Spatially varying metrics in diffeomorphic image registration Laurent Risser 1 François-Xavier Vialard 2 1 CNRS, Institut de Mathématiques de Toulouse 2 Université Paris Dauphine, CEREMADE 01/22 Laurent Risser Laurent Risser February 2015 - Vienna February 2015 - Vienna
Introduction – brief overview of the LDDMM formalism Inputs: • Template (or source) image T • Target image I n • A metric V which parameterizes how to compare the images T I n Output: • Velocity field v(t) , which encodes the optimal mapping between T and I n t ∈ [0,1] Minimized energy: [Beg et al, IJCV 2005] Registration algorithm: [Beg et al, IJCV 2005] • Initiate v(t) as equal to 0 • Perform a gradient descent with the gradient • … and the momentum where T t and I t are T and I n transported at time t though Φ , respectively Laurent Risser February 2015 - Vienna 02/22
Introduction – Motivations Influence of the kernel K : [Risser, et al, TMI 2011] Grey matter - 36 weeks Grey matter - 43 weeks Deformations obtained using different kernels K T I n “Best” K ? Joint work with F.X. Vialard and other collaborators: [Risser, et al, TMI 2011], [Bruveris, et al, SIAM MMS 2012] : Sum of Gaussian kernels [Risser et al, MedIA 2012] : Sliding motion [Schmah et al, MICCAI 2013] : Mathematical interpretation of LDDMM with spatially-varying metrics [Vialard and Risser, MICCAI 2014] : A general framework to learning spatially-varying metrics [Ongoing work] : Efficient schemes to learn spatially-varying metrics on real 3D data Laurent Risser February 2015 - Vienna 03/22
Introduction – Talk overview Talk overview: 1) From LDDMM to LIDM - interpreting spatially varying metrics in LDDMM 2) A general framework for spatially varying metrics a) Motivations b) General framework 3) Learning spatially varying metrics a) A very first strategy b) Strategy of [Vialard & Risser, MICCAI 2014] c) A new faster strategy d) Reducing the problem’s dimension 4) Results Laurent Risser February 2015 - Vienna 04/22
1) From LDDMM to LIDM - Interpreting spatially varying metrics in LDDMM Energy minimized in LDDMM and LIDM: LDDMM: Spatial (Eulerian) velocity LIDM: Convective velocity Consider point x at time 0 (in image I ) • LDDMM: motion of x at time t → driven by v(t) at point Φ t o x • LIDM: motion of x at time t → driven by v(t) at point x → Diffeomorphic registration formalism adapted to spatially varying metrics Laurent Risser February 2015 - Vienna 05/22
1) From LDDMM to LIDM - Interpreting spatially varying metrics in LDDMM As shown in [Schmah et al, MICCAI 2013], for a given metric: • Diffeomorphisms differ between LIDM and LDDMM • Final deformation is the same for LIDM and LDDMM t=0 t=0.25 t=0.5 t=0.75 t=1 → Final deformations of LDDMM registration with spatially varying metrics can be interpreted as what would be obtained using LIDM with the same metric (although the deformation path doesn’t make sense in LDDMM) Motion of x at time t → smoothed at point x and not Φ t o x • Clearly not adapted for large deformations • OK for small to reasonably large deformations Laurent Risser February 2015 - Vienna 06/22
1) From LDDMM to LIDM - Interpreting spatially varying metrics in LDDMM Results obtained using LIDM in [Schmah et al, MICCAI 2013] using very intuitive parameters: C++ code can be freely downloaded at: http://sourceforge.net/projects/utilzreg/ (with multi-kernel LDDMM, Geodesic shooting, … ) … let’s now define metrics which make sense for real images Laurent Risser February 2015 - Vienna 07/22
2) A general framework for spatially varying metrics - motivation Typical application: Registration of 3D MR brain images Laurent Risser February 2015 - Vienna 08/22
2) A general framework for spatially varying metrics - motivation 34% 13% 2% 34% 13% 2% 34% 13% 2% Spatially homogeneous smoothing kernels: • Typically, smoothing using ad-hoc Gaussian kernels (or sum of Gaussian kernels) • Pros: limited amount of kernels to tune, relatively intuitive parameters tuning. • Cons: Not necessarily adapted to the different brain structures Laurent Risser February 2015 - Vienna 08/22
2) A general framework for spatially varying metrics - motivation 34% 13% 2% 34% 13% 2% 34% 13% 2% As discussed in e.g. [Simpson et al, MICCAI 2013]: It is interesting to more or less favour the deformations according to their location. Laurent Risser February 2015 - Vienna 08/22
