Geometric Shape Features Extraction Using a Steady State Partial Differential Equation System

A unified method for extracting geometric shape features from binary image data using a steady state partial differential equation (PDE) system as a boundary value problem is presented in this paper. The PDE and functions are formulated to extract the thickness, orientation, and skeleton simultaneously. The main advantages of the proposed method is that the orientation is defined without derivatives and thickness computation is not imposed a topological constraint on the target shape. A one-dimensional analytical solution is provided to validate the proposed method. In addition, two-dimensional numerical examples are presented to confirm the usefulness of the proposed method.Comment: 31 pages, 10 figure

A 3D Reconstruction Method of Bone Shape from Un-calibrated Radiographs

학위논문 (박사)-- 서울대학교 대학원 : 공과대학 컴퓨터공학부, 2019. 2. 이제희.전산화 단층촬영(CT)은 골형상을 3차원으로 시각화하여 직관적으로 병증을 진단할 수 있다는 장점이 있다. 하지만 과도한 방사선 노출로 인해 암과 같은 부작용의 우려가 있기 때문에 EOS®와 같이 저선량으로 3차원 재건을 할 수 있는 양판 촬영 시스템이 대안으로 제시되었다. 그러나 이 시스템은 도입 비용이 높기 때문에 일부 병원이나 국가에서는 사용이 어렵다. 본 논문에서는 어느 환경의 병원에서든지 3차원 진단을 할 수 있도록 단순 방사선 영상만으로 3차원 재건을 하는 방법을 제안한다. 이 방법은 별도의 금속 보정물체 없이 단순 영상을 자가 보정하는 것에 참신성이 있다. 기술적으로는 통계형상의 모델과 영상내의 윤곽선이 일치하도록 보정과 형상재건을 번갈아 반복하는 방법을 사용한다. 또한, 위의 기술을 의료환경에 적용하기 위해 제안한 재건 방법을 포함하는 모바일 어플리케이션을 제시한다. 사용자는 모니터 화면이나 필름을 모바일 기기의 카메라로 촬영하여 입력할 수 있으며, 그래프 컷 방법을 통해 윤곽선을 자동으로 추출하고 터치 입력으로 손쉽게 윤곽선을 수정할 수 있다. 우리는 우선적으로 대퇴골 재건 모바일 어플리케이션을 개발하였고, 대퇴골 염전각 진단에 있어서 훌륭한 신뢰도와 CT와 높은 동시 타당도를 갖는 것으로 측정되었다.Computed tomography (CT) provides benefits in accurate diagnosis of bone deformity. However, the potential adverse effects of radiation exposure in CT has become a concern. To reduce radiation dose while maintaining the accuracy of diagnosis, the EOS® system has been proposed to reconstruct 3D bony shapes from calibrated bi-planar(stereo) X-ray images. However, this system requires another apparatus in addition to the conventional radiographic system, and the cost for the device installation is high and the space occupied is substantial. Purchasing an EOS® system only for 3D reconstruction may hence not be appropriate in some hospitals or countries. In this thesis, we propose a new method to reconstruct 3D bone shape only from conventional radiograph so that hospitals in any environment can perform 3D diagnosis. It has novelty in self-calibrating conventional radiograph without metal calibration object. Technically, the calibration and reconstruction were optimized by minimizing the difference between the projected contour of the bone shape and the contour of the radiographic image. To apply the above technology to a medical situation, we present a mobile application including the reconstruction method. The user can easily input a printed film or a digital image displayed on a monitor screen by taking a photograph using an embedded camera. The application provides automatic contouring with a graph-cut algorithm, but also an intuitive touch interface for modifying the contour of a radiograph. We first developed a femur reconstruction, and the measurement of femoral anteversion with the mobile application showed excellent concurrent validity and reliability.제 1장 서론 1 제 2장 배경지식 5 2.1 전산화단층 촬영 3차원 재건 기법 5 2.2 임상에서 3차원 형상의 용도 8 2.3 양판(bi-planar) 3차원 재건 시스템 11 2.4 단순 방사선 촬영 15 제 3장 통계형상모델(statistical shape model) 17 3.1 연관 논문 18 3.2 형상의 표현방법 19 3.3 두 3차원 표면의 대응점 찾기 22 3.3.1 미분 기하 25 3.3.2 열 확산 방정식과 형상의 특징 35 3.4 주성분 분석 기법(PCA) 38 3.5 뼈의 통계형상 모델 생성 방법 40 3.5.1 매끄러운 표면의 특징을 찾는 방법 41 3.5.2 대퇴골의 점 분포 모델에 외적대응 방식의 적용 42 3.5.3 거골과 대퇴골의 점 분포 모델에 내적 대응 방식의 적용 43 제4장 방사선 영상에서 골 윤곽선의 추출 48 4.1 연관 논문 49 4.2 그래프 절단을 이용한 영상 분할 49 4.2.1 사용자 입력과 최소 절단을 사용하는 방법 50 4.2.2 정규화 절단(Normalized cut) 51 4.3 골 경계선 추출 방법 53 4.3.1 그래프 최소절단 방식 영상 분할 54 4.3.2 지정한 점을 지나는 최소 가중치 경로 방식 54 4.3.3 방향성 그래프의 경로찾기 방식 58 제5장 단순 방사선 영상의 자가보정 65 5.1 Perspective N Points(PNP)문제의 반복적 해 65 5.2 통계형상모델을 보정물체로 사용하는 자가보정 방법 68 제6장 단순 방사선 3차원 재건 시스템 71 6.1 통계형상의 변형 72 6.2 볼륨 렌더링을 이용한 가상 엑스레이 생성 73 6.3 대퇴골 재건 시스템 75 6.3.1 자가보정 정확도 실험 76 6.3.2 재건 형상의 거리 오차 78 6.4 대퇴골 재건 모바일 앱 79 6.5 경비로 재건 모바일 앱 83 제7장 결론 86 Abstract 96Docto

