Lixin Liu

Athanase Papadopoulos
email: athanase.papadopoulos@math.unistra.fr

AND Weixu Su
email: suweixu@gmail.com

G Th Éret

Guillaume Théret
email: guillaume.theret71@orange.fr

ON THE CLASSIFICATION OF MAPPING CLASS ACTIONS ON THURSTON'S ASYMMETRIC METRIC

Keywords: AMS Mathematics Subject Classification: 32G15, 30F60, 57M50, 57N05 Teichmüller space, Thurston's asymmetric metric, mapping class group

We study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping classes. The study is parallel to the one made by Bers in the setting of Teichmüller space equipped with Teichmüller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmüller space equipped with the Weil-Petersson metric.

Introduction

Let S = S g,n be a connected oriented surface of finite type, of genus g with n punctures. We assume that the Euler characteristic of S is negative. We shall consider hyperbolic structures on S and any such structure will be complete and of finite area. We say that two complex structures (respectively hyperbolic structures) X and Y on S are equivalent if there is a conformal map (respectively an isometry) from X to Y which is homotopic to the identity map of S. The Teichmüller space T(S) of S is the space of complex structures (or, equivalently, the space of hyperbolic structures) on S up to equivalence.

Date: October 17, 2011.

An asymmetric metric on a set M is a nonnegative function δ on M × M which satisfies the axioms of a metric except the symmetry axiom, that is, we do not require that δ(x, y) = δ(y, x) for all x, y ∈ M .

Thurston defined in [START_REF] Thurston | Minimal stretch maps between hyperbolic surfaces[END_REF] an asymmetric metric d L on T(S) by setting for each X, Y ∈ T(S)

(1) d L (X, Y ) = inf f log L f ,
where X and Y are considered as (equivalence classes of) complete finite area hyperbolic structures on S, where the infimum is take over all homeomorphsims f : X → Y which are homotopic to the identity map of S and where L f is the Lipschitz constant of f , that is,

L f = sup x =y d Y (f (x), f (y)) d X (x, y)
for x and y in X.

We shall call this asymmetric metric the Thurston asymmetric metric.

Thurston [START_REF] Thurston | Minimal stretch maps between hyperbolic surfaces[END_REF] proved that there is a (not necessarily unique) extremal Lipschitz homeomorphsim that realizes the infimum in [START_REF] Bers | An extremal problem for quasiconformal mappings and a theorem by Thurston[END_REF], and that

d L (X, Y ) = log sup γ ℓ Y (γ) ℓ X (γ)
,

where ℓ X (γ) denotes the hyperbolic length of γ in X and γ ranges over all essential (that is, neither homotopic to a point nor a puncture) simple closed curves on S. We denote the Lipschitz constant of such an extremal Lipschitz map by L(X, Y ). Thurston showed that this function d L is indeed an asymmetric metric on Teichmüller space, that it is a Finsler metric and that any two points in Teichmüller space can be joined by a (not necessarily unique) d L -geodesic (that is, a shortest path with respect to d L ). Thurston also made a relation between extremal Lipschitz maps and a class of d Lgeodesics called stretch lines. (We shall recall the definition below.)

The Teichmüller metric is defined by

d T (X, Y ) = 1 2 inf f log K f (X, Y ),
where K f (X, Y ) is the quasiconformal dilatation of a homeomorphsim f : X → Y homotopic to the identity map of S. Using a lemma of Wolpert, we have [START_REF] Wolpert | The length spectra as moduli for compact Riemann surfaces[END_REF], d L ≤ 2d T . By definition, the topology induced by Thurston's asymmetric metric on T(S) is the one induced by its symmetrization,

1 2 {d L (X, Y ) + d L (Y, X)}
which is a genuine metic. This topology coincides with the one induced by the Teichmüller metric, see Li [START_REF] Li | Teichmüller metric and the length spectrums of Riemann surfaces[END_REF] and Papadopoulos-Théret [START_REF] Papadopoulos | On the topology defined by Thurston's asymmetric metric[END_REF] for information about the symmetrization. In an asymmetric metric space (M, δ), the left and right open balls centered at a point x and of radius ǫ, defined respectively by {y ∈ M | δ(x, y) < ǫ} and {y ∈ M | δ(y, x) < ǫ} may have completely different behaviors. Endowed with Thurston's asymmetric metric, the space T(S) is complete, and the left and right closed balls are compact. See [START_REF] Liu | On the metrics of length spectrum in Teichmüller space[END_REF], [START_REF] Papadopoulos | On the topology defined by Thurston's asymmetric metric[END_REF] and [START_REF] Papadopoulos | On Teichmüller's metric and Thurston's asymmetric metric on Teichmüller space[END_REF] for the topology induced by this asymmetric metric.

The mapping class group Mod(S) of S is the group of homotopy classes of orientation-preserving homeomorphism of S. This group acts properly discontinuously and isometrically on T(S) endowed with Teichmüller's metric or with Thurston's asymmetric metric. The quotient space is the moduli space M(S) = T(S)/Mod(S).

Thurston used his compactification of T(S) by the space of projective measured foliations to prove the following theorem (see Thurston [START_REF] Thurston | On the geometry and dynamics of diffeomorphisms of surfaces[END_REF], Theorem 4 or [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF] for details). Here, ϕ is said to be periodic if it is of finite order; ϕ is said to be reducible if there is a multicurve (i. e. a union of disjoint essential curves) on S which is globally fixed by ϕ; ϕ is said to be pseudo-Anosov if it preserves a pair of projective classes of transverse measured foliations having no singular closed leaves, multiplying the transverse measure on one by some constant K > 1 and on the other by the constant 1/K. We shall recall this classification more precisely in the following sections.

Let ϕ be an element of Mod(S). We set [START_REF] Bers | A remark on Mumford's compactness theorem[END_REF] a(ϕ) = inf

X∈T(S) d L (X, ϕ(X)).
We call a(ϕ) the translation distance of ϕ with respect to Thurston's asymmetric metric. We say that a(ϕ) is realized if there exists some X ∈ T(S) such that a(ϕ) = d L (X, ϕ(X)).

It is interesting to compare a(ϕ) with the translation distance defined by the Teichmüller metric d T :

b(ϕ) = inf X∈T(S) d T (X, ϕ(X)). Since d L ≤ 2d T , we have a(ϕ) ≤ 2b(ϕ).
Since Thurston's asymmetric metric is not symmetric, we may also consider a(ϕ -1 ) = inf

X∈T(S) d L (X, ϕ -1 (X)) = inf X∈T(S) d L (ϕ(X), X).
To motivate the work done in this paper, we recall that Bers [START_REF] Bers | An extremal problem for quasiconformal mappings and a theorem by Thurston[END_REF] studied the translation distance b(ϕ) and proved the following classification result. Theorem 1.2 (Bers [START_REF] Bers | An extremal problem for quasiconformal mappings and a theorem by Thurston[END_REF]). Let ϕ ∈ Mod(S). Then

(1) b(ϕ) = 0 and realized ⇐⇒ ϕ is periodic.

(2) b(ϕ) is not realized ⇐⇒ ϕ is reducible.

(3) b(ϕ) > 0 and realized ⇐⇒ ϕ is pseudo-Anosov.

In case [START_REF] Bers | An extremal problem for quasiconformal mappings and a theorem by Thurston[END_REF], saying that ϕ is periodic means that it is represented by a periodic homeomorphism of the surface.

Bers interpreted the realization of b(ϕ) as an extremal problem. In the case where b(ϕ) is not realized, Bers further showed that there is a generalized solution of the extremal problem which is given by a noded Riemann surface, and that the restriction of ϕ on each nonsingular component of the noded Riemann surface is either periodic or pseudo-Anosov.

Daskalopoulos and Wentworth [START_REF] Daskalopoulos | Classification of Weil-Petersson isometries[END_REF] obtained an analogous classification of elements of Mod(S) in terms of the Weil-Petersson translation distance.

In analogy with Bers' classification of elements of Mod(S) with respect to the Teichmüller metric, we classify ϕ ∈ Mod(S) into four types:

Elliptic

Parabolic Hyperbolic Pseudo-hyperbolic a(ϕ) = 0 a(ϕ) = 0 a(ϕ) > 0 a(ϕ) > 0 and realized but not realized and realized but not realized

The aim of our investigation in this paper is to analyze each of these cases.

It is easy to see that ϕ is elliptic if and only if ϕ has a fixed point in T(S). Indeed, if ϕ has a fixed point in T(S), then there is a complex structure X on S such that ϕ induces a biholomorphic automorphism of X. Using a classical result of Hurwitz, we deduce that ϕ is of finite order, therefore periodic. Conversely, if ϕ is periodic, Nielsen showed that ϕ has a fixed point in T(S). In fact, Kerckhoff proved a stronger result, namely, that the action of every finite subgroup of Mod(S) has a fixed point in T(S). (This was his solution to the so-called the Nielsen realization problem.)

A stretch line is a d L -geodesic of a special kind, defined by "stretching" along a complete geodesic lamination. We shall recall the definition in Section 4.

In this paper, we prove the following:

Theorem 1.3. If ϕ is pseudo-Anosov, then it leaves a d L -geodesic invariant.
Furthermore, for some n ∈ N, ϕ n leaves a stretch line invariant.

We also prove the following:

Theorem 1.4. If ϕ is pseudo-Anosov, then a(ϕ) = a(ϕ -1 ) = b(ϕ).
The projection on moduli space of the ϕ-invariant d L -geodesic provided by Theorem 1.3 is a closed d L -geodesic, and it is homotopic to a Teichmüller closed geodesic of the same length.

In analogy with the case of the Teichmüller metric, we prove the following:

Theorem 1.5. If ϕ is irreducible, then ϕ is either elliptic or hyperbolic.

Theorem 1.6. If ϕ is of infinite order and leaves a d L -geodesic invariant, then ϕ is hyperbolic. The translation distance a(ϕ) is attained at any point on the geodesic.

Theorem 1.7. The mapping class ϕ is of infinite order, reducible and periodic on all reduced components on S if and only if ϕ is parabolic.

To show that a reducible map ϕ ∈ Mod(S) cannot attain b(ϕ) in Teichmüller space, Bers [START_REF] Bers | An extremal problem for quasiconformal mappings and a theorem by Thurston[END_REF] used the notion of Nielsen extension of Riemann surfaces to show that for any X ∈ T(S), one can "shrink" the reduced curves to get a new Riemann surface Y satisfying d T (Y, ϕ(Y )) < d T (X, ϕ(X)). Bers' construction does not apply to the setting of Thurston's asymmetric metric. We shall see in §6 of this paper that there exists a reducible map ϕ which is hyperbolic with respect to the asymmetric d L , and therefore for which a(ϕ) is realized.

These results give quite a good picture of the behavior of the action of a mapping class with respect to Thurston's asymmetric metric, in terms of Thurston's classification. It remains to study the case where ϕ is reducible and has at least one pseudo-Anosov reduced component. In the setting of the Teichmüller metric, to show that a hyperbolic map ϕ ∈ Mod(S) (b(ϕ) > 0 and realized) leaves a Teichmüller geodesic invariant, Bers [START_REF] Bers | An extremal problem for quasiconformal mappings and a theorem by Thurston[END_REF] used the fact that any two points in Teichmüller space are connected by a unique Teichmüller geodesic. Since Thurston's asymmetric metric is not uniquely geodesic, Bers' argument does not work here. In the paper [START_REF] Daskalopoulos | Classification of Weil-Petersson isometries[END_REF] by Daskalopoulos and Wentworth, the strict convexity property of hyperbolic length functions along Weil-Petersson geodesics is used to show that a reducible map cannot leave a Weil-Petersson geodesic invariant. We don't know whether geodesic length functions are convex along d L -geodesics.

Questions 1.8. We propose the following:

(1) Is Theorem 1.4 true in general ? (2) If ϕ is reducible and has at least one pseudo-Anosov reduced component, is it pseudo-hyperbolic? Can it be hyperbolic? (3) If ϕ is hyperbolic, does it leave a d L -geodesic invariant? is it pseudo-Anosov? If ϕ is of infinite order and leaves a d L -geodesic (or even a stretch line) invariant, is it pseudo-Anosov?

In §6, we give an example, on any surface of genus ≥ 2, of a reducible mapping class which is of hyperbolic type.

It is known that closed Teichmüller geodesics in moduli space correspond to conjugacy classes of pseudo-Anosov maps. The above questions are related to the question of whether closed d L -geodesics in moduli space correspond to conjugacy classes of pseudo-Anosov maps.

Irreducible maps

In this section, we prove Theorems 1.5 and 1.6. Let S be the set of homotopy classes of essential simple closed curves on S.

A finite non-empty subset {C 1 , • • • , C k } ⊂ S is called admissible if C i = C j for all i = j and if C 1 , • • • , C k can be realized simultaneously as mutually disjoint curves. We call ϕ ∈ Mod(S) reducible if there is an admissible set {C 1 , • • • , C k } such that ϕ({C 1 , • • • , C k }) = {C 1 , • • • , C k }.
An element ϕ ∈ Mod(S) is called irreducible if there is no admissible set reduced by ϕ.

We shall use the following hyperbolic geometry estimate: Given a simple closed geodesic α of length ℓ α on a hyperbolic surface, there exists a collar neighborhood about α of width 2ω α , with sinh ω α sinh(ℓ α /2) = 1.

See Buser [START_REF] Buser | Geometry and spectra of compact Riemann surfaces[END_REF] for a reference. It follows that for small ℓ α , the length of any simple closed curve that intersects α is large. The following consequence is well-known: Lemma 2.1. There is a universal constant δ 0 > 0 such that any two distinct simple closed geodesics on a hyperbolic surface with hyperbolic length less than δ 0 are disjoint.

We also need the following theorem. See Bers [START_REF] Bers | A remark on Mumford's compactness theorem[END_REF] for a proof.

Theorem 2.2 (Mumford's compactness theorem). For each ǫ > 0, the subset M ǫ (S) = {X ∈ M(S) : ℓ(X) ≥ ǫ} ⊂ M(S) is compact.
Here ℓ(X) denotes the length of the shortest essential simple closed curve of a hyperbolic surface X. Note that this quantity is well-defined for elements X in T(S) or in M(S).

For each ǫ > 0, the subset

T ǫ (S) = {X ∈ T(S) : ℓ(X) ≥ ǫ}
is called the ǫ-thick part of T(S). Mumford's compactness theorem is equivalent to saying that for any sequence (X k ) k≥1 in an ǫ-thick part of T(S), there is a subsequence of (X k ) k≥1 , still denoted by (X k ) k≥1 , and a sequence of (ϕ k ) k≥1 in Mod(S), such that (ϕ k (X k )) k≥1 converges in T(S).

Lemma 2.3. Let ϕ ∈ Mod(S) be irreducible and X ∈ T(S). Then, with δ 0 being the constant given in Lemma 2.1, we have:

(L(X, ϕ(X))) 3g-3+n ≥ δ 0 /ℓ(X).
Proof. Suppose that

(3) (L(X, ϕ(X))) 3g-3+n < δ 0 /ℓ(X). Let α ∈ S such that ℓ X (α) = ℓ(X). Consider the curves α 1 = α, α 2 = ϕ -1 (α 1 ), • • • , α 3g-2+n = ϕ -1 (α 3g-3+n ). Since ℓ X (ϕ -1 (α)) = ℓ ϕ(X) (α) ≤ L(X, ϕ(X))ℓ X (α), each α i satisfies ℓ X (α i ) ≤ L(X, ϕ(X)) i-1 ℓ X (α). By (3), we have ℓ X (α i ) < δ 0 , i = 1, • • • , 3g -2 + n. By Lemma 2.1, the curves α 1 , α 2 , • • • , α 3g-2+n
are mutually disjoint. Since there are at most 3g -3 + n isotopy class of pairwise disjoint simple closed curves in S, there is a least number 1 ≤ j ≤ 3g -3 + n such that α j = α 1 .

Note that {α 1 , • • • , α j-1 } is admissible and

ϕ({α 1 , • • • , α j-1 }) = {α 1 , • • • , α j-1 }.
As a result, ϕ is reducible, which contradicts the assumption.

Proof of Theorem 1.5. By assumption, ϕ is irreducible. Consider a sequence (X k ) k≥1 in T(S) with ( 4) lim

k→∞ d L (X k , ϕ(X k )) = a(ϕ).
There is a constant M > 1 such that for any

k ≥ 1, L(X k , ϕ(X k )) ≤ M . By Lemma 2.3, ℓ(X k ) ≥ δ 0 (L(X k , ϕ(X k ))) 3g-3+n ≥ δ 0 M 3g-3+n .
Therefore, the sequence (X k ) lies in a thick part of T(S).

Now we use a compactness argument of Bers [START_REF] Bers | An extremal problem for quasiconformal mappings and a theorem by Thurston[END_REF] to show that a(ϕ) attains its infimum in T(S).

By Mumford's compactness theorem, there is a subsequence of (X k ) k≥1 , still denoted by (X k ) k≥1 , and a sequence (ϕ k ) k≥1 in Mod(S), such that (ϕ k (X k )) k≥1 converges to some point Y ∈ T(S). Let Y k = ϕ k (X k ). Since Mod(S) acts isometrically with respect to Thurston's asymmetric metric, we have

d L (X k , ϕ(X k )) = d L (ϕ k (X k ), ϕ k • ϕ(X k )) = d L (Y k , ϕ k • ϕ • ϕ -1 k (Y k ))
. By (4), we have [START_REF] Daskalopoulos | Classification of Weil-Petersson isometries[END_REF] lim

k→∞ d L (Y k , ϕ k • ϕ • ϕ -1 k (Y k )) = a(ϕ). Since d L (Y, ϕ k • ϕ • ϕ -1 k (Y k )) ≤ d L (Y, Y k ) + d L (Y k , ϕ k • ϕ • ϕ -1 k (Y k ))
≤ M 1 for some sufficiently large constant M 1 and since T(S) is locally compact, up to a subsequence, we can assume that (

ϕ k • ϕ • ϕ -1 k (Y k )) k≥1 converges to some Z ∈ T(S). Since (Y k ) k≥1 converges to Y , we conclude that the sequence (ϕ k • ϕ • ϕ -1 k (Y )) k≥1 converges to Z.
For any ǫ > 0, there is a number N > 0, such that for any j, k > N ,

d L (ϕ j • ϕ • ϕ -1 j (Y ), ϕ k • ϕ • ϕ -1 k (Y )) < ǫ.
Since Mod(S) acts properly discontinuously on T(S), we have, for sufficiently large k 0 ,

ϕ k • ϕ • ϕ -1 k = ϕ k 0 • ϕ • ϕ -1 k 0 , ∀ k ≥ k 0 . It follows from (5) that d L (Y, ϕ k 0 • ϕ • ϕ -1 k 0 (Y )) = a(f ).
Corollary 2.4. Suppose that ϕ ∈ Mod(S) is parabolic or pseudo-hyperbolic and suppose that (X k ) k≥1 is a sequence in T(S) with

lim k→∞ d L (X k , ϕ(X k )) = a(ϕ).
Then for any ǫ > 0, X k leaves T ǫ (S) for any sufficiently large k.

Proof. Otherwise, using Bers' compactness argument as we did in the proof of Theorem 1.5, we can show that ϕ is elliptic or hyperbolic.

Proof of Theorem 1.6. The proof follows Bers' argument in [START_REF] Bers | An extremal problem for quasiconformal mappings and a theorem by Thurston[END_REF], Theorem 5. We reproduce it here for the convenience of the reader. Suppose that a d L -geodesic through X ∈ T(S) is invariant under ϕ. Let Y be any point in T(S). Since ϕ is an isometry, we have

nd L (X, ϕ(X)) = d L (X, ϕ(X)) + d L (ϕ(X), ϕ 2 (X)) + • • • +d L (ϕ n-1 (X), ϕ n (X)) = d L (X, ϕ n (X)) ≤ d L (X, Y ) + d L (Y, ϕ n (Y )) + d L (ϕ n (Y ), ϕ n (X)) ≤ d L (X, Y ) + d L (Y, ϕ(Y ) + d L (ϕ(Y ), ϕ 2 (Y )) + • • • +d L (ϕ n-1 (Y ), ϕ n (Y )) + d L (ϕ n (Y ), ϕ n (X)) = d L (X, Y ) + nd L (Y, ϕ(Y )) + d L (Y, X). Since n is arbitrary, d L (X, ϕ(X)) ≤ d L (Y, ϕ(Y ) for any Y ∈ T(S). As a result, a(f ) = d L (X, ϕ(X)).
Bers also proved that b(ϕ) > 0 is realized if and only if ϕ leaves a Teichmüller geodesic invariant, cf. Bers [START_REF] Bers | An extremal problem for quasiconformal mappings and a theorem by Thurston[END_REF], Theorem 5. By an analysis of the Teichmüller geodesic left invariant by ϕ, he showed that the initial and terminal quadratic differentials of the Teichmüller map in the homotopy class of ϕ coincide. It follows that ϕ preserves a pair of transverse measured foliations, multiplying the transverse measure of one by K (the dilation of the Teichmüller map) and of the other one by 1/K. This means that ϕ is pseudo-Anosov. We will study pseudo-Anosov maps in Section 4.

Laminations and Thurston's stretch maps

In this section, we recall some necessary background on laminations and on Thurston's construction of strectch lines.

Fix a hyperbolic structure on the surface S. A geodesic lamination µ on S is a closed subset of S which is the union of disjoint simple geodesics (called the leaves of µ). A measured geodesic lamination is a geodesic lamination µ with a transverse invariant measure of full support µ. If a geodesic lamination µ has not all of its leaves going at both ends to cusps, then µ contains a compactly-supported sublamination admitting a transverse measure.

The stump of a geodesic lamination µ is the maximal compact sublamination of µ admitting a transverse measure.

A geodesic lamination µ on a hyperbolic surface X is complete if its complementary regions are all isometric to ideal triangles. Associated with such a pair (X, µ) is a partial (that is, supported on a subsurface) measured foliation F µ (X), satisfying the following:

(i) the foliation F µ (X) intersects µ transversely and in each ideal triangle of the complement of µ, the leaves of F µ (X) are composed of horocycles perpendicular to the boundary of the ideal triangle; (ii) the non-foliated region in each ideal triangle is bounded by three pairwise tangent leaves, as in Figure 1; (iii) The transverse measure for F µ (X) agrees with arc length on µ.

By collapsing each non-foliated region of F µ (X) onto a tripod, we get a measured foliation on S of full support, still denoted by F µ (X), which is well defined up to isotopy and which is called the horocyclic foliation corresponding to (X, µ). Note that since the hyperbolic structures we consider are complete, each puncture of X has a neighborhood isometric to a cusp. If the set of punctures of S is nonempty, each complete geodesic lamination on S has some leaves going to cusps. The corresponding horocylic foliations are standard near the cusps, meaning that in a neighborhood of each cusp, the leaves are circles homotopic to the puncture, and the total transverse measure of an arc converging to the cusp is infinite. We denote by MF(µ) the space of equivalence classes of measured foliations that are transverse to µ and standard near the cusps. Thurston [START_REF] Thurston | Minimal stretch maps between hyperbolic surfaces[END_REF] proved the following fundamental result.

Theorem 3.1. The map Φ µ : T(S) → MF(µ) : X → F µ (X) is a homeomorphism.

We also recall the following definition from Thurston [START_REF] Thurston | Minimal stretch maps between hyperbolic surfaces[END_REF]. A geodesic lamination λ is totally transverse to µ if each leaf of λ intersects µ transversely infinitely often and if each leaf of µ which does not go to a cups intersects λ transversely infinitely often. (In counting intersections, the leaves are parametrized by the reals. In particular, with this convention, a simple closed geodesic meets any transverse arc infinitely often.) By a result of Thuston [START_REF] Thurston | Minimal stretch maps between hyperbolic surfaces[END_REF], MF(µ) can be identified with the space of (equivalence classes) of measured laminations with compact support and totally transverse to µ.

From now on, we always assume that a measured geodesic lamination has compact support.

We shall say that a measured geodesic lamination λ is supported by a geodesic lamination µ if λ is a sublamination of µ.

We fix a complete geodesic lamination µ on S.

The stretch line directed by µ and passing through

X ∈ T(S) is the (image of a) path R ∋ t → X t = Φ µ -1 (e t F µ (X)).
We shall call a segment of a stretch line a stretch path. Stretch lines are geodesics for the Thurston asymmetric metric. More precisely, suppose that µ supports a measured geodesic lamination λ. Thus, for any two points X s , X t , s ≤ t on the stretch line, the Lipschitz distance d L (X s , X t ) is equal to t -s and realized by log ℓ Xt (λ) ℓ Xs (λ) .

In fact, laminations that realize the Lipschitz distance d L (X, Y ) (or, equivalently, that maximize the ratio of length) have been characterized by Thurston. A geodesic lamination µ is ratio-maximizing for X and Y if there is an L(X, Y )-Lipschitz map (homotopic to the identity map) from a neighborhood of µ in X to a neighborhood of µ in Y . A geodesic lamination µ is called chain-recurrent if for any ǫ > 0 and for any p ∈ µ there is a closed ǫ-trajectory of µ through p, that is, a closed unit speed path in the surface such that for any interval of length 1 on the path there is an interval of length 1 on some leaf of µ such that the two paths remain within ǫ of each other in the C 1 sense. By a result of Thurston [START_REF] Thurston | Minimal stretch maps between hyperbolic surfaces[END_REF], for any two distinct points X, Y ∈ T(S), there is a unique maximal (in the sense of inclusion) ratio-maximizing chain-recurrent lamination µ(X, Y ) which contains all other ratio-maximizing chain-recurrent laminations.

With the above definitions, we can state the following theorem of Thurston.

Theorem 3.2 (Thurston [START_REF] Thurston | Minimal stretch maps between hyperbolic surfaces[END_REF]). Any two points X, Y ∈ T(S) can be connected by a d L -geodesic which consists of a finite concatenation of stretch paths. Moreover, each of these stretch paths stretches along some complete lamination containing the unique ratio-maximizing chain-recurrent lamination µ(X, Y ).

Since each element ϕ of Mod(S) acts isometrically on (T(S), d L ), it maps d L -geodesics to d L -geodesics. We have the following: Lemma 3.3. Any ϕ ∈ Mod(S) maps a stretch line Φ µ -1 (e t F µ (X)) through X to a stretch line through ϕ(X), given by

Φ ϕ(µ) -1 (e t F ϕ(µ) (ϕ(X))).
Proof. It is easy to see that the image of Φ µ -1 (e t F µ (X)) under ϕ is a stretch line which passes through ϕ(X), stretched along the complete lamination ϕ(µ) and with corresponding horocyclic measured foliation ϕ(F µ (X)). Since the horocyclic measured foliation is uniquely determined by ϕ(X) and ϕ(µ), and since Φ ϕ(µ)

-1 (e t F ϕ(µ) (ϕ(X))) is also a stretch line through ϕ(X) directed by ϕ(µ), it follows that ϕ(F µ (X)) = F ϕ(µ) (ϕ(X)).

We denote by [λ] the projective class of a measured lamination (or measured foliation) λ. Such a [λ] determines a element in Thurston's boundary of Teichmüller space. We refer to Thurston [START_REF] Thurston | On the geometry and dynamics of diffeomorphisms of surfaces[END_REF] and [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF] for definitions and properties of Thurston's compactification and Thurston's boundary. Here we just recall that Thurston's compactification is homeomorphic to a closed (6g -6 + 2n)-dimensional ball whose boundary is identified with the space of projective measured foliations and that the mapping class group action on Teichmüller space extends continuously to Thurston's compactification.

The following two theorems are important for our study.

Theorem 3.4 (Papadopoulos [START_REF] Papadopoulos | On Thurston's boundary of Teichmüller space and the extension of earthquakes[END_REF]). The stretch line Φ µ -1 (e t F µ (X)) converges in the positive direction to [F µ (X)] in Thurston's boundary.

Recall that a measured geodesic lamination is said to be uniquely ergodic if its transverse measure is unique up to scalar multiples. Theorem 3.5 (Théret [14]). Suppose that the stump λ of the geodesic lamination µ is non-empty and is uniquely ergodic. Then Φ µ -1 (e t F µ (X)) converges in the negative direction to [λ] in Thurston's boundary. Now consider a pair of totally transverse measured geodesic laminations λ 1 and λ 2 . Suppose that λ 1 is uniquely ergodic. We can choose a complete geodesic lamination µ whose stump is λ 1 . By Theorem 3.1, there exists a unique X ∈ T(S) such that F µ (X) is equivalent to λ 2 . (As we noted above, we identify F µ (X) with λ 2 .) It follows from Theorem 3.4 and Theorem 3.5 that the stretch line Φ µ -1 (e t F µ (X)) converges in the positive direction to [λ 2 ] and converges in the negative direction to [λ 1 ].

We have the following consequence.

Corollary 3.6. Let λ 1 and λ 2 be a pair of totally transverse measured geodesic laminations and suppose that λ 1 is uniquely ergodic. Then there exists a stretch line which converges in the positive direction to [λ 2 ] and in the negative direction to [λ 1 ].

In general, the stretch line in Corollary 3.6 is not unique. If λ 1 is not maximal, there are different completions of λ 1 which give different stretch lines in Teichmüller space.

Pseudo-Anosov maps

In this section, we prove Theorem 1.3 and 1.4. Recall that an element ϕ ∈ Mod(S) is said to be pseudo-Anosov if it has a representative, still denoted by ϕ, which is a homeomorphism of S that preserves the projective classes of a pair of transverse measured foliations (F 1 , λ 1 ) and (F 2 , λ 2 ) and if there exists a constant K > 1 such that

ϕ((F 1 , λ 1 )) = (F 1 , 1 K λ 1 ), ϕ((F 2 , λ 2 )) = (F 2 , Kλ 2 ).
The measured foliations λ 1 and λ 2 are called the stable and unstable foliations of ϕ respectively. The quantity log K is the topological entropy of ϕ [16] [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF].

We recall some more terminology:

A measured foliation λ is minimal if no closed curve in S can be realized as a leaf of λ. Equivalently, after Whitehead moves, the foliation has only dense leaves on S. Two measured foliations are topologically equivalent if after isotopy and Whitehead moves, the leaf structure of the two foliations is the same. A theorem of Masur [START_REF] Masur | Two boundaries of Teichmüller space[END_REF] states that if a measured foliation λ is minimal and uniquely ergodic, then for any other measured foliation λ ′ satisfying i(λ, λ ′ ) = 0 we have λ ′ = cλ for some constant c > 0.

Proposition 4.1 (Thurston, see [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF]). Suppose that ϕ ∈ Mod(S) is pseudo-Anosov. Then each of the invariant foliations of ϕ is minimal and uniquely ergodic.

In what follows, we shall often pass from the measured foliations λ 1 and λ 2 to the measured laminations that represent them and vice versa.

We endow S with a hyperbolic structure and we identify λ 1 and λ 2 with the corresponding measured geodesic laminations. Since λ 1 is minimal and uniquely ergodic, each connected component of S \λ 1 is isometric to an ideal polygon or an ideal polygon with one puncture. It follows that there are finitely many different complete geodesic laminations with stump λ 1 .

Lemma 4.2. Let µ be a complete geodesic lamination with stump λ 1 . Then µ and λ 2 are totally transverse.

Proof. By Proposition 4.1, both λ 1 and λ 2 are minimal and uniquely ergodic. We replace λ 1 and λ 2 by their respective measured foliations. Up to Whitehead moves, we may assume that for both λ 1 and λ 2 , each half-leaf is dense in the surface. Then it is easy to see that each leaf of λ 2 intersects µ transversely infinitely often, and each leaf of µ which does not go to a cusp intersects λ 2 transversely infinitely often. It follows from the definition that λ 2 is totally transverse to µ. Theorem 4.3. Suppose that ϕ ∈ Mod(S) is pseudo-Anosov. Then there exists an integer n ≥ 1 such that ϕ n leaves a stretch line invariant.

Proof. Let λ 1 and λ 2 be the stable and unstable measured laminations of ϕ. Choose a complete geodesic lamination µ with stump λ 1 . By the discussion before Corollary 3.6, there is a unique stretch line, denoted by r(µ, λ 2 ), which converges in the positive direction to [λ 2 ] and converges in the negative direction to [λ 1 ].

Since ϕ fixes [λ 1 ] and [λ 2 ] on Thurston's boundary, each ϕ k (r(µ, λ 2 )) is a stretch line which also converges in the positive direction to [λ 2 ] and converges in the negative direction to [λ 1 ].

A stretch line satisfying the above conditions is uniquely determined by λ 2 and the completion of λ 1 . Since there are finitely many different completions of λ 1 , there is a integer n such that ϕ n leaves r(µ, λ 2 ) invariant.

By Theorem 1.6, ϕ n is hyperbolic. The translation distance a(ϕ n ) attains its infimum at any point on r(µ, λ 2 ). Proposition 4.4. With the above notations, a(ϕ n ) = n log K.

