Convergence of an iterative algorithm for Teichmüller maps via generalized harmonic maps
CCONVERGENCE OF AN ITERATIVE ALGORITHM FORTEICHM ¨ULLER MAPS VIA GENERALIZED HARMONIC MAPS
LOK MING LUI, XIANFENG GU AND SHING-TUNG YAU
Abstract.
Finding surface mappings with least distortion arises from many applications invarious fields. Extremal Teichm¨uller maps are surface mappings with least conformality distortion.The existence and uniqueness of the extremal Teichm¨uller map between Riemann surfaces of finitetype are theoretically guaranteed [1]. Recently, a simple iterative algorithm for computing theTeichm¨uller maps between connected Riemann surfaces with given boundary value was proposed in[11]. Numerical results was reported in the paper to show the effectiveness of the algorithm. Themethod was successfully applied to landmark-matching registration. The purpose of this paper is toprove the iterative algorithm proposed in [11] indeed converges.
Key words.
Teichm¨uller map, extremal map, quasiconformal map, harmonic energy optimiza-tion, registration.
1. Introduction.
Finding meaningful surface mappings with least distortionhas fundamental importance. Applications can be found in different areas such asregistration, shape analysis and grid generation. Conformal mapping has been widelyused to establish a good one-to-one correspondence between different surfaces, sinceit preserves the local geometry well [2, 3, 4, 5, 6, 7, 8, 9, 10]. The Riemann mappingtheorem guarantees the existence of conformal mappings between simply-connectedsurfaces. However, this fact is not valid for general Riemann surfaces. Given twoRiemann surfaces with different conformal modules, there is generally no conformalmapping between them. In this case, it is usually desirable to obtain a mappingthat minimizes the conformality distortion. Every diffeomorphic surface mapping isassociated with a unique Beltrami differential, which is a complex-valued function, µ f ,defined on the source surface. The Beltrami differential, µ f , measures the deviationof the mapping from a conformal map. Given two Riemann surfaces S and S ,there exists a unique and bijective map f : S → S , called the Teichm¨uller map ,minimizing the L ∞ norm of the Beltrami differential [1]. Therefore, the extremalTeichm¨uller map can be considered as the ‘most conformal’ map between Riemannsurfaces of the same topology, which is a natural extension of conformal mappings. Mathematically, the extremal problem for obtaininga surface mapping with least conformality distortion can be formulated as follows.Suppose ( S , σ | dz | ) and ( S , ρ | dw | ) are two Riemann surfaces of finite type, where z and w are their conformal parameters respectively. Every diffeomorphism between S and S is associated with a unique Beltrami differential. A Beltrami differential µ ( z ) d ¯ zdz on the Riemann surface S is an assignment to each chart ( U α , φ α ) of an L ∞ complex-valued function µ α , defined on local parameter z α . Then, f : S → S is saidto be a quasi-conformal mapping associated with the Beltrami differential µ ( z ) dzdz if forany chart ( U α , φ α ) on S and any chart ( V β , ψ β ) on S , the mapping f αβ := ψ β ◦ f ◦ φ − α is quasi-conformal associated with µ α ( z α ) dz α dz α .Our goal is to look for an extremal quasi-conformal mapping, which are extremalin the sense of minimizing the || · || ∞ over all Beltrami differentials corresponding toquasi-conformal mappings between S and S . The idea of extremality is to makethe supreme norm of the Beltrami differential as small as possible such that f is as‘nearly conformal’ as possible.The extremal problem can therefore be formulated as finding f : S → S that a r X i v : . [ m a t h . DG ] J u l Lui, Gu and Yau solves: f = argmin g ∈A {|| µ g || ∞ } (1.1)where A = { g : S → S : g is a diffeomorphism } .The above optimization problem (1.1) has a unique global minimizer providedthat S and S are Riemann surfaces of finite type. Also, the unique minimizer f : S → S is a Teichm¨uller map, that is, its associated Beltrami differential µ f is ofthe following form: µ f = k ¯ ϕ | ϕ | (1.2)where 0 ≤ k < ϕ is an integrable holomorphicfunction defined on S ( ϕ (cid:54) = 0). Beltrami differential of this form is said to be ofTeichm¨uller type. To solve the extremalproblem (1.1) to obtain the Teichm¨uller map between connected surfaces, an iterativealgorithm was proposed in [11], called the quasi-conformal(QC) iteration. The ulti-mate goal is to obtain the extremal map between connected (either simply-connectedor multiply-connected) surfaces with given boundary value, which minimizes the con-formality distortion. The basic idea of the iterative algorithm is to project the Bel-trami differential to the space of all Beltrami differentials of Teichm¨uller type, andcompute a quasi-conformal map whose Beltrami differential is closest to the projec-tion in the least square sense. More specifically, the QC iteration for solving (1.1) canbe described as follows. f n +1 = LBS ( µ n +1 ) , (cid:101) µ n +1 = µ n + αµ ( f n +1 , µ n ) ,µ n +1 = L ( P ( (cid:101) µ n +1 )) (1.3)where f n is the quasi-conformal map obtained at the n th iteration, ν n is the Beltramidifferential of f n and µ n is a Beltrami differential of constant modulus. LBS ( µ ) is the operator to obtain a quasi-conformal mapping whose Beltramidifferential is closest to µ in the least square sense. In other words, LBS ( µ ) = argmin f ∈A { (cid:90) S | ∂f∂ ¯ z − µ ∂f∂z | dS } (1.4) µ ( f n +1 , ν n ) denotes the Beltrami differential of f n +1 under the auxiliary metricwith respect to ν n , namely, | dz + ν n d ¯ z | ( | dz | is the original metric on S ). Moreprecisely, µ ( f n +1 , ν n ) can be explicitly computed as follows: µ ( f n +1 , ν