Geometric Algorithms for Modeling Plant Roots from Images

Roots, considered as the ”hidden half of the plant”, are essential to a plant’s health and pro- ductivity. Understanding root architecture has the potential to enhance efforts towards im- proving crop yield. In this dissertation we develop geometric approaches to non-destructively characterize the full architecture of the root system from 3D imaging while making com- putational advances in topological optimization. First, we develop a global optimization algorithm to remove topological noise, with applications in both root imaging and com- puter graphics. Second, we use our topology simplification algorithm, other methods from computer graphics, and customized algorithms to develop a high-throughput pipeline for computing hierarchy and fine-grained architectural traits from 3D imaging of maize roots. Finally, we develop an algorithm for consistently simplifying the topology of nested shapes, with a motivating application in temporal root system analysis. Along the way, we con- tribute to the computer graphics community a pair of topological simplification algorithms both for repairing a single 3D shape and for repairing a sequence of nested shapes

Curve Skeleton and Moments of Area Supported Beam Parametrization in Multi-Objective Compliance Structural Optimization

This work addresses the end-to-end virtual automation of structural optimization up to the derivation of a parametric geometry model that can be used for application areas such as additive manufacturing or the verification of the structural optimization result with the finite element method. A holistic design in structural optimization can be achieved with the weighted sum method, which can be automatically parameterized with curve skeletonization and cross-section regression to virtually verify the result and control the local size for additive manufacturing. is investigated in general. In this paper, a holistic design is understood as a design that considers various compliances as an objective function. This parameterization uses the automated determination of beam parameters by so-called curve skeletonization with subsequent cross-section shape parameter estimation based on moments of area, especially for multi-objective optimized shapes. An essential contribution is the linking of the parameterization with the results of the structural optimization, e.g., to include properties such as boundary conditions, load conditions, sensitivities or even density variables in the curve skeleton parameterization. The parameterization focuses on guiding the skeletonization based on the information provided by the optimization and the finite element model. In addition, the cross-section detection considers circular, elliptical, and tensor product spline cross-sections that can be applied to various shape descriptors such as convolutional surfaces, subdivision surfaces, or constructive solid geometry. The shape parameters of these cross-sections are estimated using stiffness distributions, moments of area of 2D images, and convolutional neural networks with a tailored loss function to moments of area. Each final geometry is designed by extruding the cross-section along the appropriate curve segment of the beam and joining it to other beams by using only unification operations. The focus of multi-objective structural optimization considering 1D, 2D and 3D elements is on cases that can be modeled using equations by the Poisson equation and linear elasticity. This enables the development of designs in application areas such as thermal conduction, electrostatics, magnetostatics, potential flow, linear elasticity and diffusion, which can be optimized in combination or individually. Due to the simplicity of the cases defined by the Poisson equation, no experts are required, so that many conceptual designs can be generated and reconstructed by ordinary users with little effort. Specifically for 1D