Sparse Adaptive Parameterization of Variability in Image Ensembles

Abstract

This paper introduces a new parameterization of diffeomorphic deformations for the characterization of the variability in image ensembles. Dense diffeomorphic deformations are built by interpolating the motion of a finite set of control points that forms a Hamiltonian flow of self-interacting particles. The proposed approach estimates a template image representative of a given image set, an optimal set of control points that focuses on the most variable parts of the image, and template-to-image registrations that quantify the variability within the image set. The method automatically selects the most relevant control points for the characterization of the image variability and estimates their optimal positions in the template domain. The optimization in position is done during the estimation of the deformations without adding any computational cost at each step of the gradient descent. The selection of the control points is done by adding a L ¹ prior to the objective function, which is optimized using the FISTA algorithm.

Appendices

Appendix A: Proof of Proposition 1

Let δ S ₀ be a small perturbation of the deformation parameters. This perturbation induces a perturbation of the system of particles δ S(t), which induces a perturbation of the position of the pixels mapped back by the inverse deformation δ y(0), which in turn induces a perturbation of the criterion δE:

$$ \delta E = (\nabla_{\mathbf{y}(0)}A )^t\delta\mathbf {y}(0) + (\nabla_{\mathbf{S}_0}L )^t \delta\mathbf{S}_0. $$

(40)

According to (21), the perturbations of the state of the system of particles δ S(t) and the pixel positions δ y(t) satisfy the linearized ODEs:

The first ODE is linear. Its solution is given by:

$$ \delta\mathbf{S}(t) = \exp \biggl(\int _0^t d_{\mathbf{S}(u)}F du \biggr)\delta \mathbf{S}_0. $$

(41)

The second ODE is linear with source term. Its solution is given by:

$$ \delta\mathbf{y}(0) = -\int _0^1 \exp \biggl(-\int_0^s \partial_1 G(u)du \biggr)\partial_2 G(s)\delta \mathbf{S}(s)ds. $$

(42)

Plugging (41) into (42) and then into (40) leads to:

$$ \nabla_{\mathbf{S}_0} E = -\int _0^1 {R_{0s}}^t \partial_2G(s)^t{V_{s0}}^t \nabla_{\mathbf{y}(0)}A ds + \nabla_{\mathbf{S}_0}L, $$

(43)

where

$$R_{st} = \exp \biggl(\int_s^t d_{\mathbf{S}(u)}F du \biggr) $$

and

$$V_{st} =\exp \biggl(-\int_s^t \partial_1 G(u) du \biggr). $$

Let us denote η(s)=−V _s0 ^t∇ _y(0) A, g(s)=∂ ₂ G(s)^t η(s) and $\xi(t) = \int_{t}^{1} {R_{0s}}^{t} g(s)ds$, so that the gradient (43) can be re-written as:

$$ \nabla_{\mathbf{S}_0} E = \int_0^1 {R_{0s}}^tg(s)ds + \nabla_{\mathbf{S}_0}L = \xi (0) + \nabla_{\mathbf{S}_0}L. $$

Now, we need to make explicit the computation of the auxiliary variables η(t) and ξ(t). By definition of V _s0, we have V ₀₀=Id and dV _s0/ds=V _s0 ∂ ₁ G(s), which implies that η(0)=−∇ _y(0) A and $\dot{\eta}(t) = -{\partial_{1} G(t)}^{t}\eta(t)$.

For ξ(t), we notice that $R_{ts} = \text{Id} - \int_{t}^{s} \frac {dR_{us}}{du} du$ $= \text{Id} + \int_{t}^{s} R_{us}d_{\mathbf {S}(u)}F(u) du$. Therefore, using Fubini’s theorem, we get:

This last equation is nothing but the integral form of the ODE given in (24).

Appendix B: Lemma: Soft Thresholding in ℝ^N

Lemma 1

Soft-thresholding in ℝ^N

Let X and X ₀≠0 two vectors in ℝ^N and F the criterion:

$$ F(X) = \Vert X\Vert + \frac{1}{2\beta} \Vert X - X_0 \Vert ^2. $$

Then F achieves its minimum for

$$ X = S_{\beta} \bigl(\Vert X_0\Vert \bigr) \frac {X_0}{\Vert X_0\Vert }, $$

where S _β (x)=max(x−β,0)−min(x+β,0).

Proof

If X≠0, F is differentiable and $\nabla_{X} F = \frac{X}{\Vert X\Vert }+(X - X_{0})/\beta$. This shows that if the minimum of F is reached for X≠0 then X is parallel to X ₀ and we can write X=λX ₀/∥X ₀∥ where λ satisfies:

$$ \lambda/\vert \lambda \vert + \bigl(\lambda- \Vert X_0\Vert \bigr)/\beta= 0. $$

If λ>0, then the minimum is reached for λ ₊=∥X ₀∥−β, which occurs only if ∥X ₀∥>β. If λ<0, then the minimum is reached for λ ₋=∥X ₀∥+β, which occurs only if ∥X ₀∥<−β. In the other cases, namely ∥X ₀∥∈[−β,β], X=0.

Since F(λ ₊ X ₀/∥X ₀∥)−F(0)<0, then the minimum of F is reached for X=(∥X ₀∥−β)X ₀/∥X ₀∥ if ∥X ₀∥>β. Since F(λ ₋ X ₀/∥X ₀∥)−F(0)>0, then the minimum of F is reached for X=(∥X ₀∥+β)X ₀/∥X ₀∥ if ∥X ₀∥<−β. If ∥X ₀∥∈[−β,β], the minimum of F is reached for X=0. These three cases are combined in a single expression using the soft-thresholding function S _β . □