Structure from motion II


 Morgan Short
 1 years ago
 Views:
Transcription
1 Structure from motion II Gabriele Bleser Thanks to Marc Pollefeys, David Nister and David Lowe
2 Introduction Previous lecture: structure and motion I Robust fundamental matrix estimation Factorization Today: structure and motion II Structure and motion loop Triangulation Drift reductionmethods Wide baseline matching (SIFT) Next lectures: dense reconstruction Lecture 3D Computer Vision 2
3 Sequential SAM Alternating estimation of camera poses and 3D feature locations (triangulation) from a (continuous) image sequence. e e F matrix estimation and factorization (last lecture) 2D feature location (from image processing) Camera pose Initialize first camera poses How? Lecture 3D Computer Vision 3
4 Sequential SAM Alternating estimation of camera poses and 3D feature locations (triangulation) from a (continuous) image sequence. 3D feature location 2D feature location (from image processing) Camera pose Initialize 3D points (triangulation) Lecture 3D Computer Vision 4
5 Sequential SAM Alternating estimation of camera poses and 3D feature locations (triangulation) from a (continuous) image sequence. 3D feature location 2D feature location (from image processing) Camera pose Estimate next camera pose Lecture 3D Computer Vision 5
6 Sequential SAM Alternating estimation of camera poses and 3D feature locations (triangulation) from a (continuous) image sequence. 3D feature location 2D feature location (from image processing) Camera pose Initialize additional 3D points Lecture 3D Computer Vision 6
7 Sequential SAM Alternating estimation of camera poses and 3D feature locations (triangulation) from a (continuous) image sequence. E.g. recursive filtering 3D feature location 2D feature location (from image processing) Camera pose Refine known 3D points with new camera poses Lecture 3D Computer Vision 7
8 Offline SAM Alternating estimation of camera poses and 3D feature locations (triangulation) from a (continuous) image sequence. 3D feature location Not sequential anymore: mostly in offline SAM, or local bundle adjustment for drift reduction 2D feature location (from image processing) Camera pose Refine known cameras with new 3D points Lecture 3D Computer Vision 8
9 We know Outline how to track features in a continuous image sequence (lecture 6) how to estimate camera poses from 2D/3D correspondences (lectures 2, 3, 5) from 2D/2D correspondences (lectures 4, 7) Now: Triangulation: estimate the 3D point X given at least 2 known cameras Cand C and 2 corresponding feature points xand x (i.e. 2 camera views). X x x C C Lecture 3D Computer Vision 9
10 Parallel cameras (simple case) x Z Then, y = y and depth Z can be computed from disparity: x d Derivation via equal triangles Note: feature displacement (disparity) is inversely proportional to depth as 0, Lecture 3D Computer Vision 10
11 Converging cameras: rectification Can be converted/rectified to parallel camera geometry: project images to plane parallel to baseline 2D homography How could this be done with methods that you know? Lecture 3D Computer Vision epipolar plane 11
12 Converging cameras: general configuration Arbitrary camera poses 3D triangulation required X j x 1j x 3j C 1 x 2j C 2 C 3 Lecture 3D Computer Vision 12
13 Triangulation Given: corresponding measured (i.e. noisy) feature points x and x and camera poses, P and P Problem: in the presence of noise, back projected rays do not intersect Rays are skew in space How could this be solved? and measured points do not lie on corresponding epipolar lines Lecture 3D Computer Vision 13
14 Reminder: direct linear transform Objective Given a sufficient number, n, of measurements determine such that Algorithm (i) Set up linear equation system (m is number of parameters): (ii) If b is zero solve with SVD. (iii) Else solve with pseudo inverse. (iv) Determine X from x. For triangulation: What is the measurement, what the unkown? Which equation is used? How many measurements are needed? Why? Lecture 3D Computer Vision 14
15 Triangulation: algebraic solution Equation for one camera view (camera pose P and feature point x) 4 entries, 3 DOF i th row of P (3x4) matrix (2x4) for one equation 3 equations, only 2 linearly independent drop third row Lecture 3D Computer Vision 15
16 Triangulation: algebraic solution Stack equations for all camera views: Minimizes algebraic error no geometrical interpretation X has 3 DOF (4 parameters due to homogeneous representation) One camera view gives two linearly independent equations Two camera views are sufficient for a minimal solution Solution for X is eigenvector of A T A corresponding to smallest eigenvalue Homogenize solution for X to obtain Euclidean vector Lecture 3D Computer Vision 16
