Stefan Sommer
Professor at Department of Computer Science, University of Copenhagen
Linear representations and random orbit model
Faculty of Science, University of Copenhagen
Stefan Sommer
Department of Computer Science, University of Copenhagen
\( \phi \)
Statistical Shape models
Session 1: (L 9-9:45) Shape analysis and actions of the diffeomorphism group
Session 2: (E 10-10:45) Landmark analysis in Theano Geometry
Session 3: (L 11-11:45) Linear representations and random orbit model
Session 4: (E 12:30-13:15) Landmarks statistics in Theano Geometry
Session 5: (L 13:30-14:15) Shape spaces of images, curves and surfaces
Session 6: (E 14:30-15:15) Analysis of continuous shapes
Shape Matching
E_{s_0,s_1}(\phi)=R(\phi)+\frac1\lambda S(\phi.s_0,s_1)
\( \phi\in\mathrm{Diff}(\Omega) \) diffeomorphism of domain \(\Omega\)
\( \phi \)
Variational problem to find optimal \(\phi\in G\):
Group and quotient
Riemannian View
R(\phi_t)=\int_0^T\|\partial_t \phi_t\|_{\phi_t}^2dt
\( \phi_t:[0,T]\to\mathrm{Diff}(\Omega) \) path of diffeomorphisms (parameter t)
\mathrm{Diff}(\Omega)
\mathrm{Id}_{\mathrm{Diff}(\Omega)}
\varphi_t
Flow Equations
Constructing diffeomorphisms:
Forward flow:
\[ \frac{d}{dt}\phi(x,t)= v(\phi(x,t),t) ,\ \forall x\in\Omega \]
\(\phi(x,0)=x\)
\(\phi_t\in G=\mathrm{Diff}(\Omega)\)
\(v_t\) is a vector field on \(\Omega\): \(v\in V\subset \mathcal{X}\)
It is the Eulerian velocity that controls the flow
Variational Problem
The family \(v_t\), \(t\in [0,T]\) determines the flow
Variational problem:
\( E_{s_0,s_1}(\phi)=\mathrm{min}_{\phi_t}R(\phi_t)+\frac1\lambda S(\phi_T.s_0,s_1) \)
\(R(\phi_t):=\int_0^T\|\partial_t \phi_t\|_{\phi_t}^2dt\), \(v_t=\partial_t\phi_t\circ\phi_t^{-1}\)
So we need a norm on \(V\) for defining \(\|\partial_t\phi_t\|_{\phi_t}=\|v_t\|\)
Velocity Fields
Norm on \(V\) is defined via a Reproducible Kernel Hilbert Space (RKHS) structure
Velocity \(v\)
Momentum \(m=Lv\) (recall: \(p\) for landmarks)
\(L\) is the momentum operator \(L:V\to V^*\)
Kernel \(K\) is the inverse momentum: \(K=L^{-1}\)
\(K\) is as well a matrix-valued function on \(\Omega\times\Omega\): \[K: \Omega\times\Omega\to\mathbb{R}^{d\times d}\]
e.g. \(K(x,y)=\alpha e^{-\frac{\|x-y\|^2}{2\sigma^2}}\)
RKHS
Kernel \(K\) is the inverse momentum: \(K=L^{-1}\)
\(K\) is as well a matrix-valued function on \(\Omega\times\Omega\): \[K: \Omega\times\Omega\to\mathbb{R}^{d\times d}\]
e.g. \(K(x,y)=\alpha e^{-\frac{\|x-y\|^2}{2\sigma^2}}\)
For \(a\in\mathbb{R}^d\), \(K(\cdot,x)a\in V\)
\[\langle K(\cdot,x)a,K(\cdot,y)b \rangle= a^TK(x,y)b \]
\(V\) is the completion of finite linear combination of \(K(\cdot,x)a\)
\(V\) is embedded in \(L^2(\Omega,\mathbb{R}^d)\)
