Coarse-To-Fine Hamiltonian Dynamics of Hierarchical Flows in Computational Anatomy

Coarse-to-Fine Hamiltonian Dynamics of Hierarchical Flows in Computational Anatomy Michael I. Miller Daniel J. Tward Johns Hopkins University Johns Hopkins University Baltimore, Maryland, USA 3400 N. Charles St., Baltimore, MD 21218 [email protected] [email protected] Alain Trouve´ Ecole Normale Superieure 45 Rue d’Ulm, 75005 Paris, France [email protected] Abstract few millimeters to centimeters). In contrast, our current ex- perimental work in Alzheimer’s disease is mapping the mi- We present here the Hamiltonian control equations for cro scales of molecular Tau-pathology measured via histol- hierarchical diffeomorphic flows of particles. We define the ogy, to the sub-millimeter scales of anatomy obtained via controls to be a series of multi-scale vector fields, each with moderate and high field MRI [13]. Towards this end we their own reproducing kernel Hilbert space norm. The hier- derive a set of Hamiltonian control equations that allow us archical control is connected across scale through succes- to directly model information across scales resulting from sive refinements that refine as they ascend the hierarchy with molecular and anatomical imaging. commensurately higher bandwidth Green’s kernels. Inter- There have been many successful approaches in the 80’s estingly the geodesic equations do not separate, with fine and 90’s for representing multi-scale data, as well as many scale motions determined by all of the particle information methods for the construction of scale-space. Our represen- simultaneously, from coarse to fine. Additionally, the hier- tation is abstract enough that it accommodates linear and archical conservation law is derived, defining the geodesics non-linear transformations including wavelets [3, 6] and and demonstrating the constancy of the Hamiltonian. We scattering functions [7] and multi-resolution pyramids for show results on one simulated example and one example signal decomposition [1, 2]. We model the controls in our from histological images of an Alzheimer’s disease brain. control system as being successively refined as the repre- We introduce the varifold action to transport the weights of sentation descends from the coarsest continuum scales of micro-scale particles for mapping to sub millimeter scale gross anatomy to the discrete, atom-like scales of cellular cortical folds. descriptions. We have been motivated by other work on multi-scale kernels, [10, 12, 15, 4, 9] although our representation differs 1. Introduction in that these multiple scale flows are not composed, rather they exist simultaneously with one another. The fact that This paper specifically focuses on a hierarchical rep- the tangent space at the identity is successively refined via resentation of diffeomorphic flows for computational a sequence of vector fields corresponding to the higher and anatomy. Our motivation is to accommodate the multiple higher dimension of the finer scales, implies that the flow scales represented, from cellular scale to meso-scale [5, 14] itself supports multiple information flows simultaneously. to millimeter [11] anatomical scales of the numerous brain These scales are not decoupled. mapping projects in which we are involved. Most current methods in computational anatomy focus on deformable 2. Shapespace as an Orbit of Varifolds brain atlases that are built from imaging data at a single scale (typically millimeter resolution clinical MRI). These Our shapes are is built via a series of generalized func- R3 atlases are mapped to new datasets using deformations reg- tions or measures spatially localized δxi ,xi ∈ , i ∈ I ularized with smooth kernels at a single scale (typically a with associated function represented by the parameter space 1 fi ∈ F. For us the function represents the amount of pro- tein or Tau tangle at micro-scale in Alzheimer’s disease for example. The mathematical model of Shapespace we use is termed mathematical varifold measures, representing mix- tures of particles given by µ = wiδxi ⊗ δfi . (1) Xi∈I Shapespace is modeled as an infinite dimensional space, generated from one or more exemplars of varifolds µ = Figure 1: Hierarchical dynamical system with states qℓ, w δ δ acted upon via the diffeomorphism group ℓ i ℓ i i xi ⊗ fi controls successively refined u = i v and flows ϕ . Pof one-to-one, onto maps, ϕ = (ϕ1,ϕ2,ϕ3) ∈ Diff (e.g. 3 P components for 3 dimensional images) with diffeomorphic action given by the varifolds at