n ) = (cid:18) ∂f n +1 ∂ ¯ z + ν n ∂f n +1 ∂z (cid:19) / (cid:18) ∂f n +1 ∂z − ν n ∂f n +1 ∂ ¯ z (cid:19) (1.5) P ( (cid:101) µ n +1 ) is the operator to project (cid:101) µ n +1 to the space of Beltrami differentialswith constant modulus. It is defined as: P ( (cid:101) µ n +1 ) = µ n + (cid:15)w n (1.6)where w n : S → C and (cid:15) : S → R + is a suitable real function on S such that | µ n + (cid:15)w n | is a constant. eichm¨uller Maps via Harmonic Energy Optimization P ( (cid:101) µ n +1 ) = (cid:32) (cid:82) S | (cid:101) µ n +1 | dS (cid:82) S dS (cid:33) (cid:101) µ n +1 | (cid:101) µ n +1 | (1.7) L is the Laplacian smoothing operator to smooth out P ( (cid:101) µ n +1 ).Both P ( ν n ), LBS ( µ n +1 ) and µ ( f n +1 , ν n ) can be easily computed. In particular,the discretization of LBS ( µ n +1 ) on a triangulation mesh can be reduced to a leastsquare problem of a linear system.When µ ( f n +1 , µ n ) is small, α can be chosen to be 1. Then, µ n + αµ ( f n +1 , µ n ) ≈ µ ( f n +1 , µ ( f n +1 ,
0) is the Beltrami differential of f n +1 under the originalmetric.. The QC iteration can be further modified as f n +1 = LBS ( µ n ) , (cid:101) µ n +1 = µ ( f n +1 , ,µ n +1 = P ( L ( (cid:101) µ n +1 )) . (1.8)The QC iteration (1.3) is very efficient. Also, numerical results reported in [11]demonstrate that the proposed iteration can compute the Teichm¨uller map accurately,even on highly irregular meshes. The algorithm was successfully applied to landmark-based registration for applications in medical imaging and computer graphics.This paper is to provide a complete analysis of the above iterative algorithm (1.3).In particular, we prove the convergence of (1.3) that f n and µ n respectively convergesto the extremal map f ∗ and its associated Beltrami differential µ ∗ , which solves theoptimization problem (1.1).We remark that although the iterative algorithm is designed for obtaining ex-tremal map between connected surfaces with given boundary values, the convergenceproof applies to general Riemann surfaces of finite type (such as high-genus closed sur-faces). In other words, the QC iteration can be applied to computing extremal mapsbetween general Riemann surfaces. For the ease of the presentation, we will restrictour discussion to the situation when both S and S are either simply-connected ormultiply-connected open surfaces. This paper is organized as follows. In Section 2, we describesome mathematical background, which is relevant to this work. In Section 3, wereformulate the extremal problem defined by (1.1) as the optimization problem of theharmonic energy, which helps us to understand the iterative algorithm (1.3) better.In Section 4, we prove the convergence of the QC iteration to our desired extremalTeichm¨uller map. A concluding remark will be given in Section 6.
2. Mathematical background.2.1. Quasi-conformal mappings and Beltrami equation.
In this section,we describe some basic mathematical concepts relevant to our algorithms. For details,we refer the readers to [12, 13].A surface S with a conformal structure is called a Riemann surface . Given twoRiemann surfaces S and S , a map f : S → S is conformal if it preserves the surfacemetric up to a multiplicative factor called the conformal factor . A generalizationof conformal maps is the quasi-conformal maps, which are orientation preservinghomeomorphisms between Riemann surfaces with bounded conformality distortion,in the sense that their first order approximations takes small circles to small ellipses Lui, Gu and Yau
Fig. 2.1 . Illustration of how the Beltrami coefficient determines the conformality distortion. of bounded eccentricity [12]. Mathematically, f : C → C is quasi-conformal providedthat it satisfies the Beltrami equation: ∂f∂z = µ ( z ) ∂f∂z . (2.1)for some complex-valued function µ satisfying || µ || ∞ < µ is called the Beltramicoefficient , which is a measure of non-conformality. µ f measures how far the map isdeviated from a conformal map. µ ≡ f is conformal. Infinitesimally,around a point p , f may be expressed with respect to its local parameter as follows: f ( z ) = f ( p ) + f z ( p ) z + f z ( p ) z = f ( p ) + f z ( p )( z + µ ( p ) z ) . (2.2)Obviously, f is not conformal if and only if µ ( p ) (cid:54) = 0. Inside the local parameterdomain, f may be considered as a map composed of a translation to f ( p ) together witha stretch map S ( z ) = z + µ ( p ) z , which is postcomposed by a multiplication of f z ( p ).All the conformal distortion of S ( z ) is caused by µ ( p ). S ( z ) is the map that causes f to map a small circle to a small ellipse. From µ ( p ), we can determine the directionsof maximal magnification and shrinking and the amount of their distortions as well.Specifically, the angle of maximal magnification is arg( µ ( p )) / | µ ( p ) | ; The angle of maximal shrinking is the orthogonal angle (arg( µ ( p )) − π ) / − | µ ( p ) | . Thus, the Beltrami coefficient µ gives us all theinformation about the properties of the map (see Figure 2.1).The maximal dilation of f is given by: K ( f ) = 1 + || µ || ∞ − || µ || ∞ . (2.3)Quasiconformal mapping between two Riemann surfaces S and S can also bedefined. Instead of the Beltrami coefficient, the Beltrami differential has to be used.A Beltrami differential µ ( z ) d ¯ zdz on the Riemann surface S is an assignment to eachchart ( U α , φ α ) of an L ∞ complex-valued function µ α , defined on local parameter z α such that µ α ( z α ) dz α dz α = µ β ( z β ) dz β dz β , (2.4)on the domain which is also covered by another chart ( U β , φ β ), where dz β dz α = ddz α φ αβ and φ αβ = φ β ◦ φ − α . eichm¨uller Maps via Harmonic Energy Optimization Fig. 2.2 . Illustration of the definition of quasi-conformal map between Riemann surfaces.
