The document discusses analytic photogrammetry techniques for determining 3D properties from 2D images. It covers exterior orientation to determine camera position and orientation, interior orientation for camera calibration, relative orientation between two cameras, and stereo triangulation for depth estimation. It also discusses nonlinear least squares optimization methods for solving the photogrammetry problems.
Camera calibration is an essential task for surface reconstruction as well as pose estimation. This is part of the computer vision course taught in Zewail City and Cairo University
New geometric interpretation and analytic solution for quadrilateral reconstr...Joo-Haeng Lee
Accepted as poster presentation for ICPR 2014, Stockholm, Sweden on August 24~28, 2014.
[Revised Version]
Title: New geometric interpretation and analytic solution for quadrilateral reconstruction
Author: Joo-Haeng Lee
Affiliation: Human-Robot Interaction Research Team, ETRI, KOREA
Abstract:
A new geometric framework, called generalized coupled line camera (GCLC), is proposed to derive an analytic solution to reconstruct an unknown scene quadrilateral and the relevant projective structure from a single or multiple image quadrilaterals. We extend the previous approach developed for rectangle to handle arbitrary scene quadrilaterals. First, we generalize a single line camera by removing the centering constraint that the principal axis should bisect a scene line. Then, we couple a pair of generalized line cameras to model a frustum with a quadrilateral base. Finally, we show that the scene quadrilateral and the center of projection can be analytically reconstructed from a single view when prior knowledge on the quadrilateral is available. A completely unknown quadrilateral can be reconstructed from four views through non-linear optimization. We also describe a improved method to handle an off-centered case by geometrically inferring a centered proxy quadrilateral, which accelerates a reconstruction process without relying on homography. The proposed method is easy to implement since each step is expressed as a simple analytic equation. We present the experimental results on real and synthetic examples.
[Submitted Version]
Title: Generalized Coupled Line Cameras and Application in Quadrilateral Reconstruction
Abstract:
Coupled line camera (CLC) provides a geometric framework to derive an analytic solution to reconstruct an unknown scene rectangle and the relevant projective structure from a single image quadrilateral. We extend this approach as generalized coupled line camera (GCLC) to handle a scene quadrilateral. First, we generalize a single line camera by removing the centering constraint that the principal axis should bisect a scene line. Then, we couple a pair of generalized line cameras to model a frustum with a quadrilateral base. Finally, we show that the scene quadrilateral and the center of projection can be analytically reconstructed from a single view when prior knowledge on the quadrilateral is available. ...
Camera Calibration from a Single Image based on Coupled Line Cameras and Rect...Joo-Haeng Lee
ICPR 2012 Paper Abstract
Title: Camera Calibration from a Single Image Based on Coupled Line Cameras and Rectangle Constraint
Author: Lee, Joo-Haeng (ETRI)
Scheduled for presentation during the Regular Session "Poster Shotgun (04): CV" (TuPSAT2), Tuesday, November 13, 2012, 08:30−09:00, Multi-Purpose Hall
21st International Conference on Pattern Recognition, November 11-15, 2012, Tsukuba International Congress Center, Tsukuba, Japan
This information is tentative and subject to change. Compiled on February 13, 2013
The Technology Research of Camera Calibration Based On LabVIEWIJRES Journal
The technology of camera calibration is most important part for machine vision detection and
location, the accuracy of calibration directly determines the processing accuracy of machine vision systems. In
this paper, we use LabVIEW and MATLAB to calibrate the internal and external parameters of the camera, at
the same time, we use dot calibration board, the circle edge is detected by Canny operator, then with the method
of circle fitting based on subpixel edge extraction, the information of dots image coordinate is extracted. The
present method reduces the difficulty of camera calibration and shortens the software development cycle, the
most important is that it has a high calibration accuracy, which can meet the actual industrial detection accuracy,
the results of experimental show that the method is feasible.
Camera calibration is an essential task for surface reconstruction as well as pose estimation. This is part of the computer vision course taught in Zewail City and Cairo University
New geometric interpretation and analytic solution for quadrilateral reconstr...Joo-Haeng Lee
Accepted as poster presentation for ICPR 2014, Stockholm, Sweden on August 24~28, 2014.
