Ruiz-Baier, A posteriori error estimates for primary and mixed finite element approximations of the deformable image registration problem. Ruiz-Baier, a posteriori error estimates for primary and mixed finite element approximations of the deformable image registration problem.
Introduction
To our knowledge, [24] is one of the first finite element analyzes for the bioconvection model. Finally, numerical experiments are presented to illustrate the performance of the technique and to confirm the expected orders.
The bioconvective flows model
From the first equation of (1.6), it is clear that the uniqueness of an eventual pressure solution of this problem (see [55] or [76]) is ensured in space. Likewise, from the condition of the total mass in the auxiliary concentration (the second equation of the second row in the system (1.6)), we see that an eventual weak solution φ of (1.6) belongs to the space.
The continuous formulation
- The augmented fully-mixed variational formulation
- The fixed point approach
- Well-definiteness of the fixed point operator
- Solvability analysis of the fixed point equation
The above analysis essentially gives us conditions for the well-posedness of the unbounded problem (1.31) or, equivalently, the well-definedness of the operator S (cf. On the other hand, the range of parameters is declared according to the guarantee of the positivity of the ellipticity constant α(Ω).
The Galerkin Scheme
The discrete framework
Similar to the continuous case, we now rewrite (1.67) as a fixed-point problem with respect to operators arising from decoupling the system. In fact, adapting the approach from Section (1.3.2), we first define Hh :=Huh ×Hϕh and introduce the operatorSh:Hh −→Hh×Huh as.
Solvability analysis
We now state the solvability of the fixed-point equation (1.70) by checking the hypotheses of Brouwer's fixed-point theorem (cf. [30, Theorem 9.9-2]). As a result of the previous two lemmas, we can state the Lipschitz-continuity of the operator Th, which represents the discrete version of Lemma 1.6.
A priori error analysis
This is why in the previous theorem we can only guarantee the existence of a discrete solution. We assume that the family {Ah}h>0 is uniformly elliptic, that is, there exists a constant α >e 0, independent of h, such that. The following lemma provides a preliminary estimate for the error(t,u)-(th,uh)k associated with system (1.76).
Therefore, the system (1.76) satisfies the hypotheses of Strang's lemma and thus a direct application of Lemma 1.13 to the specific context (1.76). 1.80) To estimate the supremum in (1.79), we handily add (t,u) and subtract from the first component of the bilinear form Aϕ−Aϕh to find.
Numerical results
Example 1: Accuracy assessment in 2D
The concepts on the right-hand side are adapted in such a way that the exact solutions are given by the smooth functions. Note that the homogeneous Dirichlet condition for the velocity, the Robin-type boundary condition for the concentration, the fluid incompressibility condition, and the zero-mean constraint for both the pressure and concentration are satisfied by the above functions. There we observe that the convergence rates are linear (in the casek= 0) and quadratic (in the case k = 1) with respect to h for all the key unknowns in their respective norms, as well as the postprocessed variables in the L2−norm.
It is also observed that the errors decrease more rapidly as we increase the order of approximation from k = 0 to k = 1.
Example 2: Accuracy assessment in 3D with concentration-dependent viscosity 39
The approximation of the velocity magnitude, the pressure and concentration is depicted in Figure 1.1, calculated with our fully mixed method on a mesh with N = 873843 degrees of freedom and k= 0. However, as we will illustrate later, it also applies - can related problems such as the image registration of the human brain. The proposal of the optical flow formulation by Horn & Schunk [61] gave rise to much mathematical analysis at the continuum level, with an increasing interest for the discrete analysis in an algorithm-specific way in [79], in the optimal-control environment within a more classical Galerkin framework [69], and more recently the variational problem has been tackled in its primal and mixed formulation in [13].
More precisely, in this last work a complete numerical analysis of the method was presented, in the special case of an elastic regularizer and a sum-of-squares-differences equality.
