1 Introduction

Visualizing diffusion tensor fields has become an important topic in many applications. However, it is still difficult to track the underlying continuous behaviors embedded in discrete diffusion tensor fields by employing existing schemes, especially around degenerate points that lead to rotational inconsistency of tensor anisotropy.

We describe our first contribution to smoothly track the continuous behaviors in diffusion tensor fields in Chapter 3 and Chapter 4 (Figure 1). This is accomplished by locating the possible degenerate points globally using a minimum spanning tree (MST) based algorithm firstly. Then we limit the size of isotropic region to avoid that these isotropic tensor samples degrade the anisotropic features of the underlying continuous behaviors in the discrete diffusion tensor fields.

The idea to locate the possible degenerate points globally in the diffusion tensor fields inspires our next contribution demonstrated in Chapter 5 (Figure 1) to control the mesh topology in quadrilateral mesh through introducing a 2D diffusion tensor field. This is because the streamlines along the two principal directions of a tensor field and a quadrilateral mesh are dual to each other. The region containing a degenerate point that is located by using our MST algorithm is dual to an extraordinary (i.e. non-degree-four) vertex in a quadrilateral mesh, while the non-degenerate region is dual to an ordinary vertex in the quadrilateral mesh. Furthermore, all-hexahedral mesh is also generated by using sweeping operations.

2 Degeneracy-Aware Interpolation of 2D Diffusion Tensor Fields

Our primary idea [1] for interpolating 2D diffusion tensor fields is to locate and resolve tensor degeneracy for tracking smooth transitions of anisotropic features inherent in the given data.

For locating the possible tensor degeneracy, we employ a minimum spanning tree (MST) strategy to group discrete tensor samples with similar orientations of the anisotropic features. For this purpose, we define the similarity between a pair of neighboring tensor samples as:

d(DS,DT)=a|CSI-CTI|+β|CSP-CTP|+γ(|ΘS,T|/(π/2))' (1)

where CSI and CSI represent the CI values of the two tensors DS and DT, and CSP and CTP represent the corresponding CP values. In addition, ΘS,T is the minimal rotation angle between the right-handed coordinate systems defined by the two sets of eigenvector directions. Here, we locate degenerate points by counting the number of degenerate pairs in one cell, where a degenerate pair is defined as a pair of tensors whose rotational angle is larger thanπ/2.

The rotational inconsistency is resolved by optimizing the rotational transformation between a pair of neighboring tensors through analyzing their associated eigenstructure.

Figure 2 shows a result where a 2D tensor field with three degenerate points is interpolated.

3 Degeneracy-Aware Interpolation of 3D Diffusion Tensor Fields

In 3D diffusion tensor fields, it becomes much more difficult to locate and resolve tensor degeneracy since the topological structure of degeneracy is much more complicated.

When constructing a minimum spanning tree in 3D space, we often encounter unwanted cases where an important pair of tensor samples are left unconnected, especially when the rotation angle between the primary eigenvectors at the two end tensor samples becomes close to 0 while those between other pairs of eigenvectors approach toπ/2. For avoiding this, we revise the previous similarity metric as the following new one [2]:

〓 (2)

where the fourth term is newly introduced to evaluate the minimum rotational angle between the two primary eigenvectors since they are the most important for fiber tracking. The fifth term is employed to discriminate between the two tensors with the same anisotropy orientation but different size.

For resolving rotational inconsistency, we minimize an objective function so as to transform each degenerate pair to non-degenerate one as well as to minimize the number of newly generated degenerate pairs.

Figure 3 shows a result where the two fibers that run around the tensor degeneracy have been tracked in a human brain dataset.

4 Quadrilateral/Hexahedral Mesh Generation based on Tensor Fields

We proposed an approach [3] to interactively controlling the mesh topology of quadrilateral meshes by introducing diffusion tensor fields into the target object.

Firstly, the Poisson equation is employed to generate the tensor field (Fig. 4(c)) by propagating the tensor anisotropic features along the boundary (Fig. 4(b)) into the interior of the target object (Fig. 4(a)). The possible degenerate points are also located by using our MST-based algorithm.

Then, the streamlines of the diffusion tensor field (Fig. 4(d)) can then be transformed into the dual graph (Fig. 4(e)) of the quadrilateral mesh (Fig. 4(f)), since the streamlines and quadrilateral mesh are dual to each other. Furthermore, our approach allows us to interactively control the mesh topology by changing the orientations of the tensor samples on the boundary of the target object.

Finally, we extend the framework of quadrilateral mesh to generate all-hexahedral mesh using sweeping method.

5 Conclusions and Future Work

In this thesis, we proposed a degeneracy-aware interpolation approach for diffusion tensor fields, which can successfully allow us to track the underlying anisotropic features such as nerve and muscle fibers. This has been achieved by using an MST-based algorithm to locate the possible degenerate points, and resolving such degeneracy by optimizing the rotation transformation between each pair of tensors.

We also introduced an approach to interactively controlling the mesh topology of quadrilateral meshes by introducing a 2D diffusion tensor field into a target object. Finally, all-hexahedral mesh is also generated when the target object can be composed through sweeping operations.

Our future extension includes the challenge to extend our framework to generate all-hexahedral meshes in the volume without constructed structure.

Figure 1: Organization of this thesis

Figure 2: Interpolating a diffusion tensor field containing three degenerate points ( represented by green circles). (a) Original tensor samples. The results with the (b) component-wise and (c) log-Euclidean. These interpolation methods cannot retain the anisotropic feature inherent in the original tensor field, as indicated by the black circle. (d) The result with geodesic-loxodrome. This usually incurs the problem of rotation inconsistency around degenerate points, as shown by the black rectangle. (e) The result obtained by our interpolation scheme. Our scheme can fully respect the high anisotropic features, and take care of the rotational inconsistency around degenerate points.