[Revised Version]
Title: New geometric interpretation and analytic solution for quadrilateral reconstruction
Author: Joo-Haeng Lee
Affiliation: Human-Robot Interaction Research Team, ETRI, KOREA
Abstract:
A new geometric framework, called generalized coupled line camera (GCLC), is proposed to derive an analytic solution to reconstruct an unknown scene quadrilateral and the relevant projective structure from a single or multiple image quadrilaterals. We extend the previous approach developed for rectangle to handle arbitrary scene quadrilaterals. First, we generalize a single line camera by removing the centering constraint that the principal axis should bisect a scene line. Then, we couple a pair of generalized line cameras to model a frustum with a quadrilateral base. Finally, we show that the scene quadrilateral and the center of projection can be analytically reconstructed from a single view when prior knowledge on the quadrilateral is available. A completely unknown quadrilateral can be reconstructed from four views through non-linear optimization. We also describe a improved method to handle an off-centered case by geometrically inferring a centered proxy quadrilateral, which accelerates a reconstruction process without relying on homography. The proposed method is easy to implement since each step is expressed as a simple analytic equation. We present the experimental results on real and synthetic examples.
[Submitted Version]
Title: Generalized Coupled Line Cameras and Application in Quadrilateral Reconstruction
Abstract:
Coupled line camera (CLC) provides a geometric framework to derive an analytic solution to reconstruct an unknown scene rectangle and the relevant projective structure from a single image quadrilateral. We extend this approach as generalized coupled line camera (GCLC) to handle a scene quadrilateral. First, we generalize a single line camera by removing the centering constraint that the principal axis should bisect a scene line. Then, we couple a pair of generalized line cameras to model a frustum with a quadrilateral base. Finally, we show that the scene quadrilateral and the center of projection can be analytically reconstructed from a single view when prior knowledge on the quadrilateral is available. ...
Camera Calibration from a Single Image based on Coupled Line Cameras and Rect...Joo-Haeng Lee
ICPR 2012 Paper Abstract
Title: Camera Calibration from a Single Image Based on Coupled Line Cameras and Rectangle Constraint
Author: Lee, Joo-Haeng (ETRI)
Scheduled for presentation during the Regular Session "Poster Shotgun (04): CV" (TuPSAT2), Tuesday, November 13, 2012, 08:30−09:00, Multi-Purpose Hall
21st International Conference on Pattern Recognition, November 11-15, 2012, Tsukuba International Congress Center, Tsukuba, Japan
This information is tentative and subject to change. Compiled on February 13, 2013
The Technology Research of Camera Calibration Based On LabVIEWIJRES Journal
The technology of camera calibration is most important part for machine vision detection and
location, the accuracy of calibration directly determines the processing accuracy of machine vision systems. In
this paper, we use LabVIEW and MATLAB to calibrate the internal and external parameters of the camera, at
the same time, we use dot calibration board, the circle edge is detected by Canny operator, then with the method
of circle fitting based on subpixel edge extraction, the information of dots image coordinate is extracted. The
present method reduces the difficulty of camera calibration and shortens the software development cycle, the
most important is that it has a high calibration accuracy, which can meet the actual industrial detection accuracy,
the results of experimental show that the method is feasible.
Corisco is a method for monocular camera orientation estimation in anthropic environments using edgels. This is my doctorate defense presentation, updated and translated to english.
Visual odometry & slam utilizing indoor structured environmentsNAVER Engineering
Visual odometry (VO) and simultaneous localization and mapping (SLAM) are fundamental building blocks for various applications from autonomous vehicles to virtual and augmented reality (VR/AR).
To improve the accuracy and robustness of the VO & SLAM approaches, we exploit multiple lines and orthogonal planar features, such as walls, floors, and ceilings, common in man-made indoor environments.
We demonstrate the effectiveness of the proposed VO & SLAM algorithms through an extensive evaluation on a variety of RGB-D datasets and compare with other state-of-the-art methods.
Conventional non-vision based navigation systems relying on purely Global Positioning System (GPS) or inertial sensors can provide the 3D position or orientation of the user. However GPS is often not available in forested regions and cannot be used indoors. Visual odometry provides an independent method to estimate position and orientation of the user/system based on the images captured by the moving user accurately. Vision based systems also provide information (e.g. images, 3D location of landmarks, detection of scene objects) about the scene that the user is looking at. In this project, a set of techniques are used for the accurate pose and position estimation of the moving vehicle for autonomous navigation using the images obtained from two cameras placed at two different locations of the same area on the top of the vehicle. These cases are referred to as stereo vision. Stereo vision provides a method for the 3D reconstruction of the environment which is required for pose and position estimation. Firstly, a set of images are captured. The Harris corner detector is utilized to automatically extract a set of feature points from the images and then feature matching is done using correlation based matching. Triangulation is applied on feature points to find the 3D co-ordinates. Next, a new set of images is captured. Then repeat the same technique for the new set of images too. Finally, by using the 3D feature points, obtained from the first set of images and the new set of images, the pose and position estimation of moving vehicle is done using QUEST algorithm.
Build Your Own 3D Scanner: 3D Scanning with Swept-PlanesDouglas Lanman
Build Your Own 3D Scanner:
3D Scanning with Swept-Planes
http://mesh.brown.edu/byo3d/
SIGGRAPH 2009 Courses
Douglas Lanman and Gabriel Taubin
This course provides a beginner with the necessary mathematics, software, and practical details to leverage projector-camera systems in their own 3D scanning projects. An example-driven approach is used throughout; each new concept is illustrated using a practical scanner implemented with off-the-shelf parts. The course concludes by detailing how these new approaches are used in rapid prototyping, entertainment, cultural heritage, and web-based applications.
Preliminary Research on Data Abnormality Diagnosis Methods of Spacecraft Prec...IJERA Editor
For precision measuring the satellite equipments, providing technical support for satellite assembly, combined
with satellite small size, complex structure, satellite equipment shapes vary, and other characteristics, presently,
indirect method that using electronic theodolite to measure cube mirror are commonly used to obtain the relative
attitude of the respective devices. But in the actual measurement process, there are measurement errors in the
measurement data. How to detect anomalies in the data is the focus of this study. This paper proposes two
methods to detect abnormal data, that is mathematical geometric method and outlier detection methods. This
paper analyzes their theoretical basis and verifies the feasibility of the two methods through part of the actual
measurement data to.
Calculation of the Curvature Expected in Photographs of a Sphere's HorizonJames Smith
A formula is derived for the curvature of the horizon's image in photos of a sphere of radius R , taken by a camera with horizontal view angle alpha from height h above the sphere's surface. The formula is validated by means of an interactive GeoGebra construction: a key angle calculated from the formula derived here is compared to the angle actually present in the construction. Using the validated formula, the amount of curvature expected to be present in a photo of the Earth's horizon from an altitude of 3 m is calculated. The result is an order of magnitude smaller than typical degrees of barrel distortion present in consumers' digital cameras. Therefore, claims that "flat horizons in photos of waterscapes prove that the Earth is flat" are untenable.
Solving the Pose Ambiguity via a Simple Concentric Circle ConstraintDr. Amarjeet Singh
Estimating the pose of objects with circle feature from images is a basic and important question in computer vision
community. This paper is focused on the ambiguity problem in pose estimation of circle feature, and a new method is proposed based
on the concentric circle constraint. The pose of a single circle feature, in general, can be determined from its projection in the image
plane with a pre-calibrated camera. However, there are generally two possible sets of pose parameters. By introducing the concentric
circle constraint, interference from the false solution can be excluded. On the basis of element at infinity in projective geometry and
the Euclidean distance invariant, cases that concentric circles are coplanar and non-coplanar are discussed respectively. Experiments
on these two cases are performed to validate the proposed method.
