CSE328 Fundamentals of Computer Graphics: Theory, Algorithms, and Applications
|
|
- Shanna Maxwell
- 6 years ago
- Views:
Transcription
1 CSE328 Fundamentals of Computer Graphics: Theor, Algorithms, and Applications Hong in State Universit of New York at Ston Brook (Ston Brook Universit) Ston Brook, New York Tel: (63) ; Fa: (63)
2 2D-3D Transformations From local, model coordinates to global, world coordinates
3 Modeling Transformations 2D-3D transformations Specif transformations for objects Allows definitions of objects in their own coordinate sstems Allows use of object definition multiple times in a scene Please pa attention to how OpenGL provides a transformation stack because the are so frequentl reused
4 Overview 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D
5 From Model Coordinates to World Coordinates (Local to Global) Model coordinates (local) World coordinates (global) CSE528 Lectures
6 Basic 2D Transformations Translation: = + t = + t Scale: = * s = * s Shear: = + h * = + h * Rotation: = *cos - *sin = *sin + *cos
7 Scaling Scaling a coordinate means multipling each of its components b a scalar Uniform scaling means this scalar is the same for all components: Non-uniform scaling: different scalars per component: How can we represent scaling in matri form?
8 Scaling Operation in Matri Form b a b a
9 Scaling Scaling operation: Or, in matri form: b a b a scaling matri
10 Rotation
11 2-D Rotation = cos() - sin() = sin() + cos()
12 2D Rotation Derivation = r cos (f) = r sin (f) = r cos (f + ) = r sin (f + ) = r cos(f) cos() r sin(f) sin() = r sin(f) sin() + r cos(f) cos() = cos() - sin() = sin() + cos()
13 2-D Rotation It is straightforward to see this procedure in matri form: cos sin sin cos Even though sin() and cos() are nonlinear functions of, is a linear combination of and is a linear combination of and
14 2D Rotation cos sin sin cos
15 Basic 2D Transformations Translation: = + t = + t Scale: = * s = * s Shear: = + h * = + h * Rotation: = *cos - *sin = *sin + *cos
16 Basic 2D Transformations Translation: = + t = + t Scale: = * s = * s Shear: = + h * = + h * Rotation: = *cos - *sin = *sin + *cos Transformations can be combined (with simple algebra)
17 Combining Transformations Transformations can be combined (with simple algebra)
18 Composite Transformations Transformations can be combined (with simple algebra)
19 Matri Representation Represent 2D transformation b a matri Multipl matri b column vector appl transformation to point
20 Overview 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D
21 Matri Representation a c b d a c b d a c b d
22 Matri Representation Represent 2D transformation b a matri a c b d Multipl matri b column vector appl transformation to point a c b d a c b d
23 Matri Representation a c b d e g f h i k j l
24 Matri Representation Transformations can be combined b multiplication Matrices are a convenient and efficient wa to represent a sequence of transformations!
25 Matri Representation Transformations combined b multiplication a c b d e g f h i k j l Matrices are a convenient and efficient wa to represent a sequence of transformations!
26
27 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Identit? 2D Scale around (,)? s s * * s s
28 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Rotate around (,)? * cos * sin * sin * cos cos sin sin cos 2D Shear? sh sh * * sh sh
29 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Mirror about Y ais? 2D Mirror over (,)?
30 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Translation? t t NO! Onl linear 2D transformations can be represented with a 22 matri
31 Linear Transformations Linear transformations are combinations of Scale, Rotation, Shear, and Mirror Properties of linear transformations: Satisfies: Origin maps to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition a d T( sp s2p2) st ( p) s2t ( p2) c b
32 Homogeneous Coordinates : How can we represent translation as a 33 matri? t t
33 Homogeneous Coordinates Homogeneous coordinates represent coordinates in 2 dimensions with a 3- vector homogeneou s coords Homogeneous coordinates appear to be far less intuitive, but the indeed make graphics operations much easier
34 Homogeneous Coordinates : How can we represent translation as a 33 matri? A: Using the rightmost column: t t Translation t t
35 Translation Eample of translation t t t t
36 Homogeneous Coordinates Add a 3rd coordinate to ever 2D point (,, w) represents a point at location (/w, /w) (,, ) represents a point at infinit (,, ) is not allowed Note that, (6,3,); (2,6,2); and (8,9,3) represent the SAME POINT in 2D Convenient coordinate sstem to represent man useful transformations
37 Basic 2D Transformations Basic 2D transformations as 33 matrices cos sin sin cos t t sh sh Translate Rotate Shear s s Scale
38 Affine Transformations Affine transformations are combinations of Linear transformations, and Translations Properties of affine transformations: Origin does not necessaril map to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition a d w b e c f w
39 Projective Transformations Projective transformations Affine transformations, and Projective warps w Properties of projective transformations: Origin does not necessaril map to origin Lines map to lines Parallel lines do not necessaril remain parallel Ratios are not preserved Closed under composition a d g b e h c f i w
40 Overview 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D
41 Matri Composition Transformations can be combined b matri multiplication w s s t t w cos sin sin cos p = T(t,t ) * R() * S(s,s ) * p
42 Matri Composition Matrices are a convenient and efficient wa to represent a sequence of transformations General purpose representation Hardware matri multipl p = (T * (R * (S*p) ) ) p = (T*R*S) * p
43 Matri Composition From local coordinates to global coordinates Be aware: order of transformations matters Matri multiplication is not commutative p = T * R * S * p
44 Matri Composition A more complicated eample: rotating 9 degrees around the mid-point of a line segment (whose coordinates are (3,2)) Can we change the order between rotation and translation?