17 Triangulation: vector solution Compute the mid point of the shortest line between the two rays Lecture 3D Computer Vision 17
18 Triangulation: minimize geometric error Estimate 3D point, which exactly satisfies the supplied camera geometry and, so it projects as where and are closest to theactualimagemeasurements. Assumes perfect camera poses! Lecture 3D Computer Vision 18
19 Triangulation: properties The smaller the angle (small baseline, big distance), the bigger the reconstruction uncertainty! X? β X? x x x x C C C C Lecture 3D Computer Vision 19
20 Structure and motion: the big problem Drift (accumulating error) How can we reduce this? Different methods in online and offline case Lecture 3D Computer Vision 20
21 Drift reduction (feature level) Reduce drift in feature tracking/matching Extend feature tracks Reacquire lost features, e.g. Advanced optical flow algorithm (lecture 6) Wide baseline matching (later today) Lecture 3D Computer Vision 21
22 Drift reduction (geometry level) Careful (precise, robust) 3D point initialization How? Triangulate over a history of measurements Enforce a minimal angle θ Use RANSAC to eliminate outliers Accept only well reconstructed points Incorporate measurement uncertainties (error propagation) E.g. simple stochastic model and WLS estimation (lecture 5): All entities modelled as Gaussian random variables Lecture 3D Computer Vision 22
23 Drift reduction (geometry level) Iterative/recursive reconstruction Refine 3D structure over time (e.g. update a 3D point each time the feature is observed) Methods: (Local) bundle adjustment Repeated triangulation Recursive filtering (e.g. extended Kalman filter) Lecture 3D Computer Vision 23
24 Results: online SAM (360 rotation) With drift reduction: error reduced by a factor of 10 Without drift reduction Lecture 3D Computer Vision 24
25 Global bundle adjustment: jointly optimize over all camera poses and 3D points (lecture 7) Offline drift reduction Estimate Residual/reprojection error Parameter vector containing all camera poses and 3D points Lecture 3D Computer Vision 25
26 Offline drift reduction Global bundle adjustment: jointly optimize over all camera poses and 3D points (lecture 7) Requires features to be matched between very different views Simple feature matching/tracking is not enough Wide baseline matching required Lecture 3D Computer Vision 26
27 Wide baseline matching Requirement to cope with larger variations between images Translation, rotation, scaling geometric Foreshortening, other distortions transformations Non diffuse reflections photometric Illumination changes Examples: Could such correspondences be found using the advanced optical flow algorithm (KLT with affine model and multi scale extension)? Lecture 3D Computer Vision 27
28 Reminder: Advanced optical flow algorithm Usually two stage approach: Stage 1: Estimate pure translation from frame to frame (coarse to fine) Stage 2: Estimate the affine transformation and illumination parameters to the initial feature appearance (use previous estimate as initial guess),,,,,,,,,, Lecture 3D Computer Vision 28
29 Lowe s SIFT features SIFT = Scale Invariant Feature Transform Extensively used for wide baseline matching Rotation and scale invariant keypoint (feature) detector and descriptor Robust against other geometric distortion and photometric changes Lecture 3D Computer Vision 29
30 Explanation: 2 nd order gradient filters Laplacian of Gaussians (LoG): gauss filtered sum of 2 nd derivatives in x and y direction Example: Example filter mask p Edges are zero crossing Edge profile 2 nd derivative Lecture 3D Computer Vision 30
31 Explanation: 2nd order gradient filters Approximate LoG with a Difference of Gaussians (DoG): 2 L Gxx x y Gyy x y (,, ) (,, ) (Laplacian) DoG G( x, y, k ) G( x, y, ) (Difference of Gaussians)
32 Explanation: 2 nd order gradient filters Example: DoG Search for zerocrossing to find edges Gauss with σ=4 Gauss with σ=6 Search for local maxima and minima to detect stable blob like image features [Mikolajczyk, 2002] Lecture 3D Computer Vision 32
33 SIFT keypoint detector: position and scale SIFT uses a pyramid of DoGs for scale invariant keypoint detection Algorithm: From the input image, construct a pyramid of DoGs Search for local maxima and minima in both position and scale space Provides candidates for scale invariant keypoints Lecture 3D Computer Vision 33
34 SIFT keypoint detector: orientation Next: assign orientation. Why? Algorithm: Create histogram of local gradient directions computed at selected positions and scales Assign canonical orientation at peak of smoothed histogram Now each keypoint specifies 2D position, scale and orientation 0 2 Lecture 3D Computer Vision 34
