Mosco convergence of time discrete geodesic paths

The weak lower semi-continuity and the coercivity ofφ_K ↦→J_K(I_K,φ_K)are shown in [213, Thm. 4]. Hence, it remains to prove weak closedness of the equality con-straint I = I_K^ext(I_K,φ_K). Since H^m(Ω,Ω)^K ↪→ C^1,α(Ω,Ω)^K, we can infer strong convergence of φ^j_K → φ_K in C^1,α(Ω,Ω)^K. Using Corollary 3.5, we conclude for every t ∈[0,1] and a.e. x∈Ωthat

I_K^ext(I_K,φ_K)(t, x) = lim

j→∞I_K^ext(︁

I_K,φ^j_K)︁

(t, x) =I(t, x) holds true.

In what follows, we always use the symbol φ_K for the minimizing set of defor-mations for givenI_K. The requirements for Mosco convergence (Definition 3.1) are the liminf-inequality (Theorem 3.12) and the limsup-inequality (Theorem 3.14).

Theorem 3.12 (liminf-inequality). Under the assumptions (W1), (W2) and (W3), the time discrete path energy J_K satisfies the liminf-inequality for J with respect to the L²([0,1], L²(Ω,H))-topology.

Proof. First, let us give a brief outline of the structure of this proof to facilitate reading. Indeed, the different steps of the proof are as follows:

1. Identification of the image and deformation families. In the first step, we retrieveI_K and φ_K from the pathI_K.

2. Lower semi-continuity of the weak material derivative. The convergence of the discrete material derivativez_K to a limiting weak material derivative z is shown and lower semi-continuity

∫︂ 1 0

∫︂

Ω

z²dxdt≤lim inf

K→∞ K

K

∑︂

k=1

∫︂

Ω

d(I_K,k−1, I_K,k◦φ_K,k)²dx is verified.

3. Lower semi-continuity of the viscous dissipation. Uniform boundedness of the velocity fieldw_K =w_K(φ_K)inK is proven, which readily implies w_K ⇀ v in L²((0,1),V). Then, the relation

∫︂ 1 0

∫︂

Ω

L[v, v] dxdt≤lim inf

K→∞ K

K

∑︂

k=1

∫︂

Ω

W(Dφ_K,k) +γ∥D^mφ_K,k∥²dx is shown.

4. Verifying admissibility of the limit. In the final step, we prove that (I, v, Y, z) is a solution of (3.16) and (3.17), where Y is the flow associated withv andI is the limit image path.

3.5 Mosco convergence of time discrete geodesic paths

1. Identification of the image and deformation vectors. Choose a sequence I_K ∈ L²([0,1], L²(Ω,H)) that converges weakly to a path I ∈ L²([0,1], L²(Ω,H)). If we exclude the trivial case lim inf_K→∞J_K(I_K) = ∞ and eventually pass to a subsequence (without relabeling), we may assume

J_K(I_K)≤ J <∞

for all K ∈ N. By definition of J_K, this directly implies I_K = I_K^ext(I_K,φ_K) with I_K = (I_K,0, . . . , I_K,K) ∈ L²(Ω,H)^K+1 and corresponding optimal deforma-tions φ_K = (φ_K,1, . . . , φ_K,K) ∈ A^Kε in (3.29), see Lemma 3.11. In particular, by incorporating Lemma 3.10, we deduce

k=1,...,Kmax ∥φ_K,k−Id∥C^1,α(Ω) ≤C max

k=1,...,K∥φ_K,k−Id∥H^m(Ω) ≤Cθ(︁

JK⁻¹)︁

≤CK⁻¹². (3.30) We denote byY_K,X_K,v_K andz_Kthe discrete quantities associated withφ_K defined in Section 3.4, which exist for K sufficiently large.

2. Lower semi-continuity of the weak material derivative. Let us remark that this step resembles the first step of the proof in the Euclidean setting replacing the squared L²-norm by the squared distance in the Hadamard manifold.