The aim of this paper is to present the essential elements of the electro-optical imaging system EOIS for space applications and how these elements can affect its function. After designing a spacecraft for low orbiting missions during day time, the design of an electro-imaging system becomes an important part in the satellite because the satellite will be able to take images of the regions of interest. An example of an electro-optical satellite imaging system will be presented through this paper where some restrictions have to be considered during the design process. Based on the optics principals and ray tracing techniques the dimensions of lenses and CCD (Charge Coupled Device) detector are changed matching the physical satellite requirements. However, many experiments were done in the physics lab to prove that the resizing of the electro optical elements of the imaging system does not affect the imaging mission configuration. The procedures used to measure the field of view and ground resolution will be discussed through this work. Examples of satellite images will be illustrated to show the ground resolution effects.
Scattering optical tomography with discretized path integralToru Tamaki
Slide of the talks:
Toru Tamaki, Scattering tomography with path integral, Séminaire A3SI (Algorithmes, architectures, analyse et synthèse d’images), Laboratoire d'Informatique Gaspard-Monge (LIGM), ESIEE Paris, Université Paris-Est, 11-June-2015.
Toru Tamaki, Scattering optical tomography with discretized path integral, Fachbereich Computerwissenschaften, Universität Salzburg, Austria, 03-December-2015.
Toru Tamaki, Scattering optical tomography with discretized path integral, Departamento de Ciências da Informação e da Decisão em Saúde Faculdade de Medicina, Universidade do Porto, Porto, Portugal, 11-December-2015.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Corisco is a method for monocular camera orientation estimation in anthropic environments using edgels. This is my doctorate defense presentation, updated and translated to english.
Visual odometry & slam utilizing indoor structured environmentsNAVER Engineering
Visual odometry (VO) and simultaneous localization and mapping (SLAM) are fundamental building blocks for various applications from autonomous vehicles to virtual and augmented reality (VR/AR).
To improve the accuracy and robustness of the VO & SLAM approaches, we exploit multiple lines and orthogonal planar features, such as walls, floors, and ceilings, common in man-made indoor environments.
We demonstrate the effectiveness of the proposed VO & SLAM algorithms through an extensive evaluation on a variety of RGB-D datasets and compare with other state-of-the-art methods.
Conventional non-vision based navigation systems relying on purely Global Positioning System (GPS) or inertial sensors can provide the 3D position or orientation of the user. However GPS is often not available in forested regions and cannot be used indoors. Visual odometry provides an independent method to estimate position and orientation of the user/system based on the images captured by the moving user accurately. Vision based systems also provide information (e.g. images, 3D location of landmarks, detection of scene objects) about the scene that the user is looking at. In this project, a set of techniques are used for the accurate pose and position estimation of the moving vehicle for autonomous navigation using the images obtained from two cameras placed at two different locations of the same area on the top of the vehicle. These cases are referred to as stereo vision. Stereo vision provides a method for the 3D reconstruction of the environment which is required for pose and position estimation. Firstly, a set of images are captured. The Harris corner detector is utilized to automatically extract a set of feature points from the images and then feature matching is done using correlation based matching. Triangulation is applied on feature points to find the 3D co-ordinates. Next, a new set of images is captured. Then repeat the same technique for the new set of images too. Finally, by using the 3D feature points, obtained from the first set of images and the new set of images, the pose and position estimation of moving vehicle is done using QUEST algorithm.
Build Your Own 3D Scanner: 3D Scanning with Swept-PlanesDouglas Lanman
Build Your Own 3D Scanner:
3D Scanning with Swept-Planes
http://mesh.brown.edu/byo3d/
SIGGRAPH 2009 Courses
Douglas Lanman and Gabriel Taubin
This course provides a beginner with the necessary mathematics, software, and practical details to leverage projector-camera systems in their own 3D scanning projects. An example-driven approach is used throughout; each new concept is illustrated using a practical scanner implemented with off-the-shelf parts. The course concludes by detailing how these new approaches are used in rapid prototyping, entertainment, cultural heritage, and web-based applications.