35 SIFT keypoint descriptor Compute from keypoint specific image patch: Centered around keypoint (as usual) Oriented and scaled according to keypoint (rotation, scale invariant extension) Based on orientation histograms: Sample image gradients over (16x16) array Create array of orientation histograms (4x4) histogram array, 8 orientations per cell 128 dimensional descriptor (normalize to length 1) Lecture 3D Computer Vision 35
36 SIFT descriptor matching Simple strategy : Compute keypoints for both images Compute pair wise distances between keypoint sets (e.g. Euclidean vector distance) Assign closest keypoints, avoid double assignments (usually many outliers) Use RANSAC for geometry estimation See references for advanced methods! Lecture 3D Computer Vision 36
37 Outlook: next lectures Dense reconstruction overall goal is 3D model generation Example: Spherical input pictures of a church Feature matching (SIFT) Solve correspondence and triangulation problem for each pixel! Sparse reconstruction Dense reconstruction Lecture 3D Computer Vision 37
38 References Hartley R. and Sturm P.: Triangulation, International Conference on Computer Analysis of Images and Patterns, Bleser et al: Real time Vision based Tracking and Reconstruction, Journal of Real Time Image Processing 2 (2007), 2 3, pp , Berlin, Heidelberg, New York : Springer Verlag. D. G. Lowe. Distinctive Image Features from Scale Invariant Keypoints. International Journal of Computer Vision, 60:91 110, November Lecture 3D Computer Vision 38
C4 Computer Vision. 4 Lectures Michaelmas Term Tutorial Sheet Prof A. Zisserman. fundamental matrix, recovering egomotion, applications.
C4 Computer Vision 4 Lectures Michaelmas Term 2004 1 Tutorial Sheet Prof A. Zisserman Overview Lecture 1: Stereo Reconstruction I: epipolar geometry, fundamental matrix. Lecture 2: Stereo Reconstruction
More informationComputer Vision  part II
Computer Vision  part II Review of main parts of Section B of the course School of Computer Science & Statistics Trinity College Dublin Dublin 2 Ireland www.scss.tcd.ie Lecture Name Course Name 1 1 2
More informationEpipolar Geometry Prof. D. Stricker
Outline 1. Short introduction: points and lines Epipolar Geometry Prof. D. Stricker 2. Two views geometry: Epipolar geometry Relation point/line in two views The geometry of two cameras Definition of the
More informationA Study on SURF Algorithm and RealTime Tracking Objects Using Optical Flow
, pp.233237 http://dx.doi.org/10.14257/astl.2014.51.53 A Study on SURF Algorithm and RealTime Tracking Objects Using Optical Flow Giwoo Kim 1, HyeYoun Lim 1 and DaeSeong Kang 1, 1 Department of electronices
More informationCamera calibration and epipolar geometry. Odilon Redon, Cyclops, 1914
Camera calibration and epipolar geometry Odilon Redon, Cyclops, 94 Review: Alignment What is the geometric relationship between pictures taken by cameras that share the same center? How many points do
More informationEpipolar Geometry and Stereo Vision
04/12/11 Epipolar Geometry and Stereo Vision Computer Vision CS 543 / ECE 549 University of Illinois Derek Hoiem Many slides adapted from Lana Lazebnik, Silvio Saverese, Steve Seitz, many figures from
More informationDepth from a single camera
Depth from a single camera Fundamental Matrix Essential Matrix Active Sensing Methods School of Computer Science & Statistics Trinity College Dublin Dublin 2 Ireland www.scss.tcd.ie 1 1 Geometry of two
More informationFeature Tracking and Optical Flow
02/09/12 Feature Tracking and Optical Flow Computer Vision CS 543 / ECE 549 University of Illinois Derek Hoiem Many slides adapted from Lana Lazebnik, Silvio Saverse, who in turn adapted slides from Steve
More informationIntroduction Epipolar Geometry Calibration Methods Further Readings. Stereo Camera Calibration
Stereo Camera Calibration Stereo Camera Calibration Stereo Camera Calibration Stereo Camera Calibration 12.10.2004 Overview Introduction Summary / Motivation Depth Perception Ambiguity of Correspondence
More informationModified Sift Algorithm for Appearance Based Recognition of American Sign Language
Modified Sift Algorithm for Appearance Based Recognition of American Sign Language Jaspreet Kaur,Navjot Kaur Electronics and Communication Engineering Department I.E.T. Bhaddal, Ropar, Punjab,India. Abstract:
More informationIntroduction to Computer Vision. Week 11, Fall 2010 Instructor: Prof. Ko Nishino
Introduction to Computer Vision Week 11, Fall 2010 Instructor: Prof. Ko Nishino The Projective Plane Why do we need homogeneous coordinates? represent points at infinity, homographies, perspective projection,
More informationStructured light systems
Structured light systems Tutorial 1: 9:00 to 12:00 Monday May 16 2011 Hiroshi Kawasaki & Ryusuke Sagawa Today Structured light systems Part I (Kawasaki@Kagoshima Univ.) Calibration of Structured light
More informationEpipolar 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.
