Recently, the research has been made progress on the accurate 3D shape restoration of objects in the real world into computer graphics. In this research, a laser range sensor is usually used to capture the 3D shape data of an object. However, the shape data is just partial in one measurement because of the view limitation of the sensor. In order to reconstruct the whole shape of the object, therefore, it is necessary to restore the neighboring status of partial data that can compose whole object shape. This restoration process is a registration among 3D data. Rough registration can be achieved manually by a user through Graphic User Interface （GUI）. But, strictly speaking, it remains the gap between data, and the result is far from being accurate. Therefore, we need to develop an algorithm to automatically minimize the gap through quantative calculation.

The conventional registration algorithms are mostly concerned with the rigid-body transformation that determines the translation and rotation parameters between a pair of the corresponding 3D data. The algorithm solves the nonlinear equation to iteratively minimize the distance between a pair of corresponding 3D data with respect to the 6 unknown parameters （3 translation and 3 rotation parameters）. This iterative minimization framework is widely known as Iterative Closest Point （ICP）. The strategy of this framework can be categorized by four issues: registration ordering, matching unit, correspondence mapping, and error metric.

Registration ordering determines the correspondence timing of registration data. Sequential registration means that just one corresponding piece of data is considered for the registration data of interest, so it cause the local discrepancies because the registration error concerned with one pair of data is inherited and accumulated to the later registration pair. Simultaneous registration means all the corresponding pieces of data is considered, and therefore it can avoid the error accumulation, distributing them among all corresponding pieces. Matching unit determines the point sampling. All point matching uses all points of data. Feature point matching uses points satisfied with some condition, for example, high curvature points. Corresponding mapping determines how the correspondence is taken with respect to the points in the data. Error metric determines what kind of amount the error between the corresponding points is, namely, euclidian distance, color distance, and so on. To design the robust registration algorithm, we investigate the registration behavior in each issue. The result is that the simultaneous ordering, all point matching, closest point-to-point distance should be adopted for the robust registration.

The registration data usually includes the random measurement noise, which causes the wrong registration result. Hence we make effort to remove their effect. Our solution is to employ M-estimator according to the lorentz function. It can detect the outlier automatically, and align only the identical part of superimposing 3D shape data. To summarize the effectiveness of our proposed registration, we compare the registration result between ours and the conventional registration, and evaluate the estimation accuracy of registration parameter.

Our proposed registration has ever made a contribution especially to the preservation, analysis, and simulation of cultural heritage and assets to provide the archaeological knowledge.

As examples making full use of the merit of our robust simultaneous registration, we then show the shape comparison and analysis by superposing 3D data of ancient Chinese bronze mirrors, containing the decorated pattern, that have the sibling relationship. The mirrors were the offering from ancient China to Yamatai state. Yamatai state is the oldest state in Japan, but archaeologists still cannot specify its location, so this is one of the major controversies. It is said that Yamatai state distributed them to the local rulers, and that they made some replicas to distribute them to their followers again. The relationship among their mirrors is called "sibling relationship". To find the sibling relationship among mirrors, we need to pay attention to the cracks commonly inherited from the identical mold, and the spreading of cracks caused by the abrasion of the mold which is worn by continuous usage. Namely, sibling mirrors have the local shape difference. Our proposed registration can detect them accurately by regarding them as outliers. As an archaeological aim of this research, we can identify the manufacturing order among sibling mirrors, and then trace the distribution root back, and as a result, we may identify the location of Yamatai state.