each level. We define the associated state ℓ ϕ · µ := wi|dϕ(xi)|δϕ(xi) ⊗ δfi . (2) process of our dynamical system t 7→ qt = (qt )ℓ∈L to Xi∈I encode the varifold action of (2), qti = (wti,xti,fi)i∈I w = |dϕ (x )|w and x = ϕ (x ), with state velocity We use varifolds and their action because they are an effi- ti t i i ti t i linear in the control t 7→ u : cient way to transport a functional feature unchanged. t The diffeomorphisms are generated from vector fields in- q˙ti =(wtidivut(xti),ut(xti), 0) . (5) dexed by time t → ut = (ut1,ut2,ut3) and integrated to generate the flows The layers in the hierarchy are coupled since the controlling vector fields uℓ are determined by successive re- ϕ˙ = u ◦ ϕ , ϕ = id . (3) t t t 0 finements vℓ, for ℓ = 0, 1,... : They extend the singular molecular varifolds of Shapes- uℓ uℓ−1 vℓ , u0 v0 , ℓ ,... (6) pace, to the dense macro-scale continuum of cellular or par- = + = = 1 ticle descriptions, to global tissue scales (for example). We make the Shapespace into a metric space by exam- 3. Shapespace as a Hierarchical Dynamical ining the measures using a measure norm. However, in or- der to accomodate arbitrary scale into the dense continuum System with Metric limit we build the underlying space of shapes into a metric Our model is a hierarchical one in which we define space by measuring within the dense space of diffeomorphic (ℓ) motions on the backround space in which the particles are a hierarchy of varifolds µ = (µ )ℓ∈L and diffeomor- (ℓ) embedded. This way, we always have a metric independent phisms ϕ = (ϕ )ℓ∈L, each corresponding to their own flow and having their action. The group becomes G = of the scale at which we look at the problem. k Rd To measure the length of the mappings of one shape to ℓ∈L Diff0 ( ) with composition of each component and Qaction componentwise on the varifolds, where for any L- another we use the distance on the diffeomorphism group (ℓ) (ℓ) uplets ϕ =(ϕ )ℓ L and ψ =(ψ )ℓ L ∈ ∈ ′ d(µ, µ )= inf ′ dDiff (id, ϕ) , (7a) (ℓ) (ℓ) ϕ:ϕ·µ=µ ϕ · ψ =(ϕ ◦ ψ )ℓ∈L , (4a) with the length metric on diffeomorphisms with the extended action of (2) becoming 1 (ℓ) (ℓ) 2 ℓ ℓ−1 2 ϕ · µ =(ϕ · µ )ℓ∈L . (4b) dDiff (id, ψ)= inf kut − ut kV ℓ dt . u:ϕ˙ t=ut◦ϕt Z0 ϕ0=id,ϕ1=ψ Xℓ The entire space of varifolds we call Shapespace, which is (7b) an orbit M = {ϕ · µ : ϕ ∈ G} generated from the varifold particle models across all scales. 4. Hamiltonian Control We represent our model as a hierarchical dynamical system, the vector fields u are the controls on the flows sat- For building correspondences from one varifold to isfying Eqn. (3). This is depicted in Figure 1. We define aonther we measure closeness between varifolds µ = ℓ ′ ′ℓ the states of our dynamical system to encode the flows of (µ )ℓ∈L, µ = (µ )ℓ∈L ∈ M via the measure norm ′ kµ − µ kM. The measure norm is defined by calcu- The Hamiltonian becomes lating the integrals of the measures against test functions . 1 h → µ(h)= h(x)µ(dx) defined by H(q,p,u) =(p|u · q) − (Lu|u) , (14) 2 R kµkM := sup |µ(h)| . (8) L ℓ ℓ ℓ ℓ ℓ ℓ ℓ h:khk=1 with ( u|u) = ℓ∈L(L v |v ) with (L v |v ) = ku − ℓ−1 2 u kV ℓ . The boundaryP term for matching in terms of the ′ 2 At every scale of the hierarchy the norm has two kernels, state U(q1) := kµ − µ(q1)k gives ℓ ℓ one for space and one for function, KS and KF (resp. on d ℓ R and F ) induced by the dot product ℓ,w 1 ∂ p = − U(q ), (15a) 1i 2 ∂wℓ 1 ℓ ′ ℓ ′ i hδx ⊗ δf ,δx′ ⊗ δf ′ i = K (x,x )K (f,f ) . (9) S F ℓ,x 1 p = − ∇xℓ U(q1) . (15b) 1i 2 i Take the index union Kℓ = J ℓ ∪ J ′ℓ of points and features then the difference of measures becomes Here, to compute the optimal value of u given q and p, it may be convenient to introduce for l ≥ 0, the mapping ℓ ′ℓ ′ µ − µ = wjδx ⊗ δf − w δx′ ⊗ δf ′ A l Rd Rd l j j j j j u 7→ (δx |u) from C0( , ) (the space of C vector fields jX∈J ℓ jX∈J ′ℓ d d ′ vanishing at the infinity) to R defined for A ∈ Ll(R , R ) Rd = αkδx˜ ⊗ δ ˜ (10) (the dual of the space of l-multilinear mappings on with k fk Rd Rd kX∈Kℓ values in ) and x ∈ as A . k with norm-square (δx |u) =(A|d u(x)) . (16) w ℓ ℓ ℓ w I2 ℓ ℓ ℓ ′ℓ 2 ′ ℓ ′ ℓ ˜ ˜ ′ kµ − µ k = αkαk KS(˜xk, x˜k )KF (fk, fk ) . We get pi div(u )(xi )wi = pi (δ ℓ |u )wi and M xi k,kX′ Kℓ ℓ,x ∈ ℓ,x ℓ pi ℓ (pi |u (xi))=(δ ℓ |u ) so that we deduce (11) xi pm,x The geodesic connection between shapes is computed by ℓ ℓ m,w m I2 i v = K m (p w δ m + δ m ) m≥ℓ i∈I i i xi xi defining the controlling vector fields to minimize the norm P P between the flowed shape and the target shape.

Load more