BackgroundIntroductionLandmark-based matchingDense MatchingApplicationsConclusionDiffeomorphismsMarc NiethammerDepartment of Computer Science, University of North Carolina, Chapel HillBackground IntroductionThis is itThis (function f ) is a diffeomorphismf : x 7→ x, x ∈ R.Let’s see what other kind of diffeomorphisms there are and whatproperties they have ...0 / 27Background IntroductionDefinitionDefinition (Diffeomorphism)A C∞mapping F : M 7→ N between C∞manifolds is adiffeomorphism if it is a homeomorphism and F−1is C∞. M and Nare diffeomorphic if there exists a diffeomorphism F : M 7→ N.“A diffeomorphism is a smooth bijective mapping with a smoothinverse.”1 / 27Background IntroductionUnderstanding the DefinitionTo understand the definition we need to clarify the following•What is a mapping?•What is a homeomorphism?•What is C∞?•What is a manifold?•What is a C∞manifold?Before we do this, let’s look at a couple of examples.2 / 27Background IntroductionExamples of DiffeomorphismsIn medical imaging, diffeomorphic mappings are often times used inconjunction with infinite-dimensional manifold (for smooth invertibleelastic mappings in registration).The concept of diffeomorphisms is much more general though.3 / 27Background IntroductionSome ExamplesAffine det(A) 6= 0Elastic deformation[Image from Ashburner]Let’s try to understand the definition now ...4 / 27Background IntroductionWhat is a homeomorphism?DefinitionHomeomorphism A function f : M 7→ N is a homeomorphism if f isbijective and f and f−1are continuous.Bijection.Continuous.Disontinuous.5 / 27Background IntroductionInterpreting a function as a coordinate transformMapping is not bijective Mapping is discontinuous1 2 3 4 5 61 2 1 2 34 5 61 2 3 5 64Homeomorphic mapping3 4 5 6What are we still missing?6 / 27Background IntroductionA Simple Example of Terrible ThingsIt seems like a homeomorphism is all we want. Because it•prevents folding and•does not allow for tearing either.What else could we ask for?The functionf : x 7→ x3, f−1: x 7→ x13, x ∈ R,is homeomorphic. But the derivative of f−1is not defined at 0.−1 −0.5 0 0.5 1−1−0.500.51−1 −0.5 0 0.5 1−1−0.500.51Will show later that this is not a diffeomorphism.7 / 27Background IntroductionWhat is C∞?DefinitionClass CkA function f is of class Ckif it is k times continuouslydifferentiable.A function is of class C∞(is smooth) if it has derivatives of all orders.f (x) = sin(x) is C∞All polynomials are C∞f (x) = x13is not C∞, because f0(x) =13x23is not def. at x = 0.A C0function is a continuous function.8 / 27Background IntroductionA Simple Example of Terrible ThingsThe functionf : x 7→ x3, x ∈ Ris not a diffeomorphism. Why?•f is C∞;dfdx= 3x2,d2fdx2= 6x,d3fdx3= 6,d4fdx4= 0, . . . .•f is a homeomorphism: f and f−1(x) = x1/3are continuous, f isbijective.•f−1is not C∞:df−1dx=13x23, not differentiable at x = 0.9 / 27Background IntroductionSimple Diffeomorphic Image TransformationsTranslationf : x 7→ x + tSimilarity Transformf : x 7→ sRxAffine Transformf : x 7→ Ax + t10 / 27Background IntroductionWhat is a (topological) manifold?DefinitionManifold [Boothby] A manifold M of dimension n is a topologicalspace with the following properties•M is Hausdorff,•M is locally Euclidean of dimension n and•M has a countable basis of open sets.Hausdorff: Points canbe separated byneighborhoods.Circle, S1, n=1.Countable basis sothat we can have ametric.11 / 27Background IntroductionWhat is differentiable (C∞) manifold?Image from Mumford.•Manifold has localcharts.•Local charts give localcoordinates.•The set of charts iscalled an atlas.If the change of coordinates from chart to chart is given by C∞functions we have a C∞manifold (and a notion of differentiability).A particularly simple C∞manifold is Euclidean space.12 / 27Background IntroductionSome Diff. Geometry [following presentation by Mumford]We•have charts/local coordinates at every point P ∈ MP 7→ (x1(P), x2(P), · · · , xn(P)•have a tangent space TPM to M which in coordinates is thevector space of infinitesimal changes(dx1, dx2, · · · , dxn)•can associate a tangent to every curve g : [0, 1] 7→ Mdg(t)dt= (· · · ,dxi(g(t))dt, · · · ) ∈ Tg(t)M•can define a metric in TPMk(dx1, · · · , dxn)k =qPni,jgi,j(P)dxidxj•can use this to define a path lengthlength(g) =R10kdg(t)dtk dt.13 / 27Background IntroductionManifold of diffeomorphisms [following pres. by Mumford]•The group Γ of diffeomorphisms is also a manifold.•Its tangent space is the linear space of vector fields on Ω andpaths x 7→ φ(x, t), t ∈ [0, 1] are described by∂φ(x, t)∂t= v(φ(x, t), t).14 / 27Background IntroductionOptimality for diffeomorphic mappingsWe want to find the diffeomorphic mapping that in some sense hasthe smallest length given some norm k · kZ10kv(x, t)k dt.The choice of norm can make a big difference.Image from Mumford.15 / 27Background Landmark-based matchingLandmark-based matchingGiven a finite set of matching landmarks (x1, x2, · · · , xn),(y1, y2, · · · , yn) in Ω find a dense diffeomorphism g in Ω.Exact matching: yi= g(xi),Inexact matching: yi≈ g(xi).16 / 27Background Landmark-based matchingBookstein SplinesMethod for landmark matching based on finding the mapping functionf (x, y ) = a0+ axx + ayy +nXi=1wiU(|pi− (x, y )T|),where U(r) = −r2log(r2)that matches landmark points to each other while minimizingIf=ZZ ∂2f∂x22+ 2∂2f∂x∂y2+∂2f∂y22!dx dy.This mapping is not necessarily diffeomorphic.17 / 27Background Landmark-based matchingExample Result for Bookstein SplinesImage from Younes.18 / 27Background Landmark-based matchingDiffeomorphic Landmark Matching [Joshi]Flowing images into each other. Mapping function h(x) = φ(x, 1)given through the ODEdφ(x, t)dt= v(φ(x, t)), t ∈ [0, 1], φ(x, 0) = x.Minimize smoothness cost subj. to landmark constraints (h(xn) = yn)ˆv(·) = argminv(·)Z10ZΩkLv(x, t)k2dx dt.This is guaranteed to give a diffeomorphic h for suitable L (forexample L = I (−∇2+ c) works).19 / 27Background Landmark-based matchingExample Result for Diff. Landmark MatchingImage from Joshi.Left: target image, middle: diff. landmark matching, right: smalldisplacement matching.20 / 27Background Landmark-based matchingLog-Euclidean Polyaffine Registration [Arsigny]Idea: Combine multiple affine transformations to one diffeomorphicmapping.Given n affine