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).
More informationHomography. Dr. Gerhard Roth
Homography Dr. Gerhard Roth Epipolar Geometry P P l P r Epipolar Plane p l Epipolar Lines p r O l e l e r O r Epipoles P r = R(P l T) Homography Consider a point x = (u,v,1) in one image and x =(u,v,1)
More informationWhat is an Edge? Computer Vision Week 4. How to detect edges? What is an Edge? Edge Detection techniques. Edge Detection techniques.
What is an Edge? Computer Vision Week 4 Edge Detection Linear filtering; pyramids, wavelets Interest Operators surface normal discontinuity depth discontinuity surface color discontinuity illumination
More informationProbabilistic Latent Semantic Analysis (plsa)
Probabilistic Latent Semantic Analysis (plsa) SS 2008 Bayesian Networks Multimedia Computing, Universität Augsburg Rainer.Lienhart@informatik.uniaugsburg.de www.multimediacomputing.{de,org} References
More informationFace Recognition using SIFT Features
Face Recognition using SIFT Features Mohamed Aly CNS186 Term Project Winter 2006 Abstract Face recognition has many important practical applications, like surveillance and access control.
More informationFast field survey with a smartphone
Fast field survey with a smartphone A. Masiero F. Fissore, F. Pirotti, A. Guarnieri, A. Vettore CIRGEO Interdept. Research Center of Geomatics University of Padova Italy cirgeo@unipd.it 1 Mobile Mapping
More informationEXPERIMENTAL EVALUATION OF RELATIVE POSE ESTIMATION ALGORITHMS
EXPERIMENTAL EVALUATION OF RELATIVE POSE ESTIMATION ALGORITHMS Marcel Brückner, Ferid Bajramovic, Joachim Denzler Chair for Computer Vision, FriedrichSchillerUniversity Jena, ErnstAbbePlatz, 7743 Jena,
More informationParallel Tracking and Mapping for Small AR Workspaces
Parallel Tracking and Mapping for Small AR Workspaces Georg Klein and David Murray Active Vision Lab, Oxford This is a PDF of the slides of the talk given at ISMAR 2007 Aim AR with a handheld camera Visual
More informationProblem definition: optical flow
Motion Estimation http://www.sandlotscience.com/distortions/breathing_objects.htm http://www.sandlotscience.com/ambiguous/barberpole.htm Why estimate motion? Lots of uses Track object behavior Correct
More informationCSE 252B: Computer Vision II
CSE 252B: Computer Vision II Lecturer: Serge Belongie Scribes: Jia Mao, Andrew Rabinovich LECTURE 9 Affine and Euclidean Reconstruction 9.1. Stratified reconstruction Recall that in 3D reconstruction from
More informationMetrics 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
More informationClassification of Fingerprints. Sarat C. Dass Department of Statistics & Probability
Classification of Fingerprints Sarat C. Dass Department of Statistics & Probability Fingerprint Classification Fingerprint classification is a coarse level partitioning of a fingerprint database into smaller
More informationGeometric Camera Parameters
Geometric Camera Parameters What assumptions have we made so far? All equations we have derived for far are written in the camera reference frames. These equations are valid only when: () all distances
More informationLeastSquares Intersection of Lines
LeastSquares Intersection of Lines Johannes Traa  UIUC 2013 This writeup derives the leastsquares solution for the intersection of lines. In the general case, a set of lines will not intersect at a
More informationImage Segmentation and Registration
Image Segmentation and Registration Dr. Christine Tanner (tanner@vision.ee.ethz.ch) Computer Vision Laboratory, ETH Zürich Dr. Verena Kaynig, Machine Learning Laboratory, ETH Zürich Outline Segmentation
More informationLecture 2: Homogeneous Coordinates, Lines and Conics
Lecture 2: Homogeneous Coordinates, Lines and Conics 1 Homogeneous Coordinates In Lecture 1 we derived the camera equations λx = P X, (1) where x = (x 1, x 2, 1), X = (X 1, X 2, X 3, 1) and P is a 3 4
More information1. Bag of visual words model: recognizing object categories
1. Bag of visual words model: recognizing object categories 1 1 1 Problem: Image Classification Given: positive training images containing an object class, and negative training images that don t Classify:
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationRandomized Trees for RealTime Keypoint Recognition
Randomized Trees for RealTime Keypoint Recognition Vincent Lepetit Pascal Lagger Pascal Fua Computer Vision Laboratory École Polytechnique Fédérale de Lausanne (EPFL) 1015 Lausanne, Switzerland Email:
More informationConvolution. 1D Formula: 2D Formula: Example on the web: http://www.jhu.edu/~signals/convolve/
Basic Filters (7) Convolution/correlation/Linear filtering Gaussian filters Smoothing and noise reduction First derivatives of Gaussian Second derivative of Gaussian: Laplacian Oriented Gaussian filters
More informationGeometric Transformations and Image Warping: Mosaicing
Geometric Transformations and Image Warping: Mosaicing CS 6640 Ross Whitaker, Guido Gerig SCI Institute, School of Computing University of Utah (with slides from: Jinxiang Chai, TAMU) faculty.cs.tamu.edu/jchai/cpsc641_spring10/lectures/lecture8.ppt
More informationA New Minimal Solution to the Relative Pose of a Calibrated Stereo Camera with Small Field of View Overlap
A New Minimal Solution to the Relative Pose of a Calibrated Stereo Camera with Small Field of View Overlap Brian Clipp 1, Christopher Zach 1, JanMichael Frahm 1 and Marc Pollefeys 2 1 Department of Computer
More informationBuild Panoramas on Android Phones
Build Panoramas on Android Phones Tao Chu, Bowen Meng, Zixuan Wang Stanford University, Stanford CA Abstract The purpose of this work is to implement panorama stitching from a sequence of photos taken
More informationBildverarbeitung und Mustererkennung Image Processing and Pattern Recognition
Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition 1. Image PreProcessing  Pixel Brightness Transformation  Geometric Transformation  Image Denoising 1 1. Image PreProcessing
More informationIMAGE FORMATION. Antonino Furnari
IPLab  Image Processing Laboratory Dipartimento di Matematica e Informatica Università degli Studi di Catania http://iplab.dmi.unict.it IMAGE FORMATION Antonino Furnari furnari@dmi.unict.it http://dmi.unict.it/~furnari
More information2View Geometry. Mark Fiala Ryerson University Mark.fiala@ryerson.ca
CRV 2010 Tutorial Day 2View Geometry Mark Fiala Ryerson University Mark.fiala@ryerson.ca 3Vectors for image points and lines Mark Fiala 2010 2D Homogeneous Points Add 3 rd number to a 2D point on image
More informationToday s Topics. Lecture 11: LoG and DoG Filters. Recall: First Derivative Filters. SecondDerivative Filters
Today s Topics Lecture : LoG and DoG Filters Laplacian of Gaussian (LoG) Filter  useful for finding edges  also useful for finding blobs! approimation using Difference of Gaussian (DoG) Recall: First
More informationOptical Tracking Using Projective Invariant Marker Pattern Properties
Optical Tracking Using Projective Invariant Marker Pattern Properties Robert van Liere, Jurriaan D. Mulder Department of Information Systems Center for Mathematics and Computer Science Amsterdam, the Netherlands
More informationVisionbased Mapping with Backward Correction
Proceedings of the 2002 IEEE/RSJ Intl. Conference on Intelligent Robots and Systems EPFL, Lausanne, Switzerland October 2002 Visionbased Mapping with Backward Correction Stephen Se David Lowe, Jim Little
More informationRobust Panoramic Image Stitching
Robust Panoramic Image Stitching CS231A Final Report Harrison Chau Department of Aeronautics and Astronautics Stanford University Stanford, CA, USA hwchau@stanford.edu Robert Karol Department of Aeronautics
More information3D 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/
More informationWii 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
More informationPart 3 tracking detectors material effects track models fitting with momentum
Part 3 tracking detectors material effects track models fitting with momentum passage of particles through matter particles traversing through a medium interact with that medium they loose energy ionization:
More informationImage Formation and Camera Calibration
EECS 432Advanced Computer Vision Notes Series 2 Image Formation and Camera Calibration Ying Wu Electrical Engineering & Computer Science Northwestern University Evanston, IL 60208 yingwu@ecenorthwesternedu
More informationAdvanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional
More informationClassifying Manipulation Primitives from Visual Data
Classifying Manipulation Primitives from Visual Data Sandy Huang and Dylan HadfieldMenell Abstract One approach to learning from demonstrations in robotics is to make use of a classifier to predict if
More informationPrinciples of inertial sensing technology and its applications in IHCI
