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Representing a distinct method of the research of fluid flows, Viscous Flows demonstrates the software of theoretical strategies and suggestions for reading and predicting fluid movement in sensible functions. by means of seriously evaluating all suitable sessions of theoretical recommendations with experimental information and/or normal numerical suggestions, it specializes in the diversity of validity of theoretical expressions instead of on their intrinsic personality.
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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.