A New Variational Model for Shape Graph Registration with Partial Matching Constraints

Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework

“…Attempting to compare two such surfaces based on model ( 21) is likely to lead to highly singular behaviour in the estimated geodesics and distances. This was already emphasized in the case of planar shapes (such as curves and shape graphs) in the authors' previous publication [55], and can be further observed in the case of 3D surfaces, as seen e.g. in Figure 7 with the formation of geometric artifacts such as the thin arc around the ear of the skull, and in Figure 12 with phalanges that shrink to almost zero volume.…”

“…Thus, our framework is well adapted to the statistical analysis of populations of shapes such as the ones appearing in biomedical applications. To further improve the robustness of our proposed methods, we also implement a weighted varifold matching framework by extending the idea proposed in the context of curves and shape graphs by [55]. The joint estimation of weights on the source surface enables this augmented model to deal more naturally with partial matching constraints or missing parts in the target shape, or differences in topology between the two shapes.…”

“…Among different possible approaches, we will rely specifically on methods derived from geometric measure theory which have been used for that particular purpose in several past works on surface registration [5,18,22,49,57], see also the recent survey [17]. In this paper, we adopt the framework of oriented varifolds introduced in [31], following an approach similar to the authors' previous works [2,5,55]. We point out, however, that the majority of the present work could be adapted without much difficulty to some of the other types of data attachment terms developed in the aforementioned papers.…”

“…Furthermore, aside from the constraint of ρ taking its values in [0, 1], the variational problem (27) does not involve any cost penalty on the weight function. Yet it would be possible to add regularizers for the weight function to the functional, including for instance the total variation norm of ρ − ρ 0 as done in [55] for planar shapes so as to promote piecewise constant weight functions. In the context of partial matching, it may also be relevant to enforce ρ to take values close to 0 or 1, which can be achieved e.g.…”

How to Register a Live onto a Liver ? Partial Matching in the Space of Varifolds

“…In Kaltenmark and Trouvé (2018) the proposed growth model introduces a first notion of partial matching incorporated to the LDDMM framework. More recently, the bijectivity constraint is circumvented by the introduction of weighted shapes: in Hsieh and Charon (2021); Sukurdeep et al (2021), authors optimize a mask defined on the shapes to exclude some subparts of these shapes (source or target). In Antonsanti et al (2021) a partial matching data fidelity term was introduced in the space of varifolds.…”