# Epipolar Geometry. Readings: See Sections 10.1 and 15.6 of Forsyth and Ponce. Right Image. Left Image. e(p ) Epipolar Lines. e(q ) q R.

Save this PDF as:

Size: px
Start display at page:

Download "Epipolar Geometry. Readings: See Sections 10.1 and 15.6 of Forsyth and Ponce. Right Image. Left Image. e(p ) Epipolar Lines. e(q ) q R."

## Transcription

1 Epipolar Geometry We consider two perspective images of a scene as taken from a stereo pair of cameras (or equivalently, assume the scene is rigid and imaged with a single camera from two different locations). p X Left Image L p q L X q Right Image R p q R L e(p ) Epipolar Lines L e(q ) Left NP Right Epipole Right NP Given a scene point X p which is imaged in the left camera at p L, where could the image of the same point be in the right camera? We denote this point as p R. The relationship between such corresponding image points turns out to be both simple and useful. Readings: See Sections 10.1 and 15.6 of Forsyth and Ponce. 2503: Epipolar Geometry c A.D. Jepson and D.J. Fleet, 2006 Page: 1

2 Epipolar Line Two Perspective Cameras: Let d L and d R be the 3D positions of the nodal points of the left and right cameras. As discussed earlier in this course, we model perspective projection by placing the image plane in front of the nodal point. A scene point X p is then imaged in the left camera at p L, which is the point of intersection of the the image plane with the line containing the scene point X p and the nodal point d L (see previous figure). Epipolar Line: If we know the left image point p L, then the corresponding scene point X p is constrained to be on a ray through this image point. The position of X p on this ray is unknown. However, the image of this whole ray is a line in the right image, namely the epipolar line e( p L ). Epipolar Plane: An alternative geometric view is to consider the 3D plane containing the image point p L along with left and right nodal points d L and d R. Then the scene point X p and the corresponding right image point p R must also be on this epipolar plane. The intersection of this epipolar plane with the right image plane provides the epipolar line e( p L ). 2503: Epipolar Geometry Page: 2

3 Epipolar Constraint Epipolar Constraint: Suppose p L is the left image position for some scene point X p. Then the corresponding point p R in the right image must lie on the epipolar line e( p L ). Right Epipole Epipolar Lines R p q R L e(p ) L e(q ) Right Image Plane Notice the epipolar line e( p L ) depends on the position of the point in the left image. For example, another image point q L generally gives rise to a different epipolar line e( q L ). Epipole: All the epipolar lines in the right image pass through a single point (possibly at infinity) called the right epipole. This point is given by the intersection of the line containing the two nodal points d L and d R with the right image plane (see figures above and on p.1). (Notice that the line containing the two nodal points must be in all the epipolar planes, and hence its image must be on all the epipolar lines.) 2503: Epipolar Geometry Page: 3

4 Constraints on Correspondence Clearly we can swap the labels left and right in the above analysis, it does not matter which image we start with. The previous analysis showed there is a mapping between points in one image and epipolar lines in the other. Such a mapping would be computationally useful since it provides strong constraints on corresponding points in two images of the same scene. For example, for each point in one image we could limit the search for a corresponding point in the second image to just the epipolar line (instead of naively searching the whole second image). Alternatively, given a set of hypothesized correspondences, we can use the epipolar constraints to identify (some) outliers. To achieve these applications we need to be able to estimate the mapping from points to epipolar lines, which is what we consider next. We begin with the case of two calibrated cameras, and then consider the uncalibrated case. 2503: Epipolar Geometry Page: 4

5 Camera Coordinates and Image Formation Here we apply the image formation model introduced earlier to the stereo imaging setup. We introduce three coordinate frames: A world coordinate frame X w, The left and right camera coordinate frames, X c L and X c R. The origins of the left and right camera coordinate frames are at their nodal points, and their z-axes are aligned with the two optical axes. Therefore, as discussed in the image formation notes, given a 3D point X L c in the left camera s coordinates, the left image of this point is at p L c = f L X L 3,c X L c = p L 1,c p L 2,c f L. (1) Here, f L is the distance between the image plane and the nodal point for the left camera. A similar expression holds for the location of the right image point. However, before applying this expression, the 3D point X c L must be transformed from the left to the right camera s coordinates. We do this via the world coordinate frame X w. 2503: Epipolar Geometry Page: 5