Proof. Since ϕ n (λ 1 ) = 1 K n λ 1 , for any X ∈ T(S), ℓ ϕ n (X) (λ 1 ) = ℓ X (ϕ -n (λ 1 )) = K n ℓ X (λ 1 ). Thus a(ϕ n ) ≥ n log K.
Take X on the stretch line r(µ, λ 2 ). Since the measure of λ 2 is "stretched" by K n , by definition of stretch line, the length of the measured lamination λ 1 is also "stretched" by K n . We have

n log K = d L (X, ϕ n (X)) ≥ a(ϕ).
The proof of Theorem 4.3 gives the following: Theorem 4.5. Suppose that ϕ ∈ Mod(S) is a pseudo-Anosov mapping class with stable and unstable measured laminations λ 1 and λ 2 and suppose that λ 1 is complete. Then ϕ leaves a unique stretch line invariant.

In the general case, we don't know whether a pseudo-Anosov mapping class leaves a stretch line invariant. We will show that such a mapping class leaves a d L -geodesic invariant. First notice the following property: Proposition 4.6. Suppose that ϕ ∈ Mod(S) is pseudo-Anosov with topological entropy log K. Then ϕ is hyperbolic and a(ϕ) = log K.

Proof. By the above results, there is a power ϕ n which fixes a stretch line and with translation length a(ϕ n ) = n log K. By the triangle inequality,

n log K ≤ d L (X, ϕ n (X)) ≤ nd L (X, ϕ(X)).
It follows that log K ≤ a(ϕ).

To show the reverse inequality, we use the following lemma.

Lemma 4.7 (Papadopoulos [START_REF] Papadopoulos | On Thurston's boundary of Teichmüller space and the extension of earthquakes[END_REF]). There is a constant C > 0 such that for any simple closed curve α and for any integer m,

i(ϕ m (λ 2 ), α) ≤ ℓ ϕ m (X) (α) ≤ i(ϕ m (λ 2 ), α) + C.
By this lemma, we have

ℓ ϕ(X) (α)) ℓ X (α) = ℓ ϕ m (X) (ϕ -1 (α)) ℓ ϕ m (X) (α) ≤ i(ϕ m (λ 2 ), ϕ -1 (α)) + C i(ϕ m (λ 2 ), α) = K m i(λ 2 , ϕ -1 (α)) + C K m i(λ 2 , α) = K m+1 i(λ 2 , α) + C K m i(λ 2 , α) .
By taking m → +∞, we have

ℓ ϕ(X) (α)) ℓ X (α) ≤ K.
Since α is arbitrary, we have

d L (X, ϕ(X)) ≤ log K.
Moreover, the infimum of a(ϕ) is attained at any point on the stretch line invariant under ϕ n . Now we show the following: Proof. By Theorem 4.3, there exists an integer n such that ϕ n leaves a stretch line r 1 invariant. Note that r i = ϕ(r i-1 ), i = 1, • • • , n -1 are also invariant by ϕ n . Choose any point X on r 1 and choose a geodesic segment between X and ϕ(X). Denote this geodesic segment by R 1 . We construct a path R in Teichmüller space by setting

R = R 1 ∪ R 2 • • • ∪ R n ∪ • • • where r k = ϕ(R k-1
). We also extend the path to the reverse direction by iteration of ϕ -1 . In this way, we get an infinite path R in Teichmüller space which is invariant by the action of ϕ.

We claim that R is a geodesic. To see this, consider the segment on R connecting ϕ -kn (X) and ϕ kn (X). The length of this segment is equal to 2kn times d L (X, ϕ(X)). On the other hand, both ϕ -kn (X) and ϕ kn (X) lie on the stretch line r 1 , and the Lipschitz distance between ϕ -kn (X) and ϕ kn (X) is equal to 2kn times log K (the entropy of ϕ). We have shown in Proposition 4.6 that d L (X, ϕ(X)) = log K. As a result, the length of the above segment on R connecting ϕ -kn (X) and ϕ kn (X) is equal to the Lipschitz distance between ϕ -kn (X) and ϕ kn (X). Since k is arbitrary, R is a d L -geodesic.

Combining Theorem 4.3 and Theorem 4.8, we obtain Theorem 1.3. Note that if ϕ is pseudo-Anosov with entropy log K, it was proved by Bers [START_REF] Bers | An extremal problem for quasiconformal mappings and a theorem by Thurston[END_REF] that b(ϕ) = log K. It follows that a(ϕ) = b(ϕ). From this, we deduce Theorem 1.4. The projection of a ϕ-invariant d L -geodesic on the moduli space is a closed d L -geodesic, which is homotopic to a closed Teichmüller geodesic with the same length.

Note that by results of Théret in [START_REF] Théret | À propos de la convergence des lignes d'étirement vers le bord de Thurston de l'espace de Teichmüller[END_REF], any two d L -geodesics left invariant by a pseudo-Anosov map are asymptotic, and two d L -geodesics left invariant by two distinct pseudo-Anosov maps are divergent. Suppose that ϕ ∈ Mod(S) is of infinite order, reducible and has no pseudo-Anosov reduced component. Since a(ϕ) ≤ 2b(ϕ) and b(ϕ) = 0, ϕ is parabolic, i. e., a(ϕ) = 0.

On the other hand, suppose that ϕ is of infinite order, reducible and has at least one pseudo-Anosov reduced component. Let Y be a pseudo-Anosov reduced component of ϕ. Up to taking a power of ϕ, we may assume that ϕ maps Y to Y . There is a pair of transverse measured foliations λ 1 and λ 2 on the surface Y such that

ϕ(λ 1 ) = 1 K λ 1 , ϕ(λ 2 ) = Kλ 2 ,
for some constant K > 1.

It follows that for any X ∈ T(S),

d L (X, ϕ(X)) ≥ log ℓ X (ϕ -1 (λ 1 )) ℓ X (λ 1 ) = log K.
As a result, a(ϕ) ≥ log K.

We deduce the following:

Theorem 5.2. ϕ is parabolic if and only if ϕ is infinite order, reducible and has no pseudo-Anosov reduced component.

Now suppose that ϕ is pseudo-hyperbolic. By Theorem 5.2, ϕ has at least one pseudo-Anosov reduced component. The maximum of the topological entropy on all of the pseudo-Anosov components gives a lower bound for a(ϕ). In fact, from Bers' results, it follows that a(ϕ) ≥ b(ϕ).

By Theorem 1.5, ϕ is reducible. Assume that ϕ is reduced by a maximal admissible set {C 1 , • • • , C k }. Up to a power of ϕ, we may assume that ϕ fixes each reduced component. Note that any power of ϕ is also pseudohyperbolic.

By definition, there exists a sequence (X k ) k≥1 in Teichmüller space such that d L (X k , ϕ(X k )) → a(ϕ). By Corollary 2.4, for any ǫ > 0, X k leaves the ǫ-thick part of T(S) for k sufficiently large. This means that ℓ(X k ) → 0.

Let Y be any pseudo-Anosov reduced component of Y . But assumption, ϕ fixes Y . Lemma 5.3. Let (α k ) k≥1 be a sequence of simple closed curves such that ℓ X k (α k ) → 0. For sufficiently large k, each α k is not contained in the interior of Y .

Proof. Suppose not and take a subsequence of (α k ) k≥1 , still denoted by (α k ), contained in the interior Y . Since ℓ X k (α k ) → 0 and d L (X k , ϕ(X k )) is uniformly bounded above, using the proof of Lemma 2.3, ϕ is reducible restricted on Y . This contradicts the assumption that Y is a pseudo-Anosov component.

A reducible and hyperbolic mapping class

In this section, we prove the following Proposition 6.1. On an arbitrary surface of genus ≥ 2, we can find a reducible mapping class which is hyperbolic.

Proof. Consider a pseudo-Anosov map ϕ on a hyperbolic surface of genus g with n > 0 boundary components, which we denote by Σ g,n in order to distinguish it from the surface S g,n with punctures. We assume that ϕ does not permute the complementary components of its stable laminations. We take two copies of this map ϕ, which we denote by ϕ 1 and ϕ 2 , and we denote the two underlying surfaces by Σ 1 and Σ 2 . We glue Σ 1 and Σ 2 in a symmetric manner along their boundary components and we obtain a homeomorphism of a closed hyperbolic surface S = S 2g+n-1,0 of genus 2g + n -1. We denote the homeomorphism ϕ 1 ∪ϕ 2 by ϕ. The surface S has an order-two symmetry ι which commutes with ϕ (i. e. ϕ • ι = ι • ϕ) and which fixes the multicurve α = ∂Σ 1 = ∂Σ 2 ֒→ S.

For j = 1, 2, let λ j ad µ j be the stable and unstable geodesic laminations of ϕ j :    ϕ j (λ j ) = kλ j ϕ j (µ j ) = 1 k µ j , λ > 1. The homeomorphism ϕ is reducible and with no Dehn twist on the reducing multicurve, and it fixes the measured geodesic laminations λ = λ 1 ∪ λ 2 and µ = µ 1 ∪ µ 2 ; cf. Figure 2. A component of S \ λ or of S \ µ may be of two types:

(1) an ideal polygon (that is, a hyperbolic polygon with cusps, or, equivalently, a surface isometric to the convex hull in H 2 of a finite number of points at infinity); (2) a crown (that is, a hyperbolic annulus with cusps on one of its boundary components). The two types of surfaces are represented in Figure 3.

The components of S \ λ or S \ µ that contain elements of the multicurve α can only be of the type [START_REF] Bers | A remark on Mumford's compactness theorem[END_REF]. We consider a component C of S \ µ and we lift it to the universal cover S. The map ϕ also lifts to S and we can choose a lift C of C such that it is fixed (setwise) by the lift ϕ of ϕ. The ideal points of C cut the circle at infinity into segments (Figure 5). Each segment between two ideal vertices of C contains in its interior exactly one fixed point; these fixed points of ϕ on S 1 ∞ are alternatively ideal vertices of lifts C µ of components C µ of S \ µ and ideal vertices of lifts C λ of components C λ of S \ λ and they are alternatively sinks and sources; cf. Thurston and Handel [START_REF] Handel | New proofs of some results of Nielsen[END_REF] and [START_REF] Casson | Automorphisms of surfaces after Nielsen and Thurston[END_REF].

Consider now a completion µ of µ such that all the extra leaves of µ are isolated. Since ϕ has no Dehn twists along its reducing multicurve, µ is fixed by ϕ. The measured geodesic lamination λ is totally transverse to µ, that is, λ ∈ ML(µ). Let F λ be a measured foliation corresponding to λ. By a result of Thurston (cf. Theorem 3.1 above), there exists a unique hyperbolic structure h on S such that F µ (h) = F λ , where F µ (h) is the horocyclic foliation. Lemma 6.2. We have F µ (ϕh) = ϕ(F µ (h)) = kF µ (h).

Proof. Since ϕ(µ = µ) and since h and ϕh are isometric hyperbolic structures on S, the leaves of the foliations F λ = F µ (h) and those of F µ (ϕh) have the same endpoints on the circle at infinity, once we lift the situation to the universal coverings. But these endpoints are those of λ and they are fixed by ϕ setwise. Hence F µ (ϕh) = F λ (here, equality forgets about the transverse measures), or, equivalently, λ µ (h) = λ µ (ϕh) = λ setwise. Let k ′ be the multiplicative factor:

λ µ (h) = k ′ λ µ (ϕh) = k ′ λ.
Then,

ℓ ϕh = ℓ h (ϕ -1 µ) = kℓ h (µ) = i(λ µ (ϕh), µ) = k ′ i(λ µ (h), µ) = k ′ ℓ h (µ).
Hence k = k ′ and the lemma is proved. Thus, ϕ fixes globally a stretch line and d T (h, ϕh) = log k, that is, ϕ acts by translation on that stretch line.

This proves proposition 6.1.

Theorem 1 . 1 (

 11 Nielsen-Thurston classification). Each ϕ ∈ Mod(S) is either periodic, reducible or pseudo-Anosov. Pseudo-Anosov mapping classes are neither periodic nor reducible.

Figure 1 .

 1 Figure 1. The horocyclic foliation.

Theorem 4 . 8 .

 48 A pseudo-Anosov map ϕ ∈ Mod(S) leaves a d L -geodesic invariant.

5 .

 5 Reducible maps By Bers' classification, we have Proposition 5.1. An element ϕ ∈ Mod(S) is of infinite order and reducible if and only if b(ϕ) is not realized. Moreover, such an element ϕ has no pseudo-Anosov reduced component if and only if b(ϕ) = 0.

  Figure 2.

Figure 3 .

 3 Figure 3. The two possible types of components of S \ λ and S \ µ.

Figure 4 .

 4 Figure 4. A crowned surface

  β

Figure 5 .

 5 Figure 5. The lift of an ideal polygon (left) and of a crown (right). β is the lift of the boundary component β of the crown.

Figure 6 .

 6 Figure 6. The dynamics on the universal cover.


Boris Gutman

Yalin Edu}

Wang

Tony Chan

Paul M Thompson

Arthur W Toga

Shape Registration with Spherical Cross Correlation

We present a framework for shape alignment that generalizes several existing methods. We assume that the shape is a closed genus zero surface. Our framework requires a diffeomorphic surface mapping to the 2-sphere which preserves rotation. Our similarity measure is a global spherical cross-correlation function of surface-intrinsic scalar attributes, weighted by the cross-correlation of the parameterization distortion. The final similarity measure may be customized according to the surfaceintrinsic scalar functions used in the application.

Introduction

Problems of shape alignment are ubiquitous in medical imaging. While many problem-specific solutions exist for particularly common cases (e.g. cortex, hippocampus) [START_REF] Shi | Direct mapping of hippocampal surfaces with intrinsic shape context[END_REF][START_REF] Tosun | Cortical surface alignment using geometry driven multispectral optical flow[END_REF][START_REF] Fischl | High-resolution intersubject averaging and a coordinate system for the cortical surface[END_REF], high quality general shape alignment remains very much an open problem. Our framework generalizes some existing methods without assuming the existence of any landmarks or data-specific features. We assume that the shape is a closed genus zero surface. Our framework requires a diffeomorphic surface mapping to the 2-sphere which preserves rotation. Our similarity measure is a global spherical cross-correlation function of surface-intrinsic scalar attributes, weighted by the cross-correlation of the parameterization distortion, sometimes known as the conformal factor. The final similarity measure may be customized according to the surface-intrinsic scalar functions used in the application. Higher order scalar functions such as mean and Gaussian curvature may be used in conjunction with low order ones, like distance to mass center, to incorporate more localized shape information in addition to global measures.

In this study we have used a global conformal mapping as the spherical parameterization, and only the distance to mass center as intrinsic scalar shape measure. Using our method, we created atlases and registered shapes from a population of hippocampi.

Previous work

There is doubtless a galaxy of existing general shape alignment methods, and due to space limitations we will only mention those most prevalent and those closest to the present work. Davies [START_REF] Davies | A minimum description length approach to statistical shape modeling[END_REF] has used an information-theoretic framework to formulate shape alignment as a minimum description length (MDL) problem. Here, each point is treated as an independent variable, while the cost function is aimed at reducing the "code length" of each shape's representation in this shape space. The beauty of this statistical approach lies in its ability to register multiple shapes simultaneously without the need to select a "target" shape. Of course, this is also its limitation, as for example when a known atlas exists and all data shapes are to be registered to it. The method requires a spherical diffeomorphism like ours.

Variants of the well-known ICP algorithm are another variety of recent developments in rigid shape alignment. Granger [START_REF] Granger | Multi-scale em-icp: A fast and robust approach for surface registration[END_REF] introduced the EM-ICP method. ICP's proclivity for terminating at suboptimal local minima is greatly reduced by treating the problem as a general expectation-maximization problem. A multiscale approach is used: at coarser scales, the blurring factor is sufficient to give crude but correct alignment, which is improved upon in later refined stages, where the algorithm approaches the original ICP. Though quite robust, the method depends on the scale of blurring factor being set correctly to avoid local minima. Thus, avoiding them is not guaranteed. Our method, by contrast, performs a global search non-iteratively. Thus, a global maximum correlation is guaranteed irrespective of the shape's original orientation without the need to tune any parameters.

Much like our algorithm, some previous methods have used spherical harmonics for rigid shape alignment. Among them are the first order ellipsoid (FOE) method, popularized by Brechbuhler [START_REF] Brechbuhler | Parameterization of closed surfaces for 3d shape description[END_REF] and used extensively in medical imaging [START_REF] Gerig | Shape versus size: Improved understanding of the morphology of brain structures[END_REF][START_REF] Gerig | Shape analysis of brain ventricles using spharm[END_REF] applications and SHREC, a recent variant of the ICP algorithm. Like ours, these methods make use of rotational properties of spherical harmonics.

