- •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
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Fig. 9.3 Global DEM from ASTER. Copyright METI/NASA, reprinted with permission from ERSDAC, http://www.ersdac.or.jp/GDEM/E/4.htm
GDEM covers land surfaces between 83◦N and 83◦S with estimated accuracy of 20 m at 95 % confidence for vertical data and 30 m at 95 % confidence for horizontal data. Although ASTER GDEM was found to contain significant anomalies and artifacts, METI and NASA decided to release it for public use with belief of its potential benefits outweighing its flaws and with expectation of improving it via the user community.
Research into DEM generation has also expanded to Mars by a combination of stereoscopic imagery and the Mars orbiter’s laser altimeter [58]. It can be foreseen that there will be the following improvements.
1.Further evaluation of geometric models to adaptively correct DEM errors caused by sensor platform attitude instability.
2.Development of more robust image matching algorithms for increasing automation and matching coverage.
For advanced stereo vision techniques that can be used for DEM generation from stereoscopic imagery, please refer to Chap. 2 of this book.
9.2.2 Accuracy Evaluation of DEMs
Quantitative evaluation of DEMs generated by satellite stereoscopic imagery can be conducted by comparing the reconstructed elevation values with GCPs collected by ground surveys or compared with corresponding DEMs generated by higher accuracy devices, such as LIDAR. In both cases, the measure of Root Mean Square Error
9 3D Digital Elevation Model Generation |
375 |
(RMSE) is used in assessing the DEM accuracy. It is defined as
RMSE = |
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n |
h |
2 |
(9.1) |
|
i |
=n |
i |
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1 |
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|
where n is the number of assessed points in evaluation, and h is the height difference between the assessed DEM and GCPs or reference DEM at point i. The standard deviation of the height difference can be calculated in Eq. (9.2).
σ = |
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i=1 |
ni − ¯ |
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(9.2) |
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n |
( h h)2 |
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where the mean height difference is given as ¯ |
1 |
n |
hi . Assuming that |
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= n |
i=1 |
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h |
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the measured data has a normal distribution, ±1σ gives 68 % level of confidence and ±2σ gives 95 % level of confidence in measurements. In the remote sensing community, the elevation accuracy is usually represented by RMSE with a level of confidence, for example, 20 m with LE95 (Linear Error with confidence of 95 %).
Possible error sources are considered in order to improve DEM accuracy. Error propagation can be tracked along the processes of DEM generation, for example, errors due to image matching and 3D reconstruction from the geometric modeling. Geometric modeling to recover elevations from stereo images is the well-known perspective projection from image coordinates to cartographic coordinates. The projection involves elementary transformations (rotations and translations), which are functions of the cameras’ interior and exterior parameters. This requires prior knowledge of the cameras, platforms, and cartographic coordinate systems. Based on such prior knowledge, a rigorous model based on collinearity conditions can be used for DEM generation. This geometric model integrates the following transformations [149, 150], where the xoy refers to the image coordinate system, rotations and translations are also referred to images, and the z-axis represents the direction perpendicular to the image common plane (usually same as the z-axis to the cartographic system).
•Rotation from the camera reference to the platform reference.
•Translation to the Earth’s center (refers to the cartographic coordinate system).
•Rotation that takes into account the platform variation over time.
•Rotation to align the z-axis with the image center on the Earth’s surface.
•Translation to the image center.
•Rotation to align the y-axis in the meridian plane.
•Rotation to have xoy (the image plane) tangential to the Earth.
•Rotation to align the x-axis in the image scan direction.
•Rotation-translation into the cartographic coordinate system.
With the geometric model and possible errors introduced by each step of the process, the integration of different distortions and their impact to the final elevation can be derived [147]. In the derivation, each of the model parameters is the combination of several correlated variables of the total viewing geometry. These include the following.
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•The orientation of the image is a combination of the platform heading due to orbital inclination, the yaw of the platform, and the convergence of the meridian.
•The scale factor in the along-track direction is a combination of the velocity, the altitude and the pitch of the platform, the detection signal time of the sensor and the component of the Earth’s rotation in the along-track direction.
•The leveling angle in the across-track direction is a combination of platform roll, the viewing angle, the orientation of the sensor and the Earth’s curvature.
Mathematical models establishing geometrical relationships between the image and cartographic coordinates can be rigorous or approximate [64]. Rigorous modeling can be applied when a comprehensive understanding of the imaging geometry exists. However in many cases, it is difficult to obtain accurate interior and exterior parameters of the imaging system due to the lack of sufficient control. Therefore, approximate modeling has been developed for real-world use. Approximate models include direct linear transformation (DLT), self-calibration DLT, rational function models, and parallel projection [64]. In analyzing the accuracy potential of DEMs generated by high-resolution satellite stereoscopic imagery, Fraser [48] pointed out that a mathematical model, such as collinearity equations, needs to be modified for different settings in a rigorous model with stereo-bundle adjustments, while in the absence of the sensors’ attitude data and sensors’ orbital parameters, approximate models are recommended.
To improve the imaging geometry, researchers have paid special attention to the B/H ratio in acquiring satellite stereo pairs. A systematic investigation was conducted by Hasegawa et al. [67]. In their research, the impact of the B/H ratio to DEM accuracy was analyzed and the conclusion was made that B/H ratios ranging from 0.5 to 0.9 give better results for automatic DEM generation from stereo pairs. Li et al. designed an accurate model of the intersection angle and B/H ratio for a spaceborne three linear array camera system [100]. It was indicated that the B/H ratio was directly related to the DEM accuracy. A favourable imaging geometry can be achieved by a B/H ratio of 0.8 or more [48]. With SPOT-5, the viewing angle can be adjusted to tune the across-track B/H ratio between 0.6 and 1.1 and the along-track B/H ratio to around 0.8 [150].
