Incorporation of a Deformation Prior in Image Reconstruction

Abstract

This article presents a method to incorporate a deformation prior in image reconstruction via the formalism of deformation modules. The framework of deformation modules allows to build diffeomorphic deformations that satisfy a given structure. The idea is to register a template image against the indirectly observed data via a modular deformation, incorporating this way the deformation prior in the reconstruction method. We show that this is a well-defined regularisation method (proving existence, stability and convergence) and present numerical examples of reconstruction from 2-D tomographic simulations and partially observed images.

Appendices

Appendix A. Algorithm

The algorithm that we computed to obtain the numerical results of Sect. 5 is available at.² It is implemented in a more generic context than the one presented here, in particular, deformation modules that are not CTG can be used. It involves concepts and notation that are not introduced in this article so in the following we will only give an overview of the implementation and not a full presentation.

The algorithm is implemented in Python and relies on the class of objects DeformationModules. It uses the Operator Discretization Library (ODL)³ in order to define discretized images and vector fields.

1.1 A.1 Deformation Modules

An abstract class DeformationModule is defined and contains the functions specifying the field generator $\zeta$ and the cost. CTG modules form a particular sub-class named TranslationBased. In the implementation we simplify slightly the definition of CTG module by defining the cost by $c :(o,h) \mapsto h^2$. Doing this, we simplify the operator $C_o :H \mapsto H$ which becomes the Identity operator. Theoretically the UEC condition is no longer satisfied, this could lead to pathological trajectories such as non integrable time-varying vector fields. However, as long as the points of the geometrical descriptors do not converge to each others during the minimisation, the UEC condition is still satisfied because the norm of the generated vector field can be lower bounded with the norm of the control. As we do not observe such a convergence in practice, we can keep this simplified cost function.

1.2 A.2 Functional Computation

The constrained indirect registration consists in the minimisation of the functional (5) with respect to the initial geometrical descriptor $a \in \mathcal {O}$ and the initial momentum $\eta \in (\mathbb {R}^n)^m$. A first step is to compute this functional. As the computation of the regularisation terms $(a, \eta _0 ) \in \mathcal {O}\times (\mathbb {R}^n)^m \mapsto \gamma R_1 (a) + \tau R_2 (\eta )$ is straightforward, we concentrate here on the computation of the attachment term $(a, \eta _0 ) \in \mathcal {O}\times (\mathbb {R}^n)^m \mapsto D(T(\varphi ^{\zeta _o (h)}_{t=1} \cdot I_0), d)^2$ where D is the $L^2$ distance on the range of the operator T. The only difficult step here is to compute the transported image $\varphi ^{\zeta _o (h)}_{t=1} \cdot I_0$. This is done by integrating (4) via an Euler scheme and by transporting simultaneously the template image, see the sketch in Algorithm 1. In the particular case of a deformation module obtained by combination of CTG modules, the complexity of this forward integration is $O (N m p^2)$ with m the total numbers of points of the compound geometrical descriptor and $p^2$ the number of pixels of the image.

1.3 A.3 Gradient Computation

As previously, the gradient of the regularisation terms $(a, \eta _0 ) \in \mathcal {O}\times (\mathbb {R}^n)^m \mapsto \gamma R_1 (a) + \tau R_2 (\eta )$ is straightforward but the gradient of the attachment term $(a, \eta _0 ) \in \mathcal {O}\times (\mathbb {R}^n)^m \mapsto D(T(\varphi ^{\zeta _o (h)}_{t=1} \cdot I_0), d)^2$ requires explanations. We use a forward–backward scheme based on the following result (which is a simplified version of a more general principle, see for instance [3]):

Proposition 8

Let $p \in \mathbb {N}$, $f :\mathbb {R}^p \mapsto \mathbb {R}^p$ a $C^j$ vector field with $j \ge 1$ and $G :q_0 \in \mathbb {R}^p \mapsto S(q(t=1))$ with $S :\mathbb {R}^p \mapsto \mathbb {R}$$C^1$ and $q :[0, 1] \mapsto \mathbb {R}^p$ defined by $q(t=0) = q_0$ and

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} {q} (t) = f(q (t) ) \,. \end{aligned}$$

(6)

Then for all $q_0 \in \mathbb {R}^p$, $\nabla G (q_0) = Z(0)$ where $Z :[0, 1] \mapsto \mathbb {R}^p$ is defined by $Z(1) = \nabla S(q(t=1))$ and

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} {Z} (t) = \mathrm {d}f (q_t) ^T Z (t) \end{aligned}$$

(7)

This is called a forward–backward scheme because it consists in a forward step where Eq. 6 is integrated, then the variable Z is initialised at $Z(1) = \nabla S(q(t=1))$ and integrated backward following Eq. 7. The computation of $\mathrm {d}f (q(t)) ^T$ can be quite hard in practice but it can be simply approximated using the following result [3]:

Proposition 9

Suppose that $p = 2 p_1$ and that there exists $\mathcal {H} :q \mapsto \mathbb {R}$ such that we can write, for $q = (o, \eta ) \in \mathbb {R}^{p_1} \times \mathbb {R}^{p_1}$, $f (q) = (\nabla _{\eta } \mathcal {H} (o, \eta ) , - \nabla _o \mathcal {H} ( o, \eta ) )$. Then for $Z = (Z_1 , Z_2) \in \mathbb {R}^{p_1} \times \mathbb {R}^{p_1}$, $\mathrm {d}f (q(t)) ^T Z = \mathrm {d}\Big ( \nabla \mathcal {H} \Big ) (o, \eta ) \cdot (- Z_2, Z_1)$.

The latter quantity is a directional derivative and can be approximated by a finit difference.

We apply these results on a discretization of $\mathcal {O}\times (\mathbb {R}^n)^m \times L^2(\Omega , \mathbb {R})$, with the function f defined in the Algorithm 1. We are currently working on a new implementation of the gradient evaluation using automatic differentiation.

B Summary of Notation

Notation	Signification
n	Dimension of the ambient space and the images (2 or 3)
$Diff^\ell _0 (\Omega )$	Space of diffeomorphisms (see Sect. 2.3)
$C^\ell _0 (\Omega , \mathbb {R}^n)$	Space of vector fields (see Sect. 2.3)
T	Forward operator
X	Space of images (reconstruction space)
Y	Data space
d	data
$\phi$	Diffeomorphism
v	vector field
$\varphi ^v$	Flow of v (see Equation (1))
$K_\sigma$	Scalar Gaussian kernel of scale $\sigma$ (see Sect. 3.1)
M	Deformation module
$\mathcal {O}$	Space of geometrical descriptors
o	Geometrical descriptor
H	Space of controls
h	Control
$h^*$	Geodesic control (see Equation (4))
$\zeta$	Field generator
c	Cost
$\eta$	Momentum (see Equation (4))