elements, a element stiffness matrices for tensor product spline cross-sections are derived, which can be used to optimize a variety of lattice structures and automatically convert them into free-form surfaces. For 2D elements, non-local trigonometric interpolation functions are used, which should significantly increase interpretability of the density distribution. To further improve the optimization, a parameter-free mesh deformation is embedded so that the compliances can be further reduced by locally shifting the node positions. Finally, the proposed end-to-end optimization and parameterization is applied to verify a linear elasto-static optimization result for and to satisfy local size constraint for the manufacturing with selective laser melting of a heat transfer optimization result for a heat sink of a CPU. For the elasto-static case, the parameterization is adjusted until a certain criterion (displacement) is satisfied, while for the heat transfer case, the manufacturing constraints are satisfied by automatically changing the local size with the proposed parameterization. This heat sink is then manufactured without manual adjustment and experimentally validated to limit the temperature of a CPU to a certain level.:TABLE OF CONTENT III I LIST OF ABBREVIATIONS V II LIST OF SYMBOLS V III LIST OF FIGURES XIII IV LIST OF TABLES XVIII 1. INTRODUCTION 1 1.1 RESEARCH DESIGN AND MOTIVATION 6 1.2 RESEARCH THESES AND CHAPTER OVERVIEW 9 2. PRELIMINARIES OF TOPOLOGY OPTIMIZATION 12 2.1 MATERIAL INTERPOLATION 16 2.2 TOPOLOGY OPTIMIZATION WITH PARAMETER-FREE SHAPE OPTIMIZATION 17 2.3 MULTI-OBJECTIVE TOPOLOGY OPTIMIZATION WITH THE WEIGHTED SUM METHOD 18 3. SIMULTANEOUS SIZE, TOPOLOGY AND PARAMETER-FREE SHAPE OPTIMIZATION OF WIREFRAMES WITH B-SPLINE CROSS-SECTIONS 21 3.1 FUNDAMENTALS IN WIREFRAME OPTIMIZATION 22 3.2 SIZE AND TOPOLOGY OPTIMIZATION WITH PERIODIC B-SPLINE CROSS-SECTIONS 27 3.3 PARAMETER-FREE SHAPE OPTIMIZATION EMBEDDED IN SIZE OPTIMIZATION 32 3.4 WEIGHTED SUM SIZE AND TOPOLOGY OPTIMIZATION 36 3.5 CROSS-SECTION COMPARISON 39 4. NON-LOCAL TRIGONOMETRIC INTERPOLATION IN TOPOLOGY OPTIMIZATION 41 4.1 FUNDAMENTALS IN MATERIAL INTERPOLATIONS 43 4.2 NON-LOCAL TRIGONOMETRIC SHAPE FUNCTIONS 45 4.3 NON-LOCAL PARAMETER-FREE SHAPE OPTIMIZATION WITH TRIGONOMETRIC SHAPE FUNCTIONS 49 4.4 NON-LOCAL AND PARAMETER-FREE MULTI-OBJECTIVE TOPOLOGY OPTIMIZATION 54 5. FUNDAMENTALS IN SKELETON GUIDED SHAPE PARAMETRIZATION IN TOPOLOGY OPTIMIZATION 58 5.1 SKELETONIZATION IN TOPOLOGY OPTIMIZATION 61 5.2 CROSS-SECTION RECOGNITION FOR IMAGES 66 5.3 SUBDIVISION SURFACES 67 5.4 CONVOLUTIONAL SURFACES WITH META BALL KERNEL 71 5.5 CONSTRUCTIVE SOLID GEOMETRY 73 6. CURVE SKELETON GUIDED BEAM PARAMETRIZATION OF TOPOLOGY OPTIMIZATION RESULTS 75 6.1 FUNDAMENTALS IN SKELETON SUPPORTED RECONSTRUCTION 76 6.2 SUBDIVISION SURFACE PARAMETRIZATION WITH PERIODIC B-SPLINE CROSS-SECTIONS 78 6.3 CURVE SKELETONIZATION TAILORED TO TOPOLOGY OPTIMIZATION WITH PRE-PROCESSING 82 6.4 SURFACE RECONSTRUCTION USING LOCAL STIFFNESS DISTRIBUTION 86 7. CROSS-SECTION SHAPE PARAMETRIZATION FOR PERIODIC B-SPLINES 96 7.1 PRELIMINARIES IN B-SPLINE CONTROL GRID ESTIMATION 97 7.2 CROSS-SECTION EXTRACTION OF 2D IMAGES 101 7.3 TENSOR SPLINE PARAMETRIZATION WITH MOMENTS OF AREA 105 7.4 B-SPLINE PARAMETRIZATION WITH MOMENTS OF AREA GUIDED CONVOLUTIONAL NEURAL NETWORK 110 8. FULLY AUTOMATED COMPLIANCE OPTIMIZATION AND CURVE-SKELETON PARAMETRIZATION FOR A CPU HEAT SINK WITH SIZE CONTROL FOR SLM 115 8.1 AUTOMATED 1D THERMAL COMPLIANCE MINIMIZATION, CONSTRAINED SURFACE RECONSTRUCTION AND ADDITIVE MANUFACTURING 118 8.2 AUTOMATED 2D THERMAL COMPLIANCE MINIMIZATION, CONSTRAINT SURFACE RECONSTRUCTION AND ADDITIVE MANUFACTURING 120 8.3 USING THE HEAT SINK PROTOTYPES COOLING A CPU 123 9. CONCLUSION 127 10. OUTLOOK 131 LITERATURE 133 APPENDIX 147 A PREVIOUS STUDIES 147 B CROSS-SECTION PROPERTIES 149 C CASE STUDIES FOR THE CROSS-SECTION PARAMETRIZATION 155 D EXPERIMENTAL SETUP 15