Principles of inertial sensing technology and its applications in IHCI Intelligent Human Computer Interaction SS 2011 Gabriele Bleser Gabriele.Bleser@dfki.de Motivation I bet you all got in touch with
More informationFourier Descriptors For Shape Recognition. Applied to Tree Leaf Identification By Tyler Karrels
Fourier Descriptors For Shape Recognition Applied to Tree Leaf Identification By Tyler Karrels Why investigate shape description? Hard drives keep getting bigger. Digital cameras allow us to capture, store,
More informationFace detection is a process of localizing and extracting the face region from the
Chapter 4 FACE NORMALIZATION 4.1 INTRODUCTION Face detection is a process of localizing and extracting the face region from the background. The detected face varies in rotation, brightness, size, etc.
More informationComputer Vision: Filtering
Computer Vision: Filtering Raquel Urtasun TTI Chicago Jan 10, 2013 Raquel Urtasun (TTIC) Computer Vision Jan 10, 2013 1 / 82 Today s lecture... Image formation Image Filtering Raquel Urtasun (TTIC) Computer
More informationSituation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli
Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing
More informationWhiteboard It! Convert Whiteboard Content into an Electronic Document
Whiteboard It! Convert Whiteboard Content into an Electronic Document Zhengyou Zhang Liwei He Microsoft Research Email: zhang@microsoft.com, lhe@microsoft.com Aug. 12, 2002 Abstract This ongoing project
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationArrowsmith: Automatic Archery Scorer Chanh Nguyen and Irving Lin
Arrowsmith: Automatic Archery Scorer Chanh Nguyen and Irving Lin Department of Computer Science, Stanford University ABSTRACT We present a method for automatically determining the score of a round of arrows
More informationImage Projection. Goal: Introduce the basic concepts and mathematics for image projection.
Image Projection Goal: Introduce the basic concepts and mathematics for image projection. Motivation: The mathematics of image projection allow us to answer two questions: Given a 3D scene, how does it
More informationDifferential Camera Tracking through Linearizing the Local Appearance Manifold
Differential Camera Tracking through Linearizing the Local Appearance Manifold Hua Yang Marc Pollefeys Greg Welch JanMichael Frahm Adrian Ilie Computer Science Department University of North Carolina
More informationEL5223. Basic Concepts of Robot Sensors, Actuators, Localization, Navigation, and1 Mappin / 12
Basic Concepts of Robot Sensors, Actuators, Localization, Navigation, and Mapping Basic Concepts of Robot Sensors, Actuators, Localization, Navigation, and1 Mappin / 12 Sensors and Actuators Robotic systems
More informationRecognition. Sanja Fidler CSC420: Intro to Image Understanding 1 / 28
Recognition Topics that we will try to cover: Indexing for fast retrieval (we still owe this one) History of recognition techniques Object classification Bagofwords Spatial pyramids Neural Networks Object
More information3D 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
More informationObject tracking & Motion detection in video sequences
Introduction Object tracking & Motion detection in video sequences Recomended link: http://cmp.felk.cvut.cz/~hlavac/teachpresen/17compvision3d/41imagemotion.pdf 1 2 DYNAMIC SCENE ANALYSIS The input to
More informationCamera geometry and image alignment
Computer Vision and Machine Learning Winter School ENS Lyon 2010 Camera geometry and image alignment Josef Sivic http://www.di.ens.fr/~josef INRIA, WILLOW, ENS/INRIA/CNRS UMR 8548 Laboratoire d Informatique,
More informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity 8G18G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More informationDirect realtime SLAM
Computer Vision Group Technical University of Munich Jakob Engel Direct RealTime SLAM Jakob Engel, Daniel Cremers David Caruso, Thomas Schöps, Vladyslav Usenko, Jörg Stückler, Jürgen Sturm Technical University
More information What is a feature?  Image processing essentials  Edge detection (Sobel & Canny)  Hough transform  Some images
Seminar: Feature extraction by André Aichert I Feature detection  What is a feature?  Image processing essentials  Edge detection (Sobel & Canny)  Hough transform  Some images II An Entropybased
More informationThe calibration problem was discussed in details during lecture 3.