An orientation preserving diffeomorphism f : S → S is called quasi-conformalassociated with µ ( z ) dzdz if for any chart ( U α , φ α ) on S and any chart ( V β , ψ β ) on S ,the mapping f αβ := ψ β ◦ f ◦ φ − α is quasi-conformal associated with µ α ( z α ) dz α dz α . SeeFigure 2.2 for an illustration. A special class of quasi-conformalmaps is called the extremal maps, which minimize the conformality distortion. Morespecifically, an extremal quasi-conformal map between S and S is extremal in thesense of minimizing the || · || ∞ over all Beltrami differentials corresponding to quasi-conformal mappings between the two surfaces. Extremal map always exists but neednot to be unique. Mathematically, an extremal quasi-conformal mapping can be de-fined as follows: Definition
Suppose S and S are connected Riemann surfaces with boundaries.Let f : S → S be a quasi-conformal mapping between S and S . f is said to be anextremal map if for any quasi-conformal mapping h : S → S isotopic to f relativeto the boundary, K ( f ) ≤ K ( h ) (2.5) It is uniquely extremal if the inequality (2.5) is strict.
Closely related to the extremal map is the
Teichm¨uller map. It is defined asfollows.
Definition
Let f : S → S be a quasi-conformal mapping. f is said to be aTeichm¨uller map associated to the integrable holomorphic function ϕ : S → C if itsassociated Beltrami differential is of the form: µ ( f ) = k ϕ | ϕ | (2.6) for some constant k < and holomorphic function ϕ (cid:54) = 0 with || ϕ || = (cid:82) S | ϕ | < ∞ . In other words, a Teichm¨uller map is a quasi-conformal mapping with uniformconformality distortion over the whole domain.Extremal map might not be unique. However, a Teichm¨uller map associatedwith a holomorphic function is the unique extremal map in its homotopic class. In
Lui, Gu and Yau particular, a Teichmuller map between two connected open surfaces with suitablegiven boundary values is the unique extremal map. The Strebel’s theorem explainsthe relationship bewtween the Teichm¨uller map and extremal map.
Definition
Boundary dilation ). The boundary dilation K [ f ] of f is defined as: K [ f ] = inf C { K ( h | S \ C ) : h ∈ F , C ⊆ S , C is compact . } (2.7) where F is the family of quasi-conformal homeomorphisms of S onto S which arehomotopic to f modulo the boundary. Theorem
Strebel’s theorem, See [14], page 319 ). Let f be an extremalquasi-conformal map with K ( f ) > . If K [ f ] < K ( f ) , then f is a Teichm¨uller mapassociated with an integrable holomorphic function on S . Hence, f is also an uniqueextremal mapping. In other words, an extremal map between S and S with suitable boundarycondition is a Teichm¨uller map. In particular, the Teichm¨uller mapping and extremalmapping of the unit disk are closely related. Theorem
See [15], page 110 ). Let g : ∂ D → ∂ D be an orientation-preservinghomeomorphism of ∂ D . Suppose further that h (cid:48) ( e iθ ) (cid:54) = 0 and h (cid:48)(cid:48) ( e iθ ) is bounded. Thenthere is a Teichm¨uller map f that is the unique extremal extension of g to D . Thatis, f : D → D is an extremal mapping with f | ∂ D = g . Thus, if the boundary correspondence satisfies certain conditions on its deriva-tives, the extremal map of the unit disk must be a Teichm¨uller map.Now, in the case when interior landmark constraints are further enforced, theexistence of unique Teichm¨uller map can be guaranteed if the boundary and landmarkcorrespondence satisfy suitable conditions. The unique Teichm¨uller map is extremal,which minimizes the maximal conformality distortion. The following theorem can bederived immediately from the Strebel’s Theorem (Theorem 2.4):
Theorem
Let { p i } ni =1 ∈ S and { q i } ni =1 ∈ S be the corresponding interiorlandmark constraints. Let f : S \ { p i } ni =1 → S \ { q i } ni =1 be the extremal map, suchthat p i corresponds to q i for all ≤ i ≤ n . If K [ f ] < K ( f ) , then f is a Teichm¨ullermap associated with an integrable holomorphic function on S \ { p i } ni =1 . Hence, f isan unique extremal map. In particular, a unique Teichm¨uller map f : D → D between unit disks with inte-rior landmark constraints enforced exists, if the boundary map f | ∂ D satisfies suitableconditions. The following theorem can be obtained directly from Theorem 2.5: Theorem
