1 Three-view geometry 3-view constraint along F Minimal algebraic sol The content described in these slides is not required in the final exam!

Slides:

Advertisements

Similar presentations

The fundamental matrix F

Advertisements

Lecture 11: Two-view geometry

The Trifocal Tensor Multiple View Geometry. Scene planes and homographies plane induces homography between two views.

Computing 3-view Geometry Class 18

MASKS © 2004 Invitation to 3D vision Lecture 7 Step-by-Step Model Buidling.

1 pb.  camera model  calibration  separation (int/ext)  pose Don’t get lost! What are we doing? Projective geometry Numerical tools Uncalibrated cameras.

Multiple View Reconstruction Class 24 Multiple View Geometry Comp Marc Pollefeys.

N-view factorization and bundle adjustment CMPUT 613.

Dr. Hassan Foroosh Dept. of Computer Science UCF

Structure from motion.

Multiple View Geometry

Computer Vision cmput 613 Sequential 3D Modeling from images using epipolar geometry and F 3D Modeling from images using epipolar geometry and F Martin.

3D reconstruction class 11

Parameter estimation class 5 Multiple View Geometry Comp Marc Pollefeys.

Epipolar geometry. (i)Correspondence geometry: Given an image point x in the first view, how does this constrain the position of the corresponding point.

Structure from motion. Multiple-view geometry questions Scene geometry (structure): Given 2D point matches in two or more images, where are the corresponding.

Lecture 21: Multiple-view geometry and structure from motion

Multiple View Geometry Marc Pollefeys University of North Carolina at Chapel Hill Modified by Philippos Mordohai.

3D reconstruction of cameras and structure x i = PX i x’ i = P’X i.

Multiple View Geometry Marc Pollefeys University of North Carolina at Chapel Hill Modified by Philippos Mordohai.

Triangulation and Multi-View Geometry Class 9 Read notes Section 3.3, , 5.1 (if interested, read Triggs’s paper on MVG using tensor notation, see.

Multiple View Geometry in Computer Vision

Assignment 2 Compute F automatically from image pair (putative matches, 8-point, 7-point, iterative, RANSAC, guided matching) (due by Wednesday 19/03/03)

CSE 803 Fall 2008 Stockman1 Structure from Motion A moving camera/computer computes the 3D structure of the scene and its own motion.

Multiple View Geometry Marc Pollefeys University of North Carolina at Chapel Hill Modified by Philippos Mordohai.

Multiple View Reconstruction Class 23 Multiple View Geometry Comp Marc Pollefeys.

Projected image of a cube. Classical Calibration.

Lec 21: Fundamental Matrix

Algorithm Evaluation and Error Analysis class 7 Multiple View Geometry Comp Marc Pollefeys.

1 Final exam of Comp300a Venue: LG1 Time: 8h30—10h30, 31 May 2003, Sat. Content: everything in these slides + projective geometry before midterm Bring:

Multiple View Geometry

Structure Computation. How to compute the position of a point in 3- space given its image in two views and the camera matrices of those two views Use.

55:148 Digital Image Processing Chapter 11 3D Vision, Geometry Topics: Basics of projective geometry Points and hyperplanes in projective space Homography.

Multi-view geometry. Multi-view geometry problems Structure: Given projections of the same 3D point in two or more images, compute the 3D coordinates.

Automatic Camera Calibration

Geometry and Algebra of Multiple Views

Epipolar geometry The fundamental matrix and the tensor

1 Preview At least two views are required to access the depth of a scene point and in turn to reconstruct scene structure Multiple views can be obtained.

Day 1 how do we represent the shape around us? course outline motivation for gathering geometry from multiple images –our eyes are two views –structure.

Lecture 04 22/11/2011 Shai Avidan הבהרה : החומר המחייב הוא החומר הנלמד בכיתה ולא זה המופיע / לא מופיע במצגת.

1 Stereo vision and two-view geometry The goal of a stereo system is to get 3D information A stereo system consists at least of two ‘converging’ cameras.

