x)(y12 + 2y1 aeg(t) + a2 eg(t)+g(2) )] dt d2 = 0
x→x1−
x→x1+
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L. Dechevsky / Nonlinear Analysis: Real World Applications 4 (2003) 203 – 221
holds, for any x ∈ [x0 ; x2 ], where y0 − y1 a = x0 g(t) ; if x ∈ [x0 ; x1 ); e dt x1
y2 − y 1 a = x2 g(t) ; if x ∈ (x1 ; x2 ]: e dt x1
Here a is the one-dimensional version of the adjusting mapping &eH (*(x)) , considered in Section 3, times the orientation parameter & = ±1 of the di$eomorphism on the beam [x1 ; x). The same construction works for n ¿ 2, x1 being the centre of a star-shaped neighbourhood of a stationary point of the solution ˜y(x) of (24), situated at x1 . 5. Concluding remarks The integral representations obtained here involve only integration over the interval, which eases numerical computations, since no multivariate quadrature formulae are needed for the approximate calculation of the integral. One way to obtain numerical approximations of the exponent mapping eT of an (n×n)-matrix would be, of course, to compute an approximation of the Jordan diagonal form of T .rst. However, there are simpler and faster ways to obtain approximations which are more numerically stable. For example, the Cayley–Hamilton theorem yields n−1 ∞ ak; 4 1 T T 4; e = (29) + k! 4! 4=0
k=n
where T is an (n × n)-matrix, the coeIcients ak; 4 are computed iteratively from ak+1; 0 = ak; n−1 a0 ; ak+1; 4 = ak; n−1 a4 + ak; 4−1 ; an; 4 = a4 ;
4 = 1; : : : ; n − 1;
k = n; n − 1; : : : ;
4 = 0; 1; : : : ; n − 1;
a4 being the coeIcients of the characteristic polynomial of T : n−1 Tn = a4 T 4 : 4=0
Then, a numerical approximation of eT can be computed from (29), after an appropriate truncation in k. In connection with the exponent mapping, we also note that when X is a space with ˜ ) via the exponent Hilbert norm there is another possible way to represent T ∈ L(X ˜ ) there mapping. Namely, if X is a complex Hilbert space, then for every T ∈ L(X ˜ exists a diagonal operator D ∈ L(X ) and two anti-symmetric operators Sj ∈ L(X ) (Sj = −Sj∗ ; j = 1; 2; Sj∗ being the Hilbert adjoint of Sj ), such that T = eS1 eD eS2 . In a real Hilbert space X the respective formula is T = &eS1 eD eS2 , where & = +1 or & = −1. This representation for invertible T was used in Theorem 3. The only part of Sections 2 and 3, where we used essentially that X is .nite dimensional, was in Section 3.3. All other results in these sections are valid for an arbitrary complex or real Banach space X .
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221
Finally, we note that developing an analogous representation technique in Sobolev m , would lead to quite di$erent results in spaces Wpm , p ¡ ∞, rather than in C m = W∞ the case of global di$eomorphisms F, because the trace of F on lower dimensional subsets like @ would in general be less regular than F on the interior of (see [7]). Acknowledgements This research has been supported by the Natural Sciences and Engineering Research Council of Canada. References [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] F.L. Bookstein, Morphometric Tools for Landmark Data: Geometry and Biology, Cambridge University Press, Cambridge, 1991. V ements d’Analyse, Vol. 1, Gauthier-Villars, Paris, 1979. [3] J. DieudonnVe, ElV [4] N. Dunford, J.T. Schwartz, Linear Operators, Vol. 1: General Theory, Wiley, New York, 1988. [5] J. Hadamard, Les Transformations Ponctuelles, Bull. Soc. Math. Fr. 34 (1906) 71–84. [6] Encyclopaedia of Mathematics, Vol. 5, Kluwer Academic Publishers, Dordrecht, 1988. [7] A. Jonsson, H. Wallin, Function Spaces on Subsets of Rn , Harwood, London, 1984. [8] A. Kneip, T. Gasser, Convergence and consistency results for self-modeling nonlinear regression, Ann. Statist. 16 (1988) 82–112. [9] A. Kneip, T. Gasser, Statistical tools to analyze data representing a sample of curves, Ann. Statist. 20 (1992) 1266–1305. [10] A. Kneip, X. Li, B. MacGibbon, J. Ramsay, Curve registration by local regression, Canad. J. Statist. 28 (2000) 19–30. [11] S. Lie, Vorlesurgen uX ber continuirliche Gruppen mit geometrischen und anderen Anwendungen, Teubner, Leipzig, 1893. [12] C. Mira, Chaotic Dynamics, World Scienti.c, Singapore, 1987. [13] S.M. Nikol’skiYZ, Approximation of Functions of Several Variables and Imbedding Theorems, Springer, New York, 1975 (translated from Russian). [14] L.S. Pontryagin, Topological Groups, Gordon & Breach, New York, 1966. [15] J.O. Ramsay, Principal di$erential analysis: data reduction by di$erential operators, J. Roy. Statist. Soc. Ser. B 58 (1996) 495–508. [16] J.O. Ramsay, Estimating smooth monotone functions, J. Roy. Statist. Soc. Ser. B 60 (1998) 365–375. [17] J.O. Ramsay, X. Li, Curve registration, J. Roy. Statist. Soc. Ser. B 60 (1998) 351–363. [18] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1964. [19] H. Salkoe, S. Chiba, Dynamic programming algorithm optimization for spoken word recognition, IEEE Trans. ASSP-26 1 (1978) 43–49. [20] B.W. Silverman, Incorporating parametric e$ects into functional principal components analysis, J. Roy. Statist. Soc. Ser. B 57 (1995) 673–689. [21] V. ThomVee, Stability theory for partial di$erence operators, SIAM Rev. 11 (1969) 152–195. [22] M.M. Vainberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Halsted Press, New York, 1973 (translated from Russian). [23] M.M. Vainberg, Functional Analysis, Prosv, Moscow, 1979 (in Russian). [24] G. Wahba, Spline Models for Observational Data, SIAM, Philadelphia, PA, 1990. [25] K. Wang, T. Gasser, Synchronizing sample curves nonparametrically, Ann. Statist. 27 (1999) 439–460.
Integral representation of local and global di$eomorphisms Lubomir Dechevsky∗ Centre of Applied Mathematics, Lulea University of Technology, S-97187 Lulea, Sweden Received 3 November 1999; accepted 7 August 2001
Abstract The identi.cation of a smooth invertible map between two closed domains in Rn is of great importance in neuroimaging and other .elds where the domains of images must be nonlinearly transformed so as to align important features. The theory for locally di$eomorphic maps is classic, but much less is known about global di$eomorphisms between closed simply connected sets. These bijectively map interiors onto interiors and boundaries onto boundaries. We use this property and the integral of the exponential map to construct representations of global di$eomorphisms for star-shaped domains. As a simple example, details are provided for maps in two dimensions. Applications are outlined for three types of inverse problems related to image registration, di$eomorphic splines, and dynamical systems. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Locally invertible continuously di$erentiable surjection; Smooth bijection; Taylor formula with integral remainder for operators; Exponential map; Banach space; Small perturbation
1. Introduction Recent results on the interface of computer graphics with medical imaging require the selection of an optimal di$eomorphism within a large class of admissible smooth mappings which are invertible between two .xed subsets of Rn (n = 1; 2; 3). This requires the development of representation theory for the di$eomorphisms acting on a “reference” subset of Rn . In this connection, the only relevant general result known to us belongs to Hadamard [5] (see also [12]), and it treats the simple case without boundary when the reference subset is Rn itself. Here we propose a di$erent approach to that of Hadamard [5] for the study of di$eomorphisms where the reference subset is ∗
Corresponding author. E-mail addresses: [email protected], [email protected] (L. Dechevsky).