Figure 3: Tracking two fibers in a human brain DT-MRI dataset, where the red point is the seed point. Several degenerate points exist between the two fibers. Interpolated results with the (a) component-wise and (b) log-Euclidean cannot fully track the two fibers due to the degeneracy in the region between the two fibers, while our scheme can successfully track the two fibers and avoid the influence of existing degeneracy. This is because we limit the size of the isotropic region while maximally respecting the anisotropy of the fibers.

Figure 4: Generating a quadrilateral mesh in the interior of the target object. (a) 2D bunny shape. (b) The diffusion tensor field along the boundary of the bunny object. (c) The generated diffusion tensor field in the interior of the bunny object. The blue point represents a detected degenerate point. (d) The streamlines of the diffusion tensor fields. (e) The dual graph of the quadrilateral mesh obtained from the streamlines in (d). (f) The quadrilateral mesh obtained from its dual graph in (e).

DT-MRI などの計測機器から得られる3次元拡散テンソル場離散サンプルデータの可視化は,脳神経繊維などの線特徴追跡の問題として,近年重要な研究対象となってきている.そして,そのような線特徴追跡には,離散サンプルからテンソル場全体を滑らかに補間する手法が必須の道具となる.しかし既存の補間手法は,テンソルの異方性特徴の回転を伴う退化点周辺では補間精度が低下してしまう.本学位論文における貢献の1つは,このような拡散テンソル場を,退化点特徴を考慮に入れて,テンソル場に内在する異方性特徴を適切に補間する手法を提案したことにある.

さらに本学位論文のもう1つの貢献は,拡散テンソル場の補間手法を,有限要素法の前処理である4辺形および6面体メッシュ生成に応用したことにある.有限要素法解析において,対象形状モデルを4辺形や6面体等の小さな基本要素形状に分割する処理は,後の解析結果の精度にも大きく影響するため重要である.しかしそれらを用いて対象形状内を適切に埋め尽くす問題は,未解決問題ととらえられているほど難しい.本学位論文では,この4辺形や6面体のメッシュ構造が,先の拡散テンソル場と双対の関係をもつことに着目し,テンソル場の補間手法を用いてメッシュ生成の問題の解決を図っている点が特筆に値する.

本学位論文は,6章から構成される.各章の内容は以下の通りである.

まず第1章では,本学位論文の全体構成について記載されている.特に,拡散テンソル場の補間に得られる異方性特徴の方向を表す流線の配置と双対の関係をなすことを示し,テンソル場の補間手法を利用することで,有限要素法のメッシュ生成問題の解決を図る本学位論文の基本概念について述べられている.

第2章では,拡散テンソル場補間の従来手法を,行列表現と固有値・固有ベクトルを用いたものに分類し,各カテゴリーの特徴について説明するとともに,4辺形・6面体メッシュ生成の従来手法については,間接法と直接法に分類し,それぞれの特質を議論している.

第3章では,2次元の拡散テンソル場の補間手法について記述している.まず,同様の異方性特徴をもつテンソルサンプル点をクラスタリングすることで,拡散テンソル場のグリッドサンプル点から退化点の存在範囲を限定する手法を提案している.そのあと,位置が同定された退化点周辺のテンソルサンプルにおいて,異方性特徴の対応付けに変更を加えることで,全体としてテンソルの異方性特徴が滑らかに補間できることを示している.

第4章では,第3章で明らかにした2次元拡散テンソル場の補間手法を,いかに3次元のテンソル場に拡張するかについて論じられている.2次元の場合に比べて,3次元の場合任意の軸がテンソルの回転軸となるため,テンソルサンプル同士の類似度の定義に変更を加えて対応することが記されている.また,実際の脳計測データにより取得された拡散テンソル場の神経繊維追跡結果について,既存手法との比較により提案手法の優位性を示すとともに,いくつかの指標により提案手法の評価も行っている.

第5章は,今までの拡散テンソル場の補間手法を,有限要素法のためのメッシュ生成にどのように応用できるかが示されている.提案されるメッシュ生成手法の手順は,最初にユーザに境界条件を指定させたのち,その境界条件から,ポアソン方程式を離散的に解くことで対象形状の内部に拡散テンソル場を生成する.そのあと,そのテンソル場の異方性特徴を追跡することで,均等に配置された流線構造を対象形状内部に生成する.最後に,この流線構造の双対構造を計算することで,最終的なメッシュ構造を生成する.本章では,4辺形メッシュの生成事例について中心に議論が展開され,6面体メッシュに関してはその簡単な拡張で対応できる事例が示されている.また,いくつかの生成メッシュの質の評価についても検討を行っている.

最後に第6章において論文を総括し,今後の課題について言及を行なっている.

なお,本学位論文の第3,4章は,高橋成雄,藤代一成との,また,第5章は,櫻井大督,高橋成雄との共同研究であるが,数理的な定式化の大部分は論文提出者によるものであり,またその手法の実装および検証もすべて行なっている.これらより,本学位論文に記載されている学術的な内容においては,論文提出者の寄与が十分であると判断する.

以上のように,拡散テンソル場の補間と有限要素法メッシュ生成の2つの別個の問題をその形状構造の双対性に基づき同じ問題として解決を図る点において,本学位論文は独創性も高く需要な研究と位置づけることができ,さらに,従来手法との比較によりその優位性も明らかにされている.審査委員会は,以上のような視点から本学位論文の貢献を高く評価し,博士号に十分値するものと判断した.

したがって,博士(科学)の学位を授与できると認める.