Tutorial: Calculation of Image Rotation for a Scanning Optical System Sergio Guevara OPTI 521 – Distance Learning College of Optical Science University.

Slides:

Advertisements

Similar presentations

Common Variable Types in Elasticity

Advertisements

Common Variable Types in Elasticity

Three Dimensional Viewing

Texture Components and Euler Angles: part 2 13th January 05

CMPE 466 COMPUTER GRAPHICS

From: McCune, B. & J. B. Grace Analysis of Ecological Communities. MjM Software Design, Gleneden Beach, Oregon

CS485/685 Computer Vision Prof. George Bebis

CS 376 Introduction to Computer Graphics 02 / 26 / 2007 Instructor: Michael Eckmann.

Texture Components and Euler Angles: part 2 June 2007

Screw Rotation and Other Rotational Forms

Euler Angles. Three Angles  A rotation matrix can be described with three free parameters. Select three separate rotations about body axesSelect three.

Vectors: planes. The plane Normal equation of the plane.

04 – Geometric Transformations Overview Geometric Primitives –Points, Lines, Planes 2D Geometric Transformations –Translation, Rotation, Scaling, Affine,

Vectors and the Geometry of Space

COS 397 Computer Graphics Svetla Boytcheva AUBG, Spring 2013.

Images in Concave Mirrors. Properties  The mirror has a reflecting surface that curves inward.  When you look at objects in the mirror, the image appears.

Section 13.4 The Cross Product. Torque Torque is a measure of how much a force acting on an object causes that object to rotate –The object rotates around.

Course 12 Calibration. 1.Introduction In theoretic discussions, we have assumed: Camera is located at the origin of coordinate system of scene.

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

Warm Up 1. Reflect the preimage using y=x as the line of reflection given the following coordinates: A(-2, 4), B(-4, -2), C(-5, 6) 2. Rotate the figure.

11.5 Day 2 Lines and Planes in Space. Equations of planes.

Jinxiang Chai CSCE441: Computer Graphics 3D Transformations 0.

Jinxiang Chai Composite Transformations and Forward Kinematics 0.

9.1—Translations Course: Geometry pre-IB Quarter: 3rd

CS-498 Computer Vision Week 7, Day 2 Camera Parameters Intrinsic Calibration  Linear  Radial Distortion (Extrinsic Calibration?) 1.

CS 325 Introduction to Computer Graphics 02 / 22 / 2010 Instructor: Michael Eckmann.

GG313 Lecture 14 Coordinate Transformations Strain.

Rotation matrices 1 Constructing rotation matricesEigenvectors and eigenvalues 0 x y.

Computer Graphics 3D Transformations. Translation.

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)

VECTORS AND THE GEOMETRY OF SPACE 12. PLANES Thus, a plane in space is determined by:  A point P 0 (x 0, y 0, z 0 ) in the plane  A vector n that is.

DOT PRODUCT CROSS PRODUCT APPLICATIONS

Graphics CSCI 343, Fall 2015 Lecture 16 Viewing I

3D Transformation A 3D point (x,y,z) – x,y, and z coordinates

Geometric Transformations UBI 516 Advanced Computer Graphics Aydın Öztürk

CSCI 425/ D Mathematical Preliminaries. CSCI 425/525 2 Coordinate Systems Z X Y Y X Z Right-handed coordinate system Left-handed coordinate system.

1 Algebra of operations Thm 1: Rotation about A thru angle  followed by translation T to axis = rotation thru same angle  about B on bisector of AA'

CSCE 552 Fall 2012 Math By Jijun Tang. Applied Trigonometry Trigonometric functions  Defined using right triangle  x y h.

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

Euler Angles This means, that we can represent an orientation with 3 numbers Assuming we limit ourselves to 3 rotations without successive rotations about.

Arizona’s First University. Command and Control Wind Tunnel Simulated Camera Design Jacob Gulotta.

2D Transformation Homogenous Coordinates Scale/Rotate/Reflect/Shear: X’ = XT Translate: X’ = X + T Multiple values for the same point e.g., (2, 3, 6)

Parametric and general equation of a geometrical object O EQUATION INPUT t OUTPUT EQUATION INPUT OUTPUT (x, y, z)

Transformation methods - Examples

Introduction To IBR Ying Wu. View Morphing Seitz & Dyer SIGGRAPH’96 Synthesize images in transition of two views based on two images No 3D shape is required.