1468-1218/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 4 6 8 - 1 2 1 8 ( 0 2 ) 0 0 0 0 5 - 6
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a star-shaped domain in Rn or Cn . As an intermediate stage, we obtain representations for the smooth locally invertible surjections, or local di$eomorphisms between subsets of Rn or Cn . We were motivated to study these local and global di$eomorphisms between subsets of Rn by problems originating in functional data analysis. This type of statistical analysis di$ers from conventional multivariate analysis in that the observations are curves or surfaces rather than vectors. An important preliminary step is often the registration or alignment of essential features of the curve or surface by suitable “smooth” bijections of the argument(s). The problem of transforming the arguments of curves so as to align various salient features has a very large literature in many di$erent .elds. The problem is referred by Silverman [20], Ramsay and Li [17] and Kneip et al. [10] as curve registration; the engineering literature uses the term time warping [19,25], and the process of registering curves for the purpose of computing average curves is called structural averaging by Kneip and Gasser [8,9]. Marker registration is the process of aligning curves by identifying the timing of certain salient features in the curves, of which the zero of acceleration during the pubertal growth spurt and optimal temperature timings are examples. However, marker registration of curves or surfaces presents some problems, especially if markers are hard to identify. Silverman [20] developed a technique for curve registration that does not require explication of markers by optimizing a global .tting criterion with respect to a parametric family of transformations or time warpings. Ramsay and Li [17] applied an arbitrarily Hexible yet computationally convenient smooth monotone transformation family developed by Ramsay [16], and thus gave a nonparametric curve registration approach. Local linear techniques for this problem were developed in [10]. Recently, however, the problem of registering surfaces and volumes has become especially important in medical imaging of the brain [2]. This higher dimensional problem is much more diIcult than the one-dimensional curve registration problem because monotone and bijective are no longer synonymous. We have paid careful attention to the unidimensional case as presented in [12,15– 17]. However, as the range of a real-valued functional is a fully ordered set, while the range of a vector-valued mapping is only partially ordered, some of the key results on real-valued functions are no longer valid for vector-valued mappings (see e.g. [13,18]) and it became necessary to develop di$erent methods here. One possible approach would be to both characterize monotone “smooth” additive transformations between subsets of Rn and to suggest a computationally convenient method for their estimation with a view to their use in the problem of surface registration. However, the additivity assumption can be quite restrictive so we prefer here to study representations of locally invertible surjections as a .rst step in the registration problem. For the mappings to be represented, continuous di$erentiability will be assumed in the sense of [18, 9.11, 9.15], [22, Chap. I, Sections 4.3–5]. This ensures that the Gateau and Frechet derivatives coincide (see [22, Theorem 2.1]). In particular, this coincidence holds in the .nite dimensional case F : ⊂ Rn → Rl or F : ⊂ Cn → Cl if all partial derivatives (@F =@x )(x0 ), = 1; : : : ; n, = 1; : : : ; l, exist and are continuous at x0 ∈
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(see [18, 9.16], [23, Append. IV]). In this paper, m times continuously di$erentiable mappings, m ¿ 1, will also be called “smooth” mappings or “C m -mappings”, for short. Continuous mappings will also be referred to as “C 0 -mappings”. In our construction of di$eomorphism representations we will make use of: the Taylor expansion with integral remainder of Frechet-di$erentiable mappings; the fact that C∞ is dense in Cm ; m ∈ N (see, e.g., [1]); the exponent mapping eT ; T ∈ L(X ) (L(X )) being the space of linear bounded operators on the Banach space X where usually X will be Rn or Cn ; and the following Theorem A (Cf., e.g., [6]). Let be a simply connected open set in Rn or Cn , n ∈ N, M is continuous on the closure M and is a C 1 -mapping and assume that F : M → F() on . Let N0 ={x ∈ : det(F (x))=0}, where F denotes the Frechet derivative of F on . Assume that N0 = ∅ or N0 is an isolated subset of and that the restriction F|@ is bijective between the boundary @ and F(@). Then, F is bijective between M M and F(). For ease of presentation, we shall develop in detail the theory of C 1 -di$eomorphisms, and then we shall revisit certain aspects of this theory in the more general context of C m -di$eomorphisms. The organization of the paper is as follows. In Section 2 we derive an integral representation for the elements of C 1 and discuss the exponent mapping. Section 3 contains the main results about the integral representation of local di$eomorphisms. We also derive integral representation theorems for global di$eomorphisms on open and closed star-shaped domains in Cn and Rn in terms of boundary mappings. The essence of these results is that we reduce the dimension of the representation problem by passing from the domain to its boundary. In Section 3.3 we show how this can be used to completely solve by induction the problem of representation of the global di$eomorphisms on the closed unit ball in Rn . In this way we construct explicitly, as a useful example, the representation in the case n = 2. We also brieHy discuss the case of C m -di$eomorphisms, m ¿ 1. In Section 4 we discuss some applications and we suggest solving ill-posed problems in brain imaging by Tikhonov regularization. We also consider the concept of “di$eomorphic splines”, and an application to the study of dynamical systems. Some concluding remarks can be found in Section 5. 2. Preliminary results 2.1. Integral representation of smooth mappings The integral-remainder Taylor expansion is valid for star-shaped sets. The set ⊂ X , X = Rn or X = Cn , is called star-shaped with respect to x0 ∈ (“with center x0 ” for short) if for every x ∈ and every s ∈ [0; 1] the convex combination (1 − s)x0 + sx is also in . A necessary and suIcient condition for this is, if for every x ∈ there exist ∈ and s ∈ [0; 1] with x = (1 − s)x0 + s, or, equivalently, for every x ∈ Rn such that there exist ∈ and s ∈ [0; 1] with x = (1 − s)x0 + s, it follows that x ∈ . A
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convex set is star-shaped with respect to every x0 ∈ . In particular, when n = 1, the notions of a convex and a star-shaped set coincide. A non-void simply connected open set 1 ⊂ X will be called an open domain; its closure M 1 in the topology of X will be called a closed domain. A set ⊂ X will ◦ be called a domain if its closure M is a closed domain. Then its open interior is an ◦
◦
open domain; @ := M \ is the boundary of ; clearly ⊂ ⊂ M holds. Let ⊂ X be a domain and Y = Rl or Y = Cl ; l ∈ N. We de.ne the space 0 C = C 0 (; Y ) of all C 0 -mappings F : → Y such that
F C 0 (; Y ) = sup F(x) Y ¡ ∞ x∈
and the space C 1 = C 1 (; Y ) of all C 1 -mappings F : → Y for which
F C 1 () = F C 0 (; Y ) + sup F (x) L(X; Y ) ¡ ∞:
(1)
x∈