6 External Calibration The external calibration parameters for the left camera provide the 3D rigid coordinate transformation from world coordinates to the left camera s coordinates (see the image formation notes) X c L = Mex[ L X w, T 1] T, (2) with M L ex a 3 4 matrix of the form M L ex = ( R L R L d L w ). (3) Here R L is a 3 3 rotation matrix and d L w is the location of the nodal point for the left camera in world coordinates. Similarly, the 3 4 matrix M R ex provides the external calibration of the right camera. 2503: Epipolar Geometry Page: 6

7 The Essential Matrix Let p w L and p w R be the left and right image points (written in world coordinates) for some given 3D point X w. Then the epipolar constraint states that these two image points and the two nodal points d w L, d w R are all coplanar. This constraint can be written as ( p w L d w L ) [( T d w L d w R ) ( p w R d ] w R ) = 0, (4) where denotes the cross-product. We rewrite this by replacing the cross-product by an equivalent matrixvector product, T p = [ T ] p, where [ T ] = 0 T 3 T 2 T 3 0 T 1 T 2 T 1 0. Also, we use (2) and (3) to write p L w d L w in terms of the left camera s coordinates as p w L d w L = (R L ) T p c L. Using the analogous expression for the right image point, we find that (4) can be rewritten as ( p L c ) T E p R c = 0, (epipolar constraint) (5) where E is the 3 3 essential matrix (or E-matrix) E = R L [ d L w d R w ] (R R ) T. (6) 2503: Epipolar Geometry Page: 7

8 Properties of the Essential Matrix Clearly, any nonzero scalar multiple of the E-matrix provides an equivalent epipolar constraint (5). From (6) it follows that the E-matrix has rank 2, with two equal nonzero singular values and one singular value at 0. Given a point p L c the corresponding point p R c where a = E T p L c. in the left image, the epipolar constraint (5) states that in the right image must be on the line a T p R c = a 1 p R 1,c + a 2 p R 2,c + a 3 f R = 0, The right epipole e R c is a null vector for E. It can be written as e R c = αm R ex[( d L w ) T, 1] T = αr R ( d L w d R w ), where α is a nonzero constant. Notice, using (6), a T e R c =( p c L ) T E e c R = α( p L c ) T R L [ d L w d R w ] (R R ) T R R ( d L w d R w )=0, so the epipole is on every epipolar line. Given a point p c R in the right image, analogous expressions give the epipolar line in the left image and the left epipole. 2503: Epipolar Geometry Page: 8

9 Internal Calibration We wish to rewrite the epipolar constraint (5) in terms of homogeneous pixel coordinates x L = (x L,y L, 1) T, where (x L,y L ) are the coordinates of an image point in terms of pixels. The internal calibration matrix Min L provides the transformation from camera coordinates to homogeneous pixel coordinates (see the image formation notes), x L = Min p L c L. (7) For example, a camera with rectangular pixels of size 1/s x by 1/s y, with nodal distance f, and piercing point (o x,o y ) (i.e., the intersection of the optical axis with the image plane provided in pixel coordinates) has the internal calibration matrix s x 0 o x /f M in = 0 s y o y /f. (8) 0 0 1/f We can use (7) to rewrite the epipolar constraint in terms of pixel coordinates. 2503: Epipolar Geometry Page: 9

10 The Fundamental Matrix By using (7) we can rewrite the epipolar constraint (5) in terms of homogeneous pixel coordinates in the left and right images as ( x L ) T F x R = 0. (9) Here the fundamental matrix (or F-matrix) is given by F = (Min L ) T E(Min R ) 1, (10) where the notation M T denotes the transpose of the inverse M. Similar to the E-matrix, the F-matrix has rank 2, but the two nonzero singular values need not be equal. The over-all scale of the F-matrix does not effect the epipolar constraint (9). So there are 7 remaining degrees of freedom in F. The right (left) null vector of F gives the homogeneous pixel coordinates for the right (left, resp.) epipole. More explicitly, for example, the epipolar constraint (9) states that, given a point x L in the left image, the corresponding point x R in the right image must be on the epipolar line a T x R = a 1 x R + a 2 y R + a 3 = 0, where a = F T x L. 2503: Epipolar Geometry Page: 10