FOE alignment uses the fact that a shape reconstructed from only the first order spherical harmonics forms an ellipsoid in object space. The method works well when the ellipsoid's three axes have distinct lengths, which largely depends on the shape itself and the degree to which the spherical parameterization preserves area. The method gives only a crude alignment and fails when two or more axes have similar lengths. Even with a proper ellipsoid, there is a symmetry problem.

SHREC [START_REF] Shen | Efficient registration of 3d spharm surfaces[END_REF] is another variation of the ICP. As in our case, the correspondence search is done iteratively on the sphere via Euler's rotation formula and icosahedral subdivision. The mapping satisfies our conditions, while the similarity measure is the RMSD. Since RMSD depends on the position of the object in space, the algorithm requires an initial pre-alignment in both spaces. Rigid Quaternion transform (RQT) is used to align shapes in R 3 after each iteration of parametric alignment. The main limitation of this algorithm is that it is not guaranteed to converge to the optimal solution (i.e. a correspondence which, when applied to RQT, minimizes RMSD). This is because the optimization in parameter space depends on the object's position in native space. Thus, though the search is more global than in the original ICP, the parameter space search is still locally biased. The original ICP suffers the same problem.

The last algorithm uses a brute force correlation: it computes a cost function anew for each rotation. To mitigate the cost of this, a hierarchical approach is used. Instead, we reduce parameter space alignment to a global refined search via the FFT. Our numerical scheme separates the effects on computation time of the level of detail used for alignment and the number of rotation samples. In SHREC, these two are tied together owing to their brute-force nature. This means that we can refine rotation space tessellation while maintaining the same level of surface detail without significantly affecting computation time. Our use of orientationinvariant shape attributes in conjunction with scale invariant cross-correlation makes our approach completely independent of changes in object position and size. SHREC, by contrast, requires volume normalization as a pre-processing step.

3 Mathematical preliminaries

Spherical harmonics

Spherical harmonics are functions f : S 2 → C which are simultaneously eigenfunctions of the Laplace-Beltrami and the angular momentum operators; they are expressed explicitly as

Y m l (θ, φ) = (2l + 1)(l -m)! 4(l + m)! P m l (cosθ)e imφ (1)
for degree and order m, l ∈ Z, |m| ≤ l , where P m l (x) is the associated Legendre polynomial. Spherical harmonics form a countable orthonormal basis for squareintegrable functions on the sphere. A projection of a function f ∈ L 2 (S 2 ) onto this basis yields the SPH coefficients

f (l, m) =< f, Y m l > (2)
where < f, g > is the usual L 2 inner product.

A key property of spherical harmonics is their behavior under a shift on the sphere. Given an element of the rotation group R ∈ SO(3), a rotated spherical harmonic is expressed as

Y m l (ω) = l n=-l Y n l (R -1 ω)D l m,n (R) (3)
where

D l m,n (R) = e -i(mα+nγ) d l m,n (β), (4)
α, β, γ are the Euler angles of R and d l m,n are irreducible representations of SO(3) [START_REF] Talman | Special functions: A group-theoretic approach[END_REF],

d l m,n (β) = t (-1) t × (l + n)!(l -n)!(l + m)!(l -m)! (l + n -t)!(l -m -t)!(t + m -n)!t! (5) ×(cos β 2 ) 2l+n-m-2t (sin β 2 ) 2t+m-n .
In particular, (3) implies that

f (ω) = g(R -1 ω) =⇒ f (l, m) = l n=-l g(l, n)D l m,n (R). ( 6
)

Discrete spherical cross correlation

Given two functions f, g ∈ L 2 (S 2 ), their spherical cross-correlation is defined as

C f,g (R) = S 2 f (ω)g(R -1 ω)dω. (7)
In the special case where

f (ω) = g(R ′-1 ω), C f,g (R) is maximized when R = R ′ ,
assuming that f is not spherically symmetric. For bandlimited functions, i.e. for those functions whose spherical harmonic coefficients vanish for all l ≥ B for some bandwidth B, the correlation becomes

C f,g (R) = B-1 l=0 l m=-l f (l, m) Λ(R)g(l, m). (8)
Here, Λ(R) is the operator associated with the rotation matrix. The expression for shifted spherical harmonic coefficients [START_REF] Brechbuhler | Parameterization of closed surfaces for 3d shape description[END_REF] implies that

C f,g (R) = l,m,n f (l, m) g(l, n)D l m,n (R). ( 9
)
This expression forms the basis of our similarity measure.

Fast cross correlation via FFT

The material presented so far has been used in the prior works we mentioned. Now, we present a simple lemma which leads to a great speed up in computing the correlation [START_REF] Shen | Efficient registration of 3d spharm surfaces[END_REF]. It suffices to make the observation that any rotation R(α, β, γ) may be expressed as a product of two rotations:

R(α, β, γ) = R 1 (α + π/2, π/2, 0)R 2 (β + π, π/2, γ + π/2). (10)
Now using the fact that we substitute ( 10) into ( 9) to obtain

D l m,n (R 1 R 2 ) = k D l m,k (R 1 )D l k,n (R 2 ), (11)
C f,g (R(α, β, γ)) = l,m,n,k f (l, m) g(l, n) (12)
×D l m,k (α + π/2, π/2, 0)D l k,n (β + π, π/2, γ + π/2) = l,m,n,k f (l, m) g(l, n)d l m,k (π/2)d l k,n (π/2
) × e i(m(α+π/2)+k(β+π)+n(γ+π/2))

= IF F T { l f (l, m) g(l, n)d l m,k (π/2)d l k,n (π/2)}(α + π/2, β + π, γ + π/2).
This simple result has been shown elsewhere [START_REF] Sorgi | Template gradient matching in spherical images[END_REF], but to the best of the authors' knowledge this is the first time it has been used for shape registration.

Shape registration with cross correlation

Similarity measure

Given a 2-manifold M ⊂ R 3 , a diffeomorphic spherical parameterization f : S 2 → M and a family of rotation-invariant shape attributes

s i : M → R, 0 < i ≤ N , let S i = s i • f. (13)
Then, given two manifolds M 1 , M 2 and their corresponding shape attribute maps {S 1,i } N i=0 , {S 2,i } N i=1 , we define our shape similarity measure as Fig. 2. Top surface from figure 1 is correlated to the bottom one, and its signal is shifted on the sphere without changing the shape's position in object space. The homologous points now appear to be in good correspondence between the surfaces.

S M1,M2 (R) = C λ1,λ2 (R) N i=1 κ i C S1,i,S2,i (R), (14)
where κ i are user-defined shape attribute weights, and λ 1 , λ 2 are spherical maps of the conformal factor of each manifold. These last two are used to mitigate the fact that scalar functions which appear similar on the sphere may in fact represent vastly differently-sized regions on the original surfaces due to varying area distortion of the spherical map.

Because we recover shifts in object space with shifts on S 2 , we require that the spherical parameterization preserve rotation in the following sense. Suppose

M 2 = R • M 1 , and f 1 , f 2 : S 2 → M 1 , M 2 are spherical maps. Then f 2 (ω) = R • f 1 (R -1 ω). (15)
Many existing parameterizations satisfy this requirement, e.g. [START_REF] Brechbuhler | Parameterization of closed surfaces for 3d shape description[END_REF][START_REF] Gu | Genus zero surface conformal mapping and its application to brain surface mapping[END_REF].

Previous methods as special cases

SHREC and FOE are special cases of our method. FOE simply takes the conformal factor λ to be constant and uses spherical harmonics up to order one only. This is equivalent to setting the bandwidth B to 2. The single shape attribute used is the Euclidian distance to the surface average value. This is roughly the same as distance to mass center, especially for area-preserving spherical maps with which FOE is typically used. SHREC minimizes RMSD, which can be reduced to spherical cross-correlation since 4πRM Thus, minimizing RMSD is equivalent to maximizing the correlation of the two shapes' spatial coordinates. These are, of course, not quite the scalar shape attributes we intend to use in our similarity measure. SHREC ′ s dependence on iterative RQT refinement for correspondence optimization makes it less robust. This is the price of using orientation-dependent features. Further, it is not clear whether the correspondence which, when applied to RQT, minimizes RMSD is truly the best correspondence. One can conceive two shapes with some patches quite similar and other very different. One may then like to align the two objects to get the best correspondence between the similar patches without regard to the different ones. In such a case, cross correlation of invariant features will achieve a better alignment. Still, SHREC could be made faster with the use of FFT-based correlation rather than a brute-force approach. SHREC requires O(B 3 N R ) operations, where B is again the bandwidth and N R the number of rotation samples. This is because recomputing D l m,n (α, β, γ) and the corresponding shifted spherical harmonic coefficients requires O(B 3 ) operations. Our method requires O(B 4 + N R log(N R )). N R is roughly also of the order B 3 even with hierarchical sampling; this means we have effectively reduced the order of operations from O(B 6 ) to O(B 4 ). This allows us to sample rotation space more finely without a significant change in execution time. It also makes using higher order coefficients and hence greater level of detail for alignment purposes feasible. We see this in experiments below.

SD 2 M1,M2 (R) = i∈{x,y,z} S 2 ||S 1,i -Λ(R)S 2,i || 2 (16) = i∈{x,y,z} S 2 [S 2 1,i + (Λ(R)S 2,i ) 2 -2S 1,i Λ(R)S 2,i ] = i∈{x,y,z} ||S 1,i || 2 2 + ||S 2,i || 2 2 -2C S1,i,S2,i (R)

Results

We used a population of 45 right hippocampal surfaces extracted from healthy elderly subjects. Our spherical parameterization was the global conformal map of Gu. et al. [START_REF] Gu | Genus zero surface conformal mapping and its application to brain surface mapping[END_REF]. Spherical harmonics were computed with the spherical FFT of [START_REF] Healy | Ffts for the 2-sphere-improvements and variations[END_REF] and cross correlation was computed with the help of the fftw library [START_REF] Frigo | A fast fourier transform compiler[END_REF]. SO(3) was sampled at 200 X 100 X 200 samples, which yielded an angle frequency of ∼ 1.8 degrees. A bandwidth of 64 was used throughout the testing. Only the distance to mass center was used as an invariant shape attribute for both populations.

As a preliminary experiment, we applied the cross correlation algorithm to a pair of hippocampal surfaces shown in figure 1. Here we see that the initial spherical maps do not provide a very good correspondence. Figure 2 shows the result of a spherical shift based on cross correlation. Figures 4 and5 show two pointwise averages of the shapes, before and after cross correlation. The improvement is obvious. For B = 64, the average running time for above experiment was 44.6 ± 3.6 seconds on a Gateway 7426GX Laptop with a 2.41 GHz AMD processor and 1 GB RAM, tested with 45 hippocampal surfaces. This includes computation of spherical harmonics, rotation matrices d l m,n and correlation. Shen [START_REF] Shen | Efficient registration of 3d spharm surfaces[END_REF] reports an average of 23.5 seconds running time on a common laptop for SHREC, while using B = 12. One would expect SHREC to take (64/12) 3 times longer for our bandwidth (see above), or on the order of 60 minutes. As a preliminary comparison, we implemented a modified version of SHREC. The only difference with the original is that at each iteration, our parameter alignment was initially done with cross correlation as described above, and subsequently refined according to the scheme outlined in [START_REF] Shen | Efficient registration of 3d spharm surfaces[END_REF]. Again, a bandwidth of 64 was used. Due to time limitations, we could not run this program on our whole dataset. We only present the results of one subject in figure 3. Here, we can see that the initial alignment determines the final result to a great degree. This example converged after only 4 iterations. Execution time was close to 7 minutes. Since the execution time reported in [START_REF] Shen | Efficient registration of 3d spharm surfaces[END_REF] was for a MATLAB implementation, while ours is in C++, and because we use a fast cross-correlation, this time is significantly lower than the 60 minute estimate.

To test the effect of higher coefficients on correlation quality, we increased the bandwidth to 128, while keeping the same rotation tessellation and correlated 6 of the 45 subjects in our HP population. Running time increased to 404 ± 10.4 seconds, while the shape distance to the target hippocampus decreased on average only 3.9 ± 9.4 %. To test for the effect of rotation sampling frequency, we also decreased N R to 100 X 50 X 100 samples, while keeping the bandwidth at 64. Running time was 38.2 ± 3.5 seconds, an insignificant improvement in speed. In another experiment, we limited our bandwidth to 16 and reran cross correlation based alignment while keeping angle resolution at 200 X 100 X 200. Execution time was reduced to about 3 seconds. We constructed hippocampal averages using FOE and two versions of cross correlation (B = 16 and B = 64) and compared results. First, a shape was selected, and all remaining shapes registered to it using each of the methods. Then, the shapes were averaged, normalized for volume (after registration, only for a fair distance comparison) and a rigid quaternion transform applied to each shape to align it both to the target subject and to the volume-normalized average shape. Our shape distance was defined as vertex-wise distance between the surfaces, weighted by product of the sum of areas of adjacent triangles in each mesh. The results are in table 1. The table does not show a notable fact: distance to the target subject was improved by our method for every subject compared to FOE. Minimal improvement was 8%, and maximal 117%. All but one subject registered with B = 64 had superior alignment to the result of using B = 16. Compared to FOE, B = 16 reduced shape distance for all but 5 subjects. Table 2 shows a summary of intra-subject differences by registration method. Using a bandwidth of 16 gives a significant improvement compared to FOE, but the results are still much improved by using a bandwidth of 64. Figures 6 -8 illustrate the hippocampal averages achieved with the three methods. Note that these averages were computed without spatially aligning the subjects to the target. Doing so would have likely given a more detailed shape. We should note here that the poor result of the FOE average is due in part to the large area distortion of the conformal map. The tail of the hippocampus is mapped to such a small region on S 2 that it is very hard to match well. We see in figure 7 that the tail suffers the most. The resulting spherical harmonic representation contains much redundant information; hence, the area distortion partially cancels out the benefits of using high order coefficients. The mapping used in [START_REF] Brechbuhler | Parameterization of closed surfaces for 3d shape description[END_REF][START_REF] Shen | Efficient registration of 3d spharm surfaces[END_REF] is by contrast area-preserving, and hence better suited for alignment. The results of [START_REF] Brechbuhler | Parameterization of closed surfaces for 3d shape description[END_REF][START_REF] Gerig | Shape analysis of brain ventricles using spharm[END_REF] look closer to the one achieved here with cross correlation. This is, however, indicative of the potential our method has when applied to area-preserving spherical maps, and we intend to experiment with this idea in the immediate future.

Conclusion

We have presented a framework for shape alignment which generalizes several existing methods. Our method is robust, fast and allows for use of greater detail in alignment than was possible before. The correspondence search is performed globally and no pre-alignment is required; thus, the result and computation time are independent of the shape's size and initial orientation. Reaching a global maximum is always guaranteed. Our method can be tailored to suit a particular application by selecting the appropriate shape features for a particular data type. We intend to experiment with various shape attributes, apply area-preserving spherical maps to our method and extend the technique to automated patch selection and matching. Lastly, all software used here is available thorugh the LONI Pipeline environment. Please contact the authors for more details.

Fig. 1 .

 1 Fig. 1. Two randomly selected hippo surfaces. The red circle is homologous to the north pole on the sphere; the blue circle, to the south pole; and the blue line, to φ = 0. The initial spherical mappings do not align these two very well.

Fig. 3 .

 3 Fig. 3. Top surface from figure 1 is initially aligned using FOE (top row) and subsequently aligned using a modified version of SHREC (bottom row). Though this is slightly different from the original SHREC, in principle the two algorithms are the same. Here, we see the local minimum problem suffered by SHREC, typical of an ICP-type algorithm: there is very little change in alignment after initialization.

Fig. 4 .

 4 Fig. 4. Average of two shapes from figure 1 before correlation.

Fig. 5 .

 5 Fig. 5. Average of two shapes from figure 1 after correlation.

Fig. 6 .

 6 Fig. 6. Average of 45 right hippocampi with cross correlation, bandwidth = 64.

Fig. 7 .

 7 Fig. 7. Average of 45 right hippocampi with FOE.

Fig. 8 .

 8 Fig. 8. Average of 45 right hippocampi with cross correlation, bandwidth = 16.

Table 1 .

 1 Weighted point-wise distance of 45 (44) hippocampal surfaces by registration method.

	Method Distance to Target Distance to Average
	FOE	4.14 ± 1.33	2.89 ± 1.05
	Cross cor.	2.64 ± 0.92	2.36 ± 0.75
	B = 16	3.38 ± 1.06	2.33 ± 0.85

Table 2 .

 2 Intra-subject improvement by registration method, in percentage of the second method's result.

	common shape FOE vs. B = 64 FOE vs. B = 16 B = 16 vs. B = 64
	target	61 % ± 26 %	25 % ± 23 %	30 % ± 18 %
	average	24 % ± 29 %	26 % ± 22 %	-6 % ± 29 %


❆②❛❝❤❡

i ε i

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t❤❡ t✇♦ s✉r❢❛❝❡s ✐s t❤❡ •❛r❣❡st ♣♦ss✐❜•❡✳ ❚❤✐s ❞✐s✲ t❛♥❝❡ ❜❡t✇❡❡♥ s❤❛♣❡s ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❤♦✇ s❤❛♣❡s ❛r❡ ♣❛r❛♠❡tr✐③❡❞ ❛♥❞ ❞♦❡s ♥♦t ❛ss✉♠❡ ♣♦✐♥t ❝♦rr❡s♣♦♥❞❡♥❝❡s ❜❡t✇❡❡♥ s❤❛♣❡s✳ ❋♦r ❝♦♠♣✉t❛t✐♦♥❛• ♣✉r♣♦s❡s✱ ✇❡ s✉♣♣♦s❡✱ ❢r♦♠ ♥♦✇ ♦♥✇❛r❞s✱ t❤❛t W ✐s ❛ r❡♣r♦❞✉❝✐❜•❡ ❦❡r♥❡• ❍✐•❜❡rt s♣❛❝❡ ✭r✳❦✳❤✳s✮ ✇✐t❤ ❦❡r♥❡• K ❬✶✹(✳ ■♥ t❤✐s s❡tt✐♥❣✱ t❤❡ s♣❛❝❡ ♦❢ ❝✉rr❡♥ts ✐s t❤❡ ❞❡♥s❡ s♣❛♥ ♦❢ ❛•• ❉✐r❛❝ ❞❡•t❛ ❝✉rr❡♥ts δ α x ✱ ✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ ❜② δ α x (ω) = ω(x), α R 3 ❢♦r ❛♥② ω ∈ W ✳ ❆ ❉✐r❛❝ ❝✉rr❡♥t ♠❛② ❜❡ s❡❡♥ ❛s ❛ t❛♥❣❡♥t ✭r❡s♣✳ ♥♦r♠❛•✮ α ❡♥t✐r❡•② ❝♦♥❝❡♥tr❛t❡❞ ❛t ♣♦✐♥t x✳ ❆•t❤♦✉❣❤ ❛ ❝✉r✈❡ ✭r❡s♣✳ ❛ s✉r❢❛❝❡✮ ❤❛s ❛♥ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ t❛♥❣❡♥ts ✭r❡s♣✳ ♥♦r♠❛•s✮✱ ❛ ♣♦•②❣♦♥❛• •✐♥❡s ✭r❡s♣✳ ❛ ♠❡s❤✮ ♠❛② ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❝✉rr❡♥ts ❜② ❛ ✜♥✐t❡ s✉♠ k δ α k x k ✇❤❡r❡ x k ✐s t❤❡ ❝❡♥t❡r ♦❢ t❤❡ s❡❣♠❡♥t ✭r❡s♣✳ ❝❡♥t❡r ♦❢ t❤❡ ♠❡s❤ ❝❡••✮ ❛♥❞ α k t❤❡ t❛♥❣❡♥t ♦❢ t❤❡ •✐♥❡ ✭r❡s♣✳ t❤❡ ♥♦r♠❛• ♦❢ t❤❡ s✉r❢❛❝❡✮ ❛t x k ✳ ▼♦r❡♦✈❡r✱ ❜❛s❡❞ ♦♥ t❤❡ |✐❡s③ t❤❡♦r❡♠✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t ❢♦r ❛♥② ❝✉rr❡♥t T ✱ t❤❡ ✈❡❝t♦r ✜❡•❞ ✭✇✐t❤✐♥ ❊q✳ ✷✳✸✮ t❤❛t ❛❝❤✐❡✈❡s ❙✉♣ ω W ≤1 |T (ω)| ❡①✐sts ❛♥❞ ✐s ✉♥✐q✉❡✳ ❲❡ ❞❡♥♦t❡ t❤✐s ✈❡❝t♦r ✜❡•❞ L -1 W (T ) ❛♥❞ ❝❛•• ✐t t❤❡ ❞✉❛• r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ T ✳ L W ✐s ✐s♦♠❡tr✐❝✿ ✐t ♣r♦✈✐❞❡s W * ✇✐t❤ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t✿ T 2 W * = T, T W * = L -1 W (T ), L -1 W (T ) W ✭✷✳✹✮ ❖♥ ❉✐r❛❝s ✇❡ ❤❛✈❡✿ L -1 W (δ α x )(y) = K(y, x)α ✭✷✳✺✮ ❚❤✐s ❡q✉❛t✐♦♥ s❤♦✇s t❤❛t K ✐s t❤❡ •r❡❡♥ ❦❡r♥❡• ♦❢ L W ✳ ❆♣♣•②✐♥❣ ❊q✳ ✷✳✹ ❛♥❞ ❊q✳ ✷✳✺ •❡❛❞s t♦✿ δ α x , δ β y W * = α t K(x, y)β ✭✷✳✻✮ ❚❤✐s ❣✐✈❡s ❜② •✐♥❡❛r✐t② ❡①♣•✐❝✐t ❛♥❞ ❡❛s✐•② tr❛❝t❛❜•❡ ❢♦r♠✉•❛ t♦ ❝♦♠♣✉t❡ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ✭❛♥❞ t❤❡♥ t❤❡ ❞✐st❛♥❝❡✮ ❜❡t✇❡❡♥ t✇♦ s❤❛♣❡s T = i δ αi xi ❛♥❞ T ′ = j δ βj yj ✿ T, T ′ W * = i j α t i K(x i , y j )β j ✳ ❚❤❡ ♠❛♣♣✐♥❣ L -1 W ✐s ❛ s✐♠✲ ♣•❡ ❝♦♥✈♦•✉t✐♦♥ ♦♣❡r❛t♦r✱ ❛s ✐••✉str❛t❡❞ ✐♥ ❋✐❣✳✷✲❛✳ ❖♥ t❤❡ ❝♦♥tr❛r②✱ ✜♥❞✐♥❣ t❤❡ ❝✉rr❡♥t T ✇❤♦s❡ ❛ss♦❝✐❛t❡❞ ✈❡❝t♦r ✜❡•❞ γ ✐s ❣✐✈❡♥ ✭T = L W (γ)✮ ✐s ❛♥ ✐••✲♣♦s❡❞ ❞❡❝♦♥✈♦•✉t✐♦♥ ♣r♦❜•❡♠ t❤❛t r❡q✉✐r❡s s♣❡❝✐✜❝ ♥✉♠❡r✐❝❛• s❝❤❡♠❡ ❛s t❤❡ ♦♥❡ ✇❡ ✇✐•• ♣r❡s❡♥t ✐♥ ❙❡❝t✐♦♥ ✸✳✸✳ ❚❤❡ ❦❡r♥❡• K ❡♥❛❜•❡s ❛•s♦ t♦ ❞❡✜♥❡ ❛♥ ❛ss♦❝✐❛t❡❞ •❛✉ss✐❛♥ ♥♦✐s❡ ♦♥ ❝✉r✲ r❡♥ts ε✳ ■♥ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥✱ ε ♠❛♣s ❡✈❡r② ❝✉rr❡♥ts t♦ ❛ •❛✉ss✐❛♥ ✈❛r✐❛❜•❡ s✉❝❤ t❤❛t Cov(ε(δ α x ), ε(δ β y )) = α t K(x, y)β✳ ■❢ ✇❡ s❡t ❛ ❣r✐❞ Λ = {x p } p=1...N ✱ t❤❡ s♣❛♥ ♦❢ (δ xp ) p∈Λ ❞❡✜♥❡s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• s♣❛❝❡ ♦❢ ❝✉rr❡♥ts✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ε ✐s ♦❢ t❤❡ ❢♦r♠ p∈Λ δ αp xp ✇❤❡r❡ (α i ) ❛r❡ ❝❡♥t❡r❡❞ •❛✉ss✐❛♥ ✈❛r✐❛❜•❡s ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① K -1 ✇❤❡r❡ K = (K(x i , x j )) i,j∈Λ ✳ ■♥ t❤✐s •❛st ❝❛s❡✱ ε ❤❛s ❛ ♣r♦❜❛❜✐•✐t② ❞❡♥✲ s✐t② ❢✉♥❝t✐♦♥ ✭♣❞❢✮ ✇❤✐❝❤ ✐s ♣r♦♣♦rt✐♦♥♥❛• t♦ exp(-ε 2 W * )✳ ❋♦r ♦✉r ❛♣♣•✐❝❛t✐♦♥s ✇❡ ✇✐•• ❝❤♦♦s❡ ❛ •❛✉ss✐❛♥ ❦❡r♥❡•✿ K(x, y) = ❡①♣(-x -y 2 /λ 2 W )Id✳ ❚♦ ♠❛❦❡ s❡♥s❡ ♦❢ t❤❡ ♠♦❞❡• ❊q✳✶✳✶✱ ♦♥❡ ♠✉st st✐•• s♣❡❝✐❢② ❤♦✇ t❤✐s ♠♦❞✲ ❡••✐♥❣ ❜❛s❡❞ ♦♥ ❝✉rr❡♥ts ♠❛② ❜❡ ❝♦✉♣•❡❞ ✇✐t❤ ❛ ❞❡❢♦r♠❛t✐♦♥ ❢r❛♠❡✇♦r❦✳ ■❢ φ ✐s ❞✐✛❡♦♠♦r♣❤✐s♠ ❛♥❞ S ❛ s✉r❢❛❝❡✱ t❤❡ ✢✉① ♦❢ ω t❤r♦✉❣❤ t❤❡ ❞❡❢♦r♠❡❞ s❤❛♣❡ φ(S)✱ ❞❡♥♦t❡❞ ❢♦r ❣❡♥❡r❛• ❝✉rr❡♥ts (φ ♯ ⋆ S)(ω)✱ ✐s ❡q✉❛•s t♦ t❤❡ ✢✉① t❤r♦✉❣❤ S ♦❢ t❤❡ ♣✉••❡❞✲❜❛❝❦ ✈❡❝t♦r ✜❡•❞ φ ♯ ⋆ ω ✇❤✐❝❤ ✐s ❣✐✈❡♥ ❜② t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s ❢♦r♠✉•❛ ✇✐t❤✐♥ t❤❡ ✢✉① ✐♥t❡❣r❛•✳ ❚❤✐s ❡♥❛❜•❡s t♦ ❞❡✜♥❡ ❛ ❣❡♥❡r❛• ♣✉s❤✲❢♦r✇❛r❞ ❛❝t✐♦♥ ♦❢ ❛ ❞✐✛❡♦♠♦r♣❤✐s♠ ♦♥ ❛♥② ❝✉rr❡♥ts✳ ❚❤✐s ❛❝t✐♦♥ r❡♣•❛❝❡s ❢♦r ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s t❤❡ ✉s✉❛• ❛❝t✐♦♥ ♦♥ ✐♠❛❣❡s✿ (φ ⋆ I)(x) = I(φ -1 (x))✳ ❚❤✐s ✐s ❤❡r❡ s•✐❣❤t•② ♠♦r❡ ❝♦♠♣•❡① s✐♥❝❡ ✇❡ ❞♦ ♥♦t tr❛♥s♣♦rt ♣♦✐♥ts ❜✉t t❛♥❣❡♥ts ♦r ♥♦r♠❛•s ✭❞✐✛❡r❡♥t✐❛• ✶ ❛♥❞ ✷✲❢♦r♠s t♦ ❜❡ ❡✈❡♥ ♠♦r❡ ♣r❡❝✐s❡✮✳ ❖♥ t❤❡ ❜❛s✐s ❡•❡♠❡♥ts✱ t❤❡ ♣✉s❤✲❢♦r✇❛r❞ ❛❝t✐♦♥ ❣✐✈❡s✿ φ ♯ ⋆ δ α x = δ dxφ(α) φ(x) ✭✷✳✼✮ ✐❢ α ✐s ❛ t❛♥❣❡♥t ♦❢ ❛ ❝✉r✈❡ ❛♥❞ φ ♯ ⋆ δ u×v x = δ dxφ(u)×dxφ(v) φ(x) ✭✷✳✽✮ ✐❢ u×v ✐s t❤❡ ♥♦r♠❛• ♦❢ ❛ s✉r❢❛❝❡✳ ❖♥❡ ♥♦t✐❝❡s t❤❛t d x φ(u)×d x φ(v) = |d x φ| d x φ -t (u× v)✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣✱ ✇❡ r❡str✐❝t t❤❡ ❞❡❢♦r♠❛t✐♦♥s φ t♦ ❜❡•♦♥❣ t♦ t❤❡ ❣r♦✉♣ ♦❢ ❞✐❢✲ ❢❡♦♠♦r♣❤✐s♠s s❡t ✉♣ ✐♥ ❬✶✺✱ ✶✻(✿ t❤❡ ❞✐✛❡♦♠♦r♣❤✐s♠s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ✐♥t❡❣r❛t✐♦♥ ♦❢ ❛ t✐♠❡✲✈❛r②✐♥❣ ✈❡❝t♦r ✜❡•❞ v t ✿ ∂ t φ t = v t • φ t ✳ ❚❤❡ ❣❡♦❞❡s✐❝ ✢♦✇s (φ vt t ) t∈[0,1] ❛r❡ ❝♦♠♣•❡t❡•② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✐♥✐t✐❛• ✈❡❝t♦r s♣❡❡❞ v 0 ✇❤✐❝❤ ❜❡•♦♥❣s t♦ ❛ r✳❦✳❤✳s✳ V ✳ ❚❤❡ ✜♥❛• ❞✐✛❡♦♠♦r♣❤✐s♠ ✐s ❞❡♥♦t❡❞ φ v0 ✳ ❍♦✇ t♦ ✜♥❞ s✉❝❤ ❛ ❞✐✛❡♦♠♦r♣❤✐s♠ t❤❛t ❜❡st ♠❛t❝❤❡s t✇♦ s❡ts ♦❢ ❝✉rr❡♥ts ✐s ❡①♣•❛✐♥❡❞ ✐♥ ❞❡t❛✐• ✐♥ ❬✶✶✱ ✶✸(✳ ✸ ❏♦✐♥t ❊st✐♠❛t✐♦♥ ♦❢ ❚❡♠♣•❛t❡ ❛♥❞ ❉❡❢♦r♠❛t✐♦♥s ✸✳✶ ❆ ❍❡✉r✐st✐❝ ▼❛①✐♠✉♠ ❆ P♦st❡r✐♦r✐ ✐♥ ■♥✜♥✐t❡ ❉✐♠❡♥s✐♦♥ ❋r♦♠ ❛ ❇❛②❡s✐❛♥ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✐♥ ❊q✳✶✳✶ ✭T i = φ i,♯ ⋆ T + ε i ✮✱ T i ❛r❡ t❤❡ ♦❜s❡r✈❛✲ t✐♦♥s✱ T ✐s ✉♥❦♥♦✇♥✱ φ i ❛r❡ ❤✐❞❞❡♥ ✈❛r✐❛❜•❡s ❛♥❞ ε i ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✐❞❡♥t✐❝❛••② •❛✉ss✐❛♥ ♥♦✐s❡ ✇✐t❤ ❦♥♦✇♥ ✈❛r✐❛♥❝❡✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡✲ ✜♥❡ •❛✉ss✐❛♥ ♣r♦❜❛❜✐•✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥s ✭♣❞❢✮ ♦♥ t❤❡ s♣❛❝❡ ♦❢ ❈✉rr❡♥ts W * ❛♥❞ ♦♥ t❤❡ s♣❛❝❡ ♦❢ ✐♥✐t✐❛• ✈❡❝t♦r ✜❡•❞s V ✿ p ε (ε) = C ε exp(-ε 2 W * /σ 2 W ) ❛♥❞ p φ (v) = C φ exp(-v 2 V /σ 2 V )✳ ■♥ t❤❛t ❝❛s❡✱ ❛ ▼❛①✐♠✉♠ ❆ P♦st❡r✐♦r✐ ✭▼❆P✮ ❡s✲ t✐♠❛t✐♦♥ ❢♦r ✐♥❞❡♣❡♥❞❡♥t ♦❜s❡r✈❛t✐♦♥s ♠❛①✐♠✐③❡s ♠❛① T N i p(T i | T )✳ ❋♦r♠❛••②✱ p(T i | T ) = p ε (T i| T , v i 0 )p φ (v i 0 )dv 0 = p ε (T i -φ v i 0 i T )p φ (v i 0 )dv 0 ❙✐♥❝❡ t❤❡ t❡r♠ ✇✐t❤✐♥ t❤❡ ✐♥t❡❣r❛• ❞❡♣❡♥❞s ♦♥ v 0 ❜② ❛ ❣❡♦❞❡s✐❝ s❤♦♦t✐♥❣ ♦❢ ❞✐✛❡♦✲ ♠♦r♣❤✐s♠s✱ t❤❡r❡ ❛r❡ ♥♦ ❝•♦s❡❞ ❢♦r♠s ❢♦r t❤✐s •✐❦❡•✐❤♦♦❞✳ ❆ ✉s✉❛• ❛♣♣r♦①✐♠❛t✐♦♥ ❝♦♥s✐sts ✐♥ r❡♣•❛❝✐♥❣ t❤❡ ✐♥t❡❣r❛• ❜② t❤❡ ♠❛①✐♠✉♠ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ✇✐t❤✐♥ t❤❡ ✐♥t❡❣r❛• ✭✐✳❡✳ ✐ts ✜rst ♠♦❞❡✮✳ ❚❤✐s •❡❛❞s t♦✿ p(T i | T ) ∼ ♠❛① v i 0 p ε (T i -φ v i 0 i T )p φ (v i 0 ) ✇❤✐❝❤ ✜♥❛••② ❣✐✈❡s✿ ( T , φ v i 0 ) = ❛r❣♠✐♥ T ,v i 0 N i=1 1 σ 2 V v i 0 2 V + 1 σ 2 W T i -φ v i 0 . T 2 W * ✭✸✳✶✮ ❝❛••❡❞ ❋❛st ❆♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ ▼♦❞❡ ✭❋❆▼✮✳ ■❢ ✇❡ ❛•r❡❛❞② ❞❡✜♥❡❞ •❛✉ss✐❛♥ ✈❛r✐❛❜•❡s ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❝✉rr❡♥ts ✭❙❡❝t✐♦♥ ✷✮✱ s✉❝❤ ✈❛r✐❛❜•❡s ❤❛✈❡ ♥♦ ♣❞❢ ❜❡❝❛✉s❡ ❯♥❜✐❛s❡❞ ❆t•❛s❡s ♦❢ ❈✉r✈❡s ❛♥❞ ❙✉r❢❛❝❡s ✼✸ W * ❛♥❞ V ❛r❡ ♦❢ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥✳ ❆ ♠♦r❡ r✐❣♦r♦✉s ▼❆P ❞❡r✐✈❛t✐♦♥ ❝♦✉•❞ ❜❡ ❞♦♥❡ ❝♦♥s✐❞❡r✐♥❣ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ v 0 ✬s ❛♥❞ ♦❢ t❤❡ T i ✬s✳ ❋♦r ✐♥st❛♥❝❡ ❝✉rr❡♥ts ♠❛② ❜❡ ♣r♦•❡❝t❡❞ ✐♥t♦ ❛ ✜①❡❞ ❣r✐❞❀ ❜✉t t❤✐s ✇♦✉•❞ r❡q✉✐r❡ t♦ ❛❞❛♣t t❤❡ r❡❣✐str❛t✐♦♥ s❝❤❡♠❡ ♦❢ ❬✶✶( t♦ ❛❝❝♦✉♥t ❢♦r s✉❝❤ ❛ ❞✐s❝r❡t✐③❛t✐♦♥✳ ▼❛r❦♦✈ ❈❤❛✐♥ ▼♦♥t❡ ❈❛r•♦ ✭▼❈▼❈✮ ❛♣♣r♦❛❝❤❡s ❢♦r s❛♠♣•✐♥❣ t❤❡ ♣♦st❡r✐♦r ❝♦✉•❞ ❜❡ ❛•s♦ ♣♦ss✐❜•❡ ❛•♦♥❣ t❤❡ •✐♥❡s ♦❢ ❬✶✼( ❜✉t t❤✐s ✐s st✐•• ❝❤❛••❡♥❣✐♥❣ ❛♥❞ ♦✉t ♦❢ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ♣❛♣❡r✳ ❆ ❛•t❡r♥❛t❡❞ ▼✐♥✐♠✐③❛t✐♦♥ Pr♦❝❡❞✉r❡ ❲❡ s♦•✈❡ ❊q✳✸✳✶ ❜② ♠✐♥✐♠✐③✐♥❣ ✐t ❛•t❡r♥❛t✐✈❡•② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t❡♠♣•❛t❡ ❛♥❞ t♦ t❤❡ ❞❡❢♦r♠❛t✐♦♥s✳ ❲❤❡♥ t❤❡ t❡♠♣•❛t❡ T ✐s ✜①❡❞✱ ❡❛❝❤ t❡r♠ ♦❢ ✸✳✶ ❝❛♥ ❜❡ ♠✐♥✐♠✐③❡❞ s❡♣❛r❛t❡•②✳ ❋♦r ❛ ❣✐✈❡♥ ♦❜s❡r✈❛t✐♦♥ T i ✱ ♠✐♥✐♠✐③✐♥❣ 1 σ 2 V v i 0 2 V + 1 σ 2 W T i -φ v i 0 i . T 2 W * ✇✐t❤ r❡s♣❡❝t t♦ v i 0 ✐s ❡①❛❝t•② ❛ r❡❣✐str❛t✐♦♥ ♣r♦❜•❡♠✱ ❛s st❛t❡❞ ❛♥❞ s♦•✈❡❞ ✐♥ ❬✶✶✱ ✶✸(✳ ❚❤✐s st❡♣ ♦❢ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ❝♦♥s✐sts t❤❡r❡❢♦r❡ ♦❢ N r❡❣✐str❛t✐♦♥s ♦❢ t❤❡ t❡♠♣•❛t❡ T t♦ ❡❛❝❤ ♦❜s❡r✈❛t✐♦♥ T i ✳ ❲❤❡♥ t❤❡ ❞❡❢♦r♠❛t✐♦♥s φ i ❛r❡ ✜①❡❞ ❢♦r ❡✈❡r② i = 1 . . . N ✱ ♠✐♥✐♠✐③✐♥❣ ✸✳✶ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t❡♠♣•❛t❡ T •❡❛❞s t♦ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢✿ J( T ) = 1 2 N i=1 φ ♯,i ⋆ T -T i 2 W * ✭✸✳✷✮ ■❢ ❛•• φ i = ■❞ ✭✐✳❡✳ ♥♦ ❞❡❢♦r♠❛t✐♦♥✮✱ t❤❡ ♠✐♥✐♠✉♠ ✐s r❡❛❝❤❡❞ ❛t t❤❡ ❡♠♣✐r✐❝❛• ♠❡❛♥✿ T = 1 N N i T i ✳ ❋♦r ❛r❜✐tr❛r② ❞❡❢♦r♠❛t✐♦♥s✱ t❤❡r❡ ✐s ♥♦ ❝•♦s❡❞ ❢♦r♠ ❛♥❞ ✇❡ ✉s❡ ❛ ❣r❛❞✐❡♥t ❞❡s❝❡♥t s❝❤❡♠❡✳ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ ❊q✳✸✳✷ ✐s ♣r❡❝✐s❡•②✿ ∇ T J = N i=1 φ * ♯,i ⋆ (φ ♯,i ⋆ T -T i ) ✭✸✳✸✮ ✇❤❡r❡ φ * ♯ ✐s t❤❡ ❛❞•♦✐♥t ❛❝t✐♦♥ ♦❢ φ ♯ ✿ φ * ♯ ⋆ T, T ′ W * = T, φ ♯ ⋆ T ′ W * ❢♦r ❛♥② ❝✉rr❡♥ts T ❛♥❞ T ′ ✳ ❚❤✐s ✇♦✉•❞ ❜❡ ❛ ♠❛tr✐① tr❛♥s♣♦s❡ ✐❢ t❤❡ ❛❝t✐♦♥ ✇❛s •✐♥❡❛r✳ ■♥ t❤✐s ♥♦♥✲•✐♥❡❛r s❡tt✐♥❣✱ st❛♥❞❛r❞ ❝♦♠♣✉t❛t✐♦♥s ❣✐✈❡ φ * ♯ ⋆ T = L W (φ ♯ ⋆ L -1 W (T ))✳ ❲✐t❤ t❤❡ ❜❛❝❦✇❛r❞ s❝❤❡♠❡✱ ❊q✳✸✳✷ ✇♦✉•❞ ❜❡✿ J( T ) = 1 2 N i=1 T -φ ♯,i ⋆ T i 2 W * ✱ ✇❤♦s❡ ♠✐♥✐♠✉♠ ❤❛s t❤❡ ❝•♦s❡❞ ❢♦r♠ T = 1 N N i=1 φ ♯,i ⋆ T i ✳ ❲❡ s❡❡ ❤❡r❡ ✇❤② t❤❡ ❛t•❛s ❜✉✐•❞✐♥❣ st❡♣ ✐s ❝♦♠♣✉t❛t✐♦♥❛••② ♠♦r❡ ❞✐✣❝✉•t ✐♥ t❤❡ ❢♦r✇❛r❞ s❡tt✐♥❣✳ ❍♦✇❡✈❡r✱ ❝♦♠♣✉t✐♥❣ t❤❡ •✐❦❡•✐❤♦♦❞ ♦❢ ❛♥② ♥❡✇ ♦❜s❡r✈❛t✐♦♥s p(T i | T ) ✇✐•• ❜❡ ♠✉❝❤ s✐♠♣•❡r ❛♥❞ ❢❛st❡r ✐♥ t❤✐s s❡tt✐♥❣✳ ❚❤❡ ✐♥♣✉t s❤❛♣❡s T i ❛r❡ s❛♠♣•❡❞ ♦❜•❡❝ts ✇❤✐❝❤ ❛r❡ ❛♣♣r♦①✐♠❛t❡❞ ❛s ✜♥✐t❡ s❡t ♦❢ ❉✐r❛❝ ❝✉rr❡♥ts✳ ❆s ✐t ✇✐•• ❛♣♣❡❛r ❢r♦♠ t❤✐s ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❝❡❞✉r❡✱ t❤❡ t❡♠♣•❛t❡ ✇✐•• ❛•s♦ ❛•✇❛②s r❡♠❛✐♥ ❛ ✜♥✐t❡ s❡t ♦❢ ❉✐r❛❝ ❝✉rr❡♥ts ❛t ❡✈❡r② ✐t❡r❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝✉rr❡♥t φ ♯,i ⋆ T -T i ✐s ♦❢ t❤❡ ❢♦r♠ k δ β i k y i k ✇❤✐❝❤ ✜♥❛••② •❡❛❞s t♦ φ ♯ i ⋆ L -1 W (φ ♯,i ⋆ T -T i ) = k (d x φ i ) t K(φ i (x), y i k )β i k ✐♥ ❝❛s❡ ♦❢ ❝✉r✈❡s✳ ❋✐♥❛••②✱ t❤❡ ✼✹ ❙✳ ❉✉rr•❡♠❛♥ ❡t ❛•✳ ✈❡❝t♦r ✜❡•❞ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❡♥❡r❣② ✸✳✷ ✐♥ ❝❛s❡ ♦❢ ❝✉r✈❡s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛t ❛♥② ♣♦✐♥t x ♦❢ t❤❡ s♣❛❝❡✿ L -1 W (∇ T J)(x) = N i=1 (d x φ i ) t k K(φ i (x), y i k )β i k ✭✸✳✹✮ ❋♦r s✉r❢❛❝❡s✱ d x φ t i ♠✉st ❜❡ r❡♣•❛❝❡❞ ❜② |d x φ| d x φ -1 ✳ ❆t t❤✐s st❛❣❡✱ ✇❡ s❡❡ t❤❛t t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t s❝❤❡♠❡ ❝♦✉•❞ ♥♦t ❜❡ ♣❡r❢♦r♠❡❞ ✇✐t❤♦✉t ❛♥ ❡✣❝✐❡♥t ♥✉♠❡r✐❝❛• ❛•❣♦r✐t❤♠ t♦ ❡st✐♠❛t❡ t❤❡ ❝✉rr❡♥t ∇J ✇❤♦s❡ ❛ss♦❝✐❛t❡❞ ✈❡❝t♦r ✜❡•❞ ✐s ❣✐✈❡♥ ❜② ❊q✳✸✳✹✳ ❚❤❡ s♣❛rs❡ ❞❡❝♦♥✈♦•✉t✐♦♥ ♠❡t❤♦❞ ♦❢ ❙❡❝t✐♦♥ ✸✳✸ ♣r❡❝✐s❡•② ♣r♦✈✐❞❡s ❛ ✜♥✐t❡ s❡t ♦❢ ❉✐r❛❝ ❝✉rr❡♥t ✇❤✐❝❤ ❛♣♣r♦①✐♠❛t❡s ∇J ❛t ❛♥② ❛r❜✐tr❛r② ❛❝❝✉r❛❝②✱ s♦ t❤❛t t❤❡ t❡♠♣•❛t❡ r❡♠❛✐♥s ❛ ✜♥✐t❡ s❡t ♦❢ ❉✐r❛❝ ❝✉rr❡♥ts ❛t ❡❛❝❤ ✐t❡r❛t✐♦♥✳ ❲❡ ✐♥✐t✐❛•✐③❡ t❤❡ ❛•❣♦r✐t❤♠ ❜② s❡tt✐♥❣ φ i = ■❞✱ T = 1 N N i=1 T i ❛♥❞ ❜② ❝♦♠✲ ♣✉t✐♥❣ γ T = L -1 W ( T ) ✈✐❛ ❛ •❛✉ss✐❛♥ ❝♦♥✈♦•✉t✐♦♥✳ ❚❤❡ ❝✉rr❡♥t T ✐s ❡♥❝♦❞❡❞ ❛s ❛ •✐st ♦❢ ✭♣♦s✐t✐♦♥✱ ✈❡❝t♦rs✮ t❤❛t ❛♣♣r♦①✐♠❛t❡ s♠❛•• s❡❣♠❡♥ts ♦r s♠❛•• tr✐❛♥❣•❡s✳ ❚❤❡ ❞❡♥s❡ ✈❡❝t♦r ✜❡•❞ γ T ✐s ❞✐s❝r❡t✐③❡❞ ❛t ❛ ✜①❡❞ ❣r✐❞✬s ♣♦✐♥ts✿ Λ = {x p } ❛♥❞ t❤❡r❡❢♦r❡ ❡♥❝♦❞❡❞ ❛s ❛♥ ✐♠❛❣❡ ♦❢ ✈❡❝t♦rs✳ ❚❤❡ ❢♦••♦✇✐♥❣ ✈❛r✐❛❜•❡ ❣r❛❞ ✐s ❛•s♦ ❛♥ ✐♠❛❣❡ ♦❢ ✈❡❝t♦rs✳ ❲❡ t❤❡♥ ✐t❡r❛t❡ t❤❡ ❢♦••♦✇✐♥❣ •♦♦♣✿ ✕ ❋♦r i = 1 . . . N ✱ φ i ← r❡❣✐str❛t✐♦♥ ♦❢ T t♦ T i ✳ ✕ ❯♥t✐• ❝♦♥✈❡r❣❡♥❝❡ ❞♦ • ❣r❛❞ = 0 • ❋♦r i = 1 . . . N ❞♦ * tr❛♥s♣♦rt t❛♥❣❡♥ts ✭♥♦r♠❛•s✮ ♦❢ T ✇✐t❤ φ i ✭❣✐✈❡s φ ♯,i ⋆ T ✮✳ * st♦r❡ t❤❡ (y i k , β i k ) s✉❝❤ t❤❛t φ ♯,i ⋆ T -T i = k δ β i k y i k * ❉❡❢♦r♠ Λ ✇✐t❤ φ i ❛♥❞ ❢♦r ❡❛❝❤ p ∈ Λ ❝♦♠♣✉t❡✿ • d xp φ i ❜② ❛ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡ • G = k K(φ i (x p ), y i k )β i k ✭❝♦♥✈♦•✉t✐♦♥✮ • ❣r❛❞(p) ← ❣r❛❞(p) + 2d xp φ t i G ✭♦r +2 d xp φ i (d xp φ i ) -1 G✮ • γ T ← γ T -τ ❣r❛❞ • ❉❡❝♦♥✈♦•✉t✐♦♥ ♦❢ γ T t♦ ❣✐✈❡ t❤❡ ♥❡✇ T ✭❙❡❡ s❡❝t✐♦♥ ✸✳✸✮✳ ✸✳✸ ❙♣❛rs❡ ❉❡❝♦♥✈♦•✉t✐♦♥ ❜② ❛ ▼❛t❝❤✐♥❣ P✉rs✉✐t ❆•❣♦r✐t❤♠ ▲✐❦❡ ❛♥② •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ ✐♥♣✉t ❝✉rr❡♥ts✱ t❤❡ ♠❡❛♥ ❝✉rr❡♥t T ✐s st♦r❡❞ ❛s t❤❡ s❡t ♦❢ ❛•• ✇❡✐❣❤t❡❞ t❛♥❣❡♥ts ✭♦r ♥♦r♠❛•s✮ ✐♥ t❤❡ ❞❛t❛❜❛s❡ T = (1/N ) N 1 T i = n 1 δ α k x k ✇❤❡r❡ n ❣r♦✇s •✐♥❡❛r•② ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ♦❜s❡r✈❛✲ t✐♦♥s✳ ❚❤✐s ❤❡❛✈② r❡♣r❡s❡♥t❛t✐♦♥ ♠❛❦❡s ❝r✐t✐❝❛• t❤❡ r❡❣✐str❛t✐♦♥ ♦❢ T r✐❣❤t ❛❢t❡r t❤❡ ✐♥✐t✐❛•✐③❛t✐♦♥ st❡♣✳ ❇✉t t❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ✐s ♦❢t❡♥ ❢❛r ❢r♦♠ ❜❡✐♥❣ ♦♣t✐♠❛•✿ ✐t ♠❛② ❜❡ ❤✐❣❤•② r❡❞✉♥❞❛♥t ❛t t❤❡ s❝❛•❡ λ W ✳ ❚♦ ✐♥t❡❣r❛t❡ t❤✐s r❡❞✉♥❞❛♥❝②✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ✈❡❝t♦r ✜❡•❞ L -1 W ( T ) ✭❋✐❣✳✷✲❛✮ ❜② ❝♦♥✈♦•✉t✐♦♥✳ ❚❤❡♥✱ ❛♥ ❛❞❡q✉❛t❡ ❞❡❝♦♥✈♦•✉t✐♦♥ s❝❤❡♠❡ ❝♦✉•❞ ❜❡ ❛♣♣•✐❡❞ t♦ ❡st✐♠❛t❡ ❛♥ ❛❞❛♣t❡❞ ❜❛s✐s ♦♥ ✇❤✐❝❤ t❤❡ tr✉❡ s♦•✉t✐♦♥ T ✐s ❞❡❝♦♠♣♦s❡❞ ✐♥ T = k ′ δ α k ′ x k ′ ✇✐t❤ ❢❛st ❞❡❝r❡❛s✐♥❣ t❡r♠s✳ ❚❤❡ ✜rst t❡r♠s ♦❢ t❤✐s s❡r✐❡s ✇✐•• ❣✐✈❡ t❤❡r❡❢♦r❡ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ T ✇✐t❤ ❛♥ ✐♥❝r❡❛s✐♥❣ ❛❝❝✉r❛❝② ✭❋✐❣✳✷✲❜✮✳ ❛✲ ▼❡❛♥ ♦❢ t✇♦ ❝✉r✈❡s ❜✲❆♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ 3 ❉✐r❛❝ ❝✉rr❡♥ts ❋✐❣✳ ✷✳ ❚♦② ❡①❛♠♣•❡ ✐♥ ✷❉ ✐••✉str❛t✐♥❣ t❤❡ s♣❛rs❡ ❞❡❝♦♥✈♦•✉t✐♦♥ s❝❤❡♠❡✳ ❛✲r✐❣❤t✿ t✇♦ ✐♥✐t✐❛• ❝✉r✈❡s ✭L1✱ L2 ✐♥ ❜•✉❡✮✱ t❤❡✐r ♠❡❛♥ ✭(L1 + L2)/2✮ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ s❡t ♦❢ ✶✾ ❉✐r❛❝ ❝✉rr❡♥ts ✐♥ r❡❞✳ ❛✲•❡❢t✿ ❛ •❛✉ss✐❛♥ ❝♦♥✈♦•✉t✐♦♥ ✭L -1 W ✮ ♦❢ t❤❡ ✶✾ ❉✐r❛❝ ❣✐✈❡s t❤❡ ❞✉❛• r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠❡❛♥ ❛s ❛ ❞❡♥s❡ ✈❡❝t♦r ✜❡•❞ ✐♥ W ✳ ❜✲ ❚❤❡ ❞❡❝♦♥✈♦•✉t✐♦♥ ♠❡t❤♦❞ ❡st✐♠❛t❡s ✐t❡r❛t✐✈❡•② ❛ s❡t ♦❢ ❉✐r❛❝ ❝✉rr❡♥ts ✭❜✲r✐❣❤t ❛❢t❡r ✸ st❡♣s✮ ✇❤♦s❡ ❝♦♥✈♦•✉t✐♦♥ r❡tr✐❡✈❡s t❤❡ ✐♥✐t✐❛• ✈❡❝t♦r ✜❡•❞ ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❛❝❝✉r❛❝②✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ tr✉❡ ✈❡❝t♦r ✜❡•❞ ❛♥❞ t❤❡ ❡st✐♠❛t❡❞ ♦♥❡ ✐s s❤♦✇♥ ✐♥ ❜✲•❡❢t✳ ■♥ ❊q✳✸✳✹✱ ✇❡ ♠❛❦❡ ❝♦♠♣✉t❛t✐♦♥s ♦♥ ✈❡❝t♦r ✜❡•❞s ❛♥❞ ✇❡ ❞♦ ♥♦t ❦♥♦✇ ❤♦✇ t♦ r❡tr✐❡✈❡ t❤❡ ❝✉rr❡♥t T t❤❛t ❝♦♠❡s ❢r♦♠ t❤❡ r❡s✉•t✐♥❣ ✈❡❝t♦r ✜❡•❞✳ ❚❤❡ s♦•✉t✐♦♥ ♠❛② r❡q✉✐r❡ t❤❡♦r❡t✐❝❛••② ❛♥ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❉✐r❛❝ ❝✉rr❡♥ts✿ T = ∞ i=0 δ αi xi ✳ ❍❡r❡ ❛❣❛✐♥✱ ✇❡ ♥❡❡❞ ❛ ❞❡❝♦♥✈♦•✉t✐♦♥ s❝❤❡♠❡ t❤❛t ❡st✐♠❛t❡s ✐t❡r❛t✐✈❡•② t❤❡ ♠♦st ✐♠♣♦rt❛♥t t❡r♠s ♦❢ t❤✐s s❡r✐❡s✳ ❚❤❡ s✉♠ ♦❢ t❤❡s❡ ✜rst t❡r♠s ✇✐•• ♣r♦✈✐❞❡ ❛ s♣❛rs❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ T ❛t ❛ ❣✐✈❡♥ ♣r❡❝✐s✐♦♥✳ ❚❤❡ ❞❡❝♦♥✈♦•✉t✐♦♥ s❝❤❡♠❡ ✇❡ ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✾✱ ✶✵(✱ ❛❞❛♣ts t♦ ♦✉r ❢r❛♠❡✇♦r❦ ❜❛s❡❞ ♦♥ ❝✉rr❡♥ts t❤❡ ♦rt❤♦❣♦♥❛• ♠❛t❝❤✐♥❣ ♣✉rs✉✐t ✭♦r✐❣✐♥❛••② ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✶✽( t♦ ❞❡❝♦♠♣♦s❡ ✐♠❛❣❡s ✐♥ ❛❞❛♣t❡❞ ✇❛✈❡•❡t ❜❛s❡s✮✳ •✐✈❡♥ ❛ ✈❡❝t♦r ✜❡•❞ γ ∈ W ✱ t❤❡ ✜rst st❡♣ ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ t❤❡ ♣♦✐♥t x 1 t❤❛t ♠❛①✐♠✐③❡s t❤❡ ♣r♦•❡❝t✐♦♥ ♦❢ L W (γ) ♦♥ δ x1 ✿ L W (γ), δ ε k x W * = γ, K(., x)ε k W = γ(x) k ✭ε k ✐s ❛♥ ♦rt❤♦♥♦r♠❛• ❜❛s✐s ♦❢ R 3 ✮✳ x 1 r❡❛❝❤❡s t❤❡r❡❢♦r❡ t❤❡ ♠❛①✐♠✉♠ ♦❢ γ✳ ❆ •✐♥❡❛r s❡t ♦❢ ❡q✉❛t✐♦♥ ❞❡t❡r♠✐♥❡s α 1 s✉❝❤ t❤❛t δ α1 x1 ✐s t❤❡ ♦rt❤♦❣♦♥❛• ♣r♦•❡❝t✐♦♥ ♦❢ L W (T ) ✐♥t♦ Span(δ x1 )✳ ❲❡ s✉❜str❛❝t t❤❡♥ K(., x 1 )α 1 ❢r♦♠ γ ❛♥❞ ✇❡ ✐t❡r❛t❡ ♦♥ t❤✐s r❡s✐❞✉❡✳ ❚❤✐s ❜✉✐•❞s ✐t❡r❛t✐✈❡•② ❛ s❡r✐❡s t❤❛t ❤❛s ❜❡❡♥ ♣r♦✈❡❞ ✐♥ ❬✶✵( t♦ ❝♦♥✈❡r❣❡ t♦ t❤❡ tr✉❡ s♦•✉t✐♦♥✳ ✹ ❊①♣❡r✐♠❡♥t❛• |❡s✉•ts ✹✳✶ ❇✉✐•❞✐♥❣ ❆t•❛s❡s ❢r♦♠ ❈✉r✈❡s ❲❡ ✉s❡ ❛ ❞❛t❛❜❛s❡ ♦❢ ❝♦rt✐❝❛• s✉•❝❛• •❛♥❞♠❛r❦s ✭✼✷ ♣❡r ❜r❛✐♥✮ ❞❡•✐♥❡❛t❡❞ ✐♥ ❛ •❛r❣❡ ♥✉♠❜❡r ♦❢ s✉❜•❡❝ts s❝❛♥♥❡❞ ✇✐t❤ ✸❉ ▼|■ ✭❛❣❡✿ ✺✶✳✽ ± ✻✳✷ ②❡❛rs✮✳ ❋r♦♠ ✸✹ s✉❜•❡❝ts ✐♥ t❤❡ ❞❛t❛❜❛s❡✱ ✇❡ ❜✉✐•❞ t❤❡ t❡♠♣•❛t❡ ❛❝❝♦r❞✐♥❣ t♦ ♦✉r ❡st✐♠❛t✐♦♥ ♠❡t❤♦❞✳ ❲❡ s❡t t❤❡ ♣❛r❛♠❡t❡rs λ W = 12♠♠✱ λ V = 25♠♠ ❛♥❞ γ = σ 2 W /σ 2 V = 0
♥♦t ❜✐❛s❡❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ♠♦❞❡• ❊q✳✶✳✶✳ t♦ ❛♥ ✉♥❜✐❛s❡❞ t❡♠♣•❛t❡ ✭✐♥ ❜•✉❡✮✳ ❚❤❡ s♣❛rs❡ ❞❡❝♦♥✈♦•✉t✐♦♥ s❝❤❡♠❡ ❡♥❛❜•❡s ❛•s♦ t♦ ❣✐✈❡ ❛ •✐❣❤t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡✿ ✇❤❡r❡❛s t❤❡ ❞❛t❛❜❛s❡ ❝♦♥t❛✐♥s ✸✷✻✹✸ s❡❣♠❡♥ts ✭♦♥ ❛✈❡r❛❣❡ ✾✻✵ s❡❣♠❡♥ts ♣❡r s✉❜•❡❝t✮✱ t❤❡ ❡st✐♠❛t❡❞ t❡♠♣•❛t❡ ✐s r❡♣r❡s❡♥t❡❞ ❜② ♦♥•② ✶✸✻✶ ❉✐r❛❝ ❝✉rr❡♥ts ✇✐t❤ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❜❡•♦✇ 5%✳ ❚❤✐s ✇♦✉•❞ ❜❡ ♦❢ ❣r❡❛t ✐♥t❡r❡st ✐♥ t❤❡ ❢✉t✉r❡✱ ❢♦r ❡①❛♠♣•❡ t♦ r❡❣✐st❡r t❤✐s t❡♠♣•❛t❡ t♦✇❛r❞ ❛♥② ♥❡✇ ❛✈❛✐•❛❜•❡ ❞❛t❛✳ ✹✳✷ ❇✉✐•❞✐♥❣ ❆t•❛s❡s ❢r♦♠ ❙✉r❢❛❝❡s ❲❡ ✉s❡ ❛ ❞❛t❛❜❛s❡ ♦❢ ✶✵ s❡❣♠❡♥t❡❞ s✉❜✲❝♦rt✐❝❛• str✉❝t✉r❡s ♦❢ t❤❡ ❜r❛✐♥ ✭❈❛✉❞❛t❡✱ P✉t❛♠❡♥✱ ••♦❜✉s P❛••✐❞✉s✱ ❆♠②❣❞❛•❛ ❛♥❞ ❍✐♣♣♦❝❛♠♣✉s✮ ✐♥ •❛r❣❡ ♥✉♠❜❡r ♦❢ ❛✉t✐st✐❝s ❛♥❞ ❤❡❛•t❤② ❝❤✐•❞r❡♥ s❝❛♥♥❡❞ ✇✐t❤ ✸❉ ▼|■ ✭❛❣❡ ✷✳✼ ± ✵✳✸✮ ❬✷✵(✳ ❋r♦♠ ✷✺ ❛✉t✐st✐❝s ❛♥❞ ✼ ❝♦♥tr♦•s✱ ✇❡ ❜✉✐•❞ t❤❡ t❡♠♣•❛t❡ ❛❝❝♦r❞✐♥❣ t♦ ♦✉r ❡st✐♠❛t✐♦♥ ♠❡t❤♦❞ ❢♦r s✉r❢❛❝❡s✳ ❲❡ ✉s❡ t❤❡ ♣❛r❛♠❡t❡rs λ W = 5♠♠✱ λ V = 20♠♠✱ γ = σ 2 W /σ 2 V = 0.001✳ ❚❤❡ s❡t ♦❢ ✶✵ s✉❜✲❝♦rt✐❝❛• str✉❝t✉r❡s ❤❛✈❡ t②♣✐❝❛••② ❛ ❞✐❛♠❡t❡r ♦❢ 60♠♠✳ ❋✐❣✉r❡s ✹✲❛✱❜ s❤♦✇ t❤❡ ❡st✐♠❛t❡❞ t❡♠♣•❛t❡ ❛❢t❡r ✷ ✐t❡r❛t✐♦♥s✳ ❊❛❝❤ ❉✐r❛❝ ❝✉rr❡♥t δ α x ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛♥ ❡q✉✐•❛t❡r❛• tr✐❛♥❣•❡ ✇❤♦s❡ ❝❡♥t❡r ♦❢ ♠❛ss ✐s x ❛♥❞ ♥♦♥✲♥♦r♠❛•✐③❡❞ ♥♦r♠❛• α✳ ❊❛❝❤ ♦❢ t❤❡ ✶✵ ♠❡s❤❡s ❤❛s ✷✽✽✵ ❝❡••s✱ s♦ t❤❛t t❤❡ t♦t❛• ♥✉♠❜❡r ♦❢ ♥♦r♠❛•s ✐s 720e 3 ✳ ❚❤❛♥❦s t♦ t❤❡ ❞❡❝♦♥✈♦•✉t✐♦♥ s❝❤❡♠❡✱ t❤❡ ❡st✐♠❛t❡❞ t❡♠♣•❛t❡ ✐s r❡♣r❡s❡♥t❡❞ ❜② ♦♥•② ✶✸✹✹ ♥♦r♠❛•s ✇✐t❤ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❜❡•♦✇ 5%✳ ■♥ ❋✐❣✳✹✲❝✱ ✇❡ s❤♦✇ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t❡♠♣•❛t❡ ♦❢ ❛✉t✐st✐❝s ❛♥❞ ❝♦♥tr♦•s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ♣r♦♣♦s❡❞ ♠❛t❝❤✐♥❣ ♣✉rs✉✐t ❛•❣♦r✐t❤♠✳ ◆♦t❡ t❤❛t t❤❡ ❛♥❛t♦♠✐❝❛• ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❜♦t❤ ❝•❛ss❡s ❛r❡ ♥♦t ♦♥•② ❝❛♣t✉r❡❞ ❜② t❤❡✐r r❡s♣❡❝t✐✈❡ t❡♠♣•❛t❡s ❜✉t ❛•s♦ ❜② t❤❡ ♠❛✐♥ ♠♦❞❡s ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥s φ i ✳ ❛✲ ❆✉t✐st✐❝s ❜✲ ❈♦♥tr♦•s ❝✲ ❉✐✛❡r❡♥❝❡ ♦♥ ❤✐♣♣♦❝❛♠♣✉s ❋✐❣✳ ✹✳ ❊st✐♠❛t❡❞ t❡♠♣•❛t❡s ❢r♦♠ ✷✺ ❛✉t✐st✐❝s ✭❛✮ ❛♥❞ ✼ ❝♦♥tr♦•s ✭❜✮ ❢♦r ✶✵ s✉❜✲❝♦rt✐❝❛• str✉❝t✉r❡s ✭♦♥❡ ♣❡r ❝♦•♦r✮✳ ■♥ ✭❝✮✱ t❤❡ ❜•✉❡ ❛rr♦✇s ❛♣♣r♦①✐♠❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❛✉t✐st✐❝s ❛♥❞ ❝♦♥tr♦•s✬ t❡♠♣•❛t❡✱ s✉♣❡r✐♠♣♦s❡❞ ✇✐t❤ t❤❡ ❤✐♣♣♦❝❛♠♣✉s ♦❢ ❛ ❝♦♥tr♦•✳ ✺ ❈♦♥❝•✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ♣r♦♣♦s❡❞ ❛ st❛t✐st✐❝❛• ♠♦❞❡• ✇❤✐❝❤ ❡st✐♠❛t❡s •♦✐♥t•② ❛ t❡♠✲ ♣•❛t❡✱ t❤❡ ❞❡❢♦r♠❛t✐♦♥s ♦❢ t❤❡ t❡♠♣•❛t❡ t♦ t❤❡ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ t❤❡ ♥♦✐s❡ ♦♥ t❤❡s❡ ♦❜s❡r✈❛t✐♦♥

