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D . The complete field is collected in the N d × 1 vector U := (U1T , . . , UdT )T . Correspondingly, F is the discrete force field built of the values of f at grid points. The differential operator A is discretized using finite differences, yielding the N d × N d matrix3 AN d,N d . 3. 9) i=0 3 To enhance readability, the index indicating the size of the matrix is omitted in the following, if not required for comprehension. 1 Image registration as a minimization problem 21 with I denoting the N d × N d identity matrix.

In analogy to many publications, γ := 1/h¯2 with h¯2 denoting the mean squared spacing of the image is used in this work [Thirion 1998; Vercauteren et al. 2007]. 13) f pas (u) := ∇R 2 + γ · (R − T ◦ ϕ)2 acts as a passive force “pulling” the template image. This term provides a computational benefit because the gradients remain fixed and do not have to be calculated in each iteration. 1. 14) was proposed. Both f act and f pas are closely related to the second order approximation of the SSD gradient [Pennec et al.

Ud )T and Δul := ∇ · ∇ul denoting the Laplace operator of a function ul . This corresponds to the diffusion equation with D being the identity. The steady-state solution for diffusion registration yields the Poisson equation Δu = f , which – from a physical point of view – can be interpreted as the stationary heat equation for a system with heat source f . 1 Stability properties of the explicit scheme To formulate discretization of the Laplace operator, the 1D case is examined first. 16) j = i, else with h denoting the pixel spacing.

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