45 Will this sequence of operations work? Matri Composition 2 3 cos(9) sin(9) sin(9) cos(9) 2 3 a a a a
46 Matri Composition After correctl ordering the matrices Multipl matrices together What results is one matri store it (on stack)! Multipl this matri b the vector of each verte All vertices easil transformed with one matri multipl
47 Overview 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D
48 3D Transformations Same idea as 2D transformations Homogeneous coordinates: (,,z,w) 44 transformation matrices w z p o n m l k j i h g f e d c b a w z
49 Basic 3D Transformations w z w z w z t t t w z z w z s s s w z z w z w z Identit Scale Translation Mirror about Y/Z plane
50 Basic 3D Transformations w z w z cos sin sin cos Rotate around Z ais: w z w z cos sin sin cos Rotate around Y ais: w z w z cos sin sin cos Rotate around X ais:
51 Reverse Rotations : How do ou undo a rotation of, R()? A: Appl the inverse of the rotation R - () = R(-) How to construct R-() = R(-) Inside the rotation matri: cos() = cos(-) The cosine elements of the inverse rotation matri are unchanged The sign of the sine elements will flip This is because the rotation matri is orthogonal matri Therefore R - () = R(-) = R T ()
52 Summar Coordinate sstems are the basis for computer graphics World vs. model coordinates (Global vs. Local) 2-D and 3-D transformations Trigonometr and geometr Matri representations Linear, affine, and projective transformations Matri operations Matri composition
Image Warping. Some slides from Steve Seitz
Image Warping http://www.jeffre-martin.com Some slides from Steve Seitz 5-463: Computational Photograph Aleei Efros, CMU, Spring 2 Image Transformations image filtering: change range of image g() = T(f())
More informationImage Warping (Szeliski Sec 2.1.2)
Image Warping (Szeliski Sec 2..2) http://www.jeffre-martin.com CS94: Image Manipulation & Computational Photograph Aleei Efros, UC Berkele, Fall 7 Some slides from Steve Seitz Image Transformations image
More informationImage Warping. Some slides from Steve Seitz
Image Warping http://www.jeffre-martin.com Some slides from Steve Seitz 5-463: Computational Photograph Aleei Efros, CMU, Fall 26 Image Warping image filtering: change range of image g() T(f()) f T f image
More informationImage Warping : Computational Photography Alexei Efros, CMU, Fall Some slides from Steve Seitz
Image Warping http://www.jeffre-martin.com Some slides from Steve Seitz 5-463: Computational Photograph Aleei Efros, CMU, Fall 2 Image Transformations image filtering: change range of image g() T(f())
More informationImage Warping CSE399b, Spring 07 Computer Vision
Image Warping CSE399b, Spring 7 Computer Vision http://maps.a9.com http://www.cs.ubc.ca/~mbrown/autostitch/autostitch.html http://www.cs.ubc.ca/~mbrown/autostitch/autostitch.html Autostiching on A9.com
More informationModeling Transformations
שיעור 3 גרפיקה ממוחשבת תשס"ח ב ליאור שפירא Modeling Transformations Heavil based on: Thomas Funkhouser Princeton Universit CS 426, Fall 2 Modeling Transformations Specif transformations for objects Allows
More informationModeling Transformations
Transformations Transformations Specif transformations for objects o Allos definitions of objects in on coordinate sstems o Allos use of object definition multiple times in a scene Adam Finkelstein Princeton
More informationModeling Transformations
Modeling Transformations Thomas Funkhouser Princeton Universit CS 426, Fall 2 Modeling Transformations Specif transformations for objects Allos definitions of objects in on coordinate sstems Allos use
More informationScene Graphs & Modeling Transformations COS 426
Scene Graphs & Modeling Transformations COS 426 3D Object Representations Points Range image Point cloud Surfaces Polgonal mesh Subdivision Parametric Implicit Solids Voels BSP tree CSG Sweep High-level