From an application point of view, errors of terrain representation (ETR) are also taken into account since these may propagate through GIS operations and affect the quality of final products which use DEMs [31]. When interpolation is needed, the way to represent the terrain surface contributes to DEM accuracy. Chen and Yue [31] developed a promising method of surface modeling based on the theorem of surface. In their work, a terrain surface was defined by the first and second fundamental coefficients with information of the surface geometric properties and its deviation from the tangent plane at the point under consideration. It was demonstrated in their work that a good criterion for DEM accuracy evaluation should have included not only errors generated in 3D reconstruction from stereoscopic geometry but also ETR at a global level. When using a DEM in an application product, ETR should be counted as an input error.
9 3D Digital Elevation Model Generation |
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Table 9.1 Characteristics of the SPOT-5 stereo-pair acquired over the study site |
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Acquisition date |
Sun angle |
Stereo |
View angle |
B/H |
Image (km) |
Pixel (m) |
No. GCPs |
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|
05 May 2003 |
52◦ |
Multidate |
+23◦ |
0.77 |
60 × 60 |
5 × 5 |
33 |
25 May 2003 |
55◦ |
across-track |
−19◦ |
|
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|
9.2.3 An Example of DEM Generation from SPOT-5 Imagery
In this section, the main steps in DEM generation from a satellite stereoscopic image pair are outlined. The example presented in this section is from [149]. The study site is the area around Quebec City, QC, Canada (47◦N, 71◦30 W). The information of the SPOT-5 stereo images in panchromatic mode is listed in Table 9.1.
A perspective projection model is established based on the geometric positions of the satellite, the camera, and the cartographic coordinate. This model links the 3D cartographic coordinate to the image coordinates, and the mathematical expression is given by Eqs. (9.3) and (9.4) [147, 148]:
|
κuu + y(1 + δγ X) − βH − H0 T = 0 |
(9.3) |
|||
X + θ cos χ |
+ αv kv + θ X − cos χ |
− kv R = 0 |
(9.4) |
||
|
H |
|
H |
|
|
where
X = (x − ay) 1 + |
z |
+ by2 + cxy |
(9.5) |
||
|
|
||||
N0 |
|||||
and |
|
|
|
||
H = z − |
x2 |
|
(9.6) |
||
2N0 |
Parameters involved in Eqs. (9.3)–(9.6) are explained as follows.
His the altitude of the point corrected for Earth curvature;
H0 |
is the satellite elevation at the image center line; |
N0 |
is the normal to the Earth; |
ais mainly a function of the rotation of the Earth;
αis the instantaneous field-of-view;
u, v |
are the image coordinates; |
ku, kv |
are the scale factors in along-track and cross-track, respectively; |
β and θ |
are a function of the leveling angles in along-track and across-track, |
|
respectively; |
T and R |
are the non-linear attitude variations ( T : combination of pitch |
|
and yaw; R: roll); |
x, y, and z |
are the ground coordinates; |
b, c, χ and δγ |
are second-order parameters, which are a function of the total ge- |
|
ometry (e.g. satellite, image, and Earth). |
378 |
H. Wei and M. Bartels |
Fig. 9.4 Left: SPOT-5 image captured on 5 May 2003. Right: DEM generated from the stereo pair. A: melting snow; B: frozen lakes; C: the St. Lawrence River with significant melting ice; D: down-hill ski stations with snow. Figure courtesy of [149]
The ground control points (GCPs) with known (x, y, z) coordinates and corresponding (u, v) image coordinates are employed for the bundle adjustment to obtain parameters in the mathematical model. The processing steps of DEM generation from SPOT-5 stereo images (see Fig. 9.4(left)) are as follows.
1.Acquisition and pre-processing of the remote sensed data (images and metadata showing configuration of image acquisition) to determine an approximate value for each parameter of the 3D projection model.
2.Collection of GCPs with their 3D cartographic coordinates and 2D image coordinates. GCPs covered the total surface with points at the lowest and highest elevation to avoid extrapolations, both in x, y and elevation.
3.Computation of the 3D projection model, initialized with the approximate parameter values and refined by an iterative least-squares bundle adjustment with the GCPs.
4.Extraction of matching points from the two stereo images by using a multi-scale normalized cross-correlation method with computation of the maximum of the correlation coefficient.
5.Computation of (x, y, z) cartographic coordinates from the matching points in a regular grid spacing using the adjusted projection model (from step 3).
The full DEM (60 km × 60 km with a 5 m grid spacing) is extracted as shown in Fig. 9.4(right). It reproduces the terrain features, such as the St. Lawrence River and a large island in the middle. The black areas correspond to mismatched areas due to radiometric differences between the multi-date images. In this case, they are a result of snow in the mountains and on frozen lakes. The quantitative evaluation is conducted by comparison of the DEM generated from the SPOT-5 stereo images to a LIDAR acquired DEM with the accuracy of 0.15 m in elevation. Accuracies of 6.5 m (LE68) and 10 m (LE90) were achieved, corresponding to an image matching error of ±1 pixel.