A Forward Model to Build Unbiased Atlases from Curves and Surfaces

Building an atlas from a set of anatomical data relies on (1) the construction of a mean anatomy (called template or prototype) and (2) the estimation of the variations of this template within the population. To avoid biases introduced by separate processing, we jointly estimate the template and its deformation, based on a consistent statistical model. We use here a forward model that considers data as noisy deformations of an unknown template. This di ers from backward schemes which estimate a template by pulling back data into a common reference frame. Once the atlas is built, the likelihood of a new observation depends on the Jacobian of the deformations in the backward setting, whereas it is directly taken into account while building the atlas in the forward scheme. As a result, a speci c numerical scheme is required to build atlases. The feasibility of the approach is shown by building atlases from 34 sets of 70 sulcal lines and 32 sets of 10 deep brain structures. 1 Forward vs. Backward Models for Template Estimation In the medical imaging eld, atlases are useful to drive the personalization of generic models of the anatomy, to analyze the variability of an organ, to characterize and measure anatomical di erences between groups, etc. Many frameworks have been proposed to build atlases from large database of medical images [1,2,3,4], much fewer were proposed for anatomical curves or surfaces [5,6]. In any case, the underlying idea remains the same: one estimates a mean anatomy (called template) and one learns how this mean model deforms within a given population. The most widely used method in medical imaging is based on a backward model that deforms every observations back to a common reference frame (See Fig.1). However, we prefer here to base our statistical estimation on a forward model, as pioneered in [7,8], which considers the observations (Ti) as noisy deformations (φi) of an unknown template (T̄ ). Formally, the forward model can be written as: Ti = φi.T̄ + εi (1.1) whereas the backward model is: φi.Ti = T̄ + εi ⇐⇒ Ti = φ−1 i T̄ + φ −1 i εi (1.2) forward scheme backward scheme Fig. 1. In the forward scheme, the physical observations (Oi) are seen as noisy deformation (φi) of unknown template (Ō). In the backward scheme, the template is an average of deformed observations. In the forward scheme the noise is removed from the observations whereas it is pulled back in the common frame with the backward scheme. The backward model considers either (Eq.1.2-left) that the template T̄ is noisy and the observations Ti free of noise or (Eq.1.2-right) that the noise added the observations (φ−1 i εi) depends on the observations via an unknown deformation. By contrast, the forward model (Eq.1.1) considers that the template is not blury, as an ideal object, and an independent and identically distributed noise εi is added to every observations. This models more accurately the physical acquisitions, whereas the backward model relies on less realistic assumptions. The observations Ti are given as discrete sampled objects. The template T̄ models an average ideal biological material and it is therefore supposed to be continuous. Since in the backward model sampled observations deform to a continuous template, an extrinsic interpolation scheme is required. By contrast, in the forward setting, the deformed template needs only to be sampled to be compared to the observations. This does not only reproduce more accurately the real physical acquisition process, but also depends on less arbitrary assumptions. Assume now that we can de ne probabilities on objects T (images, curves, surfaces, etc.) and on deformations φ. The statistical estimation of an atlas would require at least to compute p(T̄ |Ti), the probability of having the template given a training database of Ti. Once the atlas is built, one would like to know how a new observation Tk is compared to the learnt variability model: one needs to compute the likelihood of this observation given the template p(Tk|T̄ ). Because φi acts di erently in Eq.1.1 and in Eq.1.2, the computational cost of these two steps varies signi cantly. In the backward scheme, computing p(T̄ |Ti) should be simpler than computing p(Ti|T̄ ) which depends on the Jacobian of φi. It is exactly the reverse for the forward scheme. Since it is better to spend more time to build the atlas (which is done once for all) and to keep simple the test of any new available data, the forward model seems better suited even from a computational point of view. Finally, the forward model is also better understood from a theoretical point of view. For instance, the convergence of the Maximum A Posteriori (MAP) template estimation, when the number of available observations is growing, is proved for images and small deformations [7]. Such proofs for the backward model seem currently out of reach. For all these reasons, we base here our statistical estimations on the forward model. We show in this paper how the atlas building step, which is the most critical step in this paradigm, is possible in case of curves and surfaces. Compared to images, dealing with shapes requires speci c numerical scheme. We take advantage here of a sparse deconvolution scheme we introduced recently [9,10]. The paper is organized as follows. A general non parametric framework for shape statistics is introduced in Section 2. In Section 3 we detail the optimization procedure to estimate jointly a template and its deformations to the shapes. A sparse deconvolution method is presented to e ectively compute the gradient descent. In Section 4, we show templates computed from large sets of sulcal lines and sets of meshes of sub-cortical structures. 2 Non-parametric Representation of Shapes as Currents As emphasized in [11,12,5], a current is a convenient way to model geometrical shapes such as curves and surfaces. The idea is to characterize shapes via vector elds, which are used to probe them. A surface S is characterized by the ux of any vector eld ω through it: