- •Preface
- •Biological Vision Systems
- •Visual Representations from Paintings to Photographs
- •Computer Vision
- •The Limitations of Standard 2D Images
- •3D Imaging, Analysis and Applications
- •Book Objective and Content
- •Acknowledgements
- •Contents
- •Contributors
- •2.1 Introduction
- •Chapter Outline
- •2.2 An Overview of Passive 3D Imaging Systems
- •2.2.1 Multiple View Approaches
- •2.2.2 Single View Approaches
- •2.3 Camera Modeling
- •2.3.1 Homogeneous Coordinates
- •2.3.2 Perspective Projection Camera Model
- •2.3.2.1 Camera Modeling: The Coordinate Transformation
- •2.3.2.2 Camera Modeling: Perspective Projection
- •2.3.2.3 Camera Modeling: Image Sampling
- •2.3.2.4 Camera Modeling: Concatenating the Projective Mappings
- •2.3.3 Radial Distortion
- •2.4 Camera Calibration
- •2.4.1 Estimation of a Scene-to-Image Planar Homography
- •2.4.2 Basic Calibration
- •2.4.3 Refined Calibration
- •2.4.4 Calibration of a Stereo Rig
- •2.5 Two-View Geometry
- •2.5.1 Epipolar Geometry
- •2.5.2 Essential and Fundamental Matrices
- •2.5.3 The Fundamental Matrix for Pure Translation
- •2.5.4 Computation of the Fundamental Matrix
- •2.5.5 Two Views Separated by a Pure Rotation
- •2.5.6 Two Views of a Planar Scene
- •2.6 Rectification
- •2.6.1 Rectification with Calibration Information
- •2.6.2 Rectification Without Calibration Information
- •2.7 Finding Correspondences
- •2.7.1 Correlation-Based Methods
- •2.7.2 Feature-Based Methods
- •2.8 3D Reconstruction
- •2.8.1 Stereo
- •2.8.1.1 Dense Stereo Matching
- •2.8.1.2 Triangulation
- •2.8.2 Structure from Motion
- •2.9 Passive Multiple-View 3D Imaging Systems
- •2.9.1 Stereo Cameras
- •2.9.2 3D Modeling
- •2.9.3 Mobile Robot Localization and Mapping
- •2.10 Passive Versus Active 3D Imaging Systems
- •2.11 Concluding Remarks
- •2.12 Further Reading
- •2.13 Questions
- •2.14 Exercises
- •References
- •3.1 Introduction
- •3.1.1 Historical Context
- •3.1.2 Basic Measurement Principles
- •3.1.3 Active Triangulation-Based Methods
- •3.1.4 Chapter Outline
- •3.2 Spot Scanners
- •3.2.1 Spot Position Detection
- •3.3 Stripe Scanners
- •3.3.1 Camera Model
- •3.3.2 Sheet-of-Light Projector Model
- •3.3.3 Triangulation for Stripe Scanners
- •3.4 Area-Based Structured Light Systems
- •3.4.1 Gray Code Methods
- •3.4.1.1 Decoding of Binary Fringe-Based Codes
- •3.4.1.2 Advantage of the Gray Code
- •3.4.2 Phase Shift Methods
- •3.4.2.1 Removing the Phase Ambiguity
- •3.4.3 Triangulation for a Structured Light System
- •3.5 System Calibration
- •3.6 Measurement Uncertainty
- •3.6.1 Uncertainty Related to the Phase Shift Algorithm
- •3.6.2 Uncertainty Related to Intrinsic Parameters
- •3.6.3 Uncertainty Related to Extrinsic Parameters
- •3.6.4 Uncertainty as a Design Tool
- •3.7 Experimental Characterization of 3D Imaging Systems
- •3.7.1 Low-Level Characterization
- •3.7.2 System-Level Characterization
- •3.7.3 Characterization of Errors Caused by Surface Properties
- •3.7.4 Application-Based Characterization
- •3.8 Selected Advanced Topics
- •3.8.1 Thin Lens Equation
- •3.8.2 Depth of Field
- •3.8.3 Scheimpflug Condition
- •3.8.4 Speckle and Uncertainty
- •3.8.5 Laser Depth of Field
- •3.8.6 Lateral Resolution
- •3.9 Research Challenges
- •3.10 Concluding Remarks
- •3.11 Further Reading
- •3.12 Questions
- •3.13 Exercises
- •References
- •4.1 Introduction
- •Chapter Outline
- •4.2 Representation of 3D Data
- •4.2.1 Raw Data
- •4.2.1.1 Point Cloud
- •4.2.1.2 Structured Point Cloud
- •4.2.1.3 Depth Maps and Range Images
- •4.2.1.4 Needle map
- •4.2.1.5 Polygon Soup
- •4.2.2 Surface Representations
- •4.2.2.1 Triangular Mesh
- •4.2.2.2 Quadrilateral Mesh
- •4.2.2.3 Subdivision Surfaces
- •4.2.2.4 Morphable Model
- •4.2.2.5 Implicit Surface
- •4.2.2.6 Parametric Surface
- •4.2.2.7 Comparison of Surface Representations
- •4.2.3 Solid-Based Representations
- •4.2.3.1 Voxels
- •4.2.3.3 Binary Space Partitioning
- •4.2.3.4 Constructive Solid Geometry
- •4.2.3.5 Boundary Representations
- •4.2.4 Summary of Solid-Based Representations
- •4.3 Polygon Meshes
- •4.3.1 Mesh Storage
- •4.3.2 Mesh Data Structures
- •4.3.2.1 Halfedge Structure
- •4.4 Subdivision Surfaces
- •4.4.1 Doo-Sabin Scheme
- •4.4.2 Catmull-Clark Scheme
- •4.4.3 Loop Scheme
- •4.5 Local Differential Properties
- •4.5.1 Surface Normals
- •4.5.2 Differential Coordinates and the Mesh Laplacian
- •4.6 Compression and Levels of Detail
- •4.6.1 Mesh Simplification
- •4.6.1.1 Edge Collapse
- •4.6.1.2 Quadric Error Metric
- •4.6.2 QEM Simplification Summary
- •4.6.3 Surface Simplification Results
- •4.7 Visualization
- •4.8 Research Challenges
- •4.9 Concluding Remarks
- •4.10 Further Reading
- •4.11 Questions
- •4.12 Exercises
- •References
- •1.1 Introduction
- •Chapter Outline
- •1.2 A Historical Perspective on 3D Imaging
- •1.2.1 Image Formation and Image Capture
- •1.2.2 Binocular Perception of Depth
- •1.2.3 Stereoscopic Displays
- •1.3 The Development of Computer Vision
- •1.3.1 Further Reading in Computer Vision
- •1.4 Acquisition Techniques for 3D Imaging
- •1.4.1 Passive 3D Imaging
- •1.4.2 Active 3D Imaging
- •1.4.3 Passive Stereo Versus Active Stereo Imaging
- •1.5 Twelve Milestones in 3D Imaging and Shape Analysis
- •1.5.1 Active 3D Imaging: An Early Optical Triangulation System
- •1.5.2 Passive 3D Imaging: An Early Stereo System
- •1.5.3 Passive 3D Imaging: The Essential Matrix
- •1.5.4 Model Fitting: The RANSAC Approach to Feature Correspondence Analysis
- •1.5.5 Active 3D Imaging: Advances in Scanning Geometries
- •1.5.6 3D Registration: Rigid Transformation Estimation from 3D Correspondences
- •1.5.7 3D Registration: Iterative Closest Points
- •1.5.9 3D Local Shape Descriptors: Spin Images
- •1.5.10 Passive 3D Imaging: Flexible Camera Calibration
- •1.5.11 3D Shape Matching: Heat Kernel Signatures
- •1.6 Applications of 3D Imaging
- •1.7 Book Outline
- •1.7.1 Part I: 3D Imaging and Shape Representation
- •1.7.2 Part II: 3D Shape Analysis and Processing
- •1.7.3 Part III: 3D Imaging Applications
- •References
- •5.1 Introduction
- •5.1.1 Applications
- •5.1.2 Chapter Outline
- •5.2 Mathematical Background
- •5.2.1 Differential Geometry
- •5.2.2 Curvature of Two-Dimensional Surfaces
- •5.2.3 Discrete Differential Geometry
- •5.2.4 Diffusion Geometry
- •5.2.5 Discrete Diffusion Geometry
- •5.3 Feature Detectors
- •5.3.1 A Taxonomy
- •5.3.2 Harris 3D
- •5.3.3 Mesh DOG
- •5.3.4 Salient Features
- •5.3.5 Heat Kernel Features
- •5.3.6 Topological Features
- •5.3.7 Maximally Stable Components
- •5.3.8 Benchmarks
- •5.4 Feature Descriptors
- •5.4.1 A Taxonomy
- •5.4.2 Curvature-Based Descriptors (HK and SC)
- •5.4.3 Spin Images
- •5.4.4 Shape Context
- •5.4.5 Integral Volume Descriptor
- •5.4.6 Mesh Histogram of Gradients (HOG)
- •5.4.7 Heat Kernel Signature (HKS)
- •5.4.8 Scale-Invariant Heat Kernel Signature (SI-HKS)
- •5.4.9 Color Heat Kernel Signature (CHKS)
- •5.4.10 Volumetric Heat Kernel Signature (VHKS)
- •5.5 Research Challenges
- •5.6 Conclusions
- •5.7 Further Reading
- •5.8 Questions
- •5.9 Exercises
- •References
- •6.1 Introduction
- •Chapter Outline
- •6.2 Registration of Two Views
- •6.2.1 Problem Statement
- •6.2.2 The Iterative Closest Points (ICP) Algorithm
- •6.2.3 ICP Extensions
- •6.2.3.1 Techniques for Pre-alignment
- •Global Approaches
- •Local Approaches
- •6.2.3.2 Techniques for Improving Speed
- •Subsampling
- •Closest Point Computation
- •Distance Formulation
- •6.2.3.3 Techniques for Improving Accuracy
- •Outlier Rejection
- •Additional Information
- •Probabilistic Methods
- •6.3 Advanced Techniques
- •6.3.1 Registration of More than Two Views
- •Reducing Error Accumulation
- •Automating Registration
- •6.3.2 Registration in Cluttered Scenes
- •Point Signatures
- •Matching Methods
- •6.3.3 Deformable Registration
- •Methods Based on General Optimization Techniques
- •Probabilistic Methods
- •6.3.4 Machine Learning Techniques
- •Improving the Matching
- •Object Detection
- •6.4 Quantitative Performance Evaluation
- •6.5 Case Study 1: Pairwise Alignment with Outlier Rejection
- •6.6 Case Study 2: ICP with Levenberg-Marquardt
- •6.6.1 The LM-ICP Method
- •6.6.2 Computing the Derivatives
- •6.6.3 The Case of Quaternions
- •6.6.4 Summary of the LM-ICP Algorithm
- •6.6.5 Results and Discussion
- •6.7 Case Study 3: Deformable ICP with Levenberg-Marquardt
- •6.7.1 Surface Representation
- •6.7.2 Cost Function
- •Data Term: Global Surface Attraction
- •Data Term: Boundary Attraction
- •Penalty Term: Spatial Smoothness
- •Penalty Term: Temporal Smoothness
- •6.7.3 Minimization Procedure
- •6.7.4 Summary of the Algorithm
- •6.7.5 Experiments
- •6.8 Research Challenges
- •6.9 Concluding Remarks
- •6.10 Further Reading
- •6.11 Questions
- •6.12 Exercises
- •References
- •7.1 Introduction
- •7.1.1 Retrieval and Recognition Evaluation
- •7.1.2 Chapter Outline
- •7.2 Literature Review
- •7.3 3D Shape Retrieval Techniques
- •7.3.1 Depth-Buffer Descriptor
- •7.3.1.1 Computing the 2D Projections
- •7.3.1.2 Obtaining the Feature Vector
- •7.3.1.3 Evaluation
- •7.3.1.4 Complexity Analysis
- •7.3.2 Spin Images for Object Recognition
- •7.3.2.1 Matching
- •7.3.2.2 Evaluation
- •7.3.2.3 Complexity Analysis
- •7.3.3 Salient Spectral Geometric Features
- •7.3.3.1 Feature Points Detection
- •7.3.3.2 Local Descriptors
- •7.3.3.3 Shape Matching
- •7.3.3.4 Evaluation
- •7.3.3.5 Complexity Analysis
- •7.3.4 Heat Kernel Signatures
- •7.3.4.1 Evaluation
- •7.3.4.2 Complexity Analysis
- •7.4 Research Challenges
- •7.5 Concluding Remarks
- •7.6 Further Reading
- •7.7 Questions
- •7.8 Exercises
- •References
- •8.1 Introduction
- •Chapter Outline
- •8.2 3D Face Scan Representation and Visualization
- •8.3 3D Face Datasets
- •8.3.1 FRGC v2 3D Face Dataset
- •8.3.2 The Bosphorus Dataset
- •8.4 3D Face Recognition Evaluation
- •8.4.1 Face Verification
- •8.4.2 Face Identification
- •8.5 Processing Stages in 3D Face Recognition
- •8.5.1 Face Detection and Segmentation
- •8.5.2 Removal of Spikes
- •8.5.3 Filling of Holes and Missing Data
- •8.5.4 Removal of Noise
- •8.5.5 Fiducial Point Localization and Pose Correction
- •8.5.6 Spatial Resampling
- •8.5.7 Feature Extraction on Facial Surfaces
- •8.5.8 Classifiers for 3D Face Matching
- •8.6 ICP-Based 3D Face Recognition
- •8.6.1 ICP Outline
- •8.6.2 A Critical Discussion of ICP
- •8.6.3 A Typical ICP-Based 3D Face Recognition Implementation
- •8.6.4 ICP Variants and Other Surface Registration Approaches
- •8.7 PCA-Based 3D Face Recognition
- •8.7.1 PCA System Training
- •8.7.2 PCA Training Using Singular Value Decomposition
- •8.7.3 PCA Testing
- •8.7.4 PCA Performance
- •8.8 LDA-Based 3D Face Recognition
- •8.8.1 Two-Class LDA
- •8.8.2 LDA with More than Two Classes
- •8.8.3 LDA in High Dimensional 3D Face Spaces
- •8.8.4 LDA Performance
- •8.9 Normals and Curvature in 3D Face Recognition
- •8.9.1 Computing Curvature on a 3D Face Scan
- •8.10 Recent Techniques in 3D Face Recognition
- •8.10.1 3D Face Recognition Using Annotated Face Models (AFM)
- •8.10.2 Local Feature-Based 3D Face Recognition
- •8.10.2.1 Keypoint Detection and Local Feature Matching
- •8.10.2.2 Other Local Feature-Based Methods
- •8.10.3 Expression Modeling for Invariant 3D Face Recognition
- •8.10.3.1 Other Expression Modeling Approaches
- •8.11 Research Challenges
- •8.12 Concluding Remarks
- •8.13 Further Reading
- •8.14 Questions
- •8.15 Exercises
- •References
- •9.1 Introduction
- •Chapter Outline
- •9.2 DEM Generation from Stereoscopic Imagery
- •9.2.1 Stereoscopic DEM Generation: Literature Review
- •9.2.2 Accuracy Evaluation of DEMs
- •9.2.3 An Example of DEM Generation from SPOT-5 Imagery
- •9.3 DEM Generation from InSAR
- •9.3.1 Techniques for DEM Generation from InSAR
- •9.3.1.1 Basic Principle of InSAR in Elevation Measurement
- •9.3.1.2 Processing Stages of DEM Generation from InSAR
- •The Branch-Cut Method of Phase Unwrapping
- •The Least Squares (LS) Method of Phase Unwrapping
- •9.3.2 Accuracy Analysis of DEMs Generated from InSAR
- •9.3.3 Examples of DEM Generation from InSAR
- •9.4 DEM Generation from LIDAR
- •9.4.1 LIDAR Data Acquisition
- •9.4.2 Accuracy, Error Types and Countermeasures
- •9.4.3 LIDAR Interpolation
- •9.4.4 LIDAR Filtering
- •9.4.5 DTM from Statistical Properties of the Point Cloud
- •9.5 Research Challenges
- •9.6 Concluding Remarks
- •9.7 Further Reading
- •9.8 Questions
- •9.9 Exercises
- •References
- •10.1 Introduction
- •10.1.1 Allometric Modeling of Biomass
- •10.1.2 Chapter Outline
- •10.2 Aerial Photo Mensuration
- •10.2.1 Principles of Aerial Photogrammetry
