Workshop on Anatomical Models, June 16-17 th 2009 Estimation based based on on vectorized vectorized surfaces surfaces Estimation for for Craniofacial Reconstruction Reconstruction Craniofacial Yves ROZENHOLC Yves ROZENHOLC MAP5 - MAP5 - UMR CNRS 8145 MR CNRS 8145 Université Paris Descartes Université Paris Descartes joined work with F. Tilotta*, J. Glaunès, F. Richard *Ph.D thesis Support BQR 2007 from University Paris Descartes
State of the the art art State of Manual Approach Localisation of few points on the skull, Use average tissue thickness, Manual covering of the skull, Bad quality of information (dead, surgery, heterogenous data, …) Statistical Global Approach (starts only after 2003 ) Use of Sparse Template or implicit surfaces Registration of the sparse template or the full surface kPCA (skull,tissue) with missing data or PLS Statistical technics are global and do not take to local variations
Outline • The data and (sub)-mesh construction • Reproductible Kernel Hilbert Spaces for surfaces (RKHS) • Statistical framework for surfaces • Registration of surfaces using RKHS • Regression of surface using RKHS • Results
Our Framework Framework Our Build and use a large data base • more than 80 full head CT-scans. Use a local and conditionnal approach based on • individual dense meshes, around 20000 knots • « patch »: small sub-surfaces defined on the skull • extended surfaces and extended vector fields • average of extended surfaces • distance between extended surfaces
Data: : mesh and anatomical mesh and anatomical points points Data Mesh and Localization of the Anatomical Points on the Skull Mesh The closest edge on the regularized mesh is associated with the 3D coordinates found by the anatomist
Data: patch : patch Data Geodesics and Patches Patch = ordered sequence of anatomical points Bone Patch = surface defined by the patch associated points of the mesh with border defined as the geodesics between succesive points Geodesic Computation on the dense meshes : Combinaison of existing algorithms Algorithm of Surazhsky et coll. (2005) Fast Matching Algorithm (Sethian, 1999)
Data: thickness thickness Data: Skin-patches : Follow « geodesic + patch » idea to extract a skin-patch on the skin mesh: Anatomical points of the bone-patch are projected on the skin surface following the normal rays to the skull surface. Geodesics on the skin-surface are computed between projected points. The skin-patch is the skin-surface delimited.
Dealing statistically with surfaces surfaces : a : a first first idea idea Dealing statistically with Given a surface S , its normal vector field N S and a kernel k σ define Extended Normal Vector Field as S σ (x) = ∫ S k σ (|y-x|) N S (y) dS(y) ≈ Σ t ∈ S k σ (|O t -x|) n t . and, for a given function f on S define Extended Function as f σ (x) = ∫ S k σ (|y-x|) f(y) N S (y) dS(y) ≈ Σ t ∈ S f(O t ) k σ (|O t -x|) n t . Idea : Transform the surface in a 3D vector field which take into account surface geometry and distances. Interest : Allow average computations on extended objects. Problem : How to compare objects (distance, norm) ?
Surfaces Comparison using Comparison using RKHS RKHS Surfaces Given surfaces S and T with normal vector fields N S and N T Given k σ a real kernel, consider the Scalar Product between S and T << S , T >> σ := ∫ S ∫ T k σ (|y-x|) < N S (x), N T (y)> dS(x) dT(y) . For triangular meshes, if x=(a,b,c) consider O x =(a+b+c)/3 and u x =(b-a)(c-a)/2 << S , T >> σ := ∑∑ k σ (|O x -O y |) < u x , u y > where the sums are taken on all triangles x for S and y of T. Link with RKHS : Consider the RKHS (H σ , | . | σ ) of fcts from E=R 3 into R 3 associated to k σ . Define the vectorial distribution generated by S by µ S (v) = ∫ S < v(x) , N S (x) > dS(x) for every vector field v: R 3 R 3 . M The (dual) RKHS norm of µ S is || µ S || σ := sup { µ S (v) , | v | σ ≤ 1 }. Its Scalar Product S << µ S , µ T >> σ := (|| µ S || σ 2 + || µ T || σ 2 - || µ S - µ T || σ 2 )/2 satifies << µ S , µ T >> σ = << S , T >> σ .