More informationImage Warping. Many slides from Alyosha Efros + Steve Seitz. Photo by Sean Carroll
Image Warping Man slides from Alosha Efros + Steve Seitz Photo b Sean Carroll Morphing Blend from one object to other with a series of local transformations Image Transformations image filtering: change
More informationCSE528 Computer Graphics: Theory, Algorithms, and Applications
CSE528 Computer Graphics: Theor, Algorithms, and Applications Hong Qin State Universit of New York at Ston Brook (Ston Brook Universit) Ston Brook, New York 794--44 Tel: (63)632-845; Fa: (63)632-8334 qin@cs.sunsb.edu
More informationModeling Transformations Revisited
Modeling Transformations Revisited Basic 3D Transformations Translation Scale Shear Rotation 3D Transformations Same idea as 2D transformations o Homogeneous coordinates: (,,z,w) o 44 transformation matrices
More informationModeling Transformations
Modeling Transformations Michael Kazhdan (601.457/657) HB Ch. 5 FvDFH Ch. 5 Overview Ra-Tracing so far Modeling transformations Ra Tracing Image RaTrace(Camera camera, Scene scene, int width, int heigh,
More informationModeling Transformations
Modeling Transformations Michael Kazhdan (601.457/657) HB Ch. 5 FvDFH Ch. 5 Announcement Assignment 2 has been posted: Due: 10/24 ASAP: Download the code and make sure it compiles» On windows: just build
More informationComputer Graphics. Geometric Transformations
Contents coordinate sstems scalar values, points, vectors, matrices right-handed and left-handed coordinate sstems mathematical foundations transformations mathematical descriptions of geometric changes,
More informationComputer Graphics. Geometric Transformations
Computer Graphics Geometric Transformations Contents coordinate sstems scalar values, points, vectors, matrices right-handed and left-handed coordinate sstems mathematical foundations transformations mathematical
More informationTransformations II. Week 2, Wed Jan 17
Universit of British Columbia CPSC 34 Computer Graphics Jan-Apr 27 Tamara Munzner Transformations II Week 2, Wed Jan 7 http://www.ugrad.cs.ubc.ca/~cs34/vjan27 Readings for Jan 5-22 FCG Chap 6 Transformation
More informationLast Lecture. Edge Detection. Filtering Pyramid
Last Lecture Edge Detection Filtering Pramid Toda Motion Deblur Image Transformation Removing Camera Shake from a Single Photograph Rob Fergus, Barun Singh, Aaron Hertzmann, Sam T. Roweis and William T.
More informationHow is project #1 going?
How is project # going? Last Lecture Edge Detection Filtering Pramid Toda Motion Deblur Image Transformation Removing Camera Shake from a Single Photograph Rob Fergus, Barun Singh, Aaron Hertzmann, Sam
More informationImage Warping. Computational Photography Derek Hoiem, University of Illinois 09/28/17. Photo by Sean Carroll
Image Warping 9/28/7 Man slides from Alosha Efros + Steve Seitz Computational Photograph Derek Hoiem, Universit of Illinois Photo b Sean Carroll Reminder: Proj 2 due monda Much more difficult than project
More informationImage warping. image filtering: change range of image. image warping: change domain of image g(x) = f(h(x)) h(y)=0.5y+0.5. h([x,y])=[x,y/2] f h
Image warping Image warping image filtering: change range of image g() () = h(f()) h(f()) f h g h()=0.5+0.5 image warping: change domain of image g() = f(h()) f h g h([,])=[,/2] Parametric (global) warping
More informationComputer Graphics. P04 Transformations. Aleksandra Pizurica Ghent University
Computer Graphics P4 Transformations Aleksandra Pizurica Ghent Universit Telecommunications and Information Processing Image Processing and Interpretation Group Transformations in computer graphics Goal:
More informationInteractive Computer Graphics. Warping and morphing. Warping and Morphing. Warping and Morphing. Lecture 14+15: Warping and Morphing. What is.