- •10.2.1.1 Geometric Basis of Photogrammetric Measurement
- •10.2.1.2 Ground Control and Direct Georeferencing
- •10.2.2 Tree Height Measurement Using Forest Photogrammetry
- •10.2.2.2 Automated Methods in Forest Photogrammetry
- •10.3 Airborne Laser Scanning
- •10.3.1 Principles of Airborne Laser Scanning
- •10.3.1.1 Lidar-Based Measurement of Terrain and Canopy Surfaces
- •10.3.2 Individual Tree-Level Measurement Using Lidar
- •10.3.2.1 Automated Individual Tree Measurement Using Lidar
- •10.3.3 Area-Based Approach to Estimating Biomass with Lidar
- •10.4 Future Developments
- •10.5 Concluding Remarks
- •10.6 Further Reading
- •10.7 Questions
- •References
- •11.1 Introduction
- •Chapter Outline
- •11.2 Volumetric Data Acquisition
- •11.2.1 Computed Tomography
- •11.2.1.1 Characteristics of 3D CT Data
- •11.2.2 Positron Emission Tomography (PET)
- •11.2.2.1 Characteristics of 3D PET Data
- •Relaxation
- •11.2.3.1 Characteristics of the 3D MRI Data
- •Image Quality and Artifacts
- •11.2.4 Summary
- •11.3 Surface Extraction and Volumetric Visualization
- •11.3.1 Surface Extraction
- •Example: Curvatures and Geometric Tools
- •11.3.2 Volume Rendering
- •11.3.3 Summary
- •11.4 Volumetric Image Registration
- •11.4.1 A Hierarchy of Transformations
- •11.4.1.1 Rigid Body Transformation
- •11.4.1.2 Similarity Transformations and Anisotropic Scaling
- •11.4.1.3 Affine Transformations
- •11.4.1.4 Perspective Transformations
- •11.4.1.5 Non-rigid Transformations
- •11.4.2 Points and Features Used for the Registration
- •11.4.2.1 Landmark Features
- •11.4.2.2 Surface-Based Registration
- •11.4.2.3 Intensity-Based Registration
- •11.4.3 Registration Optimization
- •11.4.3.1 Estimation of Registration Errors
- •11.4.4 Summary
- •11.5 Segmentation
- •11.5.1 Semi-automatic Methods
- •11.5.1.1 Thresholding
- •11.5.1.2 Region Growing
- •11.5.1.3 Deformable Models
- •Snakes
- •Balloons
- •11.5.2 Fully Automatic Methods
- •11.5.2.1 Atlas-Based Segmentation
- •11.5.2.2 Statistical Shape Modeling and Analysis
- •11.5.3 Summary
- •11.6 Diffusion Imaging: An Illustration of a Full Pipeline
- •11.6.1 From Scalar Images to Tensors
- •11.6.2 From Tensor Image to Information
- •11.6.3 Summary
- •11.7 Applications
- •11.7.1 Diagnosis and Morphometry
- •11.7.2 Simulation and Training
- •11.7.3 Surgical Planning and Guidance
- •11.7.4 Summary
- •11.8 Concluding Remarks
- •11.9 Research Challenges
- •11.10 Further Reading
- •Data Acquisition
- •Surface Extraction
- •Volume Registration
- •Segmentation
- •Diffusion Imaging
- •Software
- •11.11 Questions
- •11.12 Exercises
- •References
- •Index
170 |
W.A.P. Smith |
Fig. 4.18 The angles used in the cotangent weights scheme for surface normal computation
The Laplacian relates absolute and differential coordinates as follows: If the long vector x RN contains the x coordinates of the vertices then Lx = Dδx , where δx RN is a long vector of the x components of the differential coordinates of the vertices. Similarly for the y and z coordinates. Note that spectral analysis of the Laplacian can be used to perform signal processing operations on an irregularly triangulated mesh [68].
