An Orbifold Framework for Point Scan = 점별 스캔을 위한 오비폴드 프레임워크

다국어 초록 (Multilingual Abstract)

This dissertation introduces a novel orbifold-based framework that resolves the
three main challenges in point scanning system such as phase shifts, mechanical-
coupling errors, and optimal frequency selection. Point scan systems, which
trace Lissajous or Rosette paths are increasingly used in LiDAR and confocal
microscopy.
Scanning paths are modeled as geodesics on a Clifford torus in S3; lifting
the dynamics to the *2222 and *22 orbifolds exposes differential geometric tools
and fits naturally into a pull-back push-forward structure.
At the heart of the framework is the SO(4) action on the torus. Phase shifts
appear as internal T 2 automorphisms and mechanical coupling error correspond
to rotations in the Grassmannian Gr(2, R4). Gauge directions are quotiented
out, leaving a base space for reconstruction.
The orbifold framework induces a general folded error functional, denoted
E(g), defined on SO(4) where g represents a system perturbation. This func-
tional quantifies internal inconsistencies within the lifted observed image by
measuring deviations from perfect symmetry across fibers generated by g. Cru-
cially, E(g) is proven to be locally convex along non-gauge directions. That
result shows that gradient descent can find the optimal point and solve the
problem.
Furthermore, the theorems established for this SO(4) framework naturally
extend to higher-dimensional frameworks, such as multiple frequency compo
nents or more complex harmonic distortions

국문 초록 (Abstract)

이 학위 논문은 위상 밀림과 기계적 결합 오류, 최적화 되지 않은 주파수 쌍의 점별
스캔의 세 문제를 해결하는 통일된 기하적 모델, 즉 새로운 오비폴드 기반의 프레
임워크를 제안한다. 이 점별 스캔은 리사주나 로제트 경로를 이용하며, 라이다부터
공초점 현미경 등에 쓰인다. 이 프레임은 스캔 경로를 S3 속 클리포드 토러스의
지오데식으로 해석하며, *2222와 *22 오비폴드의 동역학을 미분기하학적 툴을 이
용하여 해석하며, 자연스럽게 유도되는 풀백과 푸시 포워드 구조로 해석한다.
이 논문의 핵심은 토러스의 SO(4) 액션을 생각하는 것이다. 위상 밀림 현상은
토러스 내부의 자기동형사상으로 해석할 수 있으며, 기계적 결합 오류는 그라스
마니안 Gr(2,R4) 의 회전들로 생각할 수 있다. 이 방향의 운동들만이 이미지를
움직이며, 자연스럽게 주번들 구조를 형성한다. 게이지 방향은 나눠 없어지며, 복
원 가능한 주 공간을 남긴다.
이 오비폴드 프레임워크는 SO(4) 요동이 만들어 내는, E(g)로 표현되는 겹침
오류 범함수를 자연스럽게 생성한다. 이 범함수는 관찰된 이미지 내부의 불일치를
측정해서 요동 g 에 의해 만들어지는 대칭을 추측한다. 우리는 이 함수가 비-게이
지 방향으로는 국소적으로 볼록임을 증명하였다. 이를 통해 우리는 경사하강법을
적용하여 최적 해를 도출하여 문제를 해결할 수 있음을 밝혔다. 더불어, SO(4)
프레임워크에서 유도된 이 정리들은, 자연스럽게 배수파 오차들과 더 복잡한 조화
왜곡들을 해석할 수 있는 고차원 프레임워크로 확장될 수 있음을 보였다

목차 (Table of Contents)