Another example is the modeling of Fugoppe Cave and Ozuka Tumulus. Our simultaneous registration can align their whole shape data without local discrepancies. As our motivation of their modeling, we investigate how they were created. Fugoppe Cave and Ozuka Tumulus respectively has the carvings and paintings inside them, and archaeologists often ague over how ancient sculptors and painters created the carvings and paintings, since they usually imagine that it was dark inside them. The idea that ancient sculptors and painters used artificial light sources has been considered; however, this is generally suspicious since there is no firm evidence of that. In Fugoppe Cave, we consider that the natural light emitted by the sunlight could reach the interior of the cave. The reason is that the cave probably had the same entrance like the current entrance, which is large enough for the sunlight to pass through. Consider the Ozuka Tumulus that has mural using six pigments. If the tumulus was built with wall stones that were already decorated, we imagine that these stones were painted under the natural light of sunshine. But if the wall was decorated after it was built, we might suppose that the paintings were done under artificial light, such as light from taper. In the latter case, it is doubtful whether ancient artists could recognize their decoration by the taper light. Hence, we verify the possibility that they were decorated in the sunlight focusing on the shading and shadowing over the carvings and the color recognition of colors used for mural under sunlight or taper light.

Registration, including our proposed registration, is usually referred to as the rigid-body transformation. It can work just in completely the identical part. Hence, we consider the extended registration. Our extended framework enables the registration among 3D data that can deform each other through some known mathematical formula. For simple example, when comparing 3D data of two objects with the same shape but different size, we have to determine the scaling parameter in addition to the six translation and rotation parameters. It is also the case when aligning the data of a deformable object. When we replace a part of the range data, such as the cylinder, with a CAD primitive model in order to reduce the data amount or refine its shape, the parameter of the CAD primitive shape, the diameter and the height in cylinder case, should be determined from the measured data by fitting the primitive to the range data. In other words, these applications require determining more parameters concerned about shape parameter than just the 6 translation and rotation parameters, and our extended method can solve, in a unified manner, rigid-body transformation and shape parameter. In our framework, we assume that the shape changes are strictly defined by some parameterized formulation derived from the deformation mechanisms.

As this application, we estimate the shape parameter from the shape measurement data of mathematical plaster models made in Germany in the end of 19 century for educational purpose. They are a kind of cultural assets. Our motivation here was, due to no documentation about how to manufacture them, to identify the shape parameters when manufacturing their models, to estimate the deformation parameter by applying our extended registration framework to its range image and the data computed from the mathematical formula, in order to evaluate the manufacturing accuracy of the plaster model under the estimated parameter, and to remake the accurate model because the historian and the mathematicians are interested in the manufacturer's skill at those days. We actually estimate the shape parameter, and using them, we successfully reproduce the metallic mathematical replica model of the original plaster model with high accuracy. We evaluate the estimation accuracy of shape parameter.

Another application is to align the data obtained by the laser range sensor suspended beneath the balloon. We call this sensor floating laser range sensor, FLRS. In the present measurement system, a laser range sensor has to remain to be standing still on the tripod set up at the stable ground during a scanning. Since such fixed measurement takes much time and cost, we develop FLRS to provide the safer and more effective measurement of large object, such as cultural heritages. Different from the conventional 3D sensing system, this measurement result includes the distortion because of the movement during measurement, and we apply the extended registration to the distortion rectification, regarding the movement of FLRS during scanning as shape parameter. The proposed deformation registration pays attention to the significance of estimated parameter as well as the convergent registration result because the estimated parameter can be used for the system feedback. We also evaluate the estimation accuracy of FLRS movement, and the applicable limitation. We used this system for the modeling of Bayon Temple in Cambodia.

Finally, we summarize this thesis, with a conclusion, discussion and ideas about future work.