Announcements Project 3 due Thursday by 11:59pm Demos on Friday; signup on CMS Prelim to be distributed in class Friday, due Wednesday by the beginning.

Parameter estimation. 2D homography Given a set of (x i,x i ’), compute H (x i ’=Hx i ) 3D to 2D camera projection Given a set of (X i,x i ), compute.

Geometry of Multiple Views

A minimal solution to the autocalibration of radial distortion Young Ki Baik (CV Lab.) (Wed)

Raquel A. Romano 1 Scientific Computing Seminar May 12, 2004 Projective Geometry for Computer Vision Projective Geometry for Computer Vision Raquel A.

Multi-linear Systems and Invariant Theory

3D reconstruction from uncalibrated images

55:148 Digital Image Processing Chapter 11 3D Vision, Geometry Topics: Basics of projective geometry Points and hyperplanes in projective space Homography.

55:148 Digital Image Processing Chapter 11 3D Vision, Geometry Topics: Basics of projective geometry Points and hyperplanes in projective space Homography.

776 Computer Vision Jan-Michael Frahm & Enrique Dunn Spring 2013.

Reconstruction from Two Calibrated Views Two-View Geometry

Geometry Reconstruction March 22, Fundamental Matrix An important problem: Determine the epipolar geometry. That is, the correspondence between.

Structure from motion Multi-view geometry Affine structure from motion Projective structure from motion Planches : –

Parameter estimation class 5 Multiple View Geometry CPSC 689 Slides modified from Marc Pollefeys’ Comp

Lec 26: Fundamental Matrix CS4670 / 5670: Computer Vision Kavita Bala.

Matrix Addition and Scalar Multiplication

Matrix Multiplication

René Vidal and Xiaodong Fan Center for Imaging Science

Three-view geometry 3-view constraint along F Minimal algebraic sol.

Parameter estimation class 5

Epipolar geometry.

3D Photography: Epipolar geometry

Estimating 2-view relationships

3D reconstruction class 11

1.3 Vector Equations.

Automatic Panoramic Image Stitching using Invariant Features

Lecture 15: Structure from motion

Parameter estimation class 6

Presentation transcript:

1 Three-view geometry 3-view constraint along F Minimal algebraic sol The content described in these slides is not required in the final exam!

2 Where are we? 1.1-view geometry  P matrix 2.2-view geometry  P, P’  F matrix 3.3-view geometry  P, P’, P’’  T tensor The understanding of the tensor T is not required for our class!!!!

3 u O u’ O u” How about points?

4 Transferring points in 3-view It’s about re-projection or transfer from the first two views into the third one. This tensorial equation gives 9 scalar equations, 4 of which are linearly independent.

5 Minimal data for algebraic sol. of 3 views Invariants of 6 pts and projective reconstruction from 3 uncalibrated images, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 17, no. 1, 1995 Given 6 corresponding points in 3 uncalibrated images, we can compute the three projection matrices P, P’, and P’’ (similar to 7 points in 2 uncalibrated images, get P and P’ (via F)). It is important to know this result, even though we will not derive it!

6 N-view geometry No algebraically independent geometric constraints for a set of more than 3 views.

7 Bundle adjustment: practical and optimal method of 3D reconstruction for multiple views Similar equation to calibration, but 1. It is for all images indexed by k, and 2. are unknowns, xi,yi, zi, and ti are also unknowns!!! 3. A very large optimisation problem

8 Automatic computation of a projective reconstruction for a sequence projective reconstruction: 2-view, 3-view, N-view obtaining correspondences over N-views

9 Pairwise matches: compute point matches between view pairs using robust F estimation Putative correspondences: over three views from two view matches RANSAC (6-pt algo) robust estimation of three view geometry, P, P’ and P’’ Generate additional matches From 2 images to 3 images:

10 Compute all 2-view reconstructions for consecutive views Compute all 3-view reconstruction for consecutive views Extend to sequence by hierarchical merging (projective) bundle-adjustment (autocalibration) (euclidean bundle-adjustment)

11 Some real examples of reconstruction.