❋

  ♦r ❛•• t❤❡s❡ r❡❛s♦♥s✱ ✇❡ ❜❛s❡ ❤❡r❡ ♦✉r st❛t✐st✐❝❛• ❡st✐♠❛t✐♦♥s ♦♥ t❤❡ ❢♦r✇❛r❞ ♠♦❞❡•✳ ❲❡ s❤♦✇ ✐♥ t❤✐s ♣❛♣❡r ❤♦✇ t❤❡ ❛t•❛s ❜✉✐•❞✐♥❣ st❡♣✱ ✇❤✐❝❤ ✐s t❤❡ ♠♦st ❝r✐t✐❝❛• st❡♣ ✐♥ t❤✐s ♣❛r❛❞✐❣♠✱ ✐s ♣♦ss✐❜•❡ ✐♥ ❝❛s❡ ♦❢ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✳ ❈♦♠✲ ♣❛r❡❞ t♦ ✐♠❛❣❡s✱ ❞❡❛•✐♥❣ ✇✐t❤ s❤❛♣❡s r❡q✉✐r❡s s♣❡❝✐✜❝ ♥✉♠❡r✐❝❛• s❝❤❡♠❡✳ ❲❡ t❛❦❡ ❛❞✈❛♥t❛❣❡ ❤❡r❡ ♦❢ ❛ s♣❛rs❡ ❞❡❝♦♥✈♦•✉t✐♦♥ s❝❤❡♠❡ ✇❡ ✐♥tr♦❞✉❝❡❞ r❡❝❡♥t•② ❬✾✱ ✶✵(✳ ❚❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦••♦✇s✳ ❆ ❣❡♥❡r❛• ♥♦♥ ♣❛r❛♠❡tr✐❝ ❢r❛♠❡✇♦r❦ ❢♦r s❤❛♣❡ st❛t✐st✐❝s ✐s ✐♥tr♦❞✉❝❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳ ■♥ ❙❡❝t✐♦♥ ✸ ✇❡ ❞❡t❛✐• t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❝❡❞✉r❡ t♦ ❡st✐♠❛t❡ •♦✐♥t•② ❛ t❡♠♣•❛t❡ ❛♥❞ ✐ts ❞❡❢♦r♠❛t✐♦♥s t♦ t❤❡ s❤❛♣❡s✳ ❆ s♣❛rs❡ ❞❡❝♦♥✈♦•✉t✐♦♥ ♠❡t❤♦❞ ✐s ♣r❡s❡♥t❡❞ t♦ ❡✛❡❝t✐✈❡•② ❝♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t✳ ■♥ ❙❡❝t✐♦♥ ✹✱ ✇❡ s❤♦✇ t❡♠♣•❛t❡s ❝♦♠♣✉t❡❞ ❢r♦♠ •❛r❣❡ s❡ts ♦❢ s✉•❝❛• •✐♥❡s ❛♥❞ s❡ts ♦❢ ♠❡s❤❡s ♦❢ s✉❜✲❝♦rt✐❝❛• str✉❝t✉r❡s✳ ✷ ◆♦♥✲♣❛r❛♠❡tr✐❝ |❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❙❤❛♣❡s ❛s ❈✉rr❡♥ts ❆s ❡♠♣❤❛s✐③❡❞ ✐♥ ❬✶✶✱ ✶✷✱ ✺(✱ ❛ ❝✉rr❡♥t ✐s ❛ ❝♦♥✈❡♥✐❡♥t ✇❛② t♦ ♠♦❞❡• ❣❡♦♠❡tr✐❝❛• s❤❛♣❡s s✉❝❤ ❛s ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✳ ❚❤❡ ✐❞❡❛ ✐s t♦ ❝❤❛r❛❝t❡r✐③❡ s❤❛♣❡s ✈✐❛ ✈❡❝t♦r ✜❡•❞s✱ ✇❤✐❝❤ ❛r❡ ✉s❡❞ t♦ ♣r♦❜❡ t❤❡♠✳ ❆ s✉r❢❛❝❡ S ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡ ✢✉① ♦❢ ❛♥② ✈❡❝t♦r ✜❡•❞ ω t❤r♦✉❣❤ ✐t✿ S(ω) = S ω(x).(u × v)(x)dσ(x) ✭✷✳✶✮ ✇❤❡r❡ u × v ✐s t❤❡ ♥♦r♠❛• ♦❢ t❤❡ s✉r❢❛❝❡✱ (u, v) ❛♥ ♦rt❤♦♥♦r♠❛• ❜❛s✐s ♦❢ ✐ts t❛♥❣❡♥t ♣•❛♥❡✮ ❛♥❞ dσ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ t❤❡ s✉r❢❛❝❡✳ ❙✐♠✐•❛r•②✱ ❛ ❝✉r✈❡ L ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡ ♣❛t❤ ✐♥t❡❣r❛• ♦❢ ❛♥② ✈❡❝t♦r ✜❡•❞ ω ❛•♦♥❣ ✐t✿ L(ω) = L ω(x).τ (x)dx ✭✷✳✷✮ ✇❤❡r❡ τ ✐s t❤❡ t❛♥❣❡♥t ♦❢ t❤❡ ❝✉r✈❡✳ ▼♦r❡ ❣❡♥❡r❛••②✱ ✇❡ ❞❡✜♥❡ ❛ ❝✉rr❡♥t T ❛s ❛ •✐♥❡❛r ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣ ❢r♦♠ ❛ s❡t ♦❢ t❡st ✈❡❝t♦r ✜❡•❞s W t♦ R✳ ❚❤✐s ❢r❛♠❡✇♦r❦ ❡♥❛❜•❡s t♦ ❞❡✜♥❡ ❛❞❞✐t✐♦♥✱ ❞❡❢♦r♠❛t✐♦♥✱ •❛✉ss✐❛♥ ♥♦✐s❡ ♦♥ s❤❛♣❡s✱ t♦ ♠❡❛s✉r❡ ❛ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ s❤❛♣❡s✱ ✇✐t❤♦✉t ❛ss✉♠

  .01✳ ❚❤❡ ❞✐❛♠❡t❡r ♦❢ t❤❡ ❜r❛✐♥ ✐s t②♣✐❝❛••② 120♠♠✳ ❋✐❣✉r❡ ✸ s❤♦✇s t❤❡ ❡st✐♠❛t❡❞ t❡♠♣•❛t❡ ❛❢t❡r ✷ ✐t❡r❛t✐♦♥s✳ ❉✉r✐♥❣ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t✱ t❤❡ ❜✐❛s ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ♦✉r ❢♦r✇❛r❞ ♠♦❞❡• ❊q✳✶✳✶✮ ✐s r❡♠♦✈❡❞ ❢r♦♠ t❤❡ ❡♠♣✐r✐❝❛• ♠❡❛♥ ✭✐♥ r❡❞✮✱ •❡❛❞✐♥❣ ❛✲ ❙②•✈✐❛♥ ❋✐ss✉r❡ ✭▲❡❢t ❍❡♠✐s♣❤❡r❡✮ ❜✲ ❆•• 70 s✉•❝✐ ❋✐❣✳ ✸✳ ❊st✐♠❛t❡❞ t❡♠♣•❛t❡ ❢r♦♠ ✸✹ s✉❜•❡❝ts✳ ▲❡❢t✿ ✐♥✐t✐❛• ✸✹ ❙②•✈✐❛♥ ❋✐ss✉r❡ ♦❢ t❤❡ •❡❢t ❤❡♠✐s♣❤❡r❡ ✭❣r❡❡♥✮✱ t❤❡ ❡♠♣✐r✐❝❛• ♠❡❛♥ ✭r❡❞✮ ❛♥❞ t❤❡ ❡st✐♠❛t❡❞ t❡♠♣•❛t❡ ✭❜•✉❡✮✳ ■♥ ❜•❛❝❦✱ t❤❡ ♠❡❛♥ •✐♥❡s ❝♦♠♣✉t❡❞ ❢r♦♠ ❇✲s♣•✐♥❡ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❝✉r✈❡s ❬✶✾(✳ |✐❣❤t✿ ❙❛♠❡ ❝✉r✈❡s ❢♦r 70 s✉•❝✐✳ ❆•t❤♦✉❣❤ r❡s✉•ts •♦♦❦ s✐♠✐•❛r✱ ♦♥•② t❤❡ t❡♠♣•❛t❡ ✐♥ ❜•✉❡ ✐s

  s✳ ❚❤✐s ❞✐✛❡rs ❢r♦♠ ♠❡t❤♦❞s t❤❛t ✉s❡ ❡①tr✐♥s✐❝ t❡♠♣•❛t❡s✱ t❤✉s ✐♥tr♦❞✉❝✐♥❣ ❛ ❜✐❛s ✐♥ t❤❡ ❛♥❛•②s✐s ♦❢ t❤❡ ✈❛r✐❛❜✐•✐t②✳ ❖✉r ❢♦r✇❛r❞ ♠♦❞❡• ❝♦♥s✐❞❡rs ♦❜s❡r✈❛t✐♦♥s ❛s ♥♦✐s② ❞❡❢♦r♠❛t✐♦♥s ♦❢ ❛♥ ✉♥❦♥♦✇♥ t❡♠♣•❛t❡✳ ❈♦♠♣❛r❡❞ t♦ t❤❡ ♠♦r❡ ❝♦♠♠♦♥•② ✉s❡❞ ❜❛❝❦✇❛r❞ s❡tt✐♥❣✱ t❤✐s ♠♦❞❡• r❡q✉✐r❡s ❛ s♣❡❝✐✜❝ ❞❡❝♦♥✈♦•✉✲ t✐♦♥ s❝❤❡♠❡ t♦ ❞❡❛• ✇✐t❤ t❤❡ ♥♦♥✲•✐♥❡❛r ❛❝t✐♦♥ ♦❢ ❞✐✛❡♦♠♦r♣❤✐s♠s ♦♥ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✳ ❍♦✇❡✈❡r✱ ♦♥❝❡ t❤❡ ♠♦❞❡• ✐s ❜✉✐•t✱ ✇❡ ❝❛♥ ❞❡❝♦♠♣♦s❡ ❛♥② ♥❡✇ ❞❛t❛ ✐♥t♦ ❛ ❞❡❢♦r♠❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ❛♥❞ ❛ r❡s✐❞✉❛• ♥♦✐s❡ ❛♥❞ t❤❡♥ ♠❡❛s✉r❡ ❤♦✇ •✐❦❡•② t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥ ♠❛② ❜❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♠♦❞❡•✳ ❚❤✐s ♦✛❡rs ❛ ✇❛② t♦ ❝•❛s✲ s✐❢② ❞❛t❛ ❛❝❝♦r❞✐♥❣ t♦ ♣❛t❤♦•♦❣✐❡s✱ t♦ ❞❡t❡❝t ♦✉t•✐❡rs ✐♥ ❛ ❞❛t❛❜❛s❡✱ t♦ ❤✐❣❤•✐❣❤t ✇❤❡r❡ ❛ ❣✐✈❡♥ ♦❜s❡r✈❛t✐♦♥ ❞✐✛❡rs ❢r♦♠ ❛ ♠♦❞❡•✱ ❡t❝✳ ❙✉❝❤ ❛ st❛t✐st✐❝❛• ✐♥❢❡r❡♥❝❡ ✇♦✉•❞ ❜❡ ♠♦r❡ ❞✐✣❝✉•t ✇✐t❤✐♥ t❤❡ ❜❛❝❦✇❛r❞ ❢r❛♠❡✇♦r❦ s✐♥❝❡ t❤❡ •✐❦❡•✐❤♦♦❞ ♦❢ ❛♥② ♥❡✇ ♦❜s❡r✈❛t✐♦♥s ✇♦✉•❞ ✐♥✈♦•✈❡ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥s t❤❛t ✇❡ t❛❦❡ ❤❡r❡ ❞✐r❡❝t•② ✐♥t♦ ❛❝❝♦✉♥t ✇❤✐•❡ ❜✉✐•❞✐♥❣ t❤❡ ❛t•❛s✳ ◆✉♠❡r✐❝❛• ❡①♣❡r✐♠❡♥ts s❤♦✇ t❤❡ ❢❡❛s✐❜✐•✐t② ❛♥❞ r❡•❡✈❛♥❝❡ ♦❢ ♦✉r ❛♣♣r♦❛❝❤✳ ■♥ ♣❛rt✐❝✉•❛r✱ t✇♦ t❡♠♣•❛t❡s ✇❡r❡ ❡st✐♠❛t❡❞ ❢r♦♠ ❞❡❡♣ ❜r❛✐♥ str✉❝t✉r❡s ♦❢ ❛✉t✐st✐❝s ❝❤✐•❞r❡♥ ❛♥❞ ❝♦♥tr♦•s✳ ❚❤❡ r❡s✉•ts s✉❣❣❡st ❛♥❛t♦♠✐❝❛• ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❜♦t❤ ❝•❛ss❡s✳ ❍♦✇❡✈❡r✱ s✉❝❤ r❡s✉•ts s❤♦✉•❞ ❜❡ str❡♥❣t❤❡♥ ❜② r✐❣♦r♦✉s st❛t✐st✐❝❛• t❡sts✳ ❖✉r ❢✉t✉r❡ ✇♦r❦ ✇✐•• ❛•s♦ ❡✈❛•✉❛t❡ t❤❡ ♠♦❞❡• ✇✐t❤ r❡s♣❡❝t t♦ ✐ts ♣r❡❞✐❝t✐♦♥ ❛♥❞ ❝•❛ss✐✜❝❛t✐♦♥ ❝❛♣❛❜✐•✐t②✳ ❆❝❦♥♦✇•❡❞❣♠❡♥ts ❲❡ ✇❛r♠•② t❤❛♥❦ Pr♦❢✳ ❚❤♦♠♣s♦♥ ✭❯❈▲❆✮ ❢♦r ♣r♦✈✐❞✐♥❣ t❤❡ s✉•❝❛• •✐♥❡s ❞❛t❛❜❛s❡ t❤r♦✉❣❤ t❤❡ ❆s❝•❡♣✐♦s✲▲❖◆■ ❛ss♦❝✐❛t❡❞ t❡❛♠ ❇r❛✐♥❆t•❛s✱ ❛s ✇❡•• ❛s Pr♦❢✳ •❡r✐❣ ✭❯♥✐✈❡rs✐t② ♦❢ ❯t❛❤✮ ❢♦r❣✐✈✐♥❣ ❛❝❝❡ss t♦ t❤❡ ❞❛t❛❜❛s❡ ♦❢ ❞❡❡♣ ❜r❛✐♥ str✉❝t✉r❡s✳ ❚❤❡ ✇♦r❦ ✇❛s ♣❛rt•② s✉♣♣♦rt❡❞ ❜② t❤❡ ❊✉r♦♣❡❛♥ ■P ♣r♦•❡t ❍❡❛•t❤✲❡✲❈❤✐•❞ ✭■❙❚✲ ✷✵✵✹✲✵✷✼✼✹✾✮✳

❜❡ ✇r✐tt❡♥ ❛s✿