Here L(X; Y ) is the space of all bounded linear operators between X and Y , L(X ) := L(X; X ): (Since in .nite-dimensional space all norms are equivalent, we shall be assuming in this case that the norms · X and · Y are the usual Hilbert norms.) We prove the following integral representation of the elements of C 1 . ◦
Theorem 1. Let be a bounded star-shaped domain with centre x0 ∈. Then, F ∈ C 1 (; Y ) if and only if there exists a unique pair (f0 ; ), f0 ∈ Y , ∈ C 0 ◦
(; L(X; Y )), such that 1 F(x) = f0 + (x0 + t(x − x0 ))(x − x0 ) dt; 0
x ∈ ;
(2)
holds, with f0 = F(x0 ); ≡ F
◦
on ;
(3)
where the integral in (2) is the Riemann integral for vector-valued functions (see, e.g., [4, Chap. III, Section 5], [22, Chap. I, Sections 4.3–5]). Proof. The “only if ” part follows immediately from the Taylor integral-remainder expansion of the elements of C 1 . Namely, if F ∈ C 1 (; Y ), then 1 F(x) = F(x0 ) + F (x0 + t(x − x0 ))(x − x0 ) dt: (4) 0
To prove the “if ” part, we .rst assume that is smooth enough, namely, ∈ C 1 (; L(X; Y )). Now we apply Frechet di$erentiation to both sides of (2). Since [0; 1] is compact, the integrand and its Frechet derivative are uniformly continuous and Frechet di$erentiation commutes with integration in t. Here ∈ C 0 (; L(X 2 ; Y )), where L(X 2 ; Y ) is the space of bounded bilinear operators from the Cartesian product
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X 2 to Y , endowed with the uniform operator norm. We obtain 1 F (x) = [t (x0 + t(x − x0 ))(x − x0 ) + (x0 + t(x − x0 ))] dt 0
=
1
t d(x0 + t(x − x0 )) +
0
= [t(x0 + t(x − +
1
0
= (x);
x0 ))]|10
−
1
0 1
0
(x0 + t(x − x0 )) dt
(x0 + t(x − x0 )) dt
(x0 + t(x − x0 )) dt x ∈ :
The equality f0 = F(x0 ) coincides with (2) for x = x0 . This proves (3) and the “if ” part of the theorem in the partial case when is suIciently smooth. To prove the general case, we shall .rst show that the mapping 1 S = Sx : ∈ L(X; Y ) → (x0 + t(x − x0 ))(x − x0 ) dt ∈ Y 0
is continuous, uniformly in x ∈ . Indeed, by the triangle inequality, 1
Sx (1 ) − Sx (2 ) Y = [1 (x0 + t(x − x0 )) − 2 (x0 + t(x − x0 ))] 0 (x − x0 ) dt 6
0
Y
1
[1 (x0 + t(x − x0 )) − 2 (x0 + t(x − x0 ))]
(x − x0 ) Y dt 6 diam() 1 − 2 C 0 (; L(X; Y )) : From here, and from the obvious continuity of the identities IX and IL(X; Y ) in the topologies of X and L(X; Y ), respectively, the “if ” part follows for general ∈ C 0 (; L(X; Y )) by the fact that C 1 (; L(X; Y )) is dense in C 0 (; L(X; Y )). 2.2. The exponent mapping on L(X ) Assume .rst that X is Banach space over C. Analytic functions : L(X ) → L(X ) can be de.ned via Dunford’s representation [4, Chapter VII]: 1 f(T ) = f()(IX − T )−1 d; T ∈ L(X ); (5) 2 i @U where U ⊂ C is open (not necessarily connected), such that the spectrum (T ) of T is compactly contained in U, the boundary @U being a Jordan recti.able curve. Here (IX − T )−1 is the resolvent of T .
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The mapping exp z = ez , z ∈ C, is an entire analytic function; therefore, it is analytic in a neighbourhood of (T ) for any T ∈ L(X ) and (5) can be made meaningful for any T ∈ L(X ) by a suitable choice of the contour @U. The analytic expansion ∞ Tk eT = (6) k! k=0
has radius of convergence ∞. Of course, (eT )−1 = e−T ; however, eT1 +T2 = eT1 eT2 is not true in general, unless T1 and T2 commute. The mapping ez is a 2 i-periodic surjection from C to C \ {0}, and a bijection from D0 = {z ∈ C : − ¡ Im z 6 } to C \ {0}. The following lemma will be essential for the representation of local and global di$eomorphisms. ˜ ), where L(X ˜ ) denotes the Lemma 1. (a) T → eT is a surjection from L(X ) to L(X open subset of L(X ) consisting of all invertible elements of L(X ) (see [4, Chapter VII, Section 6.1; 18, 9.8]). ˜ ). (b) T → eT is a bijection from {T ∈ L(X ); (T ) ⊂ D0 } to L(X ˜ ), and that, Proof. In part (a) we have to prove that, for every T ∈ L(X ); eT is in L(X ˜ ) there exists a (not necessarily unique) T ∈ L(X ), such that S = eT . for every S ⊂ L(X The .rst of these two assertions follows from the identity eT e−T = e−T eT = IX . The second assertion follows from part (b). To prove part (b) we note that the inverse ˜ ), then 0 ∈ (S), function ln z of ez is a bijection between C \ {0} and D0 . If S ∈ L(X and, since (S) is a compact in C, there exists a neighbourhood U containing (S) such that 0 ∈ U. Then, ln S = (1=2 i) @U ln · (IX − S)−1 d, and T = ln S is such that eT = S, which implies the surjectivity part of (b) and completes the proof of (a). To prove the injectivity part of (b), assume that Tj ∈ L(X ); j = 1; 2, with (T1 ) ∪ (T2 ) ⊂ D0 , and that T1 = T2 in L(X ). If ( (T1 ) ∪ (T2 )) \ ( (T1 ) ∩ (T2 )) = ∅, then, since for any analytic function on (T1 ) ∪ (T2 ) the identity (f(T )) = f( (T )) holds, and since ez is analytic and bijective from D0 to C \ {0}, we obtain ( (eT1 ) ∪ (eT2 )) \ ( (eT1 ) ∩ (eT2 )) = ∅, hence, eT1 = eT2 in L(X ). It remains to consider the more diIcult case when the sets (T1 ) and (T2 ) coincide, but T1 and T2 have di$erent invariant subspaces corresponding to the elements of their common spectrum. Assume again that T1 = T2 in L(X ), i.e., T1 − T2 L(X ) = 0; denote Sj = eTj ; j = 1; 2, and suppose that S1 − S2 L(X ) = 0. Then, by Hilbert’s identity for resolvents and the triangle inequality, 1 −1 −1
T1 − T2 L(X ) = ln [(IX − S1 ) − (IX − S2 ) ] d 2 @U L(X ) 1 −1 −1 = ln [(IX − S1 ) (S1 − S2 )(IX − S2 ) ] d 2 @U L(X ) 2
6
1
ln(·) H1 (@U) max (IX − Sj )−1 L(X ) S1 − S2 L(X ) : ∈@U 2 j=1
(7)
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Here (T1 ) = (T2 ) ⊂ U ⊂ D0 , and U can be chosen bounded, because (T1 ) is compact. Hence, @U can be chosen recti.able and of .nite length. So chosen, @U is also a compact in C. The space Hp (@); 1 6 p 6 ∞, is the Hardy space on @U and its norm is denoted by · Hp (@U) . Then, ln(·) H1 (@U) ¡ ∞ holds, because ln(·) ∈ H∞ (@U) and the length of @U is .nite. Now, max∈@U (IX − Sj )−1 L(X ) ¡ ∞ holds, because
(IX −Sj )−1 L(X ) is continuous in and @U is compact. Hence, by (7), T1 −T2 L(X ) = 0, which is a contradiction. This proves the injectivity part of (b) and completes the proof of the lemma. Now let us consider the case when X is a Banach space over the reals. In this ˜ ) is not connected, but is the union of two case there is one essential di$erence: L(X ˜ disjoint connected open sets L+ (X ) and L˜− (X ) such that ±IX ∈ L˜± (X ). Thus, the ˜ ) exponent mapping represents only the elements of L˜+ (X ); to represent the whole L(X when X is a real Banach space, an additional orientation parameter & = ±1 must be introduced. 3. Main results In this section X = Rn or Cn and ⊂ X is star-shaped bounded domain with centre ◦
x0 ∈. 3.1. Integral representation of local C 1 -di9eomorphisms We consider those F ∈ C 1 (; X ) which are local di$eomorphisms, that is, they are ◦
locally invertible from to subsets of X , i.e., F (x)−1 exists in L(X ) for every x ∈. Global di$eomorphisms from to subsets of X are included, of course, but the class of local di$eomorphisms is essentially larger [18, Chap. 9, Problem 12]. Corollary 1. F ∈ C 1 (; X ) is a local C 1 -di9eomorphism from to a subset of X if and only if there exists a couple (f0 ; G); f0 ∈ X; G ∈ C 0 (; L(X )), such that 1 eG(x0 +t(x−x0 )) (x − x0 ) dt; x ∈ (8) F(x) = f0 + 0
and F(x0 ) = f0 ; F ≡ eG n
on :
(9)
G
When X = R , in (8), (9) e must be factored by the orientation parameter & = ±1. If (G(x)) ⊂ D0 for every x ∈ , then representation (8) and (9) is 1–1. Proof. The proof follows from part (a) of Theorem 1 and Lemma 1. The integral representation (8) and (9) has been previously proposed by Ramsay [15–17] for representing global di$eomorphisms in the case X = R.