11 Estimating the Fundamental Matrix Given corresponding image points {( x k L, x k R)}K k=1 the F-matrix. we wish to estimate Gold Standard Approach: Suppose the noise in the point positions x µ k, for µ = L, R is independent and normally distributed with mean zero and covariance Σ µ k. (Note that there is no noise in the third component of x µ k.) That is, x µ k = m µ k + n µ k, (11) where m µ k is the true position of the point and n µ k is the mean zero noise. Then the (maximum likelihood) problem is to find F R 3 3 along with m µ k for k =1,...,K and µ = L, R, such that the following objective function is minimized: O K ( x µ k m µ k )T (Σ µ k ) ( x µ k m µ k ) (12) µ {L,R} k=1 where (Σ µ k ) denotes the pseudo-inverse. We minimize this objective function O subject to the epipolar constraints: ( m L k )T F m R k = 0, k = 1,...,K, (13) rank(f) = 2 (14) Thus O is a quadratic objective function for the m µ k s, with nonlinear constraints (13) and (14). 2503: Epipolar Geometry Page: 11

12 Alternative Estimation Approaches We would like to be able to avoid a nonlinear optimization problem. The cost of this will be to obtain a noiser estimate of the F -matrix than the one provided by the previous gold standard approach. An initial simplification is to ignore the noise in x k L for the purpose of estimating the epipolar line e( m L k ). That is, we say the corresponding right image point x k R should be close to e( xl k ) instead of e( ml k ). This epipolar line e( x L k ) can be written as ( n T,c) x R =0, (15) where ( ) n 1 = c (I 2 0)F T x k L F T x k L. (16) 2 The normalization in (16) simply ensures n is the unit normal to the epipolar line e( x k L ). Therefore d( x R k,e( x L k )) ( n T,c) x R, (17) is the perpendicular distance between x R k L and the epipolar line e( x k ). We could try to minimize the sum of squares of these epipolar distances d( x k R,e( x k L )) for k =1,...,K. However, due to the normalization factor in (16), the objective function is not quadratic in the unknown F. 2503: Epipolar Geometry Page: 12

13 Algebraic Error Consider the reweighted epipolar distance objective function O(F ) = K k=1 k=1 w( x L k )d 2 ( x R k,e( x L k )) K [ ] ( x L k ) T F x k R 2. (18) Here the weights w( x k L ) are chosen to provide a quadratic objective function O(F ). That is, w( x L k )= (I 2 0)F T x L k 2 2. (19) This objective function corresponds to the algebraic error in the noiseless epipolar constraint (9). In terms of maximum likelihood estimation, Equation (18) is appropriate when the variances of the error in algebraic constraints (9) are roughly constant (and the means are zero). If the variances deviate significantly from this, then we will get poor estimates for F. Indeed, without any rescaling (which we discuss next), this approach provides excessively noisy estimates of F. 2503: Epipolar Geometry Page: 13

14 Renormalized 8-Point Algorithm Hartley (PAMI, 1997) introduced the following algorithm. Given corresponding points {( x k L, x k R)}K k=1 with K 8, 1. Recenter and rescale the image points using M µ, µ = L, R, such that s µ 0 b µ 1 M µ = 0 s µ b µ 2, (20) with 1 K 1 K K M µ x µ k = (0, 0, 1) T, (21) k=1 K [M µ x µ k (0, 0, 1)T ] 2 = (σ2 1,σ2 2, 0)T, (22) k=1 where σ1 2 + σ2 2 =2. Here [...]2 denotes the square of each element. Rescale the image points using r µ k = M µ x µ k for k =1,...,Kand µ = L, R. 2. Minimize the objective function O( ˆF ) O( ˆF ) K k=1 [ ( r L k ) T ˆF r R k ] 2. (23) Note this is a linear least squares problem for the elements of ˆF. (Continued on next page.) 2503: Epipolar Geometry Page: 14

15 Renormalized 8-Point Algorithm (Cont.) 3. Project ˆF to the nearest rank 2 matrix (with the error measured in the Frobenius norm): (a) Form the SVD of ˆF = UΣV T. In general Σ = diag[σ1,σ 2 2,σ 2 3] 2 with σi 2 σ2 i+1 for i =1, 2. (b) Reset σ 3 = 0. (c) Assign ˆF to be UΣV T. 4. Undo the normalization of the image points, F = (M L ) T ˆFM R (24) This algorithm has been found to provide reasonable estimates for the F -matrix given correspondence data with small amounts of noise (see Hartley and Zisserman, Multiple View Geometry in Computer Vision, Camb. Univ. Press., 2000). It is not robust to outliers. In order to deal with outliers, we apply the Random Sample Consensus (RANSAC) algorithm to the estimation of the F -matrix. 2503: Epipolar Geometry Page: 15

16 RANSAC Algorithm for the F-Matrix Suppose we are given corresponding points {( x k L, x k R)}K k=1, which may include outliers. Let ɛ>0 be an error tolerance, and T be the number of trials to do. Loop T times: 1. Randomly select 8 pairs ( x L k, x R k ). 2. Use the renormalized algorithm to solve for F using only the eight selected pairs of points. 3. Compute perpendicular errors d( x R k,e( x L k )) and d( x L k,e( x R k )),see (16) and (17) for 1 k K. 4. Identify inliers In = {k : d( x L k,e( x R k )) <ɛand d( x R k,e( x L k )) <ɛ, 1 k K}. 5. If the number of inliers In is the largest seen so far, remember the current estimate of F and the inlier set In. End loop. 6. Solve for F using all pairs with k In (i.e., all inliers). Re-solve for the inlier set In as done in steps 3 and 4 above. Can iterate step 6 until the set of inliers In does not change. 2503: Epipolar Geometry Page: 16

17 RANSAC: How Many Trials? Suppose our data set consists of a fraction p inliers, and 1 p outliers. How many trials T should be done so that we can be reasonably confident that at least one sampled data set of size d =8was all inliers? The probability of choosing d =8inliers from such a population is roughly p d when K>>d(it is exactly p d if we sample with replacement). So the probability that a given trial of RANSAC fails to select d inliers is 1 p d. Therefore, the probability that RANSAC failed to have any trial with d inliers is (1 p d ) T. In other words, the probability P 0 that at least one of the RANSAC trials will be a success is P 0 =1 (1 p d ) T Given an estimate for the fraction of inliers p in the data set, we could then choose T such that P 0 > 0.95, say. That is, T>log(1 P 0 )/ log(1 p d ). For example, for 70% inliers and d =8, we require T > 50. Alternatively, if we only have 50% inliers, the same formula states that T should be chosen to be at least : Epipolar Geometry Page: 17

18 Example Given local image features, RANSAC was used to fit the F -matrix. Here have choosen random colours to circle image features. The same colour is then used for the corresponding point in the other image, and also for the epipolar lines generated from these two points. Note: 1. By construction, each point lies close to the epipolar line generated by its corresponding point in the other image. 2. A visual sanity check can be obtained by sampling other points on one epipolar line, and checking that they also appear somewhere along the corresponding epipolar line. This must be the case since, when the F-matrix is correct, both epipolar lines correspond to the intersection of the scene with the epipolar plane. (Compare the current fit with the result of a poor fit shown on p.19.) 3. The intersection of the epipolar lines corresponds to the epipole in each image. The nodal point of the second camera is on the line (in world coordinates) containing the nodal point of the first camera and the epipole in the first image. 2503: Epipolar Geometry Page: 18

19 Poorly Fitted F-Matrix The same local image features were used as in the previous example, andransacwasusedtofitthef -matrix (but with only 10 trials). The solution it found is displayed below: Note: 1. The feature points are still near the corresponding epipolar lines. Here 82% of the data points are within 4 pixels of the corresponding epipolar line. In contrast, the solution on the previous page achieved 94%. 2. However, the visual sanity check fails. This is most apparent for (proposed) epipolar planes which intersect the scene over a large range of depths. For example, consider the (proposed) epipolar planes which cut across the tower at the top of the image and at least one of the buildings in front. 2503: Epipolar Geometry Page: 19

### Wii Remote Calibration Using the Sensor Bar

Wii Remote Calibration Using the Sensor Bar Alparslan Yildiz Abdullah Akay Yusuf Sinan Akgul GIT Vision Lab - http://vision.gyte.edu.tr Gebze Institute of Technology Kocaeli, Turkey {yildiz, akay, akgul}@bilmuh.gyte.edu.tr

### CS 534: Computer Vision 3D Model-based recognition

CS 534: Computer Vision 3D Model-based recognition Ahmed Elgammal Dept of Computer Science CS 534 3D Model-based Vision - 1 High Level Vision Object Recognition: What it means? Two main recognition tasks:!

### Projective Reconstruction from Line Correspondences

Projective Reconstruction from Line Correspondences Richard I. Hartley G.E. CRD, Schenectady, NY, 12301. Abstract The paper gives a practical rapid algorithm for doing projective reconstruction of a scene

### Epipolar Geometry and Visual Servoing

Epipolar Geometry and Visual Servoing Domenico Prattichizzo joint with with Gian Luca Mariottini and Jacopo Piazzi www.dii.unisi.it/prattichizzo Robotics & Systems Lab University of Siena, Italy Scuoladi

### STATISTICS AND DATA ANALYSIS IN GEOLOGY, 3rd ed. Clarificationof zonationprocedure described onpp. 238-239

STATISTICS AND DATA ANALYSIS IN GEOLOGY, 3rd ed. by John C. Davis Clarificationof zonationprocedure described onpp. 38-39 Because the notation used in this section (Eqs. 4.8 through 4.84) is inconsistent

### Single Image 3D Reconstruction of Ball Motion and Spin From Motion Blur

Single Image 3D Reconstruction of Ball Motion and Spin From Motion Blur An Experiment in Motion from Blur Giacomo Boracchi, Vincenzo Caglioti, Alessandro Giusti Objective From a single image, reconstruct:

### Time Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication

Time Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication Thomas Reilly Data Physics Corporation 1741 Technology Drive, Suite 260 San Jose, CA 95110 (408) 216-8440 This paper

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.

An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points

### Daniel F. DeMenthon and Larry S. Davis. Center for Automation Research. University of Maryland

Model-Based Object Pose in 25 Lines of Code Daniel F. DeMenthon and Larry S. Davis Computer Vision Laboratory Center for Automation Research University of Maryland College Park, MD 20742 Abstract In this

### Monash University Clayton s School of Information Technology CSE3313 Computer Graphics Sample Exam Questions 2007

Monash University Clayton s School of Information Technology CSE3313 Computer Graphics Questions 2007 INSTRUCTIONS: Answer all questions. Spend approximately 1 minute per mark. Question 1 30 Marks Total

### Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they

### 4.3 Least Squares Approximations

18 Chapter. Orthogonality.3 Least Squares Approximations It often happens that Ax D b has no solution. The usual reason is: too many equations. The matrix has more rows than columns. There are more equations

### Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition

Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition 1. Image Pre-Processing - Pixel Brightness Transformation - Geometric Transformation - Image Denoising 1 1. Image Pre-Processing

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### Using Many Cameras as One

Using Many Cameras as One Robert Pless Department of Computer Science and Engineering Washington University in St. Louis, Box 1045, One Brookings Ave, St. Louis, MO, 63130 pless@cs.wustl.edu Abstract We

### 8. Linear least-squares

8. Linear least-squares EE13 (Fall 211-12) definition examples and applications solution of a least-squares problem, normal equations 8-1 Definition overdetermined linear equations if b range(a), cannot

### An Efficient Solution to the Five-Point Relative Pose Problem

An Efficient Solution to the Five-Point Relative Pose Problem David Nistér Sarnoff Corporation CN5300, Princeton, NJ 08530 dnister@sarnoff.com Abstract An efficient algorithmic solution to the classical

### ( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### Metrics on SO(3) and Inverse Kinematics

Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### CS3220 Lecture Notes: QR factorization and orthogonal transformations

CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss

### 521493S Computer Graphics. Exercise 2 & course schedule change

521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 16-18 in TS128 Question 2.1 Given two nonparallel,

### Segmentation of building models from dense 3D point-clouds

Segmentation of building models from dense 3D point-clouds Joachim Bauer, Konrad Karner, Konrad Schindler, Andreas Klaus, Christopher Zach VRVis Research Center for Virtual Reality and Visualization, Institute

: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### An Iterative Image Registration Technique with an Application to Stereo Vision

An Iterative Image Registration Technique with an Application to Stereo Vision Bruce D. Lucas Takeo Kanade Computer Science Department Carnegie-Mellon University Pittsburgh, Pennsylvania 15213 Abstract

### Solution Guide III-C. 3D Vision. Building Vision for Business. MVTec Software GmbH

Solution Guide III-C 3D Vision MVTec Software GmbH Building Vision for Business Machine vision in 3D world coordinates, Version 10.0.4 All rights reserved. No part of this publication may be reproduced,

### Section 2.4: Equations of Lines and Planes

Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y

### 9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

### CHAPTER FIVE. 5. Equations of Lines in R 3

118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a

### THE problem of visual servoing guiding a robot using

582 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 13, NO. 4, AUGUST 1997 A Modular System for Robust Positioning Using Feedback from Stereo Vision Gregory D. Hager, Member, IEEE Abstract This paper

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson

JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3

### Factor analysis. Angela Montanari

Factor analysis Angela Montanari 1 Introduction Factor analysis is a statistical model that allows to explain the correlations between a large number of observed correlated variables through a small number

### Multivariate Normal Distribution

Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues

### DYNAMIC RANGE IMPROVEMENT THROUGH MULTIPLE EXPOSURES. Mark A. Robertson, Sean Borman, and Robert L. Stevenson

c 1999 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or

### (a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,

Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Master s Theory Exam Spring 2006

Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem

### B4 Computational Geometry

3CG 2006 / B4 Computational Geometry David Murray david.murray@eng.o.ac.uk www.robots.o.ac.uk/ dwm/courses/3cg Michaelmas 2006 3CG 2006 2 / Overview Computational geometry is concerned with the derivation

### CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

### Content. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11

Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter

### Manifold Learning Examples PCA, LLE and ISOMAP

Manifold Learning Examples PCA, LLE and ISOMAP Dan Ventura October 14, 28 Abstract We try to give a helpful concrete example that demonstrates how to use PCA, LLE and Isomap, attempts to provide some intuition

### Dimensionality Reduction: Principal Components Analysis

Dimensionality Reduction: Principal Components Analysis In data mining one often encounters situations where there are a large number of variables in the database. In such situations it is very likely

### Quick Guide for Data Collection on the NIU Bruker Smart CCD

Quick Guide for Data Collection on the NIU Bruker Smart CCD 1. Create a new project 2. Optically align the crystal 3. Take rotation picture 4. Collect matrix to determine unit cell 5. Refine unit cell

### NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

### ME 115(b): Solution to Homework #1

ME 115(b): Solution to Homework #1 Solution to Problem #1: To construct the hybrid Jacobian for a manipulator, you could either construct the body Jacobian, JST b, and then use the body-to-hybrid velocity

### Two-Frame Motion Estimation Based on Polynomial Expansion

Two-Frame Motion Estimation Based on Polynomial Expansion Gunnar Farnebäck Computer Vision Laboratory, Linköping University, SE-581 83 Linköping, Sweden gf@isy.liu.se http://www.isy.liu.se/cvl/ Abstract.

### Notes on Elastic and Inelastic Collisions

Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just

### Imputing Values to Missing Data

Imputing Values to Missing Data In federated data, between 30%-70% of the data points will have at least one missing attribute - data wastage if we ignore all records with a missing value Remaining data

### Non-Inferiority Tests for One Mean

Chapter 45 Non-Inferiority ests for One Mean Introduction his module computes power and sample size for non-inferiority tests in one-sample designs in which the outcome is distributed as a normal random

### Factorization Theorems

Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

### 1.5 Equations of Lines and Planes in 3-D

40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3-D Recall that given a point P = (a, b, c), one can draw a vector from

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### Lecture 8: Signal Detection and Noise Assumption

ECE 83 Fall Statistical Signal Processing instructor: R. Nowak, scribe: Feng Ju Lecture 8: Signal Detection and Noise Assumption Signal Detection : X = W H : X = S + W where W N(, σ I n n and S = [s, s,...,

### LINES AND PLANES CHRIS JOHNSON

LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3-space, as well as define the angle between two non-parallel planes, and determine the distance

### SAS Software to Fit the Generalized Linear Model

SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Network quality control

Network quality control Network quality control P.J.G. Teunissen Delft Institute of Earth Observation and Space systems (DEOS) Delft University of Technology VSSD iv Series on Mathematical Geodesy and

### Geometric Algebra Computing Analysis of point clouds 27.11.2014 Dr. Dietmar Hildenbrand

Geometric Algebra Computing Analysis of point clouds 27.11.2014 Dr. Dietmar Hildenbrand Technische Universität Darmstadt Literature Book Foundations of Geometric Algebra Computing, Dietmar Hildenbrand

### ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

### CS 5614: (Big) Data Management Systems. B. Aditya Prakash Lecture #18: Dimensionality Reduc7on

CS 5614: (Big) Data Management Systems B. Aditya Prakash Lecture #18: Dimensionality Reduc7on Dimensionality Reduc=on Assump=on: Data lies on or near a low d- dimensional subspace Axes of this subspace

### Probabilistic Latent Semantic Analysis (plsa)

Probabilistic Latent Semantic Analysis (plsa) SS 2008 Bayesian Networks Multimedia Computing, Universität Augsburg Rainer.Lienhart@informatik.uni-augsburg.de www.multimedia-computing.{de,org} References

Name: University of Chicago Graduate School of Business Business 41000: Business Statistics Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper for the formulas. 2. Throughout

### (Quasi-)Newton methods

(Quasi-)Newton methods 1 Introduction 1.1 Newton method Newton method is a method to find the zeros of a differentiable non-linear function g, x such that g(x) = 0, where g : R n R n. Given a starting

### x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

### Terrain Traversability Analysis using Organized Point Cloud, Superpixel Surface Normals-based segmentation and PCA-based Classification

Terrain Traversability Analysis using Organized Point Cloud, Superpixel Surface Normals-based segmentation and PCA-based Classification Aras Dargazany 1 and Karsten Berns 2 Abstract In this paper, an stereo-based

### Statistical Models in R

Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Structure of models in R Model Assessment (Part IA) Anova

### The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression

The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonal-diagonal-orthogonal type matrix decompositions Every

### 3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture 6. 2013 Steve Marschner (with previous instructors James/Bala)

3D Tranformations CS 4620 Lecture 6 1 Translation 2 Translation 2 Translation 2 Translation 2 Scaling 3 Scaling 3 Scaling 3 Scaling 3 Rotation about z axis 4 Rotation about z axis 4 Rotation about x axis

### We shall turn our attention to solving linear systems of equations. Ax = b

59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system

### What can be seen in three dimensions with uncalibrated stereo rig?

What can be seen in three dimensions with uncalibrated stereo rig? an Olivier D. Faugeras INRIA-Sophia, 2004 Route des Lucioles, 06560 Valbonne, France Abstract. This paper addresses the problem of determining

### Automatic Labeling of Lane Markings for Autonomous Vehicles

Automatic Labeling of Lane Markings for Autonomous Vehicles Jeffrey Kiske Stanford University 450 Serra Mall, Stanford, CA 94305 jkiske@stanford.edu 1. Introduction As autonomous vehicles become more popular,

### V-PITS : VIDEO BASED PHONOMICROSURGERY INSTRUMENT TRACKING SYSTEM. Ketan Surender

V-PITS : VIDEO BASED PHONOMICROSURGERY INSTRUMENT TRACKING SYSTEM by Ketan Surender A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science (Electrical Engineering)