A straightforward computation shows

∫︂ 1 0

∫︂

Ω

z_K² dxdt=

K

∑︂

k=1

∫︂ t_K,k t_K,k−1

∫︂

Ω

K²d(︁

I_K,k−1◦x_K,k, I_K,k◦φ_K,k◦x_K,k)︁2

dxdt

=

K

∑︂

k=1

∫︂ tK,k

tK,k−1

∫︂

Ω

K²d(I_K,k−1, I_K,k◦φ_K,k)²detDy_K,kdxdt. (3.31) Next, we want to bound the difference of detDy_K,k and 1 in the L^∞-norm. Thus, we have

Dy_K,k(t, x) =1+K(t−t_K,k−1)(Dφ_K,k(x)−1).

Then, the Lipschitz continuity of the determinant on the ballB_r(1)with associated radius r= sup_Kmax_k=1,...,K∥φ_K,k−Id∥^Hm(Ω)<∞ implies

∥detDy_K,k−1∥^L∞([tK,k−1,tK,k)×Ω) ≤C∥φ_K,k−Id∥C^1,α(Ω). Hence, we can deduce from (3.30) and t_K,k−t_K,k−1 =K⁻¹ that

⃓

⃓

⃓

⃓

K

∑︂

k=1

K²

∫︂ tK,k

tK,k−1

∫︂

Ω

d(I_K,k−1, I_K,k◦φ_K,k)²(detDy_K,k−1) dxdt

⃓

⃓

⃓

⃓

≤δJC max

k=1,...,K∥φ_K,k−Id∥C^1,α(Ω) ≤δJCK⁻¹².

Taking into account the definition of z_K in (3.24), this ultimately leads to

Klim→∞

∫︂ 1 0

∫︂

Ω

z_K² dxdt= lim

K→∞K

K

∑︂

k=1

∫︂

Ω

d(I_K,k−1, I_K,k◦φ_K,k)²dx.

This also shows the uniform boundedness of z_K ∈ L²((0,1), L²(Ω)), which implies the existence of a weakly convergent subsequence with limit z ∈ L²((0,1), L²(Ω)). Hence, using the weak lower semi-continuity of the norm, we get

∫︂ 1 0

∫︂

Ω

z²dxdt ≤lim inf

K→∞

∫︂ 1 0

∫︂

Ω

z_K² dxdt= lim inf

K→∞ K

K

∑︂

k=1

∫︂

Ω

d(I_K,k−1, I_K,k◦φ_K,k)²dx.

3. Lower semi-continuity of the viscous dissipation. We highlight that this step differs from the corresponding step appearing in [33] due to the modification of the assumption (W2), where the overall structure persists.

Note that the velocity fields v_K = v_K(φ_K) are not necessarily in L²((0,1),V). First, we show that the sequence w_K = w_K(φ_K) ∈ L²((0,1),V) is uniformly bounded in L²((0,1),V). To see this, we assume that K is large enough such that max_k=1,...,K∥Dφ_K,k−1∥C⁰(Ω) < r_W, see (W2), which is possible due to (3.30).

Then, using Korn’s inequality, the Poincaré inequality as well as (W2), we obtain

∫︂ 1 0

∫︂

Ω∥w_K∥²dxdt≤C

K

∑︂

k=1

∫︂ t_K,k t_K,k−1

∫︂

Ω

K²∥(Dφ_K,k)^sym−1∥²dxdt

≤CK

K

∑︂

k=1

∫︂

Ω

W(Dφ_K,k)

C_W,1 dx≤ CJ C_W,1,

∫︂ 1 0

∫︂

Ω∥D^mwK∥²dxdt=

K

∑︂

k=1

∫︂ tK,k

t_K,k−1

∫︂

Ω

K²∥D^m(φ_K,k−Id)∥²dxdt

=

K

∑︂

k=1

K

∫︂

Ω

∥D^mφ_K,k∥²dx≤ J γ.

The Gagliardo–Nirenberg inequality implies that the sequence w_K is uniformly bounded in L²((0,1),V). By passing to a subsequence (again labeled in the same way), we can deduce wK ⇀ v∈L²((0,1),V) for K → ∞.

It remains to verify the lower semi-continuity of the sum of the approximate Riemannian distances in (3.14), i.e.,

∫︂ 1 0

∫︂

Ω

L[v, v] dxdt≤lim inf

K→∞ K

K

∑︂

k=1

∫︂

Ω

W(Dφ_K,k) +γ∥D^mφ_K,k∥²dx.

The second order Taylor expansion aroundt_K,k−1 of t↦→W(1+ (t−t_K,k−1)Dw_K,k) evaluated at t=t_K,k yields

W(Dφ_K,k) =W(1) +K⁻¹DW(1)(DwK,k) + 1

2K²D²W(1)(DwK,k, DwK,k) +rK,k

=K⁻²(︁_λ

2tr(ε[w_K,k])²+µtr(ε[w_K,k]²))︁

+r_K,k, (3.32)

where r_K,k denotes a remainder. Here, the lower order terms vanish due to (3.12) and the last equality follows from (3.13). By Taylor’s theorem, the definition

3.5 Mosco convergence of time discrete geodesic paths of w_K,k = K(φ_K,k −Id) and the growth estimate given in (3.30), it follows that the remainder satisfies ∥r_K,k∥ ≤CK⁻³∥Dw_K,k∥³. Then,

K

K

∑︂

k=1

∫︂

Ω

W(Dφ_K,k) +γ∥D^mφ_K,k∥²dx

=K⁻¹

K

∑︂

k=1

∫︂

Ω λ

2tr(ε[w_K,k])²+µtr(ε[w_K,k]²) +γ∥D^mw_K,k∥²dx+K

K

∑︂

k=1

∫︂

Ω

r_K,kdx, and the remainder is of order K⁻¹². To see this, we apply (3.30), Lemma 3.10 and the uniform bound on the energy to deduce

K

K

∑︂

k=1

∫︂

Ω∥r_K,k∥dx≤CK

K

∑︂

k=1

∫︂

Ω

K⁻³∥Dw_K,k∥³dx

≤CK max

k=1,...,K∥φ_K,k−Id∥C¹(Ω) K

∑︂

k=1

∥φ_K,k−Id∥²H^m(Ω)

≤CKθ(︁

JK⁻¹)︁

K

∑︂

k=1

θ(︁

R(I_K,k−1, I_K,k, φ_K,k))︁2

≤CK¹²

K

∑︂

k=1

R(I_K,k−1, I_K,k, φ_K,k)≤CJK⁻¹².

Finally, a standard weak lower semi-continuity argument [83, Thm. 3.20] shows

lim inf

K→∞ K

K

∑︂

k=1

∫︂

Ω

W(Dφ_K,k) +γ∥D^mφ_K,k∥²dx

= lim inf

K→∞

∫︂ 1 0

∫︂

Ω λ

2 tr(ε[w_K])²+µtr(ε[w_K]²) +γ∥D^mw_K∥²dxdt

≥

∫︂ 1 0

∫︂

Ω λ

2tr(ε[v])²+µtr(ε[v]²) +γ∥D^mv∥²dxdt,

which implies weak lower semi-continuity of the path energy for {I_K}^K∈N.

4. Verifying admissibility of the limit. Finally, it remains to verify that (I, v, Y, z) for a suitable Y is a solution of (3.16) and (3.17). We have already pointed out that the manifold-valued metamorphosis energy functional necessitates a variational inequality, which results in significant modifications of this step com-pared to [33].

Let Y˜ denote the solution of

d

dtY˜ (t, x) =v(t, Y˜ (t, x)) for (t, x)∈[0,1]×Ω,

Y˜ (0, x) =x for x∈Ω, (3.33)

which exists due to Theorem 3.6. Furthermore, (3.23) and the uniform bound-edness of w_K ∈ L²((0,1),V) imply that the sequence v_K is uniformly bounded in L²((0,1), C^1,α(Ω)). Incorporating Remark 3.7, we infer that Y_K is uniformly bounded in C⁰([0,1], C^1,α(Ω)), and by exploiting Hölder’s inequality we can even show that the sequence is uniformly bounded in C^0,1/2([0,1], C^1,α(Ω)). Hence, by using the compact embedding of Hölder spaces, the sequenceY_K converges strongly to someY in C^0,β([0,1], C^1,β(Ω)) for β = ¹₂min(¹₂, α).

It remains to verify thatY˜ =Y. To this end, the solutions of (3.33) correspond-ing to w_K are denoted by Y˜

K. Then,

⃦⃦Y−Y˜⃦

⃦C⁰([0,1]×Ω) ≤ ∥Y−Y_K∥C⁰([0,1]×Ω)+⃦

⃦Y_K−Y˜_K⃦

⃦C⁰([0,1]×Ω)+⃦

⃦Y˜_K−Y˜⃦

⃦C⁰([0,1]×Ω). Here, the first term converges to zero as shown above and the last term converges to zero by the continuous dependence of Y˜

K onw_K discussed in Theorem 3.6. Then, we can estimate as follows

∥Y_K−Y˜_K∥C⁰([0,1]×Ω) ≤C

K

∑︂

k=1

∫︂ tK,k

tK,k−1

∥w_K,k(s, x_K,k(s,·))−w_K,k(s,·)∥C⁰(Ω)ds

≤C

K

∑︂

k=1

∫︂ tK,k

t_K,k−1

∥wK,k(s,·)∥^Hm(Ω)∥yK,k(s,·)−Id∥C⁰(Ω)ds

≤C∥wK∥L²((0,1),H^m(Ω)) max

k=1,...,K∥φ_K,k−Id∥C⁰(Ω).

Here, the first inequality is deduced from Remark 3.7. Further, to derive the second inequality, we exploit Lipschitz continuity of x ↦→ w_K,k(s, x_K,k(s, x))−w_K,k(s, x), where the Lipschitz constant is bounded byC∥w_K,k(s,·)∥^Hm(Ω), and apply the coor-dinate transformy_K,k(s,·). The uniform control ofw_K and (3.30) imply Y =Y˜ and by Hölder’s inequalityY ∈C^0,1/2([0,1], C^1,α(Ω)). Finally,X_K is uniformly bounded inC^0,1/2([0,1], C^1,α(Ω)) due to Remark 3.7. Thus, (3.16) is fulfilled.

Next, note that for s, t∈[0,1]we obtain

∫︂

Ω

d(︁

I_K(t, Y_K(t, x)), I_K(s, Y_K(s, x)))︁2

dx≤

∫︂

Ω

(︂∫︂ s t

z_K(r, Y_K(r, x)) dr)︂2

dx

≤ |s−t|⃓

⃓

⃓

∫︂

Ω

∫︂ s t

z_K(r, Y_K(r, x))²drdx

⃓

⃓

⃓. By uniform boundedness ofzKinL²((0,1), L²(Ω)), we getIK◦YK ∈A¹

2,L,|detDY|for some appropriateL. Next, we show weak convergence of a subsequence of I_K◦Y_K toI ◦Y ∈A¹

2,L,|detDY|. To this end, we observe lim sup

K→∞

d₂(I_K, I)² = lim sup

K→∞

∫︂ 1 0

∫︂

Ω

d(︁

I_K(t, Y_K(t, x)), I(t, Y_K(t, x)))︁2

|detDY_K|dxdt

= lim sup

K→∞

∫︂ 1 0

∫︂

Ω

d(︁

I_K(t, Y_K(t, x)), I(t, Y(t, x)))︁2

|detDY|dxdt.

3.5 Mosco convergence of time discrete geodesic paths For the first equality, we incorporate the transformation formula, the second equality follows from the uniform convergence of DY_K, the metric triangle inequality and the convergence of I(t, Y_K(t, x)) to I(t, Y(t, x)), see Lemma 3.4. To sum up, this proves weak convergence of I_K ◦Y_K according to (3.2), and by Theorem 3.3, the limit is also contained in A1

2,L,|detDY|.

Finally, it remains to verify (3.17). Assume there exist s < t∈ [0,1] such that the set

B :={︂

x∈Ω : d(︁

I(s, Y(s, x)), I(t, Y(t, x)))︁

>

∫︂ t s

z(r, Y(r, x)) dr}︂

has positive Lebesgue measure. From the joint convexity of the metricd(·,·)and the continuity of point evaluation in time, we infer that I ↦→ ∫︁

Bd(I(s, x), I(t, x)) dx is continuous and convex onA1

2,L,|detDY|. Now, this implies weak lower semi-continuity of the mapping, see [13, Lem. 3.2.3], and we obtain

∫︂

B

d(︁

I(s, Y(s, x)), I(t, Y(t, x)))︁

dx

≤lim inf

K→∞

∫︂

B

d(︁

I_K(s, Y_K(s, x)), I_K(t, Y_K(t, x)))︁

dx

≤lim inf

K→∞

∫︂

B

∫︂ t s

z_K(r, Y_K(r, x)) drdx=

∫︂

B

∫︂ t s

z(r, Y(r, x)) drdx,

where the last equality follows from the weak convergence ofzK combined with the strong convergence ofY_K, which also implies the weak convergence ofz_K◦Y_K. This yields a contradiction and concludes the proof of the liminf-inequality.

In what follows, we prove existence of a recovery sequence and thus establish Mosco convergence. As a preparation, we prove that the infimum in (3.15) is actu-ally attained, where we exploit some results of the proof of Theorem 3.12.

Proposition 3.13. For I ∈ L²([0,1], L²(Ω,H)) with J(I) < ∞, the infimum in (3.15)is attained, i.e., there exists a tuple (v, z)∈ C(I) satisfying(3.16)and(3.17). Proof. We first observe that the functional (v, z) ↦→ ∫︁1

0

∫︁

ΩL[v, v] + ¹_δz²dxdt is weakly lower semi-continuous and coercive on C(I), cf. [28]. SinceC(I) is a subset of a reflexive Banach space, it suffices to prove weak closedness of C(I) to obtain existence of an optimal tuple (v, z)∈ C(I).

Let{(v_k, z_k)}k∈N ∈ C(I)be a weakly convergent sequence with limit(v, z). Due to Theorem 3.6, the corresponding flowsY_kandY given by (3.16) exist andY_k →Y in C⁰([0,1]×Ω). Further, weak convergence of v_k implies uniform boundedness of {v_k}k∈NinL²((0,1), C^1,α(Ω)). Thus, the reasoning in the paragraph following (3.33) implies that a subsequence of{Y_k}^k∈Nconverges strongly toY inC^0,β([0,1], C^1,β(Ω)) for β = ¹₂min(¹₂, α).

Finally, Lemma 3.4 implies I(t, Y_k(t, x)) → I(t, Y(t, x)) in L²([0,1], L²(Ω,H)) and the last part of the proof of Theorem 3.12 shows that (I, Y, z) is a solution of (3.17).

Theorem 3.14 (Recovery sequence). Let I_A, I_B ∈L²(Ω,H) be fixed input images and let I ∈ L²([0,1], L²(Ω,H)) be an image path with I(0) = I_A and I(1) = I_B. Then there exists a recovery sequence {I_K}^K∈N with I_K(0) = I_A and I_K(1) = I_B for all K ∈ N such that the limsup-inequality in Definition 3.1 with respect to the L²([0,1], L²(Ω,H))-topology is valid.

Proof. We proceed in three steps, which follow the usual general guideline to show existence of recovery sequences in the context of Γ-convergence:

1. Construction of the recovery sequence. 2. Verification of the limsup-inequality.

3. Identification of the recovery sequence limit.

1. Construction of the recovery sequence. Compared to [33], our construc-tion avoids the approximaconstruc-tion of v and defines the deformations directly. Due to Proposition 3.13, there exist optimal (v, Y, z) corresponding to I satisfying (3.16) and (3.17). Incorporating the flowY, we define for given K ∈N a vector of diffeo-morphisms φ_K = (φ_K,1, . . . , φ_K,K)∈H^m(Ω,Rⁿ)^K by

φ_K,k =Y_tK,k−1(t_K,k,·),

where Y_a(b,·) := Y(b, Y⁻¹(a,·)) ∈ H^m(Ω)^K with a, b ∈ [0,1]. This expression coincides with the evaluation at t= 1of the flow corresponding to the velocity field v_a,b(t, x) := (b−a)v(a+ (b−a)t, x), i.e., the solution of

Ẏ_a,b(t, x) = v_a,b(t, Y_a,b(t, x)) for(t, x)∈[0,1]×Ω,

Y_a,b(0, x) = x forx∈Ω.

Here, v is the velocity field whose existence is postulated in Proposition 3.13. Next, we bound the C¹(Ω)-norm of the displacements as follows

max

k∈{1,...,K}∥φ_K,k−Id∥C¹(Ω)

≤ sup

s,t∈[0,1]

|t−s|≤K⁻¹

∥Y_s(t,·)−Id∥C¹(Ω) ≤ sup

s,t∈[0,1]

|t−s|≤K⁻¹

C

∫︂ 1

0 ∥v_s,t(r, Y_s,t(r,·))∥^Hm(Ω)dr

≤ sup

s,t∈[0,1]

|t−s|≤K⁻¹

C

⃓

⃓

⃓

∫︂ t s

∥v(r,·)∥H^m(Ω)dr

⃓

⃓

⃓≤CK⁻¹² sup

s,t∈[0,1]

|t−s|≤K⁻¹

⃓

⃓

⃓

∫︂ t s

∥v(r,·)∥²H^m(Ω)dr

⃓

⃓

⃓

1 2.

(3.34) For the third inequality, we exploit the estimate

∥v(t, Y(t,·))∥^Hm(Ω)≤C∥v(t,·)∥^Hm(Ω), (3.35)

3.5 Mosco convergence of time discrete geodesic paths which follows from [49, Lem. 3.5] and an extension argument as shown in Theo-rem 3.6, and use the transformation formula. The last inequality is implied by the Cauchy–Schwarz inequality.

Choosing K sufficiently large ensures φ_K ∈ A^Kε and consequently we can apply the temporal extension from Section 3.4. Finally, the recovery sequence is defined asI_K =I_K^ext(I_K,φ_K), where

I_K = (I_K,0, I_K,1, . . . , I_K,K) =(︁

I(t_K,0,·), . . . , I(t_K,K,·))︁

.

2. Verification of the limsup-inequality. Note that this step shares some sim-ilarities with the corresponding step in [33] with modifications necessitated by the manifold structure and the different construction. In the following, all terms in the discrete energy J_K(I_K,φ_K) are estimated separately. For any k = 1, . . . , K, we infer using (3.17), Jensen’s inequality and (3.34) that

∫︂

Ω

d(︁

I_K,k−1, I_K,k ◦φ_K,k)︁2

dx

=

∫︂

Ω

d(︁

I_K,k−1◦Y(t_K,k−1, x), I_K,k◦Y(t_K,k, x))︁2

det(︁

DY(t_K,k−1, x))︁

dx

≤

∫︂

Ω

(︃∫︂ tK,k

t_K,k−1

z(s, Y(s, x)) ds )︃2

det(︁

DY(t_K,k−1, x))︁

dx

≤K⁻¹

∫︂ tK,k

tK,k−1

∫︂

Ω

z²(s, x) det(︁

DY_s(t_K,k−1, x))︁

dxds

≤K⁻¹(︁

1 +CK⁻¹²)︁

∫︂ tK,k

tK,k−1

∫︂

Ω

z²dxds. (3.36)

Recall that w_K,k = K(φ_K,k −Id). Now, the same Taylor argument as in (3.32) implies

∫︂

Ω

W(Dφ_K,k) +γ∥D^mφ_K,k∥²dx≤K⁻²

∫︂

Ω

L[w_K,k, w_K,k] dx+CK⁻³

∫︂

Ω∥Dw_K,k∥³dx.

(3.37) Summing over the second term on the right hand side and taking into account (3.34), we obtain

K

∑︂

k=1

∫︂

Ω∥Dw_K,k∥³dx≤CK³

K

∑︂

k=1

∥φ_K,k−Id∥³_C¹_(Ω) ≤CK³².

A direct application of Jensen’s inequality shows that the lower order term satisfies

∫︂

Ω

L[w_K,k, w_K,k] dx

=

∫︂

Ω

L [︃

K

∫︂ t_K,k t_K,k−1

v(t, Y_tK,k−1(t, x)) dt, K

∫︂ t_K,k t_K,k−1

v(t, Y_tK,k−1(t, x)) dt ]︃

dx

≤

∫︂

Ω

K

∫︂ tK,k

t_K,k−1

L[︁

v(t, Y_tK,k−1(t, x)), v(t, Y_tK,k−1(t, x))]︁

dtdx.

