Representation of Surfaces with Normal Cycles and Application to Surface Registration

Normal and equivolumetric coordinate systems for cortical areas

“…Here, is a reparametrization-invariant discrepancy measure between (unparametrized) surfaces. Several versions of this cost function have been introduced, based on representations of surfaces as currents [ 12 ], varifolds [ 13 , 14 ] or normal cycles [ 15 ].…”

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

“…We now describe how to construct the key ingredient in the relaxed model outlined above, namely, an effective and simple to compute data attachment term Γ which gives a notion of discrepancy between unparametrized surfaces. 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].…”

Normal and Equivolumetric Coordinate Systems for Cortical Areas

“…Here, D is a reparametrizationinvariant discrepancy measure between (unparametrized) surfaces. Several versions of this cost function have been introduced, based on representation of surfaces as currents [20], varifolds [7,13] or normal cycles [19]. Assume that S 0 and S 1 are triangulated surfaces and that the cost function D is replaced by a discrete approximation, still denoted D. Then, the optimization problem can be reduced to one tracking explicitly the evolution of the vertices of the triangulation, using the reproducing kernel of V denoted as K. This kernel is a matrix-valued function of two variables x, y ∈ R 3 such that, for all α, y ∈ R 3 , the vector field x → K(x, y)α belongs to V and for all v ∈ V ,…”