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From the point of view of the theory of Lie groups and Lie algebrae, Eq. (8) provides a transparent connection between the elements F of a global group of local di$eomorphisms and the elements G of the Lie algebra of this group. If the group is local in the sense of classical Lie theory [11,14] (that is, if only those F in a suIciently small neighbourhood of the restriction of IX onto are considered, so that G is close enough to OL(X ) on for the equality eG = IX + G to hold with suIcient accuracy), then, for the case X = Cn , (8) is linearized to 1 F(x) = x + (F(x0 ) − x0 ) + G(x0 + t(x − x0 ))(x − x0 ) dt; x ∈ ; (10) 0
with the additional orientation factor & = ±1 in the case of X = Rn . 3.2. Integral representation of local and global C 1 -di9eomorphisms on star-shaped domains Our aim in this subsection is to derive an integral representation for the global di$eomorphisms from an open (closed) star-shaped domain ⊂ X onto another open (closed), not necessarily star-shaped, domain ) ⊂ X . To achieve this, we .rst note that every global di$eomorphism F : → F() = ) can be extended by continuity to a ◦ ◦ M which maps surjectively M onto ), M onto ) and @ onto @). M de.ned on , mapping F, In fact, the last property is possessed, more generally, by any open homeomorphism (a mapping is called open, if it maps open sets onto open sets). Local di$eomorphisms F are certainly continuous; by the inverse function theorem (see [3], [18, 9.17–18], [23, Append. V]) they are also open mappings. This suIces to ensure that FM is a ◦ ◦ surjection from M onto )M and from onto ). This does not guarantee, however, that the local di$eomorphism FM is a surjection from @ onto @) (a counterexample is given in [5]). Thus, global di$eomorphisms from M onto )M fall in a special class ◦ ◦ M onto ), and @ of local di$eomorphisms: those which map surjectively M onto ), onto @). Our approach will, therefore, be to .rst represent this speci.c class of local di$eomorphisms F in terms of F and the trace F|@ in order to characterize all global ◦
di$eomorphisms on . Further narrowing of the trace class will give a representation M of the global di$eomorphisms on . The de.nition of a star-shaped set and of a domain and its boundary imply easily that ◦ if is a star-shaped domain with centre x0 ∈, then, for every x ∈ M \ {x0 }, there exist a unique x1 ∈ @ and a unique s ∈ [0; 1] with x = (1 − s)x0 + x1 . This correspondence de.nes a surjection * : M \ {x0 } → @ which is constant on every ray beginning at x0 : *(x0 + s1 (x − x0 )) = *(x0 + s2 (x − x0 )), where s1 ¿ 0; s2 ¿ 0; s1 = s2 , are such that M j = 0; 1. In particular, it suIces to consider only the value s = 1 x0 + sj (x − x0 ) ∈ ; which corresponds to *(x); x ∈ M \ {x0 }. For example, if is the ball with centre x0 and radius + ¿ 0, then *(x) = x0 + +(x − x0 )= x − x0 X . Besides *(x), we also consider the continuous surjective boundary mapping g : @ → @), and the composition - =g◦* M 0} which, in view of the properties of * and g, is a continuous surjection from \{x onto @).
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◦
Lemma 2. Let F : → X be a local di9eomorphism from onto a domain F() ⊂ ◦ ◦ X , and assume that FM maps M onto F(), onto F() and @ onto @F(). Let ◦
◦
◦
◦
) ⊂ X be a domain, such that ) ∩ F() = ∅ and f0 ∈) ∩ F(), where f0 = F(x0 ); x0 is a centre of . Let g be any continuous surjection of @ onto @). Then, for ˜ ) (T ∈ L˜+ (X ) in the case of real X ), such every x1 ∈ @ there exists T = Tx1 ∈ L(X that g(x1 ) − f0 = Tx1 (F(x1 ) − f0 ). Moreover, the dependence of Tx1 on x1 ∈ @ is continuous at F(x1 ) − f0 , that is, for any x1 ∈ @ and for any sequence x1; ∈ @ with
x1 − x1; X → 0; → +∞, it follows that (Tx1; − Tx1 )(F(x1 ) − f0 ) X → 0, where
F(x1 ) − f0 X = 0. ◦
◦
Proof. F(x1 )−f0 = 0 and g(x1 )−f0 = 0, because f0 ∈)∩F(), while F(x1 ) ∈ @F(), by assumption about F, and g(x1 ) ∈ @), by de.nition of g. This easily implies the ˜ ). Moreover, if X is a real Hilbert space, then Tx1 can be chosen existence of Tx1 ∈ L(X to be x1 Sx1 , where Sx1 is rotation on the positive angle between F(x1 )−f0 and g(x1 )− f0 , and x1 = g(x1 )−f0 X = F(x1 )−f0 X ¿ 0. Therefore, Tx1 ∈ L˜+ (X ). Hence, in both cases—whether X is complex or real—there exists H : @ → L(X ), such that Tx1 = eH (x1 ) . Moreover, for real Hilbert space X there exists an anti-symmetric operator-valued function H : @ → L(X ); H (x1 ) = −H (x1 ) , and there exists : @ → (0; ∞), such that Tx1 = (x1 )eH (x1 ) , x1 ∈ @. To prove continuity of Tx1 , we note .rst that max (x1 ) =
x1 ∈@
g(x1 ) − f0 X diam()) 6 ¡ ∞;
F(x1 ) − f0 X dist(f0 ; @F())
˜ ) in such a way that maxx1 ∈@ Tx1 L(X ) ¡ ∞. Now therefore, Tx1 can be chosen in L(X continuity of Tx1 at F(x1 ) − f0 follows from the bound
(Tx1; − Tx1 )(F(x1 ) − f0 ) X = g(x1; ) − g(x1 ) − Tx1; (F(x1; ) − F(x1 )) X 6 g(x1; ) − g(x1 ) X + max T L(X ) F(x1; ) ∈@
− F(x1 ) X → 0: Since we shall be always applying Tx1 to the corresponding argument F(x1 ) − f0 , we can assume, without loss of generality, that Tx1 = eH (*(x)) where x1 = *(x), M 0 }; L(X )). H ∈ C 0 (@; L(X )), H ◦ * ∈ C 0 (\{x ◦
Lemma 3. Assume that ⊂ X is a bounded star-shaped domain with centre x0 ∈, and that ) ⊂ X is a bounded domain. Let *(x); g(x1 ) and -(x)=g(*(x)) be as de:ned M 0 }; x1 ∈ @. Then, F ∈ C 1 (; M X ) is a local C 1 -di9eomorphism from M above, x ∈ \{x ◦ ◦ onto )M which maps surjectively onto ) and @ onto @) and for which F (x0 ) = 0 ◦
holds, if and only if there exists a 4-tuple (f0 ; g; G; 0), in which f0 ∈); g : @ → @)
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L. Dechevsky / Nonlinear Analysis: Real World Applications 4 (2003) 203 – 221 ◦
◦
is a continuous surjection, G ∈ C 0 (; L(X )), 0 ∈ C 0 (\{x0 }; R), limx→x0 0(x) = +∞, M such that, for x ∈ , F(x) = f0 +
0
1
eH (*(x)) eG(x0 +t(x−x0 ))−0(x0 +t(x−x0 ))IX (x − x0 ) dt;
