A variational model with fractional-order regularization term arising in registration of diffusion tensor image

In this paper, a new variational model with fractional-order regularization term arising in registration of diffusion tensor image(DTI) is presented. Moreover, the existence of its solution is proved to ensure that there is a regular solution for this model.


Introduction
Let Ω ⊂ R 3  (1.1) In DTI registration, T and D are viewed as two images defined on Ω, where T is called floating image and D is called target image. The goal of registration is to find a 1-to-1 spatial transformation h : Ω → Ω such that T • h(·) is close to D(·) in some sense. On the other hand, in order to keep T • h(·) align with spatial transformation, reorientation of T • h(·) must be additionally considered. For this purpose, Alexander [1] put forward two reorientation strategies: finite strain(FS) strategy and preservation principle direction(PPD) strategy. Based on FS strategy, Li [8] introduced a new operator "⋄" defined by With the help of this operator, the DTI registration model(cf. [6]) can be formulated as v = arg min vH (v), (1.3) whereH(v) = τ 0 Lv(·, t) 2 L 2 (Ω) dt + T ⋄ h(·) − D(·) 2 L 2 (Ω) , L : [H 3 0 (Ω)] 3 → [L 2 (Ω)] 3 is a linear differential operator satisfing Lv(·, t) 2 and J = ∇ x η(τ ; 0, x) (cf. [6]). Here the existence of function h −1 is given by (ii) in Lemma 3.5 and the definition of (JJ T ) − 1 2 can refer to Appendix in [6]. In [6], authors prove that there exists a solution to variational problem (1.3)-(1.5) on some suitable space. Note that almost all the DTI registration model [8,14] have employed integerorder derivatives in linear differential operator L. In fact, during the last decades, it has been showed that many problems involving science and engineering can be modeled more accurately by employing fractional-order derivatives [11,13,16] than integer-order derivatives. Motivated by this fact, the aim of this paper is to employ fractional-order derivatives in DTI registration model.
Based on definition of Riemann-Liouville derivative in [4], for x = (x 1 , x 2 , x 3 ) ∈ Ω and function f : Definition 1.1. For α > 0 and function g : Ω → R, define semi-norms , and norms Based on Definition 1.1, define space F α L,0 (Ω) and F α R,0 (Ω) as the closure of C ∞ 0 (Ω) under the norm · F α L,0 (Ω) and · F α R,0 (Ω) , respectively. Definition 1.2. For α > 0 and u ∈ L 1 (R 3 ), define the semi-norm and norm where here and in what follows, u(ξ) = 1 endowing with the following inner product and norm Based on the above notations and definitions, the variational model with fractional-order regularization term arising in registration of diffusion tensor image(DTI) can be formulated as v = arg min v∈F H(v), (1.6) where (Ω) and h(x) is defined by (1.5). Another purpose of this paper is to give a rigid proof on the existence of solution to (1.6). As to this problem, we have the following result: then there also exists a global minimizer to H(v) on space (Ω) for any t ∈ [0, τ ] and i = 1, 2, 3}, endowing with the following inner product and norm 2 Equivalence of F α L,0 (Ω), F α R,0 (Ω) and H α 0 (Ω) In [6], authors impose the condition (1.4) on L such that v(·, t) ∈ [H 3 0 (Ω)] 3 ֒→ [C 1 (Ω)] 3 which ensures the existence and uniqueness of solution to (1.5). As the basic space of this paper, F α L,0 (Ω) and F α R,0 (Ω) are also needed to embedded into C 1 (Ω). Otherwise, the uniqueness of solution to (1.5) can not be guaranteed [12].
First, we introduce some definitions.
By Lemma 2.5, we obtain that F α S,0 (Ω) and H α 0 (Ω) are equivalent. Based on above Lemmas, we give the main result of this section.

(3.13)
Case 2(α ≥ 3.5). Since F α L,0 (Ω) = H α 0 (Ω) ֒→ C 2 (Ω)[2, Theorem 4.57], then (3.14) This concludes (ii). By (1.6), we know H is a functional about v and η, where v and η are constrained by (1.5). In this paper, we write H as a functional only about v. Therefore, (1.5) should admit a unique solution. Otherwise, the definition of functional H is ambiguous. As to the well-define of H, we have the following result. Proof. Based on (i) in Lemma 3.3, this conclusion can be proved in a similar way with Lemma 2.2 in [6].
As to the existence of h −1 : Ω → Ω, we have the following result.  (ii).If we denote n k as the sequence number of a weakly convergent subsequence {v n k (x, s)} with weak limit v(x, s), then  Proof.(i). By Lemma 3.2, we know · F is a norm and F is a separable Hilbert space. This implies (i) for the fact that any closed ball in a separable Hilbert space is a weakly compact set.
(ii). Since · F is a norm, by the lower weak semi-continuity of norm, we obtain (3.18).

(iii). By Lemma 3.4, we know that equations (3.19) and (3.20) have a unique solution
Before we give a proof of Theorem 1.1, let's recall the following Lemma. By (ii) in Lemma 3.6, we know that respectively. Based on these notations, here we claim that the functional The proof of claim (3.27) can be divided into following five steps.
(3.33) By (3.33), we know that Θ n k Θ T n k , ΘΘ T ∈ SP D(3). Let λ (1) n k > 0, λ (1) ≥ λ (2) ≥ λ (3) > 0 be the eigenvalues of Θ n k Θ T n k , ΘΘ T , respectively. What's more, by (3.33), we obtain that In a similar way, we can obtain that By singularity decomposition theorem [7], we can find two 3 × 3 orthogonal matrix U n k , V n k n k , λ n k , the columns of U n k , V n k are orthogonal eigenvectors of Θ n k Θ T n k and Θ T n k Θ n k , respectively.
, the columns of U , V are orthogonal eigenvectors of ΘΘ T and Θ T Θ, respectively.
Similarly , we know that A −1 ≤ M 2 . By (3.36), we obtain that since A − A n k = Θ n k Θ T n k − ΘΘ T k − → 0 by (3.32).