• Abstract i
• I Introduction and Motivation 1
• Chapter 1 Introduction 2
• 1.1 What is the scanning 2
• 1.1.1 Scanning Paradigms 2
• 1.1.2 Lissajous Trajectories 3
• 1.1.3 Rosette Trajectories 4
• 1.1.4 Fill Factor: Quantifying Coverage 4
• 1.1.5 Key points 5
• 1.2 Industrial Context 6
• 1.2.1 Biomedical Imaging 6
• 1.2.2 LiDAR and Remote Sensing 7
• 1.2.3 Surface Metrology and Atomic-Force Microscopy 7
• 1.2.4 Cross-Domain Metric Summary 7
• 1.3 Three Core Problems 8
• 1.3.1 Overview 8
• 1.3.2 Phase Control and Mechanical Error 9
• 1.3.3 Frequency Choice and Fill Factor 9
• 1.3.4 Coupled Impact and Thesis Objective 10
• 1.4 Research Goals 11
• 1.4.1 Statement of Purpose 11
• 1.4.2 Goal AGuessing phase and Error Calibration 11
• 1.4.3 Goal BOptimal Frequency Choice 12
• 1.5 Dissertation Structure 12
• Chapter 2 Contribution 14
• 2.1 Folding framework Paradigm 14
• 2.2 Unified Solutions to Three Core Problems 16
• 2.2.1 Master Error Functional 17
• 2.2.2 Phase Search 17
• 2.2.3 Mechanical error Calibration 18
• 2.2.4 Frequency Selection 19
• 2.3 High-Dimensional Extension 19
• 2.3.1 Multi-harmonic error the SO(4m) Lift 19
• 2.3.2 n-Dimensional Folding Confocal LiDAR Example 20
• 2.4 Summary of Contributions 22
• II Mathematical Foundation 24
• Chapter 3 An Orbifold Framework 25
• 3.1 Definition of the scanning system 25
• 3.2 Curves and the Clifford torus 27
• 3.2.1 The Clifford Torus 27
• 3.2.2 The Curve Geodesic Transformation Map 27
• 3.2.3 The Projection Theorem for Lissajous Curves 28
• 3.2.4 The Projection Theorem for Rosette Patterns 28
• 3.3 PhaseSpace and Orbifold covering 29
• 3.3.1 Phase map and lifting of an image 29
• 3.3.2 Orbifold charts and good-ness 30
• 3.3.3 The regular 4fold covering 30
• 3.3.4 BV stability of the lift 31
• 3.3.5 Symmetryaveraged error functional 31
• 3.4 SO(4) Motion in Phase Space 32
• 3.4.1 Six infinitesimal generators 32
• 3.4.2 Coincidence multiplicity for arbitrary g 32
• 3.4.3 Generalized error functional 33
• 3.4.4 Generatorwise behavior 33
• 3.4.5 Outlook to Chapter 4 (Optimization) 33
• 3.5 A Grassmannian Viewpoint 34
• 3.5.1 Gauge rotations generated by L13 and L24 34
• 3.5.2 The Grassmann manifold Gr(2, R4) 34
• 3.5.3 Error functional reduced to the Grassmannian 35
• 3.5.4 Geometric intuition & Optimization message 35
• Chapter 4 Problem Setting 37
• 4.1 Formal Setting 37
• 4.1.1 Domain the Clifford torus T 37
• 4.1.2 Sheet (folding) structure 38
• 4.1.3 Function space BV(T) 38
• 4.1.4 Group action small rotations in (SO(4)) 38
• 4.1.5 Folding-aware error functional 39
• 4.1.6 Local convexity assumption 39
• 4.2 Phase Shifting (Phase Alignment) 40
• 4.2.1 Folding-aware error restricted to H 40
• 4.2.2 Example I Constant image (fully symmetric) 41
• 4.2.3 Example II One-axis step image (only x varies) 41
• 4.2.4 Example III Two-axis quadrant image (x and y both
• vary) 42
• 4.3 Mechanical Coupling Error Correction 43
• 4.3.1 Lie-algebra decomposition 43
• 4.3.2 Grassmann manifold as the effective space 43
• 4.3.3 Minimization problem on Gr(2, R4) 43
• 4.3.4 Strict-convexity remark 44
• 4.3.5 Summary 44
• 4.4 Frequency Optimization 45
• 4.4.1 Why we optimize on the phase torus, not in image space 45
• 4.4.2 A coverage metric: the maximal gap 45
• 4.4.3 One-dimensional recall: the Three-Gap Theorem 45
• 4.4.4 Gapminimizing objective 46
• 4.4.5 Why smaller gaps imply better fill factor 46
• 4.4.6 Choosing ω 47
• III Main Theorem and Applications 48
• Chapter 5 Convexity Theorem 49
• 5.1 Error Functional Setting and Lemmas 49
• 5.1.1 Standing hypotheses and notation 49
