1

Hyperbolic Harmonic Mapping for SurfaceRegistration

Rui Shi, Wei Zeng, Zhengyu Su, Yalin Wang, Xianfeng Gu

F

Abstract—Automatic computation of surface correspondencevia harmonic map is an active research field in computer vision,computer graphics and computational geometry. It may helpdocument and understand physical and biological phenomenaand also has broad applications in biometrics, medical imagingand motion capture. Although numerous studies have beendevoted to harmonic map research, limited progress has beenmade to compute a diffeomorphic harmonic map on generaltopology surfaces with landmark constraints. This work conquerthis problem by changing the Riemannian metric on the targetsurface to a hyperbolic metric, so that the harmonic mappingis guaranteed to be a diffeomorphism under landmark con-straints. The computational algorithms are based on the Ricciflow method and the method is general and robust. We applyour algorithm to study constrained surface registration problemwhich applied to both medical and computer vision applications.Experimental results demonstrate that, by changing the Rie-mannian metric, the registrations are always diffeomorphic, andachieve relative high performance when evaluated with somepopular surface registration evaluation standards.

1 INTRODUCTION

Harmonic mapping has been commonly appliedfor brain cortical surface registration. Physically, aharmonic mapping minimizes the “stretching en-ergy”, and produces smooth registration. The har-monic mappings between two hemsiphere corticalsurfaces, which were modeled as genus zero closedsurfaces, are guaranteed to be diffeomorphic, andangle-preserving [1]. Furthermore, all such kind ofharmonic mappings differ by the Mobius trans-formation group. Numerically, finding a harmonicmapping is equivalent to solve an elliptic partialdifferential equation, which is stable in the com-putation and robust to the input noises.

Unfortunately, harmonic mappings with con-straints may not be diffeomorphic any more, andproduces invalid registrations with flips. In order to

overcome this shortcoming, in this work we proposea novel brain registration method, which is based onhyperbolic harmonic mapping. Conventional regis-tration methods map the template brain surface tothe sphere or planar domain [1], [2], then computeharmonic mappings from the source brain to thesphere or planar domain. When the target domainsare with complicated topologies, or the landmarks,the harmonic mappings may not be diffeomrophic.In contrast, in our work, we slice the brain surfacesalong the landmarks, and assign a unique hyperbolicmetric on the template brain, such that all theboundaries become geodesics, harmonic mappingsare established and guaranteed to be diffeomorphic.

2 THEORETIC BACKGROUNDHyperbolic Harmonic Map: Suppose S is anoriented surface with a Riemannian metric g. Onecan choose a special local coordinates (x, y), theso-called isothermal parameters, such that g =σ(x, y)(dx2+dy2) = σ(z)dzdz, where the complexparameter z = x + iy, dz = dx + idy. An atlasconsisting of isothermal parameter charts is calledan conformal structure.

Definition 2.1 (Harmonic Map): The harmonicenergy of the mapping is defined as E(f) =∫Sρ(z)(|wz|2 + |wz|2)dxdy. If f is a critical point

of the harmonic energy, then f is called a harmonicmap.The necessary condition for f to be a harmonic mapis the Euler-Lagrange equation wzz +

ρwρwzwz ≡ 0.

The theory on the existence, uniqueness and reg-ularity of harmonic maps have been thoroughlydiscussed in [3]. The following theorem lays downthe theoretic foundation of our proposed method.

Theorem 2.2: [3] Suppose f : (S1,g1) →(S2,g2) is a degree one harmonic map, furthermore

2

the Riemann metric on S2 induces negative Gausscurvature, then for each homotopy class, the har-monic map is unique and diffeomorphic.

Ricci Flow: Ricci flow deforms the Riemannianmetric proportional to the curvature, such that thecurvature evolves according to a heat diffusion pro-cess and eventually becomes constant everywhere.

Definition 2.3 (Ricci Flow): Hamilton’s surfaceRicci flow is defined as dgij

dt= −2Kgij.

Theorem 2.4 (Hamilton): Let (S,g) be compact.If χ(S) < 0, then the solution to Ricci Flowequation exists for all t > 0 and converges to ametric of constant curvature.

Fundamental Group and Fuchs Group: Let Sbe a surface, all the homotopy classes of loops formthe fundamental group (homotopy group), denotedas π1(S). A surface S with a projection map p :S → S is called the universal covering space ofS. The Deck transformation ϕ : S → S satisfiesϕ◦p = p and form a group Deck(S). Let γ be a loopon the hyperbolic surface, then its homotopy class[γ] corresponds to a unique Mobius transformationϕγ . As the Gauss curvature of S is negative, in eachhomotopy class [γ], there is a unique geodesic loopgiven by the axis of ϕγ .

Hyperbolic Pants Decomposition: Given a topo-logical surface S, it can be decomposed to pairs ofpants. Each pair of pants is a genus zero surfacewith three boundaries. If the surface is with ahyperbolic metric, then each homotopy class hasa unique geodesic loop. Suppose a pair of hy-perbolic pants with three boundaries {γi, γj, γk},which are geodesics. Let {τi, τj, τk} be the shortestgeodesic paths connecting each pair of boundaries.The shortest paths divide the surface to two identicalhyperbolic hexagons with right inner angles. Whenthe hyperbolic hexagon with right inner angles isisometrically embedded on the Klein disk model, itis identical to a convex Euclidean hexagon.

3 ALGORITHMS

We explain our registration algorithm pipeline asillustrated in Alg. 1 and Fig. 1:

REFERENCES

[1] X. Gu and et al. Genus zero surface conformal mapping and itsapplication to brain surface mapping. IEEE Trans. Med. Imaging,2004.

Algorithm 1 Brain Surface Registration Algo-rithm Pipeline.

1. Slice the cortical surface along the landmarkcurves.2. Compute the hyperbolic metric using Ricciflow.3. Hyperbolic pants decomposition, isometricallyembed them to Klein model.4. Compute harmonic maps using Euclidean met-rics between corresponding pairs of pants, withconsistent curvature based boundary matchingconstraints computed by the DWT algorithm.5. Use nonlinear heat diffusion to improve themapping to a global harmonic map on Poincaredisk model.

Fig. 1. Algorithm Pipeline (suppose we have2 brain surface M and N as input): (a). Oneof the input brain models M , with landmarksbeing cut open as boundaries. (b). Hyperbolicembedding of the M on the Poincare disk. (c).Decompose M into multiple pants by cuting thelandmarks into boundaries, and each pant is fur-ther decomposed to 2 hyperbolic hexagons. (d).Hyperbolic hexagons on Poincare disk becomeconvex hexagons under the Klein model, thena one-to-one map between the correspondentparts of M and N can be obtained. Then we canapply our hyperbolic heat diffusion algorithm toget a global harmonic diffeomorphism.

[2] A. A. Joshi and et al. A parameterization-based numericalmethod for isotropic and anisotropic diffusion smoothing on non-flat surfaces. Trans. Img. Proc., 2009.

[3] R. M. Schoen and S.-T. Yau. Lectures on harmonic maps.International Press, 1997, Mathematics.