Reproducing Kernel Hilbert Hilbert Spaces Spaces (RKHS): (RKHS): Reproducing Kernel Let ( H, || . || H ) be a Hilbert space of functions from a set E into R . H is a RKHS if the linear map f f(x) from H into R is continuous for every x of E . Riesz representation theorem: for every x in E there exists an element K x of H such that : f(x) = < f , K x > H for all f in H . K(x,y) = K x (y) is called the reproducting kernel for the Hilbert space H. Moore-Aronszajn theorem: Given a symmetric positive definite kernel K (x,y) on E there is a unique Hilbert space H of functions on E into R for which K is a RKHS M S (for H) . Example : k σ ( |y - x| ) where | . | is a norm on E and k σ is a classical kernel on R,
A Gaussian statistical framework Gaussian statistical framework for « for « surfaces surfaces » » A Given a kernel k σ the RKHS construction defined an Hilbert space ( H σ , || . || σ ) which contains all the vectorial distributions µ S for all surfaces S of R 3 . In a classical way, we define Gaussian random variables of vectorial distributions X ~ N( µ , Σ ) on H σ by <X, h> ~ N(< µ , h> , < Σ .h,h>) for all h in H σ E ||X|| σ 2 = Tr Σ It is then possible, given a sample of vectorial distributions, to derive classical theorems for estimation (LLN, TCL, …). Regression : if one observe (X i , Y i ) i=1..n , from a joint Gaussian distribution of vectorial distributions, one can provide estimation of E(Y / X=X 0 ) Remember that for surface S with normal vector fields N S µ S (v) = ∫ S < v(x) , N S (x) > dS(x) for every vector field vector field v on R 3 .
Transport and and registration of surfaces registration of surfaces with with RKHS RKHS Transport Semi-rigid Registration of the patches : patch registration uses translation + rotation + dilatation For a given diffeomorphisme φ of the space R 3 the transport of µ S by φ satisfies φ .µ S = µ φ . S . The registration of T on S is then defined as φ ST = arg φ min || µ S - µ φ . T || σ where φ is translation + rotation + dilatation. Because of semi-rigid registration φ ST = φ TS One can define a pseudo-distance between extended surfaces d σ (S, T) = ||S - φ ST .T|| σ = || φ TS .S - T|| σ d σ (S, T) = 0 ≠> S = T
Regression via RKHS for surfaces via RKHS for surfaces with with registration registration Regression Given learning database with n individuals defined by their patch-skull surfaces S i and their skin-surface P i . Let us denote µ 1 … µ n the vectorial distributions associated to the skin surfaces P 1 … P n . Call φ 0i the « measure » registration of the known but dry patch-skull to dress S 0 on the known dressed skulls S i of the « local » learning database. Skin Surface Estimation Let us consider the empirical mean µ = (w 01 φ 01 . µ 1 + … + w 0n φ 0n . µ n )/(w 01 + … + w 0n ) and for the closest skin surface P of the learning database ψ = arg min φ || φ .µ P - µ || σ + λ || φ || defines directly the estimated skin-surface ψ . P . Weights are chosen uniform or for example as w 0i = 1 / d σ (S 0 , S i ) = 1 / || µ 0 - φ 0i . µ i || .
Regression via RKHS for surfaces via RKHS for surfaces with with registration registration Regression Given S 1 … S n n registred surfaces, we consider their associated vectorial distributions denoted µ 1 … µ n We can define the empirical mean distribution µ = ( µ 1 +…+ µ n )/n Unfortunately this does not defined a surface. We propose to choose the « closest » surface S in S 1 … S n which minimize || µ i - µ || σ and to find an elastic (or fluid) registration ψ such that ψ = arg min φ || φ .µ S - µ || σ + λ || φ || where || φ || is a norm on the registration space of interest. As ψ .µ S = µ ψ . S our estimated surface is the transported surface ψ . S Remark : An easiest way could be to consider the median of the chosen surfaces. Remember that for surface S with normal vector fields N S µ S (v) = ∫ S < v(x) , N S (x) > dS(x) for every vector field vector field v on R 3 .
Registration of the bone-patch with nearest neighbors selection w 0i = 1 / d σ (S 0 , S i ) if d σ (S 0 , S i ) ≤ c min j d σ (S 0 , S i ) and 0 if not.
Associated skin-patch for the selected bone-patch
Using the closest extended surface to the estimate one, apply a non rigid registration to match the estimated extended surface
Selected Criterion 1.4 x minimal distance 10 selected individuals
Selected Criterion 1.4 x minimal distance 10 selected individuals Average Error = 1.20 mm
Selected Criterion 1.3 x minimal distance 7 selected individuals Average error = 0.99 mm
Selected Criterion 1.2 x minimal distance 5 selected individuals Average error = 1.37 mm
Mathematical Treatment of of the the Data: Data: Results Results Mathematical Treatment Using direct approach on skin-surface Estimated risk by cross-validation = 0.99 mm range = 0,21 to 2.41 mm * * Upper bound due to artefact Estimated Skin-Surface Real Skin-Surface Cross-Validation : Leave one out. The risk is computed on all 48 young women .
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