(11)
where f0 = F(x0 ); -(x) = g(*(x)) = F(*(x)), F (x) = eH (*(x)) eG(x)−0(x)IX ;
◦
x ∈ \ {x0 };
F (x0 ) = 0:
(12)
Here H (*(x)) is as de:ned above. Equivalently, making the boundary conditions explicit,
x − x0 X
x − x0 X f0 + F(x) = 1 − -(x)
*(x) − x0 X
* − x0 X 1
x − x0 X + ; t eH (*(x)) eG(x0 +t(*(x)−x0 ))−0(x0 +t(*(x)−x0 )) K
*(x) − x0 X 0 (x − x0 ) dt;
(13)
M with kernel K is de:ned by K(2; t) = 1=2 − 1, 0 6 t ¡ 2 6 1, and by where x ∈ , K(2; t) = −1; 0 ¡ 2 6 t 6 1. Proof. Recalling that *(x) = *(x0 + t(x − x0 )), and substituting this into (11) we see that (11) and (12) coincide with (8) and (9). Hence, the only thing to verify is that F(*(x)) = -(x). In fact, by de.nition of H (*(x)), eH (*(x))
1
0
eG(x0 +t(*(x)−x0 ))−0(x0 +t(*(x)−x0 ))IX (*(x) − x0 ) dt = -(x) − f0 :
(14)
On the other hand, computing (11) at *(x) and comparing the result with (14) yields F(*(x)) = -(x). Formula (13) follows from (11) and (14) and an appropriate change of variable in the integral of (11). In fact, (13) is the Lagrange expansion of F with simple knots at x0 and -(x), and with integral remainder. When X = Rn , the product of exponents in (11)–(14) must be factored by & = ±1. Now we are in a position to prove the main result of this subsection. Theorem 2. Under the conditions of Lemma 3, F, with F (x0 ) = 0, represented via (11) and (12), is a global C 1 -di9eomorphism between ◦
◦
(a) the open domains and ), if and only if the continuous surjection g : @ → @) is a bijection;
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M if and only if F is de:ned and continuous on @, (b) the closed domains M and ), condition (a) about g is satis:ed and, additionally, g is C 1 -smooth. Proof. Part (a) follows from Lemma 3 and Theorem A in Section 1. The assumptions in part (b) are equivalent to continuous di$erentiability of F : M → )M on @, hence, part (b) follows from part (a). 3.3. Representation of boundary mappings in Rn Theorem 2(b) reduces the problem of representation of global C 1 -di$eomorphisms from M onto )M by one dimension, the new problem being to represent the global C 1 -di$eomorphisms from @ onto @). When X = Rn , this reduction suIces to solve the n-dimensional problem, by n consecutive reductions. In the case n=1, because of the full ordering on the real line, the Lagrange mean-value theorem holds, and monotonicity is equivalent to bijectivity. Therefore, for n = 1 the class of global C 1 -di$eomorphisms coincides with the whole class of local di$eomorphisms between these intervals, and can be represented easily (see Proposition 1). Consider now the unit circle in R2 . By polar change of variables, the unit circle is mapped di$eomorphically onto an interval. By utilizing the already available representation of C 1 -di$eomorphisms on the interval, and then applying inverse polar change of variables, we achieve a representation of the global C 1 -di$eomorphisms on the unit circle. Hence, by Theorem 2(b), we also obtain a representation of the global C 1 -di$eomorphisms on the unit disk in R2 . Thus, the problem is solved for n = 2. To solve it for n = 3, we reiterate the procedure. By spherical change of variables, we map the unit sphere onto a rectangle in R2 ; by inverse polar change of variables this rectangle is mapped onto the unit disk in R2 ; the already available representation on the unit disk is utilized and, going back via polar and inverse spherical change of variables to the unit sphere, we represent the C 1 -di$eomorphisms on the unit sphere, hence, by Theorem 2(b) the problem can be solved also for n = 3. And so on, the main tools at each iteration being Theorem 2(b) and the hyperspherical change of variables (r; 1 ; : : : ; n−1 ) → (x1 ; : : : ; x n ): x1 = r cos 1 ; x = r
−1
4=1
x = r
−1
sin 4 cos ; = 2; : : : ; n − 1; sin 4 ;
4=1
r ¿ 0; ∈ [0; ], = 1; : : : ; n − 2; n−1 ∈ [0; 2 ), with Jacobian J = r n−1 n−2 n−1−
. For r = 1 this change of variables maps di$eomorphically the unit
=1 sin sphere in Rn onto the hyperrectangle [0; ]n−2 × [0; 2 ) ⊂ Rn−1 . Below we give the results of this construction for n = 2, but .rst let us look into the simple case n = 1.
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Proposition 1. Let X = R, = M = [x0 − +; x0 + +], ) = )M = [y0 − r; y0 + r], x0 ∈ R, M = )M is a global di9eomorphism from M y0 ∈ R, + ¿ 0, r ¿ 0. Then, F : M → F() onto )M if and only if there exists a couple (G; &), in which G ∈ C 0 ([x0 − +; x0 + +]; R), & ∈ {−1; 1}, such that x eG(t) dt x0 −+ F(x) = y0 + &r −1 + 2 x0 ++ (15) ; x ∈ [x0 − +; x0 + +]: eG(t) dt x0 −+ Proof. When x varies on the ray { ∈ R: ¿ x0 − +}, the operator eH (*) in (11)–(14) reduces to a real constant c ¿ 0, it is unique and can be computed explicitly, which yields (15). The following is the main result of this subsection. Consider the canonical isometry between R2 and C, de.ned by (1; 0) → 1, (0; 1) → i, so that to every (x1 ; x2 ) ∈ R2 corresponds x1 + i · x1 ∈ C, given via its polar coordinates rei . M Theorem 3. Let X =R2 be endowed with the usual Hilbert norm. Let = M =)= )= ◦ 2 1 M 2 {x ∈ R : x X 6 1}, let F ∈ C (; R ), be a surjection on , , and @, and assume that F (0) = 0. Then, F is a C 1 -di9eomorphism on M if and only if there exists an 8-tuple (r0 ; 0 ; ˜1 ; ˜2 ; 8˜ 1 ; 8˜ 2 ; *; &), with the following properties r0 ∈ [0; 1), 0 ∈ [0; 2 ), ˜ j ∈ C 0 (\{0}; R), 8˜ j ∈ C 0 (; [0; 2 )), j = 1; 2, * ∈ C 0 ([0; 2 ); R), & ∈ {−1; 1}; for j de:ned by j (rei ) = ˜j (x1 ; x2 ), where x1 + ix2 = rei , there holds limr→0+ j (rei ) = −∞, for any in [0; 2 ); 8j are de:ned by 8j (rei ) = 8˜ j (x), x = (x1 ; x2 ); A1 ( ) + iB1 ( ) = 0 for ∈ [0; 2 ) where A1 ; B1 are de:ned in (18) and (19) and below Ar ( ) + iBr ( ) ig( ) − r0 ei 0 ) (e A1 ( ) + iB1 ( )
F(rei ) = r0 ei 0 + r holds, where
g( ) = 1 + & −1 + 2 Ar ( ) =
1
0
{e1 (rte
− e2 (rte Br ( ) =
0
1
i
)
{e1 (rte
+ e2 (rte r ∈ [0; 1], ∈ [0; 2 ).
i
)
i
)
0 2 0
e*(t) dt e*(t) dt
;
)
(17)
cos 81 (rtei ) cos [82 (rtei ) + ]
sin 81 (rtei ) sin [82 (rtei ) + ]} dt; i
(16)
(18)
sin 81 (rtei ) cos [82 (rtei ) + ]
cos 81 (rtei ) sin [82 (rtei ) + ]} dt;
(19)
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Proof. Assume .rst that F is a C 1 -global di$eomorphism. Then, F (x), x=(x1 ; x2 ) ∈ , can be written as ˜ 0 cos 8˜ 1 (x) −sin 8˜ 1 (x) e1 (x) F (x) = & ˜ sin 8˜ 1 (x) cos 8˜ 1 (x) 0 e2 (x) cos 8˜ 2 (x) −sin 8˜ 2 (x) ; (20) sin 8˜ 2 (x) cos 8˜ 2 (x) where ˜1 ; ˜2 ; 8˜ 1 ; 8˜ 2 have the properties described in statement of the theorem. The ◦
condition F (0) = 0 determines the behaviour of 1 and 2 at 0. F(0) ∈ , since F ◦
maps onto itself. Under a polar change of variables x = rei in the identity F(x) = 1 F(0) + 0 F (tx)x dt, and after using (20) and some computations, we have 1 (21) F (tx)x dt = r(Ar ( ) + iBr ( )); 0
where the equality is understood as a correspondence by way of the isometry between R2 and C. We also have F(0) = r0 ei 0 for some r0 ¡ 1, 0 ∈ [0; 2 ). Now we see that also A1 ( ) + iB1 ( ) = 0 holds, uniformly in ∈ [0; 2 ), because x1 = ei ∈ @, A1 ( )+iB1 ( )=F(x1 )−F(0) and F(x1 ) ∈ @ because F maps @ onto itself. Moreover, F is a C 1 -di$eomorphism on @, which by Proposition 1 proves the existence of g and * with the implied properties and (17) holds. Therefore, eig( ) = r0 ei 0 + A1 ( ) + iB1 ( )
(22)
and in (22) we can divide by A1 ( ) + iB1 ( ). Substituting the result in (21), we obtain (16). This completes the proof of the necessary part. To prove suIciency, we note that eig( ) = r0 ei 0 for every ∈ [0; 2 ), since r0 ¡ 1. Therefore, (A1 ( ) + iB1 ( ))=(eig( ) − r0 ei ) exists. Hence, by inverse polar change of variables, (eig( ) − r0 ei )=(A1 ( )+iB1 ( )) corresponds to an invertible operator, which we write as eH (*(x)) . By the same inverse polar change, after computations, (16) becomes (11), where one can .nd G(x); 0(x), such that eG(x)−0(x)IX is the RHS of (20). Therefore, F as determined by the representation 8-tuple via (16), is a local C 1 ◦
di$eomorphism which is surjective on and on @. Proposition 1 and (17) imply that it is also a C 1 -smooth bijective mapping on @. By Theorem 2(b), F is a C 1 -di$eomorphism. The proof is complete. We note that analogous results to those in 3.2 and 3.3 hold when is star shaped with respect to x0 ∈ @. Then, the restriction F (x0 )=0 can be avoided. This restriction can also be avoided in the present context, if we consider di$eomorphisms on the “punctured” domain \ {x0 }. 3.4. C m -di9eomorphisms, m ∈ N Detailed consideration of this topic will not be possible here; however, we give an outline of some basic details.
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Given X and Y are Banach spaces, the 4th Frechet derivative of F(X ), F (4) (x), takes its values in the space L(X 4 ; Y ) of 4-multilinear operators, acting from the Cartesian product X × · · · × X = X 4 to Y , and bounded with respect to the uniform operator 4
norm [22, Chap. I, Section 4]. The respective Taylor expansion, with F (0) := F, is [22, Theorem 4.7] m−1 1 F (k) (x0 )(x − x0 )k F(x) = k! k=0
+ for
1 (m − 1)!
1
0
(1 − t)m−1 F (m) (x0 + t(x − x0 ))(x − x0 )m dt;
F ∈ C m (; Y ) = F ∈ C 0 (; Y ) : F C m (; Y ) = F C 0 (; Y )
+
m
sup F 4 (x) L(X 4 ; Y ) ¡ ∞
4=1 x∈
:
The notation F (4) (x )(x − x0 )4 has the meaning that the 4-polylinear operator F (4) (x ) is applied to (x − x0 )4 := (x − x0 ; : : : ; x − x0 ) ∈ X 4 [22, Chapter 4, Section 4]. For local C m -di$eomorphisms the basic representation remains, of course, (8) and (9), but now the function G itself is also to be represented. For m = 2; 3; : : :, m−2 1 1 G(x) = H4 (x − x0 )4 + 4! (m − 2)! 4=0
0
1
(1 − t)m−2 H (x0 + t(x − x0 ))(x − x0 )m−1 dt;
(23)
where H4 ∈ L(X 4+1 ; X )=L(X 4 ; L(X )), 4 =0; : : : ; m−2, and H ∈ C m−1 (; L(X m ; X )) are the parameters of the representation, together with f0 =F(x0 ) ∈ X . In the case of global di$eomorphisms, it can be shown that if F is C m -smooth and @ is C m−1 -smooth, Lemma 2 about the adjusting mapping Tx1 = eH (*(x)) can be strengthened to prove that Tx1 is in certain sense C m−1 -smooth, too. This means that the function GH (x) de.ned by eGH (x) = eH (*(x)) eG(x) is also C m−1 -smooth, hence, for @ smooth enough and m ¿ 1, the study of global C m -di$eomorphisms can be based on the representation of local C m -di$eomorphisms, similarly to the case m = 1. Instead of substituting (23) for G in (8), in practice one would prefer to integrate by parts appropriately in (8). Because of the fast convergence of series (6) in the norm topology of L(X ), given G ∈ C m−1 (; X ), it is always possible to integrate m − 1 times by parts in (8). The derivatives (d k =dt k )[eG(x0 +t(x−x0 )) ] can be computed in terms of analytic series, by Frechet di$erentiation in (5) or (6). For example, let us compute A ∈ L(X 2 ; X ) : A(x − x0 )2 := (d=dt[eG(x0 +t(x−x0 )) (x − x0 )]|t=0 in terms of H0 = G(x0 ) and
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217
H1 = G (x0 ). Frechet di$erentiation in (5) yields, by invoking Hilbert’s identity, −1 −1 Ah − 1 e (IX − H0 ) (H1 h)(IX − H0 ) d = 0; (H0 ) ⊂ U 2 i @U
L(X )
for any h ∈ X . This computation can be done also directly in (6), with the help of the identity [21, p. 156], T1j − T2j =
j−1
=0
T1j− (T1 − T2 )T2 ;
T1 ∈ L(X );
and the result is k−1 ∞ 1 k− −1
H0 (H1 h)H0 Ah − k! k=0
=0
T2 ∈ L(X )
=0
L(X )