Interactive Computer Graphics Warping and morphing Lecture 14+15: Warping and Morphing Lecture 14: Warping and Morphing: Slide 1 Lecture 14: Warping and Morphing: Slide 2 Warping and Morphing What is Warping
More informationWarping, Morphing and Mosaics
Computational Photograph and Video: Warping, Morphing and Mosaics Prof. Marc Pollefes Dr. Gabriel Brostow Toda s schedule Last week s recap Warping Morphing Mosaics Toda s schedule Last week s recap Warping
More informationGeneral Purpose Computation (CAD/CAM/CAE) on the GPU (a.k.a. Topics in Manufacturing)
ME 29-R: General Purpose Computation (CAD/CAM/CAE) on the GPU (a.k.a. Topics in Manufacturing) Sara McMains Spring 29 lecture 2 Toda s GPU eample: moldabilit feedback Two-part mold [The Complete Sculptor
More informationTransformations. Examples of transformations: shear. scaling
Transformations Eamples of transformations: translation rotation scaling shear Transformations More eamples: reflection with respect to the y-ais reflection with respect to the origin Transformations Linear
More informationImage Warping, mesh, and triangulation CSE399b, Spring 07 Computer Vision
http://grail.cs.washington.edu/projects/rotoscoping/ Image Warping, mesh, and triangulation CSE399b, Spring 7 Computer Vision Man of the slides from A. Efros. Parametric (global) warping Eamples of parametric
More informationTransformations III. Week 2, Fri Jan 19
Universit of British Columbia CPSC 34 Computer Graphics Jan-Apr 2007 Tamara Munzner Transformations III Week 2, Fri Jan 9 http://www.ugrad.cs.ubc.ca/~cs34/vjan2007 Readings for Jan 5-22 FCG Chap 6 Transformation
More informationCS4670: Computer Vision
CS467: Computer Vision Noah Snavely Lecture 8: Geometric transformations Szeliski: Chapter 3.6 Reading Announcements Project 2 out today, due Oct. 4 (demo at end of class today) Image alignment Why don
More information3-Dimensional Viewing
CHAPTER 6 3-Dimensional Vieing Vieing and projection Objects in orld coordinates are projected on to the vie plane, hich is defined perpendicular to the vieing direction along the v -ais. The to main tpes
More informationEditing and Transformation
Lecture 5 Editing and Transformation Modeling Model can be produced b the combination of entities that have been edited. D: circle, arc, line, ellipse 3D: primitive bodies, etrusion and revolved of a profile
More information1. We ll look at: Types of geometrical transformation. Vector and matrix representations
Tob Howard COMP272 Computer Graphics and Image Processing 3: Transformations Tob.Howard@manchester.ac.uk Introduction We ll look at: Tpes of geometrical transformation Vector and matri representations
More informationCS559: Computer Graphics
CS559: Computer Graphics Lecture 8: 3D Transforms Li Zhang Spring 28 Most Slides from Stephen Chenne Finish Color space Toda 3D Transforms and Coordinate sstem Reading: Shirle ch 6 RGB and HSV Green(,,)
More informationCS770/870 Spring 2017 Transformations
CS770/870 Spring 2017 Transformations Coordinate sstems 2D Transformations Homogeneous coordinates Matrices, vectors, points Coordinate Sstems Coordinate sstems used in graphics Screen coordinates: the
More information4. Two Dimensional Transformations
4. Two Dimensional Transformations CS362 Introduction to Computer Graphics Helena Wong, 2 In man applications, changes in orientations, sizes, and shapes are accomplished with geometric transformations
More informationToday s class. Geometric objects and transformations. Informationsteknologi. Wednesday, November 7, 2007 Computer Graphics - Class 5 1
Toda s class Geometric objects and transformations Wednesda, November 7, 27 Computer Graphics - Class 5 Vector operations Review of vector operations needed for working in computer graphics adding two
More informationHomogeneous Coordinates
COMS W4172 3D Math 2 Steven Feiner Department of Computer Science Columbia Universit New York, NY 127 www.cs.columbia.edu/graphics/courses/csw4172 Februar 1, 218 1 Homogeneous Coordinates w X W Y X W Y
More informationThe 3-D Graphics Rendering Pipeline
The 3-D Graphics Rendering Pipeline Modeling Trival Rejection Illumination Viewing Clipping Projection Almost ever discussion of 3-D graphics begins here Seldom are an two versions drawn the same wa Seldom
More informationDetermining the 2d transformation that brings one image into alignment (registers it) with another. And
Last two lectures: Representing an image as a weighted combination of other images. Toda: A different kind of coordinate sstem change. Solving the biggest problem in using eigenfaces? Toda Recognition
More informationWhat and Why Transformations?