2) A general framework for spatially varying metrics - motivation 34% 13% 2% Smoothing with preferential directions has also been well explored in classic registration algorithms (e.g. optical flow) Laurent Risser February 2015 - Vienna 08/22
2) A general framework for spatially varying metrics - motivation Local relations Strategy we will explore in this talk: • Learning relations between the motion of pairs of points • Using these relations to smooth the deformations Laurent Risser February 2015 - Vienna 09/22
2) A general framework for spatially varying metrics - motivation Local relations Long distance relations Slighly related motion No relation Strategy we will explore in this talk: • Learning relations between the motion of pairs of points • Using these relations to smooth the deformations Laurent Risser February 2015 - Vienna 09/22
2) A general framework for spatially varying metrics - motivation Local relations Long distance relations Direction based relations Slighly related motion Motion in opposite directions No relation Strategy we will explore in this talk: • Learning relations between the motion of pairs of points • Using these relations to smooth the deformations Laurent Risser February 2015 - Vienna 09/22
2) A general framework for spatially varying metrics – general overview Same registration algorithm as in [Beg et al, IJCV 2005], except the smoothing part. • Initiate v(t) as equal to 0 • Perform a gradient descent with the gradient • … and the momentum Mathematical property the smoothing kernel must ensure: The Hilbert space of v t has to be embedded in the Banach space of C 1 vector fields Instead of a spatially-homogeneous K , we propose to use: where: ^ • K is a spatially-homogeneous kernel (typically Gaussian) • M is a positive-semidefinite operator Laurent Risser February 2015 - Vienna 10/22
2) A general framework for spatially varying metrics – general overview How to smooth the momentum P t at point with K ? Smoothing P t with K : ^ • Convolution with K • Smoothing with M ^ • Convolution with K again Laurent Risser February 2015 - Vienna 11/22
2) A general framework for spatially varying metrics – general overview How to smooth the momentum P t at point with K ? Smoothing P t with K : ^ • Convolution with K • Smoothing with M ^ • Convolution with K again Same properties in the image domain Same properties in directions x, y, z Laurent Risser February 2015 - Vienna 11/22
2) A general framework for spatially varying metrics – general overview How to smooth the momentum P t at point with K ? Smoothing P t with K : ^ • Convolution with K • Smoothing with M ^ • Convolution with K again Positive relation Negative relation Different properties in the image domain Different properties in directions x, y, z → Local deformations in a given direction may be related to deformations other directions and other locations Laurent Risser February 2015 - Vienna 11/22
2) A general framework for spatially varying metrics – general overview How to smooth the momentum P t at point with K ? Smoothing P t with K : ^ • Convolution with K • Smoothing with M ^ • Convolution with K again Same properties in the image domain Same properties in directions x, y, z Laurent Risser February 2015 - Vienna 11/22
2) A general framework for spatially varying metrics – general overview Consider a pair of voxels: Matrix M : ... ... ... 1 N N+1 2N 2N+1 3N p j = ( x j , y j , z j ) 1 ... u x (p j ) u y (p j ) u z (p j ) i M( u y (p j ) , u x (p i ) ) … M( u x (p j ) , u x (p i ) ) u x (p i ) ... N N+1 ... N+i … M( u x (p j ) , u y (p i ) ) … u y (p i ) ... 2N 2N+1 ... 2N+i … … … u z (p i ) ... 3N p i = ( x i , y i , z i ) Smoothing a 3D vector field Γ t with M : Vectorize Γ t → Matrix/vector multiplication → Recompose the vector field Important note: Ideally, M should be normalized Laurent Risser February 2015 - Vienna 12/22
3) Learning matrix M – A first strategy Consider a template image T defined on a (discrete) domain Ω and N reference images I n … T I 1 I 2 I N First strategy: Register T on the N images I n with M equals to Identity (equivalent to [Beg et al, IJCV 2005] ) 1. Average the estimated velocity fields v n (t,x) on the temporal axis → v n (x) 2. Compute the matrix M using the N velocity fields v n (x) 3. Pros: • Relatively simple to implement • Relations taken into account Cons: • No regularisation, and therefore no prior, on M ^ • Particularly dependent on K • At some locations, there may have no information Laurent Risser February 2015 - Vienna 13/22
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