Let g : ∂ D → ∂ D be an orientation-preserving homeomorphismof ∂ D . Suppose further that h (cid:48) ( e iθ ) (cid:54) = 0 and h (cid:48)(cid:48) ( e iθ ) is bounded. Let { p i } ni =1 ∈ D and { q i } ni =1 ∈ D be the corresponding interior landmark constraints. Then there is aTeichm¨uller map f : D \{ p i } ni =1 → D \{ q i } ni =1 matching the interior landmarks, whichis the unique extremal extension of g to D . That is, f : D \ { p i } ni =1 → D \ { q i } ni =1 isan extremal Teichm¨uller map with f | ∂ D = g matching the interior landmarks. eichm¨uller Maps via Harmonic Energy Optimization Our iterative algorithm to compute Teichm¨uller maps isclosely related to harmonic maps. Let ( S , σ | dz | ) and ( S , ρ | dw | ) be two Riemannsurfaces of finite type, where z and w refer to the local conformal coordinate on thesurface S and S .For a Lipschitz map f : ( S , σ | dz | ) → ( S , ρ | dw | ), we define the energy E ( f ; σ, ρ )of the map w to be E harm ( f ; σ, ρ ) = (cid:90) S (cid:107) df (cid:107) dv ( σ ) = (cid:90) S ρ ( w ( z )) σ ( z ) ( | w z | + | w ¯ z | ) σ ( z ) dzd ¯ z. (2.8)Therefore E harm ( f ; σ, ρ ) = (cid:90) S ρ ( w ( z ))( | w z | + | w ¯ z | ) dzd ¯ z. (2.9)It depends on the metric structure of the target surface ρ | dw | and the conformalstructure σ | dz | of the source.A critical point of this functional is called a harmonic map. We will focus onthe situation where we have fixed the homotopy class f : S → S of maps into thecompact target S with non-positive curvature K ( w ) ≤ f ( σ, ρ ) : ( S , σ ) → ( S , ρ ) in the homotopy class of f . If f is harmonic, then f z ¯ z + (log ρ ) z f z f ¯ z ≡ . (2.10)The pull back metric on S induced by f is given by f ∗ ( ρ ( w ) | dw | ) = ρ ( f z dz + f ¯ z d ¯ z )( ¯ f z d ¯ z + f ¯ z dz ) (2.11)Then the Hopf differential is Φ( f ) := ρ ( f ( z )) f z f ¯ z dz . (2.12)It can be shown that f is harmonic if and only if its Hopf differential is a holomorphicquadratic differential.
3. Quasi-conformal iteration.
Before giving a complete analysis of the con-vergence of the QC iteration, we reformulate the extremal problem (1.1) as the opti-mization problem of the harmonic energy, in order to better understand the iterativealgorithm.Consider two connected open surfaces S and S with boundaries, which are ofthe same topology. S and S can either be simply-connected or multiply-connected.Suppose σ | dz | and ρ | dw | are the Riemannian metric on S and S respectively.Assume ( S , ρ | dw | ) has non-positive Gaussian curvature K ( w ) everywhere. Let f : S → S be any quasi-conformal mapping between S and S . In the homotopic class[ f ] of f , there exists a unique Teichm¨uller map, f ∗ . f ∗ is also extremal within thehomotopic class [ f ]. More specifically, the homotopic class [ f ] can be defined as:[ f ] = { g : S : S : g | ∂S = f | ∂S } . (3.1)We have, || µ f ∗ || ∞ ≤ || µ g || ∞ for all g ∈ [ f ], where µ f ∗ and µ g are the Beltramidifferentials of f ∗ and g respectively. Lui, Gu and Yau
Consider the space of all admissible Beltrami differentials on S , which is denotedby B ( S , S ). Every Beltrami differential µ ∈ B ( S , S ) induces a conformal structure g ( µ ) on S , namely, g ( µ ) = | dz + µd ¯ z | (3.2)Suppose µ , µ ∈ B ( S , S ), we say that they are globally equivalent , if there is abiholomorphic mapping f : ( S , g ( µ )) → ( S , g ( µ )) such that f is homotopic to theidentity map of S . The equivalence class of µ is represented by [ µ ]. Each globalequivalence class of Beltrami differentials has a unique representative of Teichm¨ullerform. We denote the space of all Beltrami differentials of Teichm¨uller form by T ( S , S ) := { µ ∈ B ( S , S ) : | µ | is a constant . } . (3.3)We can now define an energy functional E BC on B ( S , S ). For any µ ∈ B ( S , S ),there exists a unique harmonic map f ( µ, ρ ) : ( S , g ( µ )) → ( S , ρ | dw | ) ∈ [ f ] , whichis solely determined by µ and ρ | dw | . The value of E BC ( µ ) can then be defined asthe harmonic energy of f ( µ, ρ ). That is, E BC ( µ ) = E harm ( f ( µ, ρ )) = (cid:90) S || df ( µ, ρ ) || (3.4) E BC : B ( S , S ) → R is a smooth function. Lemma 3.1.