Preliminary Research on Data Abnormality Diagnosis Methods of Spacecraft Prec...IJERA Editor
For precision measuring the satellite equipments, providing technical support for satellite assembly, combined
with satellite small size, complex structure, satellite equipment shapes vary, and other characteristics, presently,
indirect method that using electronic theodolite to measure cube mirror are commonly used to obtain the relative
attitude of the respective devices. But in the actual measurement process, there are measurement errors in the
measurement data. How to detect anomalies in the data is the focus of this study. This paper proposes two
methods to detect abnormal data, that is mathematical geometric method and outlier detection methods. This
paper analyzes their theoretical basis and verifies the feasibility of the two methods through part of the actual
measurement data to.
Calculation of the Curvature Expected in Photographs of a Sphere's HorizonJames Smith
A formula is derived for the curvature of the horizon's image in photos of a sphere of radius R , taken by a camera with horizontal view angle alpha from height h above the sphere's surface. The formula is validated by means of an interactive GeoGebra construction: a key angle calculated from the formula derived here is compared to the angle actually present in the construction. Using the validated formula, the amount of curvature expected to be present in a photo of the Earth's horizon from an altitude of 3 m is calculated. The result is an order of magnitude smaller than typical degrees of barrel distortion present in consumers' digital cameras. Therefore, claims that "flat horizons in photos of waterscapes prove that the Earth is flat" are untenable.
Solving the Pose Ambiguity via a Simple Concentric Circle ConstraintDr. Amarjeet Singh
Estimating the pose of objects with circle feature from images is a basic and important question in computer vision
community. This paper is focused on the ambiguity problem in pose estimation of circle feature, and a new method is proposed based
on the concentric circle constraint. The pose of a single circle feature, in general, can be determined from its projection in the image
plane with a pre-calibrated camera. However, there are generally two possible sets of pose parameters. By introducing the concentric
circle constraint, interference from the false solution can be excluded. On the basis of element at infinity in projective geometry and
the Euclidean distance invariant, cases that concentric circles are coplanar and non-coplanar are discussed respectively. Experiments
on these two cases are performed to validate the proposed method.
The aim of this paper is to present the essential elements of the electro-optical imaging system EOIS for space applications and how these elements can affect its function. After designing a spacecraft for low orbiting missions during day time, the design of an electro-imaging system becomes an important part in the satellite because the satellite will be able to take images of the regions of interest. An example of an electro-optical satellite imaging system will be presented through this paper where some restrictions have to be considered during the design process. Based on the optics principals and ray tracing techniques the dimensions of lenses and CCD (Charge Coupled Device) detector are changed matching the physical satellite requirements. However, many experiments were done in the physics lab to prove that the resizing of the electro optical elements of the imaging system does not affect the imaging mission configuration. The procedures used to measure the field of view and ground resolution will be discussed through this work. Examples of satellite images will be illustrated to show the ground resolution effects.
Scattering optical tomography with discretized path integralToru Tamaki
Slide of the talks:
Toru Tamaki, Scattering tomography with path integral, Séminaire A3SI (Algorithmes, architectures, analyse et synthèse d’images), Laboratoire d'Informatique Gaspard-Monge (LIGM), ESIEE Paris, Université Paris-Est, 11-June-2015.
Toru Tamaki, Scattering optical tomography with discretized path integral, Fachbereich Computerwissenschaften, Universität Salzburg, Austria, 03-December-2015.
Toru Tamaki, Scattering optical tomography with discretized path integral, Departamento de Ciências da Informação e da Decisão em Saúde Faculdade de Medicina, Universidade do Porto, Porto, Portugal, 11-December-2015.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
MATATAG CURRICULUM: ASSESSING THE READINESS OF ELEM. PUBLIC SCHOOL TEACHERS I...NelTorrente
In this research, it concludes that while the readiness of teachers in Caloocan City to implement the MATATAG Curriculum is generally positive, targeted efforts in professional development, resource distribution, support networks, and comprehensive preparation can address the existing gaps and ensure successful curriculum implementation.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
Delivering Micro-Credentials in Technical and Vocational Education and TrainingAG2 Design
Explore how micro-credentials are transforming Technical and Vocational Education and Training (TVET) with this comprehensive slide deck. Discover what micro-credentials are, their importance in TVET, the advantages they offer, and the insights from industry experts. Additionally, learn about the top software applications available for creating and managing micro-credentials. This presentation also includes valuable resources and a discussion on the future of these specialised certifications.
