Lecture 3 Jitendra Malik

Slides:

Advertisements

Similar presentations

Invariants (continued).

Advertisements

Computer Graphics Lecture 4 Geometry & Transformations.

Primitives Behaviour at infinity HZ 2.2 Projective DLT alg Invariants

Introduction A Euclidean group consists of 2 types of transformations: Uniform translations T(b). Uniform rotations R n (  ). Goals: Introduce techniques.

This presentation is the intellectual property of Christine Markstrum Chapter 7 Transformations.

Rotations California Standards for Geometry

Recovering metric and affine properties from images

Camera Models A camera is a mapping between the 3D world and a 2D image The principal camera of interest is central projection.

2D Geometric Transformations

Multiple View Geometry Projective Geometry & Transformations of 2D Vladimir Nedović Intelligent Systems Lab Amsterdam (ISLA) Informatics Institute,

The 2D Projective Plane Points and Lines.

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

Geometry of Images Pinhole camera, projection A taste of projective geometry Two view geometry:  Homography  Epipolar geometry, the essential matrix.

2D/3D Geometric Transformations CS485/685 Computer Vision Dr. George Bebis.

Projective geometry- 2D Acknowledgements Marc Pollefeys: for allowing the use of his excellent slides on this topic

Ch. 2: Rigid Body Motions and Homogeneous Transforms

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

Uncalibrated Geometry & Stratification Sastry and Yang

Projective 2D geometry Appunti basati sulla parte iniziale del testo

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

Math 310 Sections Isometry. Transformations Def A transformation is a map from the plane to itself that takes each point in the plane to exactly.

Previously Two view geometry: epipolar geometry Stereo vision: 3D reconstruction epipolar lines Baseline O O’ epipolar plane.

CS223b, Jana Kosecka Rigid Body Motion and Image Formation.

Projective 2D geometry course 2 Multiple View Geometry Comp Marc Pollefeys.

ME/ECE Professor N. J. Ferrier Forward Kinematics Professor Nicola Ferrier ME Room 2246,

Geometric Transformation. So far…. We have been discussing the basic elements of geometric programming. We have discussed points, vectors and their operations.

Homogeneous Coordinates (Projective Space) Let be a point in Euclidean space Change to homogeneous coordinates: Defined up to scale: Can go back to non-homogeneous.

Autonomous Navigation for Flying Robots Lecture 2.2: 2D Geometry

CSE 681 Review: Transformations. CSE 681 Transformations Modeling transformations build complex models by positioning (transforming) simple components.

Transformations Jehee Lee Seoul National University.

Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations.

1 Chapter 2: Geometric Camera Models Objective: Formulate the geometrical relationships between image and scene measurements Scene: a 3-D function, g(x,y,z)

Comparing Two Motions Jehee Lee Seoul National University.

 Composition of Transformation- 2 or more transformations are combined to form a single transformation  The composition of 2 (or more) isometries is.

Transformations on the Coordinate Plane Mr. J. Grossman.

7.5 Composition Transformations California Standards for Geometry 17: Prove theorems using coordinate geometry 22: Know the effect of rigid motions on.

INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 1)

CS 325 Introduction to Computer Graphics 02 / 19 / 2010 Instructor: Michael Eckmann.

9-4 Compositions of Isometries. Isometry: a transformation that preserves distance or length (translations, reflections, rotations) There are 4 kinds.

9-4 COMPOSITIONS OF ISOMETRIES. Isometries Isometry: transformation that preserves distance, or length. Translations, reflections, and rotations are isometries.

 Complete the Summary of Transformations handout. Translation of h units horizontally and y units vertically Reflection over the y-axis Reflection over.

MASKS © 2004 Invitation to 3D vision Lecture 6 Introduction to Algebra & Rigid-Body Motion Allen Y. Yang September 18 th, 2006.

Modeling Transformation

Projective 2D geometry course 2 Multiple View Geometry Comp Marc Pollefeys.

Computer Graphics Lecture 15 Fasih ur Rehman. Last Class Combining Transformations Affine versus Rigid body Transformations Homogenous Transformations.

CS682, Jana Kosecka Rigid Body Motion and Image Formation Jana Kosecka

9.4 Composition of Transformations

Ch. 2: Rigid Body Motions and Homogeneous Transforms

Sect. 7.1 Rigid Motion in a Plane

Lesson 7.1 Rigid Motion in a Plane.

3. Transformation

Section 12-4 Compositions of Reflections SPI 32D: determine whether the plane figure has been translated given a diagram.

Transformations.

Review: Transformations

Composition of Isometries & Dilations

Homogeneous Coordinates (Projective Space)

Translation Rotation reflection Dilation Pre Image Image Rigid Motions

Epipolar geometry.

Lecture 3: Camera Rotations and Homographies

CPSC 452 Spatial Descriptions and Coordinate Transform

Uncalibrated Geometry & Stratification

Rigid Body Transformations

Course 7 Motion.

Reflections in Coordinate Plane

9.1 TRANSFORMAIONS.

Geometrical Transformations

Transformations 2 University of British Columbia

Rigid Body Transformations

Rotation and Orientation: Fundamentals

Presentation transcript:

Lecture 3 Jitendra Malik On Transformations Lecture 3 Jitendra Malik

Pose and Shape

Rotations and reflections are examples of orthogonal transformations

Rigid body motions (Euclidean transformations / isometries) Theorem: Any rigid body motion can be expressed as an orthogonal transformation followed by a translation.

Orthogonal Matrices

Orthogonal Matrices in 2D

Orthogonal Matrices in 3D

Parameterizing Rotations in 3D

The solution to this differential equation uses the matrix exponential

The composition of two isometries is an isometry

Affine transformations Definition: An affine transformation is a nonsingular linear transformation followed by a translation.

Some examples of affine transforms…

Number of parameters required to specify isometry vs. affine transform In 2D In 3D

Invariants under transformation (Properties that remain unchanged) Lengths Angles Area … Parallelism Midpoints

The big picture ... But are affine transforms as general as we need to be?

Projective Transformations Under perspective projection, parallel lines can map to lines that intersect. Therefore, this cannot be modeled by an affine transform! Projective transformations are a more general family which includes affine transforms and perspective projections. Projective transformations are linear transformations using homogeneous coordinates. We will study them later in the course.