The energy functional E BC : T ( S , S ) → R is bounded below by E BC ( µ ) ≥ (cid:90) S ρ ( w ) dudv (3.5) where w = u + iv . The equality holds if and only if ( S , g ( µ )) is conformally equiv-alent to ( S , ρ | dw | ) . And the harmonic map f ( µ, ρ ) : ( S , g ( µ )) → ( S , ρ | dw | ) is aconformal mapping.Proof . Let z = x + iy be the local coordinate of ( S , g ( µ )). The Jacobian of themapping f ( µ, ρ ) : ( S , g ( µ )) → ( S , ρ | dw | ) is given by J ( z ) = | w z | − | w ¯ z | . (3.6)Therefore, J ( z ) dxdy = ( | w z | − | w ¯ z | ) dxdy = dudv. (3.7)The harmonic energy is given by E harm ( f ( µ, ρ )) = E BC ( µ ) = (cid:90) S ρ ( w )( | w z | + | w ¯ z | ) dxdy = (cid:90) S ρ ( w ) | w z | + | w ¯ z | | w z | − | w ¯ z | dudv, (3.8)where | w z | + | w ¯ z | | w z | − | w ¯ z | = 1 + | w ¯ z w z | − | w ¯ z w z | = 1 + | µ | − | µ | = 1 + k − k = 12 (cid:18) k − k + 1 − k k (cid:19) = 12 (cid:18) K + 1 K (cid:19) and (3.9) eichm¨uller Maps via Harmonic Energy Optimization k = | µ | , ≤ k ≤ , K = 1 + k − k , K ≥ . (3.10)Hence, E BC ( µ ) = 12 (cid:90) S ρ ( w ) (cid:18) K + 1 K (cid:19) dudv ≥ (cid:90) S ρ ( w )(2) dudv = (cid:90) S ρ ( w ) dudv. (3.11)Equality holds if and only if K ≡
1, namely, k ≡
0. This implies f ( µ, ρ ) is a conformalmapping. Theorem 3.2.
The global minimizer of the energy functional E BC : T ( S , S ) → R is the Beltrami differerntial associated to the unique Teichm¨uller map between ( S , σ | dz | ) and ( S , ρ | dw | ) in the homotopic class [ f ] of f .Proof . Let µ ∗ be the Beltrami differential of the Teichm¨uller map ˜ f . It sufficesto show that ˜ f : ( S , g ( µ ∗ )) → ( S , ρ | dw | ) is a conformal mapping.To see this, let ˜ f ∗ ( ρ | dw | ) denote the pull back metric. Then,˜ f ∗ ( ρ | dw | ) = e λ ( ˜ f ( z )) | df ( z ) | . (3.12)Under the pull back metric, the mapping ˜ f : ( S , ˜ f ∗ ( ρ | dw | )) → ( S , ρ | dw | ) isisometric. We have d ˜ f ( z ) = ∂ ˜ f ( z ) ∂z dz + ∂ ˜ f ( z ) ∂ ¯ z d ¯ z = ∂ ˜ f ( z ) ∂z ( dz + µ ∗ d ¯ z ) . (3.13)Hence, ˜ f ∗ ( ρ | dw | ) = e λ ( ˜ f ( z )) | ∂ ˜ f ( z ) ∂z | | dz + µ ∗ d ¯ z | . (3.14)So, ˜ f ∗ ( ρ | dw | ) = e λ ( ˜ f ( z )) − λ ( z ) | ∂ ˜ f ( z ) ∂z | g ( µ ∗ ). f ∗ ( ρ | dw | ) is conformal to g ( µ ∗ ).We conclude that ˜ f : ( S , g ( µ ∗ )) → ( S , ρ | dw | ) is conformal. According to Theo-rem 3.1, the Beltrami differential associated to ˜ f is the global minimizer of E BC : T ( S , S ) → R .In other words, finding the extremal Teichm¨uller map, f ∗ , is equivalent to mini-mizing the energy functional E BC . During the QC iteration, the Beltrami differential µ n is iteratively adjusted and a new map is obtained by f n = LBS ( µ n ). It turns out LBS ( µ n ) is equivalent to computing the harmonic map f ( µ n , ρ ). It can be explainedin more details as follows. Lemma 3.3.
Suppose µ ∈ T ( S , S ) . The mapping f := LBS ( µ ) is a harmonicmap between ( S , g ( µ )) and ( S , ρ | dw | ) .Proof . Let ζ be the coordinates of S with respect to the metric g ( µ ). Let h bethe harmonic map between ( S , g ( µ )) and ( S , g ( ρ )). Then h is a critical point of thefollowing harmonic energy: E harm ( h ) = (cid:90) S ρ ( h ( ζ ))( | h ζ | + | h ¯ ζ | ) dxdy Lui, Gu and Yau
Since f := LBS ( µ ), according to the definition, f is the critical point of the followingenergy functional: E LBS ( f ) = (cid:90) S ρ ( f ( z ))( | f ¯ z − µf z | ) dxdy We will show that the above two energy functionals have the same set of criticalpoints.Note that dζ = dz + µd ¯ z , then d ¯ ζ = d ¯ z + ¯ µdz. (3.15)We obtain dz = 11 − | µ | ( dζ − µd ¯ ζ ); d ¯ z = 11 − | µ | ( − ¯ µdζ + d ¯ ζ ) . (3.16)Hence, dz ∧ d ¯ z = 11 − | µ | dζ ∧ d ¯ ζ ; h ¯ ζ = 11 − | µ | ( h ¯ z − µh z ) . (3.17)Now, the Jacobian J h of h and the Jacobian J f of f are given by J h = | h ζ | − | h ¯ ζ | ; J f = | f z | − | f ¯ z | (3.18)Hence, E harm ( h ) = (cid:90) S ρ ( h ( ζ ))(2 | h ¯ ζ | + J h ) idζ ∧ d ¯ ζ = (cid:90) S − | µ | ρ ( h ( z )) | h ¯ z − µh z | idz ∧ d ¯ z + (cid:90) S ρ ( h ( ζ )) J h idζ ∧ d ¯ ζ (3.19)Since µ ∈ T ( S , S ), | µ | is a constant. Thus, E harm ( h ) = 21 − | µ | (cid:90) S ρ ( h ( z )) | h ¯ z − µh z | idz ∧ d ¯ z + A (3.20)where A is the surface area of S . We conclude that E harm and E LBS has the sameset of critical points. Since f is a critical point of E LBS , f is also a critical point of E harm . Hence, f is a harmonic map between ( S , g ( µ )) and ( S , ρ | dw | ).The Beltrami differential µ n ∈ T ( S , S ) is iteratively adjusted during the QCiteration. In the next section, we will prove that E BC ( µ n ) monotonically decreasesto the global minimizer of E BC .