Extended primal formulation in abstract form
- Setting of the problem
- Analysis of the continuous formulation
- Analysis of the discrete scheme
- The rates of convergence
To this end, first let N be the kernel of the operator induced by B, that is. In this way, applying now the discrete version of the Babuˇska-Brezzi theory (cf. [49, Theorem 2.4]) and using from (A3) thatk∇D(zh)k ≤ MD, we complete the proof. Hh ×Q be the resulting unique solution of the discrete scheme (2.22) when the functional Guh is replaced there by Gu.
On the other hand, the linear character of the discrete problem (2.23) easily implies that the difference (ubh,bλh),ρbh.
Extended mixed formulation and application to elastic energies
- Setting of the problem
- Analysis of the continuous formulation
- Analysis of the discrete scheme
- A priori error analysis
There exists a positive constant γV such that. 3.19)), it follows from (2.56) and the fact that all the norms in Q are equivalent, that there exists a positive constantcE, dependent only on Q, such that. 2.74) Furthermore, there exists a positive constant C, which depends only on αK, βB, γV, and the norms of the operators induced by a and b, such that. This follows from a simple application of Theorem 2.6. 2.76) The Lipschitz continuity of the operatorT is established in the following lemma. Indeed, if vh−ΠQvh 6=0, we know from the first part of the proof of [50, Lemma 4.1] that there exists ζh ∈ Hσh such that div(ζh) = Ph(vh −ΠQvh) and kζhkdiv;Ω ≤ CeNkvh−ΠQvhk0,Ω, where Ph :L2(Ω) → Huh is the orthogonal projection, and CeN is a positive constant independent of h.
Moreover, there exists a positive constant C depending on only one αK, βeB, γV, and the norms of the operators induced by a and b such that.
Implementation of the methods
The derivation of the relationship between kun+1−unk2V and kun−un−1k2V is analogous to that in the proof of Lemma 2.14, except for a small modification. In fact, since the time steps are different, the time derivatives create new terms that cancel , i.e. Finally, the condition relating the subsequent time steps ∆tn+1 and ∆tn is obtained by imposing
The above formulation and its accompanying analysis are straightforwardly applicable to the mixed case, the only difference being that, while the H1 inner product is used in the regularization terms for the primal case, the L2 one is used for the mixed approach.
Numerical examples
- Example 1: Convergence
- Example 2: To extend or not to extend
- Example 3: Translations in the quasi-incompressible case
- Example 4: Rotations in the quasi-incompressible case
- Example 5: Application to the image registration of the human brain
This test was performed for the same translation sample settings, but with the rotation images with C= 20 and a= 0.4. This approach can be interpreted as a semi-implicit formulation of the proximal point algorithm [84], recently extended to a more general class of proximal operators by using forward-backward splitting [45]. The formulation of the optical flow problem put forward by Horn & Schunk [61] leads to a more rigorous mathematical analysis of the continuous formulation of the DIR problem, which contrasts with the lack of rigorous numerical analysis of its discrete counterpart, recently developed in the variation formulation [79] in an algorithm specific way and also in the optimal control setting within a more classical Galerkin framework [69].
In this chapter, we propose an a posteriori mesh refinement scheme specifically tailored to primal and mixed formulations of the DIR problem.
Mathematical formulation of the deformable image registration problem
Continuous and discrete weak formulations of DIR
- DIR primal formulation
- DIR mixed formulation
- The primal Galerkin finite-element scheme
- The mixed Galerkin finite-element scheme
We begin by defining an auxiliary field as the tilted symmetric gradient components of the displacement field. Then, the strong form of the mixed-registration BVP associated with (3.3) becomes: Find u,σ and ρ such that. Analogously to the continuous case, we consider the auxiliary problem: Given zh∈Hh, finduh∈Hh such that.
This allows us to define the discrete operator Th : Quh → Quh given by Th(zh) := uh, where uh is the unique displacement from (3.23), and then we rewrite the nonlinear discrete problem as: Find uh ∈Quh such that .
Residual-based a posteriori error estimators
Preliminaries
We first let Eh be the set of all edges of the triangulation Th, and given K ∈ Th, let E(K) be the set of its edges. We also determine for each edge e∈ Eh a unit normal vector νe:= (ν1,ν2)t and let:= (−ν2,ν1)t be the corresponding fixed unit tangential vector along e. The main techniques used below to prove efficiency include localization technique based on element bubble and edge bubble functions.