2D transformations What and Wh Transformations? What? : The geometrical changes of an object from a current state to modified state. Changing an object s position (translation), orientation (rotation)
More informationProf. Feng Liu. Winter /05/2019
Prof. Feng Liu Winter 2019 http://www.cs.pd.edu/~fliu/courses/cs410/ 02/05/2019 Last Time Image alignment 2 Toda Image warping The slides for this topic are used from Prof. Yung-Yu Chuang, which use materials
More informationGLOBAL EDITION. Interactive Computer Graphics. A Top-Down Approach with WebGL SEVENTH EDITION. Edward Angel Dave Shreiner
GLOBAL EDITION Interactive Computer Graphics A Top-Down Approach with WebGL SEVENTH EDITION Edward Angel Dave Shreiner This page is intentionall left blank. 4.10 Concatenation of Transformations 219 in
More information2D Image Transforms Computer Vision (Kris Kitani) Carnegie Mellon University
2D Image Transforms 16-385 Computer Vision (Kris Kitani) Carnegie Mellon University Extract features from an image what do we do next? Feature matching (object recognition, 3D reconstruction, augmented
More information(x, y) (ρ, θ) ρ θ. Polar Coordinates. Cartesian Coordinates
Coordinate Sstems Point Representation in two dimensions Cartesian Coordinates: (; ) Polar Coordinates: (; ) (, ) ρ θ (ρ, θ) Cartesian Coordinates Polar Coordinates p = CPS1, 9: Computer Graphics D Geometric
More informationVector Algebra Transformations. Lecture 4
Vector Algebra Transformations Lecture 4 Cornell CS4620 Fall 2008 Lecture 4 2008 Steve Marschner 1 Geometry A part of mathematics concerned with questions of size, shape, and relative positions of figures
More information1/29/13. Computer Graphics. Transformations. Simple Transformations
/29/3 Computer Graphics Transformations Simple Transformations /29/3 Contet 3D Coordinate Sstems Right hand (or counterclockwise) coordinate sstem Left hand coordinate sstem Not used in this class and
More information2D transformations and homogeneous coordinates
2D transformations and homogeneous coordinates Dr Nicolas Holzschuch Universit of Cape Ton e-mail: holzschu@cs.uct.ac.za Map of the lecture Transformations in 2D: vector/matri notation eample: translation,
More informationCS 335 Graphics and Multimedia. Geometric Warping
CS 335 Graphics and Multimedia Geometric Warping Geometric Image Operations Eample transformations Straightforward methods and their problems The affine transformation Transformation algorithms: Forward
More informationGEOMETRIC TRANSFORMATIONS AND VIEWING
GEOMETRIC TRANSFORMATIONS AND VIEWING 2D and 3D 1/44 2D TRANSFORMATIONS HOMOGENIZED Transformation Scaling Rotation Translation Matrix s x s y cosθ sinθ sinθ cosθ 1 dx 1 dy These 3 transformations are
More informationMath background. 2D Geometric Transformations. Implicit representations. Explicit representations. Read: CS 4620 Lecture 6
Math background 2D Geometric Transformations CS 4620 Lecture 6 Read: Chapter 2: Miscellaneous Math Chapter 5: Linear Algebra Notation for sets, functions, mappings Linear transformations Matrices Matrix-vector
More informationToday. Today. Introduction. Matrices. Matrices. Computergrafik. Transformations & matrices Introduction Matrices
Computergrafik Matthias Zwicker Universität Bern Herbst 2008 Today Transformations & matrices Introduction Matrices Homogeneous Affine transformations Concatenating transformations Change of Common coordinate
More informationImage warping/morphing
Image warping/morphing Digital Visual Effects, Spring 2007 Yung-Yu Chuang 2007/3/20 with slides b Richard Szeliski, Steve Seitz, Tom Funkhouser and Aleei Efros Image warping Image formation B A Sampling
More informationUses of Transformations. 2D transformations Homogeneous coordinates. Transformations. Transformations. Transformations. Transformations and matrices
Uses of Transformations 2D transformations Homogeneous coordinates odeling: position and resie parts of a comple model; Viewing: define and position the virtual camera Animation: define how objects move/change
More informationMore on Transformations. COS 426, Spring 2019 Princeton University
More on Transformations COS 426, Spring 2019 Princeton Universit Agenda Grab-bag of topics related to transformations: General rotations! Euler angles! Rodrigues s rotation formula Maintaining camera transformations!
More information3D Geometry and Camera Calibration
3D Geometr and Camera Calibration 3D Coordinate Sstems Right-handed vs. left-handed 2D Coordinate Sstems ais up vs. ais down Origin at center vs. corner Will often write (u, v) for image coordinates v
More informationComputer Graphics. 2D transformations. Transforma3ons in computer graphics. Overview. Basic classes of geometric transforma3ons
Transforma3ons in computer graphics omputer Graphics Transforma3ons leksandra Piurica Goal: introduce methodolog to hange coordinate sstem Move and deform objects Principle: transforma3ons are applied
More informationAffine and Projective Transformations
CS 674: Intro to Computer Vision Affine and Projective Transformations Prof. Adriana Kovaska Universit of Pittsburg October 3, 26 Alignment problem We previousl discussed ow to matc features across images,
More information3D Coordinates & Transformations
3D Coordinates & Transformations Prof. Aaron Lanterman (Based on slides b Prof. Hsien-Hsin Sean Lee) School of Electrical and Computer Engineering Georgia Institute of Technolog 3D graphics rendering pipeline
More informationCS770/870 Spring 2017 Transformations
CS770/870 Spring 2017 Transformations Coordinate sstems 2D Transformations Homogeneous coordinates Matrices, vectors, points 01/29/2017 1 Coordinate Sstems Coordinate sstems used in graphics Screen coordinates:
More informationWarping. 12 May 2015
Warping 12 May 2015 Warping, morphing, mosaic Slides from Durand and Freeman (MIT), Efros (CMU, Berkeley), Szeliski (MSR), Seitz (UW), Lowe (UBC) http://szeliski.org/book/ 2 Image Warping Image filtering:
More informationGeometric Model of Camera
Geometric Model of Camera Dr. Gerhard Roth COMP 42A Winter 25 Version 2 Similar Triangles 2 Geometric Model of Camera Perspective projection P(X,Y,Z) p(,) f X Z f Y Z 3 Parallel lines aren t 4 Figure b
More informationGeometric Transformations
CS INTRODUCTION TO COMPUTER GRAPHICS Geometric Transformations D and D Andries an Dam 9/9/7 /46 CS INTRODUCTION TO COMPUTER GRAPHICS How do we use Geometric Transformations? (/) Objects in a scene at the
More informationCS 2770: Intro to Computer Vision. Multiple Views. Prof. Adriana Kovashka University of Pittsburgh March 14, 2017
CS 277: Intro to Computer Vision Multiple Views Prof. Adriana Kovashka Universit of Pittsburgh March 4, 27 Plan for toda Affine and projective image transformations Homographies and image mosaics Stereo
More informationTransformations using matrices
Transformations using matrices 6 sllabusref eferenceence Core topic: Matrices and applications In this cha 6A 6B 6C 6D 6E 6F 6G chapter Geometric transformations and matri algebra Linear transformations
More informationMotivation. What we ve seen so far. Demo (Projection Tutorial) Outline. Projections. Foundations of Computer Graphics
Foundations of Computer Graphics Online Lecture 5: Viewing Orthographic Projection Ravi Ramamoorthi Motivation We have seen transforms (between coord sstems) But all that is in 3D We still need to make
More informationImage Metamorphosis By Affine Transformations
Image Metamorphosis B Affine Transformations Tim Mers and Peter Spiegel December 16, 2005 Abstract Among the man was to manipulate an image is a technique known as morphing. Image morphing is a special
More informationCS F-07 Objects in 2D 1
CS420-2010F-07 Objects in 2D 1 07-0: Representing Polgons We want to represent a simple polgon Triangle, rectangle, square, etc Assume for the moment our game onl uses these simple shapes No curves for
More informationPolynomials. Math 4800/6080 Project Course
Polnomials. Math 4800/6080 Project Course 2. The Plane. Boss, boss, ze plane, ze plane! Tattoo, Fantas Island The points of the plane R 2 are ordered pairs (x, ) of real numbers. We ll also use vector
More informationMatrix Transformations. Affine Transformations
Matri ransformations Basic Graphics ransforms ranslation Scaling Rotation Reflection Shear All Can be Epressed As Linear Functions of the Original Coordinates : A + B + C D + E + F ' A ' D 1 B E C F 1
More informationMEM380 Applied Autonomous Robots Winter Robot Kinematics
MEM38 Applied Autonomous obots Winter obot Kinematics Coordinate Transformations Motivation Ultimatel, we are interested in the motion of the robot with respect to a global or inertial navigation frame
More informationTo Do. Motivation. Demo (Projection Tutorial) What we ve seen so far. Computer Graphics. Summary: The Whole Viewing Pipeline
Computer Graphics CSE 67 [Win 9], Lecture 5: Viewing Ravi Ramamoorthi http://viscomp.ucsd.edu/classes/cse67/wi9 To Do Questions/concerns about assignment? Remember it is due tomorrow! (Jan 6). Ask me or
More informationCT5510: Computer Graphics. Transformation BOCHANG MOON
CT5510: Computer Graphics Transformation BOCHANG MOON 2D Translation Transformations such as rotation and scale can be represented using a matrix M.., How about translation? No way to express this using
More informationM y. Image Warping. Targil 7 : Image Warping. Image Warping. 2D Geometric Transformations. image filtering: change range of image g(x) = T(f(x))
Hebrew Universi Image Processing - 6 Image Warping Hebrew Universi Image Processing - 6 argil 7 : Image Warping D Geomeric ransormaions hp://www.jere-marin.com Man slides rom Seve Seiz and Aleei Eros Image
More informationMust first specify the type of projection desired. When use parallel projections? For technical drawings, etc. Specify the viewing parameters
walters@buffalo.edu CSE 480/580 Lecture 4 Slide 3-D Viewing Continued Eamples of 3-D Viewing Must first specif the tpe of projection desired When use parallel projections? For technical drawings, etc.
More informationTo Do. Outline. Translation. Homogeneous Coordinates. Foundations of Computer Graphics. Representation of Points (4-Vectors) Start doing HW 1
Foundations of Computer Graphics Homogeneous Coordinates Start doing HW 1 To Do Specifics of HW 1 Last lecture covered basic material on transformations in 2D Likely need this lecture to understand full
More information1.1 Horizontal & Vertical Translations
Unit II Transformations of Functions. Horizontal & Vertical Translations Goal: Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related
More information3D graphics rendering pipeline (1) 3D graphics rendering pipeline (3) 3D graphics rendering pipeline (2) 8/29/11
3D graphics rendering pipeline (1) Geometr Rasteriation 3D Coordinates & Transformations Prof. Aaron Lanterman (Based on slides b Prof. Hsien-Hsin Sean Lee) School of Electrical and Computer Engineering
More informationGraphics and Interaction Transformation geometry and homogeneous coordinates
433-324 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationCOMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates
COMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationLast week. Machiraju/Zhang/Möller/Fuhrmann
Last week Machiraju/Zhang/Möller/Fuhrmann 1 Geometry basics Scalar, point, and vector Vector space and affine space Basic point and vector operations Sided-ness test Lines, planes, and triangles Linear
More informationMAN-522: COMPUTER VISION SET-2 Projections and Camera Calibration
MAN-522: COMPUTER VISION SET-2 Projections and Camera Calibration Image formation How are objects in the world captured in an image? Phsical parameters of image formation Geometric Tpe of projection Camera
More informationCS Computer Graphics: Transformations & The Synthetic Camera
CS 543 - Computer Graphics: Transformations The Snthetic Camera b Robert W. Lindeman gogo@wpi.edu (with help from Emmanuel Agu ;-) Introduction to Transformations A transformation changes an objects Size
More information[ ] [ ] Orthogonal Transformation of Cartesian Coordinates in 2D & 3D. φ = cos 1 1/ φ = tan 1 [ 2 /1]
Orthogonal Transformation of Cartesian Coordinates in 2D & 3D A vector is specified b its coordinates, so it is defined relative to a reference frame. The same vector will have different coordinates in
More informationTransformations II. Arbitrary 3D Rotation. What is its inverse? What is its transpose? Can we constructively elucidate this relationship?