4. Proof of convergence.
In this section, we prove the convergence of theQuasi-conformal iteration to the desired Teichm¨uller map.
Lemma 4.1.
Suppose µ ∈ B ( S , S ) is deformed by µ → µ + (cid:15)ν ∈ B ( S , S ) . Then, the variation of E BC satisfies: E BC ( µ + (cid:15)ν ) ≤ E BC ( µ ) − Re (cid:90) S (cid:15) Φ( f ( µ, ρ )) ν dz µ ∧ d ¯ z µ − i + O ( (cid:15) ) . eichm¨uller Maps via Harmonic Energy Optimization where z µ is the coordinates of S under the metric g ( µ ) .Proof . Let ζ be the coordinate of S under the metric g ( µ + (cid:15)ν ). For simplicity,let z = z µ . Then, we have dz = dζ − (cid:15)νd ¯ ζ ; d ¯ z = d ¯ ζ − (cid:15) ¯ νdζ. (4.1)The area element with respect to z is given by dz ∧ d ¯ z = dζ ∧ d ¯ ζ − (cid:15)νd ¯ ζ ∧ d ¯ ζ − (cid:15) ¯ νdζ ∧ dζ + (cid:15) | ν | d ¯ ζdζ. (4.2)Hence, dz ∧ d ¯ z = dζ ∧ d ¯ ζ + (cid:15) | ν | d ¯ ζdζ. (4.3)Similarly, dζ ∧ d ¯ ζ = dz ∧ d ¯ z + (cid:15) | ν | d ¯ zdz. (4.4)Let w = f ( µ, ρ ). Then, dw = w ζ dζ + w ¯ ζ d ¯ ζ = w z dz + w ¯ z d ¯ z = w z ( dζ − (cid:15)νd ¯ ζ ) + w ¯ z ( d ¯ ζ − (cid:15) ¯ νdζ ) , (4.5)Therefore, w ζ w ζ = ( w z − (cid:15) ¯ νw ¯ z )( w z − (cid:15)νw ¯ z )= | w z | + (cid:15) | ν | | w ¯ z | − (cid:15)νw z w ¯ z − (cid:15) ¯ νw z w ¯ z . (4.6)Similarly, w ¯ ζ w ¯ ζ = ( w ¯ z − (cid:15)νw z )( w ¯ z − (cid:15) ¯ νw z )= | w ¯ z | + (cid:15) | ν | | w z | − (cid:15)νw ¯ z w z − (cid:15) ¯ νw z w ¯ z . (4.7)As a result, we get E BC ( µ + (cid:15)ν ) ≤ E harm ( w ) = (cid:90) S ρ ( w ( ζ ))( | w ζ | + | w ¯ ζ | ) dζ ∧ d ¯ ζ − i = (cid:90) S ρ ( w ( z ))( | w z | + | w ¯ z | ) dz ∧ d ¯ z − i − Re (cid:90) S (cid:15)ρ ( w ( z )) w z w ¯ z ν dz ∧ d ¯ z − i + O ( (cid:15) )= E BC ( µ ) − Re (cid:90) S (cid:15) ρ ( w ( z )) w z w ¯ z ν dz ∧ d ¯ z − i + O ( (cid:15) )= E BC ( µ ) − Re (cid:90) S (cid:15) Φ( f ( µ, ρ )) ν dz µ ∧ d ¯ z µ − i + O ( (cid:15) ) . (4.8)This completes the proof of the inequality. Theorem 4.2.
Suppose µ ∈ T ( S , S ) . For any α > , there exists w ∈ B ( S , S ) and (cid:15) : S → R such that:(i) µ + (cid:15)w ∈ T ( S , S ) ; Lui, Gu and Yau (ii) | (cid:15) ( p ) w ( p ) | < α and | w ( p ) | = | Φ( f ( µ, ρ ))( p ) | for all p ∈ S ;(iii) (cid:82) S (cid:15)w Φ( f ( µ, ρ )) dz µ ∧ d ¯ z µ − i ≥ .Proof . Let ˜ k = | µ | and ν = Φ( f ( µ, ρ )). Pick β ∈ R + such that: β sup p ∈ S | ν ( p ) | < α/ . (4.9)Consider ˜ µ = µ + βν .Suppose: Ω = { p ∈ S : arg( ν ) = arg( µ ) } ;Ω = { p ∈ S : arg( ν ) = − arg( µ ) } . (4.10)Let: γ = (cid:90) Ω | ν | dz µ ∧ d ¯ z µ − i − (cid:90) Ω | ν | dz µ ∧ d ¯ z µ − i . (4.11)If γ >
0, choose ˜ k < k < sup p ∈ S | ˜ µ ( p ) | .If γ <
0, choose inf p ∈ S | ˜ µ ( p ) | < k < ˜ k .If γ = 0 (including Ω = Ω = ∅ ), choose inf p ∈ S | ˜ µ ( p ) | < k < sup p ∈ S | ˜ µ ( p ) | .Let: r = k ˜ µ | ˜ µ | ; w = r − µ | r − µ | | ν | and (cid:15) = | r − µ || ν | . (4.12)By definition, µ + (cid:15)w = r = k ˜ µ | ˜ µ | ∈ T ( S , S ). Hence, (i) is satisfied.Now, | w ( p ) | = | ν ( p ) | = | Φ( f ( µ, ρ ))( p ) | for all p ∈ S . (4.13)Also, | (cid:15) ( p ) w ( p ) | = | r − µ |≤ | r − ˜ µ | + | ˜ µ − µ | = | r − ˜ µ | + | βν | < α α α. (4.14)Thus, (ii) is also satisfied.Finally, it is easy to check that: (cid:90) S \ (Ω ∪ Ω ) (cid:15)w Φ( f ( µ, ρ )) dz µ ∧ d ¯ z µ − i ≥ . (4.15)Now, if γ > (cid:90) Ω (cid:15)w Φ( f ( µ, ρ )) dz µ ∧ d ¯ z µ − i + (cid:90) Ω (cid:15)w Φ( f ( µ, ρ )) dz µ ∧ d ¯ z µ − i = (cid:90) Ω ( k − ˜ k ) | ν | dz µ ∧ d ¯ z µ − i − (cid:90) Ω ( k − ˜ k ) | ν | dz µ ∧ d ¯ z µ − i = ( k − ˜ k ) γ > . (4.16) eichm¨uller Maps via Harmonic Energy Optimization γ < (cid:90) Ω (cid:15)w Φ( f ( µ, ρ )) dz µ ∧ d ¯ z µ − i + (cid:90) Ω (cid:15)w Φ( f ( µ, ρ )) dz µ ∧ d ¯ z µ − i = − (cid:90) Ω (˜ k − k ) | ν | dz µ ∧ d ¯ z µ − i + (cid:90) Ω (˜ k − k ) | ν | dz µ ∧ d ¯ z µ − i = − (˜ k − k ) γ > . (4.17)We conclude that (cid:82) S (cid:15)w Φ( f ( µ, ρ )) dz µ ∧ d ¯ z µ − i ≥ Theorem 4.3.