By using (3.34) and|tr(AB)| ≤ |tr(A)|+|tr(A(B−1))|for A, B ∈R^n×n multiple times, we can estimate the part corresponding to the first summand of L, see (3.3), as follows

∫︂

Ω

∫︂ t_K,k t_K,k−1

tr(︂

D(︁

v(t, Y_tK,k−1(t, x)))︁)︂2

dtdx

=

∫︂

Ω

∫︂ tK,k

t_K,k−1

tr(︁

Dv(t, Y_tK,k−1(t, x))DY_tK,k−1(t, x))︁2

dtdx

≤

∫︂

Ω

∫︂ tK,k

t_K,k−1

tr(︁

Dv(t, Y_tK,k−1(t, x)))︁2

+ tr(︁

Dv(t, Y_tK,k−1(t, x))(1−DY_tK,k−1(t, x)))︁2

+ 2⃓

⃓tr(︁

Dv(t, Y_tK,k−1(t, x)))︁

tr(︁

Dv(t, Y_tK,k−1(t, x))(1−DY_tK,k−1(t, x)))︁⃓

⃓ dtdx

≤

∫︂

Ω

∫︂ t_K,k t_K,k−1

tr(︁

Dv(t, Y_tK,k−1(t, x)))︁2

+C(︁

1 +∥v(t,·)∥³H^m(Ω)

)︁K⁻¹² dtdx

≤

∫︂

Ω

∫︂ t_K,k t_K,k−1

tr(ε[v])²+C(︁

1 +∥v(t,·)∥³H^m(Ω)

)︁K⁻¹² dtdx.

For the last inequality, we additionally used the transformation formula and (3.34).

The second term in L is estimated analogously

∫︂

Ω

∫︂ tK,k

t_K,k−1

tr(︁

ε[︁

v(t, Yt_K,k−1(t, x))]︁2)︁

dtdx

≤

∫︂

Ω

∫︂ tK,k

t_K,k−1

tr(ε[v]²) +C(︁

1 +∥v(t,·)∥³H^m(Ω)

)︁K⁻¹² dtdx.

It remains to bound the higher order term appearing in the definition of L. To this end, we use (3.35) and the bound ∥f g∥H^m˜ ≤ C∥f∥^Hm∥g∥H^m˜ for f ∈ H^m(Ω), g ∈H^m˜(Ω) and any 0≤m˜ ≤m, see [166, Lem. 2.3], which results in the following estimates

|v(t, Y_tK,k−1(t,·))|^Hm(Ω)

≤ |Dv(t, Y_tK,k−1(t,·))|H^m−1(Ω)+∥Dv(t, Y_tK,k−1(t,·))D(Y_tK,k−1(t,·)−Id)∥H^m−1(Ω)

≤ |Dv(t, Yt_K,k−1(t,·))|H^m−1(Ω)+C⃦

⃦v(t,·)∥^Hm(Ω)

⃦⃦Yt_K,k−1(t,·)−Id∥^Hm(Ω)

≤ |Dv(t, Y_tK,k−1(t,·))|^Hm−1(Ω)+C⃦

⃦v(t,·)∥^Hm(Ω)K⁻¹².

Iterating this argument and applying a change of variables, we obtain for the last term of L

∫︂ t_K,k t_K,k−1

|v(t, Y_tK,k−1(t,·))|²H^m(Ω)dt≤

∫︂ t_K,k t_K,k−1

|v(t,·)|²H^m(Ω)+C⃦

⃦v(t,·)∥²H^m(Ω)K⁻¹² dt.

(3.38)

3.5 Mosco convergence of time discrete geodesic paths By combining the estimate (3.36) with (3.37)–(3.38) for the second inequality below, we get

J^K(I_K)≤K

K

∑︂

k=1

∫︂

Ω

W(Dφ_K,k) +γ|D^mφ_K,k|²+ ¹_δd(︁

I_K,k−1, I_K,k◦φ_K,k)︁2

dx

≤

K

∑︂

k=1

(︃∫︂ tK,k

t_K,k−1

∫︂

Ω

L[v, v] +CK⁻¹|DwK,k|³+¹_δ(︁

1 +CK⁻¹²)︁

z²dxdt )︃

≤

∫︂ 1 0

∫︂

Ω

L[v, v] + ¹_δz²dxdt+CK⁻¹² +C¹_δK⁻¹² =J(I) +O(K⁻¹²), which readily implies the limsup-inequality.

3. Identification of the recovery sequence limit. It remains to verify the convergence I_K →I inL²([0,1], L²(Ω,H)) asK → ∞. To see this we estimate

∫︂ 1 0

∫︂

Ω

d(︁

I(s, Y(s, x)), I_K^ext(I_K,φ_K)(s, Y_K(s, x)))︁2

dxds

=

K

∑︂

k=1

∫︂ tK,k

tK,k−1

∫︂

Ω

d(︁

I(s, Y(s, x)), I_K^ext(I_K,φ_K)(s, Y_K(s, x)))︁2

dxds

≤C

K

∑︂

k=1

(︃∫︂ tK,k

t_K,k−1

∫︂

Ω

K⁻²z²(s, Y(s, x)) dxds +

∫︂ tK,k

t_K,k−1

∫︂

Ω

d(︁

IK,k−1(Y(tK,k−1, x)), I_K^ext(IK,φ_K)(s, YK(s, x)))︁2

dxds )︃

≤CK⁻²∥z(t, Y(t, x))∥²L²((0,1)×Ω).

Here, we combined (3.17) with the Cauchy–Schwarz inequality to obtain an es-timate for the term d(I(s, Y(s, x)), IK,k(Y(tK,k−1, x))) in the first inequality and used the definition of I_K^ext, see (3.21), together with (3.1), (3.17) and the Cauchy–

Schwarz inequality in the second inequality. Due to the convergence ofY_K toY and Corollary 3.5, this readily implies the claimed convergence I_K^ext(IK,φ_K)→I.

We conclude this section with the desired convergence statement for discrete geodesic paths.

Theorem 3.15 (Convergence of discrete geodesic paths). Let IA, IB ∈ L²(Ω,H) and suppose that the assumptions (W1), (W2) and (W3) hold true. For every K ∈N let I_K be a minimizer of J_K subject to I_K(0) =I_A and I_K(1) =I_B. Then, a subsequence of {IK}^K∈N converges weakly in L²([0,1], L²(Ω,H)) to a minimizer of the continuous path energyJ asK → ∞, and the associated sequence of discrete energies converges to the minimal continuous path energy.

Proof. Using a comparison argument with v ≡0and z(t, x) = d₂(I_A(x), I_B(x)), we deduce that the path energy J_K is bounded by J = ¹_δd₂(I_A, I_B)². For optimal

vectors of images I_K and deformations φ_K in the definition of J_K, see (3.29), we apply the temporal extension construction from Section 3.4. In particular, J_K(I_K,φ_K) ≤ J for all K ∈ N. Using (3.22) and (3.31), we conclude that z_K is uniformly bounded inL²((0,1)×Ω). Next, Remark 3.7 together with (3.23) and (3.30) imply uniform boundedness of Y_K, X_K in C⁰([0,1], C^1,α(Ω)). Incorporating (3.17), we obtain for f_a(x) =a with a∈ H that

d₂(I_K(t,·), f_a)≤C(︁

d₂(︁

I_K(t, Y_K(t,·)), I_A)︁

+ d₂(I_A, f_a))︁

≤C(︁

∥z_K∥^L2((0,1)×Ω)+ 1)︁

. Therefore,{I_K}^K∈Nis uniformly bounded inL^∞([0,1], L²(Ω,H))and a subsequence converges weakly to some I ∈L²([0,1], L²(Ω,H))in L²([0,1], L²(Ω,H)).

Now, we follow the usual argument and assume that there exists an image path I˜ ∈ L²([0,1], L²(Ω,H)) with corresponding optimal tuple (I˜, v˜, Y˜, z), which exists due to Proposition 3.13, satisfying (3.16) and (3.17) such that

J[I˜]<J[I]. (3.39)

By Theorem 3.14, we get existence of a sequence {I˜_K}^K∈N ⊂ L²((0,1), L²(Ω,H)) satisfying lim sup_K→∞J_K[I˜

K] ≤ J[I˜]. Thus, we obtain applying Theorem 3.12 that

J[I]≤lim inf

K→∞ J_K[I_K]≤lim sup

K→∞

J_K[I˜_K]≤J[I˜], (3.40) which contradicts (3.39). Hence, I minimizes the continuous path energy over all admissible image paths. Finally, the discrete path energies converge to the limiting path energy along a subsequence, i.e., lim_K→∞J_K[I_K] =J[I], which again follows from (3.40) by usingI˜ =I.

Im Dokument Deformation and transport of image data (Seite 77-88)