for any h ∈ X . 4. Applications Here we outline the applications of our results to three types of inverse problems. Detailed considerations will be given elsewhere. 4.1. Smooth registration of images As mentioned in the introduction, this problem is the main motivation of our research. We present an outline of a technique, based on our results concerning representation of di$eomorphisms, which can be helpful in resolving the multidimensional problem of registration without markers, that is, in the case when the markers are the connected parts of the stationary manifold, {(x; f(x)): x ∈ ; f (x) = 0}, of the “smooth image”, f : → R, where f is the gradient of f. By a slight abuse of terminology, we use the term “C 1 -smooth reference image” for any g : → R of interest, where ⊂ Rn , n = 1; 2; 3, is a closed domain, and where g has continuous partial derivatives on up to order m ¿ 1. The relevance of the di$eomorphism-based approach is justi.ed by the following. Let us consider a “C 1 -smooth reference image” g de.ned on , the unit disk of 2 R . At a .nite subset of distinct points {x }N =1 , x ∈ , we have a corresponding set of N noisy observations {Y }N =1 , where it is known that Y = f(x ) + & . Here f is a C 1 -smooth function on (to be registered with respect to the “C 1 -smooth reference image” g), and the errors & are independent, identically distributed random variables with mean zero. The problem is to .nd a C 1 -smooth image f and a C 1 -di$eomorphism F on such that f(x) = g(F(x)) holds uniformly in x ∈ . We must .nd an optimal compromise between .tting the data and the smoothness of the image. Using the representation for F given in Theorem 3, with &=+1, it is possible to solve the above inverse stochastic problem by the following method of Tikhonov regularization: .nd optimal admissible values for the parameters of the representation of the C 1 -di$eomorphism F
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L. Dechevsky / Nonlinear Analysis: Real World Applications 4 (2003) 203 – 221
such that for .xed t ¿ 0, 2 ¿ 0 and = ¿ 0 the minimum of the following functional: N
2
w (Y − g(F(x ))) + t
2
j 2L2 ()
+2
j=1
=1
2
8k 2L2 () + = * 2L2 ([0;2 ))
k=1
is achieved, with j ; 8k and * as in (17)–(19). Here L2 is a weighted Lebesgue N space of square integrable functions; the weights w ¿ 0 are such that =1 w = 1; mi for example, we could use wi = 1=(3V ) =1 Vi , where Vi are the areas of those triangles in the Voronoi triangulation of the convex hull of {xk }Nk=1 which have xi as a vertex, mi being the number of all such triangles, V being the area of the convex hull of {xk }Nk=1 . The choice of the optimal regularization parameters t = tN∗ , 2 = 2∗N , = = =N∗ is subject to statistical considerations (see, e.g., [24]) and tN∗ → 0, 2∗N → 0, =N∗ → 0 as N → ∞. 4.2. Di9eomorphic splines Consider a bounded closed domain D ⊂ Rn with C 1 -smooth boundary, and a .nite cover {Uk }lk=1 of D, where Uk ⊂ D is a closed, say, convex domain, k = 1; : : : ; l. Consider a corresponding set {Vk }lk=1 of closed domains Vk ⊂ Rn . Based on the idea of Theorem 2(b), a representation of global di$eomorphisms Fk : Uk → Vk , k = 1; : : : ; l, can be considered, so that varying the parameters of the representation of Fk , k=1; : : : ; l, leads to a representation of any possible atlas of C 1 -manifolds on D over the .xed ◦
◦
j
k
family of maps {(Uk ; Vk )}lk=1 . For every j; k, such that U ∩ U = ∅ the change Fjk ◦
◦
◦
◦
j
k
j
k
of variables between Fk (U ∩ U ) and Fj (U ∩ U ) is continuously di$erentiable and can be obtained explicitly from the parameters of the representations of Fk and Fj ; for example, the Jacobian is −1
det(Fjk ()) = &j &k eTr Gj (Fk ◦
◦
j
k
())−Tr Gk (Fk−1 ())
= 0;
where ∈ Fk (U ∩ U ), Fk = eGk , Fj = eGj , and Tr G is the trace of G. Let Sk be the set of nk representation parameters corresponding to Fk , k = 1; : : : ; l. l We call k=1 Sk a C 1 -di$eomorphic spline. We see the potential importance of this concept in the fact that for every -dimensional manifold M ⊂ D, 1 6 6 n, such that its maps are {(Uk ; Vk )}, Uk =Uk ∩M , Vk ⊂ R , with corresponding C 1 -di$eomorphisms k : Uk → Vk , there exists a (generally non-unique) C 1 -di$eomorphic spline with corresponding {Vk }lk=1 , Vk ⊂ Rn and C 1 -di$eomorphisms {Fk }lk=1 , Fk : Uk → Vk , such that Fk (Uk )=Vk and Fk |Uk =k . This induces a representation of an arbitrary -dimensional manifold M ⊂ D whose maps are generated by an appropriate di$eomorphic spline on D. This construction can be applied to study the following inverse problem which is a generalization of the problem considered in 4.1. Consider a bounded closed domain D ⊂ R3 and a C 1 -smooth reference surface M0 ⊂ D. Consider also a scattered set of noisy data {X4 }N4=1 , X4 ∈ D. Find a C 1 -smooth surface M ⊂ D which is C 1 -di$eomorphic to M0 and which is “optimally near” to the data {X4 } and “optimally
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219
smooth”, with respect to a certain criterion. The proposed type of method for solving this problem is again via Tikhonov regularization based on a penalization criterion. 4.3. Systems of di9erential equations and Lie groups of di9eomorphisms Consider a dynamical system d y4 = B4 (x; ˜y); x ∈ R; y˜ = (y1 ; : : : ; yn ) ∈ Rn ; 1 6 4 6 n; dx allowing a fundamental set of solutions, i.e., having a superposition formula ˜ y 1 (x); : : : ; ˜y k (x); C1 ; : : : ; Cn ); ˜y(x) = F(˜
(24)
(25)
expressing the general solution ˜y(x) in terms of a .nite number k of particular solutions ˜y i (x) and n signi.cant constants C1 ; : : : ; Cn , specifying the initial conditions. A theorem by Lie [11] states that a necessary and suIcient condition for a fundamental set of solutions and a superposition formula (25) to exist is that the RHS of (24) should lie in a .nite-dimensional Lie algebra, i.e., r B4 (x; y) = X (x)Y4 (y); (26) Z =
n 4=1
=1
Y4 (y)
@ ; @y4
[Zi ; Zj ] =
r
ij Z ;
(27)
=1
where [·; ·] is the Lie bracket and ij , i; j; = 1; : : : ; r, are the structural constants of the r-dimensional Lie algebra. Our results about representation of local (and global) di$eomorphisms can be used in the design of dynamical systems (24) satisfying (25)–(27) whose solutions have additional special properties, by imposing appropriate constraints on the functions X (x) in (26). Let us consider one simple model example for the case n = 1. Lie has shown that this case is very simple: the most general single Eq. (24) which can be solved by superposition (25) is the Riccati equation y = A(x) + B(x)y + C(x)y2 :
(28)