1 2 The calibration problem was discussed in details during lecture 3. 3 Once the camera is calibrated (intrinsics are known) and the transformation from the world reference system to the camera reference
More informationLecture 19 Camera Matrices and Calibration
Lecture 19 Camera Matrices and Calibration Project Suggestions Texture Synthesis for InPainting Section 10.5.1 in Szeliski Text Project Suggestions Image Stitching (Chapter 9) Face Recognition Chapter
More informationStereo Vision (Correspondences)
Stereo Vision (Correspondences) EECS 59808 Fall 2014! Foundations of Computer Vision!! Instructor: Jason Corso (jjcorso)! web.eecs.umich.edu/~jjcorso/t/598f14!! Readings: FP 7; SZ 11; TV 7! Date: 10/27/14!!
More informationDistinctive Image Features from ScaleInvariant Keypoints
Distinctive Image Features from ScaleInvariant Keypoints David G. Lowe Computer Science Department University of British Columbia Vancouver, B.C., Canada lowe@cs.ubc.ca January 5, 2004 Abstract This paper
More information2. Norm, distance, angle
L. Vandenberghe EE133A (Spring 2016) 2. Norm, distance, angle norm distance angle hyperplanes complex vectors 21 Euclidean norm (Euclidean) norm of vector a R n : a = a 2 1 + a2 2 + + a2 n = a T a if
More information3D Scanner using Line Laser. 1. Introduction. 2. Theory
. Introduction 3D Scanner using Line Laser Di Lu Electrical, Computer, and Systems Engineering Rensselaer Polytechnic Institute The goal of 3D reconstruction is to recover the 3D properties of a geometric
More informationBlind Deconvolution of Barcodes via Dictionary Analysis and Wiener Filter of Barcode Subsections
Blind Deconvolution of Barcodes via Dictionary Analysis and Wiener Filter of Barcode Subsections Maximilian Hung, Bohyun B. Kim, Xiling Zhang August 17, 2013 Abstract While current systems already provide
More informationAn 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 CarnegieMellon University Pittsburgh, Pennsylvania 15213 Abstract
More informationEfficient visual search of local features. Cordelia Schmid
Efficient visual search of local features Cordelia Schmid Visual search change in viewing angle Matches 22 correct matches Image search system for large datasets Large image dataset (one million images
More informationPolygonal Approximation of Closed Curves across Multiple Views
Polygonal Approximation of Closed Curves across Multiple Views M. Pawan Kumar Saurabh Goyal C. V. Jawahar P. J. Narayanan Centre for Visual Information Technology International Institute of Information
More informationFACTS  A Computer Vision System for 3D Recovery and Semantic Mapping of Human Factors
FACTS  A Computer Vision System for 3D Recovery and Semantic Mapping of Human Factors Lucas Paletta, Katrin Santner, Gerald Fritz, Albert Hofmann, Gerald Lodron, Georg Thallinger, Heinz Mayer 2 Human
More informationMonash 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
More informationAndroid Ros Application
Android Ros Application Advanced Practical course : Sensorenabled Intelligent Environments 2011/2012 Presentation by: Rim Zahir Supervisor: Dejan Pangercic SIFT Matching Objects Android Camera Topic :
More informationModule 6: Pinhole camera model Lecture 30: Intrinsic camera parameters, Perspective projection using homogeneous coordinates
The Lecture Contains: Pinhole camera model 6.1 Intrinsic camera parameters A. Perspective projection using homogeneous coordinates B. Principalpoint offset C. Imagesensor characteristics file:///d /...0(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2030/30_1.htm[12/31/2015
More informationA Learning Based Method for SuperResolution of Low Resolution Images
A Learning Based Method for SuperResolution of Low Resolution Images Emre Ugur June 1, 2004 emre.ugur@ceng.metu.edu.tr Abstract The main objective of this project is the study of a learning based method
More informationImage Segmentation Preview Segmentation subdivides an image to regions or objects Two basic properties of intensity values Discontinuity Edge detection Similarity Thresholding Region growing/splitting/merging
More informationDETECTION OF PLANAR PATCHES IN HANDHELD IMAGE SEQUENCES
DETECTION OF PLANAR PATCHES IN HANDHELD IMAGE SEQUENCES Olaf Kähler, Joachim Denzler FriedrichSchillerUniversity, Dept. Mathematics and Computer Science, 07743 Jena, Germany {kaehler,denzler}@informatik.unijena.de