Suppose S and S are open Riemann surfaces with boundariesof the same topology. Given a smooth boundary correspondence h : ∂S → ∂S , theQuasi-conformal (QC) iteration (1.3) converges to the unique extremal map, which isalso a Teichm¨uller map.Proof . Suppose the pair ( f n , µ n ) is obtained at the n th iteration. The QC iterationfirst compute a new quasi-conformal map by f n +1 = LBS ( µ n ). According to Lemma3.3, f n +1 is a harmonic map between ( S , g ( µ n )) and ( S , ρ | dw | ). The Beltramidifferential ν n +1 of f n +1 can be computed by (cid:101) µ n +1 = µ n + βµ ( f n +1 , µ n ). µ ( f n +1 , µ n )denotes the Beltrami differential of f n +1 under the auxiliary metric with respect to µ n ,namely, | dz + µ n d ¯ z | . A new Beltrami differential can then be obtained by projecting (cid:101) µ n +1 onto T ( S , S ) to get P ( ν n ) = µ n + (cid:15)w n . (4.18)Here, w n : S → C and (cid:15) : S → R is a suitable real function on S such that | µ n + (cid:15)µ ( f n +1 , µ n ) | ≡ k , where k is a positive constant.According to Theorem 4.2, by choosing a suitable k , we can assume that (cid:90) S (cid:15)w n Φ( f ( µ, ρ )) dz µ ∧ d ¯ z µ − i ≥ . (4.19) P ( ν n ) is then smoothed out by the Laplacian operator L with the constraint thatit still preserves Equation 4.19. We get that E BC ( µ n +1 ) − E BC ( µ n ) = − Re (cid:90) S (cid:15) Φ( f ( µ n , ρ )) w n dz µ ∧ d ¯ z µ − i + O ( (cid:15) ) ≤ E ( µ n ) is monotonically decreasing. According to Lemma 3.1, E is boundedfrom below. Hence, E ( µ n ) converges. Also, the QC iteration is essential the gradientdescend algorithm of E BC and it converges at the critical point µ ∗ = k ∗ e iθ . That is,Φ( f ( µ ∗ , ρ )) = 0. In this case, g ( µ ∗ ) is conformal to ρ and hence f ( µ ∗ , ρ ) is a quasi-conformal map with Beltrami differential µ ∗ . Furthermore, at the critical point, theLaplacian L of the Beltrami differential is zero. We conclude that θ is harmonic. Since θ is harmonic, we can find its harmonic conjugate r such that r + iθ is holomorphic.Define ϕ = e r − iθ , which is also holomorphic. Then, µ ∗ = k ∗ ϕ | ϕ | is of Teichm¨uller type.Since µ ∗ is of Teichm¨uller type, f ( µ ∗ , ρ ) must be a Teichm¨uller map. Now, given asmooth boundary correspondence h : ∂S → ∂S , there exists a unique Teichm¨ullermap which is an extremal map. We conclude that f ( µ, ρ ) is the unique extremalTeichm¨uller map.4 Lui, Gu and Yau
Fig. 4.1 . Two simply-connected domains. (A) a unit disk D (B) an arbitrary simply-connecteddomain. Fig. 4.2 . Extremal Teichm¨uller map between two simply-connected domains as shown in Figure4.1(A) and (B), with given boundary correspondence.
5. Numerical experiments.
Although the numerical testing is not the mainfocus of this work, we demonstrate some numerical results in this section for thecompleteness of the paper. The results agree with our theoretical findings.
Example 1.