For more detailed information on delivering micro-credentials in TVET, visit this https://tvettrainer.com/delivering-micro-credentials-in-tvet/
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Ch14.ppt
1. Digital Camera and Computer Vision Laboratory
Department of Computer Science and Information Engineering
National Taiwan University, Taipei, Taiwan, R.O.C.
Chapter 14
Analytic Photogrammetry
Presented by 王夏果 and Dr. Fuh
R94922103@ntu.edu.tw
0937384214
2. DC & CV Lab.
NTU CSIE
Analytic Photogrammetry
Make inferences about :
3D position
Orientation
Length of the observed 3D object parts
in a world reference frame from
measurements of one or more 2D-
perspective projections of a 3D object
3. DC & CV Lab.
NTU CSIE
Analytic Photogrammetry (cont.)
These inference problems can be construed
as nonlinear least-square problems
Iteratively linearize the nonlinear functions
from an initially given approximate solution
4. DC & CV Lab.
NTU CSIE
Photogrammetry
Provide a collection of methods for
determining the position and orientation of
cameras and range sensors in the scene and
relating camera positions and range
measurements to scene coordinates
GIS: Geographic Information System
GPS: Global Positioning System
6. DC & CV Lab.
NTU CSIE
Exterior Orientation
Determine position and orientation of camera
in absolute coordinate system from
projections of calibration points in scene
The exterior orientation of the camera is
specified by all parameters of camera pose,
such as perspectivity center position, optical
axis direction.
9. DC & CV Lab.
NTU CSIE
Interior Orientation
Determine internal geometry of camera
The interior orientation of camera is specified
by all the parameters that determines the
geometry of 3D rays from measured image
coordinates
10. DC & CV Lab.
NTU CSIE
Interior Orientation (cont.)
The parameters of interior orientation relate
the geometry of ideal perspective projection
to the physics of a camera.
Parameters: camera constant, principal point,
lens distortion, …
11. DC & CV Lab.
NTU CSIE
Interior Orientation (cont.)
With interior and external orientation, we can
complete specify the camera orientation.
12. DC & CV Lab.
NTU CSIE
Relative Orientation
Determine relative position and orientation
between 2 cameras from projections of
calibration points in scene
Calibrate relation between two cameras for
stereo
Relates coordinate systems of two cameras
to each other, not knowing 3D points
themselves, only their projections in image
13. DC & CV Lab.
NTU CSIE
Relative Orientation (cont.)
Assume interior orientation of each camera
known
Specified by 5 parameters: 3 rotation angles,
2 translations
15. DC & CV Lab.
NTU CSIE
Absolute Orientation
Determine transformation between 2
coordinate systems or position and
orientation of range sensor in absolute
coordinate system from coordinates of
calibration points
Convert depth measurements in viewer-
centered coordinates to absolute coordinate
system for the scene
16. DC & CV Lab.
NTU CSIE
Absolute Orientation (cont.)
Orientation of stereo model in world
reference frame
Determine scale, 3 translations, 3 rotations
Recovery of relation between two coordinate
system
22. DC & CV Lab.
NTU CSIE
World Frame to Camera Frame
(x, y, z)’ in world frame represented by
(p, q, s)’ in camera frame:
23. DC & CV Lab.
NTU CSIE
Pinhole Camera Projection
Pinhole camera with image at distance f
from camera lens, projection:
where f is a camera constant, related to
focal length of lens
24. DC & CV Lab.
NTU CSIE
Principal Point
Origin of measurement image plane
coordinate
Represented by (u0, v0)
26. DC & CV Lab.
NTU CSIE
Perspective Projection Equations
(cont.)
Show that the relationship between the
measured 2D-perspective projection
coordinates and the 3D coordinates is a
nonlinear function of u0, v0, x0, y0, z0, ω, ψ,
and κ
29. DC & CV Lab.
NTU CSIE
Nonlinear Least-Square
Solutions (cont.)