Utah School of Computing Fall 25 Transformations II CS46 Computer Graphics From Rich Riesenfeld Fall 25 Arbitrar 3D Rotation What is its inverse? What is its transpose? Can we constructivel elucidate this
More informationTwo Dimensional Viewing
Two Dimensional Viewing Dr. S.M. Malaek Assistant: M. Younesi Two Dimensional Viewing Basic Interactive Programming Basic Interactive Programming User controls contents, structure, and appearance of objects
More informationTo Do. Demo (Projection Tutorial) Motivation. What we ve seen so far. Outline. Foundations of Computer Graphics (Fall 2012) CS 184, Lecture 5: Viewing
Foundations of Computer Graphics (Fall 0) CS 84, Lecture 5: Viewing http://inst.eecs.berkele.edu/~cs84 To Do Questions/concerns about assignment? Remember it is due Sep. Ask me or TAs re problems Motivation
More informationTo Do. Course Outline. Course Outline. Goals. Motivation. Foundations of Computer Graphics (Fall 2012) CS 184, Lecture 3: Transformations 1
Fondations of Compter Graphics (Fall 212) CS 184, Lectre 3: Transformations 1 http://inst.eecs.berkele.ed/~cs184 Sbmit HW b To Do Start looking at HW 1 (simple, bt need to think) Ais-angle rotation and
More informationWhat does OpenGL do?
Theor behind Geometrical Transform What does OpenGL do? So the user specifies a lot of information Ee Center Up Near, far, UP EE Left, right top, bottom, etc. f b CENTER left right top bottom What does
More informationPhoto by Carl Warner
Photo b Carl Warner Photo b Carl Warner Photo b Carl Warner Fitting and Alignment Szeliski 6. Computer Vision CS 43, Brown James Has Acknowledgment: Man slides from Derek Hoiem and Grauman&Leibe 2008 AAAI
More informationImage warping , , Computational Photography Fall 2017, Lecture 10
Image warping http://graphics.cs.cmu.edu/courses/15-463 15-463, 15-663, 15-862 Computational Photography Fall 2017, Lecture 10 Course announcements Second make-up lecture on Friday, October 6 th, noon-1:30
More informationGeometric Transformations
Geometric Transformations CS 4620 Lecture 9 2017 Steve Marschner 1 A little quick math background Notation for sets, functions, mappings Linear and affine transformations Matrices Matrix-vector multiplication
More informationNotes. University of British Columbia
Notes Drop-bo is no. 14 You can hand in our assignments Assignment 0 due Fri. 4pm Assignment 1 is out Office hours toda 16:00 17:00, in lab or in reading room Uniersit of Uniersit of Chapter 4 - Reminder
More informationIMGD The Game Development Process: 3D Modeling and Transformations
IMGD - The Game Development Process: 3D Modeling and Transformations b Robert W. Lindeman (gogo@wpi.edu Kent Quirk (kent_quirk@cognito.com (with lots of input from Mark Clapool! Overview of 3D Modeling
More information3D Computer Vision II. Reminder Projective Geometry, Transformations. Nassir Navab. October 27, 2009
3D Computer Vision II Reminder Projective Geometr, Transformations Nassir Navab based on a course given at UNC b Marc Pollefes & the book Multiple View Geometr b Hartle & Zisserman October 27, 29 2D Transformations
More informationGeometric Transformations Hearn & Baker Chapter 5. Some slides are taken from Robert Thomsons notes.
Geometric Tranformation Hearn & Baker Chapter 5 Some lie are taken from Robert Thomon note. OVERVIEW Two imenional tranformation Matri repreentation Invere tranformation Three imenional tranformation OpenGL
More informationMatrix Representations
CONDENSED LESSON 6. Matri Representations In this lesson, ou Represent closed sstems with transition diagrams and transition matrices Use matrices to organize information Sandra works at a da-care center.
More informationComputer Graphics. 2D Transforma5ons. Review Vertex Transforma5ons 2/3/15. adjust the zoom. posi+on the camera. posi+on the model
/3/5 Computer Graphics D Transforma5ons Review Verte Transforma5ons posi+on the model posi+on the camera adjust the zoom verte shader input verte shader output, transformed /3/5 From Object to World Space
More informationHigh Dimensional Rendering in OpenGL
High Dimensional Rendering in OpenGL Josh McCo December, 2003 Description of Project Adding high dimensional rendering capabilit to the OpenGL graphics programming environment is the goal of this project
More informationUnit 2: Function Transformation Chapter 1
Basic Transformations Reflections Inverses Unit 2: Function Transformation Chapter 1 Section 1.1: Horizontal and Vertical Transformations A of a function alters the and an combination of the of the graph.
More informationProjections. Brian Curless CSE 457 Spring Reading. Shrinking the pinhole. The pinhole camera. Required:
Reading Required: Projections Brian Curless CSE 457 Spring 2013 Angel, 5.1-5.6 Further reading: Fole, et al, Chapter 5.6 and Chapter 6 David F. Rogers and J. Alan Adams, Mathematical Elements for Computer
More information