The model problem is: .nd the constraints on A(x), B(x), C(x) in (28) so that the solution y(x) of (25) is de.ned on the interval [x0 ; x2 ], increases from x0 to x1 , decreases from x1 to x2 , takes values y(xj ) = yj , j = 0; 1; 2, and is C 1 -smooth on (x0 ; x2 ). Here −∞ ¡ x0 ¡ x1 ¡ x2 ¡ ∞ and y0 ; y1 ; y2 are a priori given, with max(y0 ; y2 ) ¡ y1 . To solve this problem, we employ representation (15) in Proposition 1. The resulting constraint on A(x), B(x), C(x) in (28) is that on (x0 ; x1 ) ∪ (x1 ; x2 ) there exists a continuous real-valued function g(x) which has a vertical asymptote at x1 : lim g(x)= lim g(x)= −∞, and for which x x [ − aeg(x) + A(x) + B(x)(y1 + aeg(t) ) x1
x1
+ C(x)(y12 + 2y1 aeg(t) + a2 eg(t)+g(2) )] dt d2 = 0
x→x1−
x→x1+
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L. Dechevsky / Nonlinear Analysis: Real World Applications 4 (2003) 203 – 221
holds, for any x ∈ [x0 ; x2 ], where y0 − y1 a = x0 g(t) ; if x ∈ [x0 ; x1 ); e dt x1
y2 − y 1 a = x2 g(t) ; if x ∈ (x1 ; x2 ]: e dt x1
Here a is the one-dimensional version of the adjusting mapping &eH (*(x)) , considered in Section 3, times the orientation parameter & = ±1 of the di$eomorphism on the beam [x1 ; x). The same construction works for n ¿ 2, x1 being the centre of a star-shaped neighbourhood of a stationary point of the solution ˜y(x) of (24), situated at x1 . 5. Concluding remarks The integral representations obtained here involve only integration over the interval, which eases numerical computations, since no multivariate quadrature formulae are needed for the approximate calculation of the integral. One way to obtain numerical approximations of the exponent mapping eT of an (n×n)-matrix would be, of course, to compute an approximation of the Jordan diagonal form of T .rst. However, there are simpler and faster ways to obtain approximations which are more numerically stable. For example, the Cayley–Hamilton theorem yields n−1 ∞ ak; 4 1 T T 4; e = (29) + k! 4! 4=0
k=n
where T is an (n × n)-matrix, the coeIcients ak; 4 are computed iteratively from ak+1; 0 = ak; n−1 a0 ; ak+1; 4 = ak; n−1 a4 + ak; 4−1 ; an; 4 = a4 ;
4 = 1; : : : ; n − 1;
k = n; n − 1; : : : ;
4 = 0; 1; : : : ; n − 1;
a4 being the coeIcients of the characteristic polynomial of T : n−1 Tn = a4 T 4 : 4=0
Then, a numerical approximation of eT can be computed from (29), after an appropriate truncation in k. In connection with the exponent mapping, we also note that when X is a space with ˜ ) via the exponent Hilbert norm there is another possible way to represent T ∈ L(X ˜ ) there mapping. Namely, if X is a complex Hilbert space, then for every T ∈ L(X ˜ exists a diagonal operator D ∈ L(X ) and two anti-symmetric operators Sj ∈ L(X ) (Sj = −Sj∗ ; j = 1; 2; Sj∗ being the Hilbert adjoint of Sj ), such that T = eS1 eD eS2 . In a real Hilbert space X the respective formula is T = &eS1 eD eS2 , where & = +1 or & = −1. This representation for invertible T was used in Theorem 3. The only part of Sections 2 and 3, where we used essentially that X is .nite dimensional, was in Section 3.3. All other results in these sections are valid for an arbitrary complex or real Banach space X .
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221
Finally, we note that developing an analogous representation technique in Sobolev m , would lead to quite di$erent results in spaces Wpm , p ¡ ∞, rather than in C m = W∞ the case of global di$eomorphisms F, because the trace of F on lower dimensional subsets like @ would in general be less regular than F on the interior of (see [7]). Acknowledgements This research has been supported by the Natural Sciences and Engineering Research Council of Canada. References [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] F.L. Bookstein, Morphometric Tools for Landmark Data: Geometry and Biology, Cambridge University Press, Cambridge, 1991. V ements d’Analyse, Vol. 1, Gauthier-Villars, Paris, 1979. [3] J. DieudonnVe, ElV [4] N. Dunford, J.T. Schwartz, Linear Operators, Vol. 1: General Theory, Wiley, New York, 1988. [5] J. Hadamard, Les Transformations Ponctuelles, Bull. Soc. Math. Fr. 34 (1906) 71–84. [6] Encyclopaedia of Mathematics, Vol. 5, Kluwer Academic Publishers, Dordrecht, 1988. [7] A. Jonsson, H. Wallin, Function Spaces on Subsets of Rn , Harwood, London, 1984. [8] A. Kneip, T. Gasser, Convergence and consistency results for self-modeling nonlinear regression, Ann. Statist. 16 (1988) 82–112. [9] A. Kneip, T. Gasser, Statistical tools to analyze data representing a sample of curves, Ann. Statist. 20 (1992) 1266–1305. [10] A. Kneip, X. Li, B. MacGibbon, J. Ramsay, Curve registration by local regression, Canad. J. Statist. 28 (2000) 19–30. [11] S. Lie, Vorlesurgen uX ber continuirliche Gruppen mit geometrischen und anderen Anwendungen, Teubner, Leipzig, 1893. [12] C. Mira, Chaotic Dynamics, World Scienti.c, Singapore, 1987. [13] S.M. Nikol’skiYZ, Approximation of Functions of Several Variables and Imbedding Theorems, Springer, New York, 1975 (translated from Russian). [14] L.S. Pontryagin, Topological Groups, Gordon & Breach, New York, 1966. [15] J.O. Ramsay, Principal di$erential analysis: data reduction by di$erential operators, J. Roy. Statist. Soc. Ser. B 58 (1996) 495–508. [16] J.O. Ramsay, Estimating smooth monotone functions, J. Roy. Statist. Soc. Ser. B 60 (1998) 365–375. [17] J.O. Ramsay, X. Li, Curve registration, J. Roy. Statist. Soc. Ser. B 60 (1998) 351–363. [18] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1964. [19] H. Salkoe, S. Chiba, Dynamic programming algorithm optimization for spoken word recognition, IEEE Trans. ASSP-26 1 (1978) 43–49. [20] B.W. Silverman, Incorporating parametric e$ects into functional principal components analysis, J. Roy. Statist. Soc. Ser. B 57 (1995) 673–689. [21] V. ThomVee, Stability theory for partial di$erence operators, SIAM Rev. 11 (1969) 152–195. [22] M.M. Vainberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Halsted Press, New York, 1973 (translated from Russian). [23] M.M. Vainberg, Functional Analysis, Prosv, Moscow, 1979 (in Russian). [24] G. Wahba, Spline Models for Observational Data, SIAM, Philadelphia, PA, 1990. [25] K. Wang, T. Gasser, Synchronizing sample curves nonparametrically, Ann. Statist. 27 (1999) 439–460.