Maximum likelihood solution: β1, …, βM
maximize Prob(α1, …, αk | β1, …, βM )
In other words, this solution minimizes
least-squares criterion:
where
30. DC & CV Lab.
NTU CSIE
First-Order Taylor Series
Expansion
First-order Taylor series expansion of gk
taken around βt:
31. DC & CV Lab.
NTU CSIE
First-Order Taylor Series
Expansion (cont.)
32. DC & CV Lab.
NTU CSIE
Exterior Orientation Problem
Determine the unknown rotation and
translation that put the camera reference
frame in the world reference frame.
33. DC & CV Lab.
NTU CSIE
Exterior Orientation Problem
(cont.)
34. DC & CV Lab.
NTU CSIE
One Camera Exterior Orientation
Problem
Known: (xn, yn, zn)’ and (un, vn)’
(un, vn)’ is the corresponding set of 2D-
perspective projections, n = 1, …, N
Unknown: (ω,ψ,κ) and (x0, y0, z0)’
35. DC & CV Lab.
NTU CSIE
Other Exterior Orientation
Problem
Camera calibration problem: unknown
position of camera in object frame
Object pose estimation problem: unknown
object position in camera frame
Spatial resection problem in
photogrammetries: 3D positions from 2D
orientation
36. DC & CV Lab.
NTU CSIE
Nonlinear Transformation For
Exterior Orientation
37. DC & CV Lab.
NTU CSIE
Standard Solution
By chain rule,
40. DC & CV Lab.
NTU CSIE
Standard Solution (cont.)
41. DC & CV Lab.
NTU CSIE
Auxiliary Solution
Not iteratively adjust the angles directly
Reorganize the calculation such that we
iteratively adjust the three auxiliary
parameters of a skew symmetric matrix
associated with the rotation matrix
Then, we determine the adjustment of the
angles
43. DC & CV Lab.
NTU CSIE
Quaternion Representation
From any skew symmetric matrix,
we can construct a rotation matrix R by
choosing scalar d: R = (dI + S)(dI - S)-1
which guarantees that R’R = I
44. DC & CV Lab.
NTU CSIE
Quaternion Representation (cont.)
Expanding the equation for R:
parameters a, b, c, d can be constrained to
satisfy a2 + b2 + c2 + d2 = 1
45. DC & CV Lab.
NTU CSIE
Quaternion Representation (cont.)
47. DC & CV Lab.
NTU CSIE
Relative Orientation
The transformation from one camera station
to another can be represented by a rotation
and a translation
The relation between the coordinates, rl and
rr of a point P can be given by means of a
rotation matrix and an offset vector
49. DC & CV Lab.
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Relative Orientation (cont.)
Relative orientation is typically with the
determination of the position and
orientation of one photograph with respect
to another, given a set of corresponding
image points
50. DC & CV Lab.
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Relative Orientation (cont.)
Relative orientation specified by five
parameters: (yR - yL), (zR - zL), (ωR - ωL),
(ψR - ψL), (κR - κL)
Assumption:
Camera interior orientation known
Image positions expressed to identical scale and
with respect to principal point
51. DC & CV Lab.
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Standard Solution
Let Q’L and Q’R be the rotation matrices with
the exterior orientation of the left and the
right image:
52. DC & CV Lab.
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Standard Solution (cont.)
fR: distance between right image plane and
right lens
fL: distance between left image plane and
left lens
From perspective collinearity equation
53. DC & CV Lab.
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Standard Solution (cont.)
Hence,
where
54. DC & CV Lab.
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Quaternion Solution
Instead of determining the relative orientation
of the right image with respect to the left
image, we aligns a reference frame having its
x-axis along the line from the left image lens
to the right image lens
55. DC & CV Lab.
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Quaternion Solution (cont.)