• 5.1.2 Folded squarederror functional 50
• 5.1.3 Change locus of the folding multiplicity 50
• 5.2 Global C2 Regularity of the Folded Functional 52
• 5.2.1 Functional Definition and Assumptions 52
• 5.2.2 Regularity Theorem 56
• 5.2.3 Example: Clifford torus with SO(4) 60
• 5.3 Apply to our framework 61
• 5.3.1 Lifted data on the Clifford torus 61
• 5.3.2 Folding fiber 62
• 5.3.3 Reference embedding (zero phase) 62
• 5.3.4 Liealgebra split of SO(4) 62
• 5.3.5 Foldingaware error functional 63
• 5.4 Main Convexity Theorems 63
• 5.4.1 Global Minimum Theorem 63
• 5.4.2 Gauge Invariant Theorem 67
• 5.4.3 Locally Convex Theorem 69
• 5.4.4 Corollaries of Convexity Theorem 72
• Chapter 6 Application to Main Problems 75
• 6.1 Motivation and Problem Statement 75
• 6.2 Sensor-Free Phase Estimation 77
• 6.2.1 Problem statement 77
• 6.2.2 Error functional 77
• 6.2.3 Convexity and uniqueness 78
• 6.3 Mechanical Coupling Calibration 79
• 6.3.1 Problem statement 79
• 6.3.2 Error functional 80
• 6.3.3 Convexity and uniqueness 81
• 6.3.4 Gauge fixing and effective degrees of freedom 82
• 6.4 Frequency Optimization 83
• 6.4.1 Problem statement 83
• 6.4.2 ThreeGap formula 84
• 6.4.3 Bounds via continued fractions 85
• 6.4.4 Optimal largest gap 86
• IV High dimensional Extension 88
• Chapter 7 High Frequency Extension 89
• 7.1 Motivation and Problem Statement 89
• 7.2 Setup 91
• 7.2.1 Torus Embedding and Coordinates 91
• 7.2.2 Rotations Acting on the Scan Path 91
• 7.2.3 GaugeEffective Decomposition 91
• 7.2.4 The Generalized Error Functional Structure 92
• 7.3 Gauge actions for the SO(4m) Action 92
• 7.3.1 Identification of the Gauge Lie Algebra 93
• 7.3.2 Effective Subspace 94
• 7.4 Convexity Analysis for SO(4m) 94
• 7.4.1 Main Convexity Theorem for SO(4m) 94
• 7.4.2 Consequences for Optimization 96
• 7.5 Expressivity of the SO(4m) Geodesic Model 97
• 7.5.1 Harmonics as orthogonal 2planes 97
• 7.5.2 Approximation theorem 98
• 7.5.3 Dimensional bookkeeping 99
• Chapter 8 k-axis extension 100
• 8.1 Motivation and Problem Statement 100
• 8.2 Mathematical Setting for the k-Dimensional Case 102
• 8.2.1 Torus Embedding and Coordinates 102
• 8.2.2 Rotation Group Action 102
• 8.2.3 Gauge and Effective Subspaces 102
• 8.2.4 Notational Summary 103
• 8.3 Convexity of the Error Functional on fGr(k, R2k) 103
• 8.3.1 Error Functional Restated and C2 Regularity 103
• 8.3.2 First and Second Variations (Hessian) 104
• 8.3.3 Main Theorem 106
• 8.3.4 Implications 107
• 8.4 Frequency Optimization 107
• 8.4.1 Why choose the frequency vector carefully 107
• 8.4.2 Small denominators and the Diophantine condition 108
• 8.4.3 No single "golden ratio'' for k 2: Badly Approximable (BA) vectors 108
• 8.4.4 Practical heuristics for picking a BA vector 108
• 8.5 Summary 109
• V Limitation and Future-work 110
• Chapter 9 Noise Robustness 111
• 9.1 Motivation 111
• 9.2 Type A Noise: Pixel-Value Perturbation 112
• 9.2.1 Set-up and Notation 112
• 9.2.2 Expected Discrete TV of White Noise 113
• 9.2.3 Relating sigma to PSNR 113
• 9.2.4 Effect on the Convexity Margin 114
• 9.2.5 Coarse-to-Fine Remedy 114
• 9.3 Type B Noise: DomainShift Perturbations 114
• 9.3.1 Model 114
• 9.3.2 Meansquare error of the displacement 115
• 9.3.3 Firstorder approximation 116
• 9.3.4 Displacementdefined PSNR 116
• 9.3.5 Erosion of the convexity margin 116
• 9.3.6 Critical displacementPSNR 116
• 9.3.7 Mitigation in practice 117
• Chapter 10 Future work 118
• 10.1 PhaseSpace Sampling and PushForward Reconstruction 118
• 10.1.1 Data Model 118
• 10.1.2 Geodesic Sampling Gaps 118