We first test the algorithm to compute the extremal Teichm¨ullermap between two simply-connected domains Ω and Ω . Ω is chosen to be the unitdisk D as shown in Figure 4.1(A). Ω is deformed to an arbitrary simply-connectedshape Ω as shown in (B). The boundary correspondence h of Ω and Ω is given. Wecompute the extremal Teichm¨uller map f : Ω → Ω such that f | ∂ Ω = h using theproposed QC iterations. The obtained map is visualized using texture map as shownin Figure 4.2. The small circles on the source domain is mapped to small ellipseson the target domain with the same eccentricity. Figure 4.3(A) shows the energy E ( µ n ) := E BC ( µ n ) − A (Ω ) versus each iterations in the QC iterations, where A (Ω )is the area of Ω . The energy monotonically decreases to 0, which agrees with Theorem4.3. (B) shows the histogram of the norm of the optimal Beltrami differential µ ∗ . Itaccumulates at 0.33, which illustrates that the obtained map is indeed a Teichm¨ullermap. Since µ ∗ is of Teichm¨uller type, its argument must be harmonic. (C) showsthe histogram of the Laplacian of arg ( µ ∗ ). It accumulates at 0, meaning that theargument of µ ∗ is indeed harmonic. Example 2.
In our second example, we test our algorithm to compute theextremal Teichmuller map between two punctured unit disks. Figure 5.1(A) and eichm¨uller Maps via Harmonic Energy Optimization Fig. 4.3 . (A) shows the energy E ( µ n ) := E BC ( µ n ) − A (Ω ) per iterations during the QCiterations of Example 1. (B) shows the histogram of the norm of the optimal Beltrami coefficient µ ∗ . (C) shows the histogram of the Laplacian of arg ( µ ∗ ) Fig. 5.1 . Two punctured unit disks. (A) and (B) show two unit disks, each with 6 punctures. (B) show two unit disks, each with 6 punctures. Denote the source domain byΩ := D \ { p i } i =1 , and denote the target domain by Ω := D \ { q i } i =1 . The bound-ary correspondence of ∂ D is chosen to be the identity map. Using the QC iteration,we compute the extremal Teichm¨uller map f : Ω → Ω such that f | ∂ D = id and f ( p i ) = q i for 1 ≤ i ≤
6. The obtained map is visualized using texture map as shownin Figure 5.2. The small circles on the source domain is mapped to small ellipseson the target domain with the same eccentricity. Figure 5.3(A) shows the energy E ( µ n ) := E BC ( µ n ) − A (Ω ) versus each iterations in the QC iterations, where A (Ω )is the area of Omega . The energy monotonically decreases to 0, which agrees with ourtheoretical finding. (B) shows the histogram of the norm of the Beltrami differential.It accumulates at 0.6, which illustrates that the obtained map is indeed a Teichm¨ullermap. (C) shows the histogram of the Laplacian of arg ( µ ∗ ). It accumulates at 0,meaning that the argument of µ ∗ is indeed harmonic. Example 3.
Finally, we test the QC iterations to compute the extremal Te-ichm¨uller map between two triply-connected domains Ω and Ω , each with 6 punc-tures. As shown in Figure 5.4(A), Ω is chosen to be unit disk with three inner disksand six points removed (denote it by { p i } i =1 ). Ω is chosen to be unit disk with threeinner regions (with arbitrary shapes) and six points removed (denote it by { q i } i =1 ),as shown in (B). Again, the boundary correspondence h : ∂ Ω → ∂ Ω is given. Usingthe QC iterations, we compute the extremal Teichm¨uller map f : Ω → Ω such that6 Lui, Gu and Yau
Fig. 5.2 . Extremal Teichm¨uller map between two punctured unit disks as shown in Figure5.1(A) and (B), with given boundary correspondence. .
Fig. 5.3 . (A) shows the energy E ( µ n ) := E BC ( µ n ) − A (Ω ) per iterations during the QCiterations of Example 2. (B) shows the histogram of the norm of the optimal Beltrami coefficient µ ∗ . (C) shows the histogram of the Laplacian of arg ( µ ∗ ) f | ∂ Ω = h and f ( p i ) = q i for 1 ≤ i ≤
6. The obtained map is visualized using texturemap as shown in Figure 5.5. The small circles on the source domain is mapped tosmall ellipses on the target domain with the same eccentricity. Figure 5.6(A) showsthe energy E ( µ n ) := E BC ( µ n ) − A (Ω ) versus each iterations in the QC iterations,where A (Ω ) is the area of Ω . The energy monotonically decreases to 0, which agreeswith our theoretical finding. (B) shows the histogram of the norm of the Beltramidifferential. It accumulates at 0.42, which illustrates that the obtained map is in-deed a Teichm¨uller map. (C) shows the histogram of the Laplacian of arg ( µ ∗ ). Itaccumulates at 0, meaning that the argument of µ ∗ is indeed harmonic.
6. Conclusion.
This paper gives the convergence proof of the iterative algorithmproposed in [11] to compute the extremal Teichm¨uller map between Riemann surfacesof finite type. The iterative algorithm, which is named as quasi-conformal (QC)iteration, can be formulated as the optimization process of the harmonic energy. Withthis formulation, the QC iteration can be considered as the gradient descent of theharmonic energy under the auxiliary metric given by the Beltrami differentials.In the future, we will further improve the efficiency of the iterative scheme tooptimize the harmonic energy. The proposed framework will also be further extendedto compute Teichm¨uller maps between high-genus surfaces (genus ≥ REFERENCESeichm¨uller Maps via Harmonic Energy Optimization Fig. 5.4 . Two triply-connected domains, each with 6 punctures. (A) and (B) show the sourcedomain and target domain respectively.
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