The relative orientation is then determined by
the angles (ωR, ψR, κR), which rotate the right
image into this reference frame, and the
angles (ωL, ψL, κL), which rotate the left
image into this reference frame
56. DC & CV Lab.
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Interior Orientation
A camera is specified by:
Camera constant f: distance between image
plane and camera lens
Principal point (up, vp): intersection of optic axis
with image plane in measurement reference
frame located on image plane
Geometric distortion characteristics of the lens;
assuming isotropic around the principal point
61. DC & CV Lab.
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Stereo (cont.)
Parallax: deplacement in perspective
projection by position translation
(x, y, z): 3D point position
(uL, vL): perspective projection on left image
of stereo pair
(uR, vR): perspective projection on right image
of stereo pair
bx: baseline length in x-axis
65. DC & CV Lab.
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Stereo (cont.)
Relation is close to being useless in
real-world, because
Observed perspective projections are subject to
measurement errors so that vL ≠ vR for corresponding
points
Left and right camera frames may have slightly different
orientations
When two cameras used, almost always fR ≠ fL
67. DC & CV Lab.
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Relationship Between Coordinate
System
The relationship between two coordinate
systems is easy to find if we can measure the
coordinates of a number of points in both
systems
68. DC & CV Lab.
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Relationship Between Coordinate
System(cont.)
It takes three measurements to tie two
coordinate systems together uniquely
A single measurement leaves three degrees of
freedom motion
A second measurement removes all but one
degree of freedom
Third measurement rigidly attaches two
coordinate systems to each other
70. DC & CV Lab.
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2D-2D Pose Detection Problem
Determine from matched points more precise
estimate of rotation matrix R and translation t
such that yn = Rxn + t, n = 1, …, N
Determine R and t that minimize weighted
sum of residual errors:
71. DC & CV Lab.
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3D-3D Absolute Orientation
We must determine rotation matrix R and
translation vector t satisfying
Constrained least-squares problem to
minimize
72. DC & CV Lab.
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3D-3D Absolute Orientation
(cont.)
The least-square problem can be modeled by
a mechanical system in which corresponding
points in the two coordinate systems are
attached to each other by means of springs
The solution to the least-squares problem
corresponds to the equilibrium position of the
system, which minimizes the energy stored in
the springs
74. DC & CV Lab.
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Robust M-Estimation
Least-squares techniques are ideal when
random data perturbations or measurement
errors are Gaussian distribution
We need some robust techniques for
nonlinear regression
76. DC & CV Lab.
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Robust M-Estimation (cont.)
or
77. DC & CV Lab.
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Robust M-Estimation (cont.)
ρ:
Symmetric
Positive-defined function
Has unique minimum at zero
Chosen to be less increasing than square
78. DC & CV Lab.
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Robust M-Estimation (cont.)
79. DC & CV Lab.
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Error Propagation
If we have the input parameter x1, …, xN ,
and random errors Δx1, …, ΔxN , the quantity
y depends on input parameters through
known function f: y = f(x1, …, xN ) will become
y + Δy= f(x1 +Δx1, …, xN +ΔxN )
80. DC & CV Lab.
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Error Propagation Analysis
Determines expected value and variance of
y + Δy
Known information about Δx1, …, ΔxN :
mean and variance
81. DC & CV Lab.
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Implicit Form
A known function f has the form:
f(x1, …, xN, y) = 0
The quantities (x1 +Δx1, …, xN +ΔxN ) are
observed, and the quantity y + Δy is
determined to satisfy
f(x1 +Δx1, …, xN +ΔxN , y + Δy ) =0
82. DC & CV Lab.
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Implicit Form: General Case
General case: y is not a scalar but a L × 1
vector β
x1, …, xN : are K N × 1 vectors representing
true values
x1 +Δx1, …, xK +ΔxK : are K N × 1 vectors
representing noisy observed values
Δx1, …, ΔxK : random perturbations
β: a L × 1 vector representing unknown true
parameters
83. DC & CV Lab.
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Implicit Form: General Case
Noiseless model:
With noisy observations, the idealized model:
84. DC & CV Lab.
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Summary
We have shown how to:
Take a nonlinear least-squares problem
Linearize it
Solve by iteratively solving successive linearized
least-squares problems