### Multivariate Analysis of Variance (MANOVA): I. Theory

Gregory Carey, 1998 MANOVA: I - 1 Multivariate Analysis of Variance (MANOVA): I. Theory Introduction The purpose of a t test is to assess the likelihood that the means for two groups are sampled from the

### Factor Analysis. Principal components factor analysis. Use of extracted factors in multivariate dependency models

Factor Analysis Principal components factor analysis Use of extracted factors in multivariate dependency models 2 KEY CONCEPTS ***** Factor Analysis Interdependency technique Assumptions of factor analysis

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### Assessment. Presenter: Yupu Zhang, Guoliang Jin, Tuo Wang Computer Vision 2008 Fall

Automatic Photo Quality Assessment Presenter: Yupu Zhang, Guoliang Jin, Tuo Wang Computer Vision 2008 Fall Estimating i the photorealism of images: Distinguishing i i paintings from photographs h Florin

### THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages

### THE CONTROL OF A ROBOT END-EFFECTOR USING PHOTOGRAMMETRY

THE CONTROL OF A ROBOT END-EFFECTOR USING PHOTOGRAMMETRY Dr. T. Clarke & Dr. X. Wang Optical Metrology Centre, City University, Northampton Square, London, EC1V 0HB, UK t.a.clarke@city.ac.uk, x.wang@city.ac.uk

### Computational Optical Imaging - Optique Numerique. -- Deconvolution --

Computational Optical Imaging - Optique Numerique -- Deconvolution -- Winter 2014 Ivo Ihrke Deconvolution Ivo Ihrke Outline Deconvolution Theory example 1D deconvolution Fourier method Algebraic method

### Applications to Data Smoothing and Image Processing I

Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is

### Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

### REAL TIME 3D FUSION OF IMAGERY AND MOBILE LIDAR INTRODUCTION

REAL TIME 3D FUSION OF IMAGERY AND MOBILE LIDAR Paul Mrstik, Vice President Technology Kresimir Kusevic, R&D Engineer Terrapoint Inc. 140-1 Antares Dr. Ottawa, Ontario K2E 8C4 Canada paul.mrstik@terrapoint.com

### Automatic Calibration of an In-vehicle Gaze Tracking System Using Driver s Typical Gaze Behavior

Automatic Calibration of an In-vehicle Gaze Tracking System Using Driver s Typical Gaze Behavior Kenji Yamashiro, Daisuke Deguchi, Tomokazu Takahashi,2, Ichiro Ide, Hiroshi Murase, Kazunori Higuchi 3,

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Fast and Robust Normal Estimation for Point Clouds with Sharp Features

1/37 Fast and Robust Normal Estimation for Point Clouds with Sharp Features Alexandre Boulch & Renaud Marlet University Paris-Est, LIGM (UMR CNRS), Ecole des Ponts ParisTech Symposium on Geometry Processing

### 3D Model based Object Class Detection in An Arbitrary View

3D Model based Object Class Detection in An Arbitrary View Pingkun Yan, Saad M. Khan, Mubarak Shah School of Electrical Engineering and Computer Science University of Central Florida http://www.eecs.ucf.edu/

### Lines and Planes 1. x(t) = at + b y(t) = ct + d

1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b

### 1 Interest rates, and risk-free investments

Interest rates, and risk-free investments Copyright c 2005 by Karl Sigman. Interest and compounded interest Suppose that you place x 0 (\$) in an account that offers a fixed (never to change over time)

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### Localization of Mobile Robots Using Odometry and an External Vision Sensor

Sensors 21, 1, 3655-368; doi:1.339/s143655 OPEN ACCESS sensors ISSN 1424-822 www.mdpi.com/journal/sensors Article Localization of Mobile Robots Using Odometry and an External Vision Sensor Daniel Pizarro,

### A Negative Result Concerning Explicit Matrices With The Restricted Isometry Property

A Negative Result Concerning Explicit Matrices With The Restricted Isometry Property Venkat Chandar March 1, 2008 Abstract In this note, we prove that matrices whose entries are all 0 or 1 cannot achieve