本論文は、レーザーレンジセンサなどにより得られる部分三次元形状データから、全体形状を構成する際に用いられる形状データ間の位置合わせとその拡張手法について提案している。本論文は8章からなり、第1章は本論文のテーマである三次元形状データの位置合わせの研究に関する背景及び目的について述べられている。第2章では関連研究について述べ、これらが目指す目的と手法について分類し、それらと本論文の研究との位置づけを記載している。第2章を受け、第3章では本論文の目的である初期位置やデータノイズにロバストな位置合わせアルゴリズムの設計について述べている。第4章、第5章では、ロバストな位置合わせ手法の考古学の分野への応用について記載している。第6章の冒頭では、この位置合わせ手法の拡張手法として、形状合わせも含んだ位置合わせ手法について提案している。後半ではこのフレームワークを用いて、石膏模型の計測データから形状パラメタを推定した研究について記している。第7章では同様に拡張フレームワークを用いて、浮遊型レーザーレンジセンサから得られた三次元形状データの歪み補正を行った研究について示している。第8章では、本論文のまとめと今後の課題について述べられている。以下で各章の内容について述べる。

近年、レーザーレンジセンサなどにより実世界に存在する物体の正確な三次元形状データが取得できるようになった。これらのデータは対象物の部分形状データであるため、同一形状を含む部分形状データの間でそれらの隣接関係を正しく復元する必要があり、この処理が三次元形状データの位置合わせである。実世界の物体の正確な形状モデリングは位置合わせの精度によるので、本論文では正確な位置合わせについて追求している。また、通常の位置合わせは回転・平行移動による剛体変換によって行われるが、既知の変形式を介することによって形状合わせが可能な場合は、形状合わせも含めた位置合わせを行えるような拡張手法について提案している。

第2章では、剛体変換による位置合わせの関連研究について述べている。これらを、データ間の位置合わせ順序・対応点サンプリング・対応の取り方・誤差指標・データノイズへの対処の5つの観点から分類し、正確な位置合わせ手法を確立するために各々の項目に対しどのような戦略を選択すべきかについて言及している。

第3章では第2章を受け、データ間の初期位置やデータノイズにロバストな位置合わせ手法の設計について述べている。この手法はデータ間の最近傍点間距離の重み付け二乗和を誤差関数とし、この誤差関数を位置合わせのパラメタに関して最小化する問題として実装されている。この手法は通常、計算コストが非常に高いため、無駄な計算を排除することによる効率化の工夫についても述べられている。また、実際に設計した位置合わせ手法の精度評価が先行研究と比較して行われており、これにより設計した手法のロバスト性が証明されている。

第4章、第5章では設計されたロバスト位置合わせ手法を考古学に応用し、文化財の形状解析や考古学における仮定をシミュレーションした例を示している。これらの研究は考古学における仮定の検証方法を提示し、実際に検証することで新たな知見が得られた例であり、情報考古学のはしりともいえる先駆的な応用例であるといえる。

第6章の冒頭では、ロバスト位置合わせ手法を拡張し、位置合わせと同時に既知の変形式を介して形状合わせを行う手法を提案している。本論文の提案手法の前提として、変形式が既知であり、かつ変形のメカニズムに厳密に規定されているものとしている。データ間の最近傍点間距離を最小化する剛体変換と同様に、提案手法では位置合わせと形状のパラメタに関して統一的な枠組みで誤差関数を最小化することを行っている。第6章の後半ではこの拡張フレームワークを用いて、文化財である石膏模型の計測データから形状パラメタを推定した研究について述べてある。また、推定した形状パラメタの精度を評価するため、合成データを用いたシミュレーション実験を行っており、これにより推定されたパラメタが正確であることが証明されている。

第7章では拡張フレームワークを用いて、気球につり下げられたレーザーレンジセンサの位置合わせを行った研究について述べられている。このセンサは通常のセンサと異なり、計測中に動いてしまうため、得られる形状データには歪みが発生している。提案手法ではこの歪みをセンサの計測中の運動を表すパラメタで定義することにより、歪みを補正しつつ位置合わせすることが可能となっている。後半では歪み補正パラメタの精度評価のため、合成データを用いたシミュレーション実験を行っており、その結果から正確に歪みが補正されていることが証明されている。歪み補正パラメタを推定することは計測中のセンサの運動を推定することであり、このパラメタがハードウェアのシステム構成にもフィードバックされている。こうして、提案手法により新たな三次元形状の計測システムを構築することが可能となった。