RKHS
\(V\) is the completion of finite linear combination of \(K(\cdot,x)a\)
\(V\) is embedded in \(L^2(\Omega,\mathbb{R}^d)\)
vector \(w\in L^2(\Omega,\mathbb{R}^d)\) gives covector
\(v\mapsto\int_\Omega w(x)^Tv(x)dx\)
velocity vectors \(v_t\) are often smooth
momenta \(m_t=Lv_t\) are often rough
e.g. Dirac deltas:
\(v_t=\sum_{i=1}^NK(\cdot,x_i)a_i\)
\(m_t=\sum_{i=1}^Na_i\otimes\delta_{x_i}(\cdot)\) i.e. landmark momenta \(p_i=a_i\otimes\delta_{x_i}(\cdot)\)
Smooth vs. Rough
Geodesic Equation
The norm \(\|\partial_t\phi_t\|_{\phi_t}^2\) determines a Riemannian metric (that's why we draw \(G\) curved)
Riemannian geodesics, length/energy minimizing:
\[\phi_t=\mathrm{argmin}_{\phi_t}\int_0^T\|\partial_t\phi_t\|_{\phi_t}^2dt\]
\(v_0=\partial_t\phi_t|_{t=0}\) determines evolution
\(\mathrm{Exp}:V\to G\) endpoint map
\(\mathrm{Log}:G\to V\) initial velocity map
G=\mathrm{Diff}(\Omega)
\mathrm{Id}_{\mathrm{Diff}(\Omega)}
\varphi_t
Shape Geodesics
\(\pi\) is a Riemannian submersion
it induces a Riemannian metric on \(\mathcal{S}\)
Geodesics on \(\mathcal{S}\) lift to
horizontal geodesics on \(G\)
Horizontal geodesics on \(G\)
project to geodesics on \(\mathcal{S}\)
Those were the landmark geodesics
But we can do this for any shape space
Exp/Log maps are defined on \(\mathcal{S}\) as well
Random Orbit
\(V\) is a vector space (infinite dimensional)
We can perform regular statistics on \(V\)
Normal distribution \(N(0,\Sigma)\) maps to distribution \(\mathrm{Exp}_*N(0,\Sigma)\) on \(G\) - random orbits
Data \(s_1,\ldots,s_N\) can be analysed
in \(V\): \(v_i=\mathrm{Log}(s_i)\)
G=\mathrm{Diff}(\Omega)
\mathrm{Id}_{\mathrm{Diff}(\Omega)}
\varphi_t
\mathrm{Exp}:V\to
Statistics
Template estimation (mean shape/image):
\[s_0=\mathrm{argmin}_{s\in\mathcal{S}}\sum_{i=1}^nd(s,s_i)^2\]
Geodesic distance on \(\mathcal{S}\):
\[d(s,s_i)=\argmin_{\phi_t,\phi_T.s=s_i}\int_0^T\|v_t\|^2dt\]
Tangent space Principal Component Analysis:
\[\mathrm{PCA}(\mathrm{Log}(s_0,s_1),\ldots,\mathrm{Log}(s_0,s_n))\]
\partial_t \phi_t = F(\phi_t)\ \to\ d\phi_t=F(\phi_t)dt\color{blue}{+\sigma(\phi_t) dW_t}
\mathrm{Diff}(\Omega)
\mathrm{Id}_{\mathrm{Diff}(\Omega)}
\phi_t
Stochastic Flows
e.g. Riemannian Brownian motion
Landmark Statistics in Theano
Exercise session:
- open notebook courses/shape19-E2.ipynb
- sample random initial momenta and forward integrate to create a distribution of shapes (replace example code that samples from a Riemannian Brownian motion)
- find Frechet mean of the distribution
- perform PCA on landmark tangent space
hint: use a low number of landmarks to reduce computation time
Linear representations and random orbit model
By Stefan Sommer