More informationAccurate and robust image superresolution by neural processing of local image representations
Accurate and robust image superresolution by neural processing of local image representations Carlos Miravet 1,2 and Francisco B. Rodríguez 1 1 Grupo de Neurocomputación Biológica (GNB), Escuela Politécnica
More informationCSCI 445 Amin Atrash. Ultrasound, Laser and Vision Sensors. Introduction to Robotics L. Itti & M. J. Mataric
Introduction to Robotics CSCI 445 Amin Atrash Ultrasound, Laser and Vision Sensors Today s Lecture Outline Ultrasound (sonar) Laser rangefinders (ladar, not lidar) Vision Stereo vision Ultrasound/Sonar
More informationLecture 9: Shape Description (Regions)
Lecture 9: Shape Description (Regions) c Bryan S. Morse, Brigham Young University, 1998 2000 Last modified on February 16, 2000 at 4:00 PM Contents 9.1 What Are Descriptors?.........................................
More informationProjective Geometry. Projective Geometry
Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,
More informationCS 534: Computer Vision 3D Modelbased recognition
CS 534: Computer Vision 3D Modelbased recognition Ahmed Elgammal Dept of Computer Science CS 534 3D Modelbased Vision  1 High Level Vision Object Recognition: What it means? Two main recognition tasks:!
More information3D POINT CLOUD CONSTRUCTION FROM STEREO IMAGES
3D POINT CLOUD CONSTRUCTION FROM STEREO IMAGES Brian Peasley * I propose an algorithm to construct a 3D point cloud from a sequence of stereo image pairs that show a full 360 degree view of an object.
More informationJiří Matas. Hough Transform
Hough Transform Jiří Matas Center for Machine Perception Department of Cybernetics, Faculty of Electrical Engineering Czech Technical University, Prague Many slides thanks to Kristen Grauman and Bastian
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Information Meeting 15 Dec 2009 by Marianna Pronobis Content of the Meeting 1. Motivation & Objectives 2. What the course will be about (Stefan) 3. Content of
More informationOptical Flow as a property of moving objects used for their registration
Optical Flow as a property of moving objects used for their registration Wolfgang Schulz Computer Vision Course Project York University Email:wschulz@cs.yorku.ca 1. Introduction A soccer game is a real
More informationMachine Vision Basics: Optics Part Two
Machine Vision Basics: Optics Part Two Webinar Gregory Hollows Director, Machine Vision Solutions Edmund Optics, Inc. Celia Hoyer Product Marketing Vision Systems Cognex Corporation Agenda Quick Review
More informationComputer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D Welcome everybody. We continue the discussion on 2D
More information2D Geometric Transformations. COMP 770 Fall 2011
2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrixvector multiplication Matrixmatrix multiplication
More informationChapter 24. Wave Optics
Chapter 24 Wave Optics Wave Optics The wave nature of light is needed to explain various phenomena. Interference Diffraction Polarization The particle nature of light was the basis for ray (geometric)
More informationAdvanced Techniques for Mobile Robotics Camera Calibration. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Advanced Techniques for Mobile Robotics Camera Calibration Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz What is Camera Calibration? A camera projects 3D world points onto the 2D image
More informationRelating Vanishing Points to Catadioptric Camera Calibration
Relating Vanishing Points to Catadioptric Camera Calibration Wenting Duan* a, Hui Zhang b, Nigel M. Allinson a a Laboratory of Vision Engineering, University of Lincoln, Brayford Pool, Lincoln, U.K. LN6
More informationPartBased Recognition
PartBased Recognition Benedict Brown CS597D, Fall 2003 Princeton University CS 597D, PartBased Recognition p. 1/32 Introduction Many objects are made up of parts It s presumably easier to identify simple
More information8 th Grade Math Curriculum/7 th Grade Advanced Course Information: Course 3 of Prentice Hall Common Core
8 th Grade Math Curriculum/7 th Grade Advanced Course Information: Course: Length: Course 3 of Prentice Hall Common Core 46 minutes/day Description: Mathematics at the 8 th grade level will cover a variety
More information