From differential geometry it is known that an infinitesimal curvilinear integral about a point on a smooth surface is related to the mean curvature H (vi ) at vi and the surface normal ni :
|
1 |
|
|
γlim0 |
|
(vi − v)dl(v) = −H (vi )ni , |
(4.28) |
γ |
|||
| |→ |
| | |
v γ |
|
where γ is a closed simple surface curve l(v) around vi . By rewriting the differential coordinate vector, it can be seen that a discrete approximation to this integral can be made:
|
1 |
|
|
|
|
Adj(i) |
|
(vi − vj ) ≈ −H (vi )ni . |
(4.29) |
|
j Adj(i) |
|
||
Hence, the direction of the differential coordinate vector approximates the surface normal and the magnitude is proportional to the mean curvature [68]. This provides an alternative means to surface normal calculation for a mesh which is motivated by differential geometry. A final alternative proposed by Meyer et al. [42] is based on cotangent weights:
cot |
= |
1 |
1 |
(cot αij + cot βij )(vi − vj ), |
(4.30) |
|
δi |
|
|
|
|||
| i | j Adj(i) |
2 |
|||||
where | i | is the size of the Voronoi cell of i and αij and βij denote the two angles opposite the edge {i, j } (see Fig. 4.18).
4.6 Compression and Levels of Detail
Broadly speaking, the storage requirements for 3D data can be reduced in two ways: storing a 3D representation in less space (compression) or deriving a lower resolu-
4 Representing, Storing and Visualizing 3D Data |
171 |
tion representation that approximates the original data. We have already seen some representations that lead to space efficient storage of 3D data. For example, subdivision surfaces require only the storage of a low resolution base mesh and subdivision rule, while the octree representation uses adaptive resolution depending on the complexity of the volume. Compression of 3D data in general [1, 49] is a more challenging problem than is solved by the relatively mature technologies available for compression of audio, image and video data. Techniques for 3D data compression can be categorized into three classes of approach:
Mesh-based methods. These methods involve traversal of a polygonal mesh and encoding of the mesh structure in a manner that can be compressed. An example of such an approach is topological surgery [69]. This method proceeds by quantizing vertex positions within the desired accuracy. A vertex spanning tree is then used to predict the position of each vertex from 2, 3 or 4 of its ancestors in the tree. Finally, the correction vectors required to recover the original positions are entropy encoded. Another popular approach is based on a spectral analysis of the mesh Laplacian [27]. An eigendecomposition of the Laplacian matrix (see Sect. 4.5.2) yields an orthonormal set of basis vectors onto which the geometry signals (vectors of x, y and z components) can be projected. By discarding high frequency components of the decomposition (i.e. those eigenvectors with small eigenvalues), the mesh can be compressed such that only high frequency detail is lost.
Progressive and hierarchical methods. We have already seen subdivision surfaces that fall into this category. Another important approach is the compressed progressive mesh [24]. In the original progressive mesh, a mesh is encoded as a form of reverse simplification. The opposite of an edge collapse operation is a vertex split, in which a vertex is divided into two and additional edges and faces are added to the mesh. A progressive mesh represents a high resolution mesh as a series of vertex split operations starting from a base mesh. Although this allows progressive transmission of increasingly detailed mesh data, there is a storage overhead, which means space requirements increase. Pajarola and Rossignac [47] showed how this representation can be compressed.
Imaged-based methods. A 3D surface may be represented in image space. This can either be native to the representation (in the case of a range image or bump map) or via a process known as surface parameterization or surface flattening [59]. Once represented as an image, any existing image compression algorithm may be applied to the data. However, it is important to note that the objectives of lossy image compression may not yield correspondingly high quality results in the surface domain since the redundancies in the two data sources may not be the same. One example of an image-based approach is geometry images [20]. Geometry is captured as a 2D array of quantized points. To transform a mesh to this representation, an arbitrary mesh is cut along a network of edge paths and the resulting single chart is parameterized onto a square. Face connectivity is implicit in the representation and bump maps or texture can be stored in the same parameterization. Compressing the resulting data using an image wavelet-encoder allows dramatic reductions in storage requirements without severely affecting the resulting geometry.