Date of download: 6/21/2016 Copyright © 2016 SPIE. All rights reserved. Deviation angles of a single prism: α = 0.2 rad, n = 1.5. Figure Legend: From:

Modeling Transformation

Extended Work on 3D Lines and Planes. Intersection of a Line and a Plane Find the point of intersection between the line and the plane Answer: (2, -3,

Date of download: 7/9/2016 Copyright © 2016 SPIE. All rights reserved. Photo of measurement boat and unit. Figure Legend: From: Development of shape measurement.

Projection Our 3-D scenes are all specified in 3-D world coordinates

3D Viewing cgvr.korea.ac.kr.

LESSON 90 – DISTANCES IN 3 SPACE

Translations 9.2 Content Standards

11 Vectors and the Geometry of Space

Haosheng Hu College of Optical Sciences University of Arizona

Notation of lines in space

College of Optical Sciences

A movement of a figure in a plane.

A movement of a figure in a plane.

By the end of Week 3: You would learn how to solve many problems involving lines/planes and manipulate with different coordinate systems. These are.

4-4 Geometric Transformations with Matrices

Five-Minute Check (over Lesson 9–1) CCSS Then/Now New Vocabulary

TRANSFORMATIONS Translations Reflections Rotations

Reflections in Coordinate Plane

CSCE441: Computer Graphics 2D/3D Transformations

11 Vectors and the Geometry of Space

Maps one figure onto another figure in a plane.

Five-Minute Check (over Lesson 3–1) Mathematical Practices Then/Now

Presentation transcript:

Tutorial: Calculation of Image Rotation for a Scanning Optical System Sergio Guevara OPTI 521 – Distance Learning College of Optical Science University of Arizona

Brief Overview of Scanning Optical Systems  Uses: Image Scanning  IR Imaging Systems Laser Scanning  Laser Printers  Laser Projectors Microvision’s SHOW Projector  Artifacts Image Distortion Image Rotation Figure from US Patent #4,106,845

Steps to Analyze Image Rotation 1. Sketch the system including: a) Mirror locations b) Mirrors’ axis of rotation. c) Input beam. d) Image plane. 2. Choose a global coordinate system. 3. Define all objects in the global coordinate system. a) Define mirror matrices for mirrors in global coordinate system. b) Define axis of rotation for mirrors in global coordinate system. 4. Find image for non-rotated case. 5. Add rotation to mirrors. 6. Find image in rotated case. 7. Project rotated and non-rotated images onto image plane. 8. Compare the projected images of the rotated case with the non-rotated case to determine rotation of image.

Sketch the System Figure from US Patent #4,106,845 Top View

Sketch the System Figure from US Patent #4,106,845 Front ViewRight View

Choose Global Coordinate System Top ViewRight View

Define Input Beam & Image Plane in Global Coordinates  Input Beam is Parallel to Z-axis.  Image Plane lies on the X-Z plane.

Define Mirrors in Global Coordinates  Find Mirror’s Normal Unit Vectors Mirror 1 normal unit vector: Mirror 2 normal unit vector:  Create Mirror Matrices Mirror 1 Matrix: Mirror 2 Matrix:

Define Rotation of Mirrors in Global Coordinates  Mirror Rotation  Rotation Matrices Mirror 1 Rotation Matrix: (where α is the angle of rotation) Mirror 2 Rotation Matrix: (where β is the angle of rotation)

System Model  Input thru Mirror For no image rotation  Projection onto Image Plane

Calculation of Rotation  Is a comparison of the initial image with no mirror rotation to an image with mirror rotation.  Use the definition of the cross product:  Rotation Equation:

Calculations

Questions?

Rotation Matrices via Euler Parameters  Euler Parameters where the axis of rotation is a unit vector,, and the angle of rotation about that axis is,.  Rotation Matrix in Einstein Notation where δ ij is the Kronecker delta, and ε ijk is the permutation symbol.

Further Resources  How to perform mirror rotations in Zemax a-Scanning-Mirror/Page1.html a-Scanning-Mirror/Page1.html  Euler Parameters  Line onto Plane Projection /plane/lineOnPlane/index.htm /plane/lineOnPlane/index.htm /line/projections/index.htm. /line/projections/index.htm