Given K ∈ Th and e ∈ E(K), let ψK and ψe be the usual triangle-bubble and edge-bubble functions [89, eq.
A posteriori error analysis for the primal finite-element scheme
From the definition of Rh(w−wh), integrating by parts on each K∈ Th, and adding and subtracting a suitable term, we can write. 3.29) Then choose what is the Cl´ement interpolant of w, that iswh :=Ih(w), the approximation properties of Ih (cf. In this way, applying the Cauchy-Schwarz inequality to each term (3.29), and using (3.30) together with the Lipschitz continuity of Fu (eq.
A posteriori error analysis for the mixed finite-element scheme
By applying Cauchy-Schwarz inequality and noting that σh : η = 12(σh −σth) : η, together with the condition (3.5), we can determine. In this section we give upper bounds depending on the actual errors for the seven terms that define the local indicator ΨK2 (cf. For this we start, analogous to [43, Section 4.3], with the first three that appear there, more precisely, since div (σ) = αfu in Ω, we have that.
Upper bounds for terms involving only the C−1σh+ρh tensor are established in the following result.
Applications and performance assessment
Numerical implementation
The Picard iterations with pseudo-time steps are placed inside the adaptive refinement loop, which consists of solving, estimating, marking, and refining.
Example 1: Registration of smooth synthetic images
However, such clear scaling is not required for a posteriori estimation in the mixed method. We also show in panels (i,j,k) the approximate solutions (Frobenius norm of the stress, displacement magnitude and Frobenius norm of the rotation matrix) generated by the mixed method at the final level of refinement. The numerical convergence of the primary and mixed DIR methods are shown in Figure 3.2(a) and Figure 3.2(b), respectively.
No differences were observed in the number of Picard iterations required by the uniform and adaptive improvement strategies.
Example 2: Registration of smooth synthetic images with high gradients
A remarkable improvement in convergence is observed for the specific case of the adaptive refinement scheme using second-order element interpolations. Convergence rates for the primary DIR method using uniform and adaptive refinement are reported in Table 3.3, where we see that the case of adaptive refinement using second-order elements results in convergence rates reaching k = 2, which is notoriously higher then the convergence speed ofk= 1.5 achieved by the primal method under uniform refinement. In the case of the mixed method, adaptive refinement always results in better convergence than uniform refinement for the displacement, stress, and rotation fields, see Figure 3.4(b).
Table 3.4 reports the convergence rates of the mixed method, where we note that adaptive refinement always results in rates that are greater than those obtained with uniform refinement.
Example 3: Registration of brain medical images
The tolerance for the Picard scheme is increased to 1e-04, and for the mixed method, the ratio of the refinement density is determined by the constant γ ratio = 0.1. Convergence percentages and Picard iteration count for the estimated displacements uh produced by the first and second order primal method; and tabulated according to the resolution level, under uniform (a) and adaptive mesh refinement guided by Θ, with γ ratio = 0.01 ((b) also shows the rescaled effectiveness index). Adaptive mesh refinement in the acquisition of medical brain images. a) Mesh after four steps of adaptive refinement using the error indicator Θ for the primary DIR method;.
An average of 17 fixed point iterations is required for the primary approximations and 25 for the mixed scheme.
Example 4: Registration of binary images under large deformation
CPU time (in [s]) for each step of the adaptive finite element method for the DIR problem, measured for the primal and mixed methods, starting from coarse meshes. Ruiz-Baier, A posteriori error estimates for primal and mixed finite element approximations of the deformable image registration problem, Preprint 2018-50, Centro de Investigaci´on en Ingenier´ıa Mate´matica (CI2MA), Universidad de Concepci´on , Chile, (2018 ). Oyarz'ua, Analysis of an extended fully mixed formulation for coupling the Stokes and heat equations, ESAIM: Mathematical Modeling and Numerical Analysis p.
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