Geometric Image Transformations


 Pearl Adams
 2 years ago
 Views:
Transcription
1 Geometric Image Transformations Part One 2D Transformations Spatial Coordinates (x,y) are mapped to new coords (u,v) pixels of source image > pixels of destination image Types of 2D Transformations Affine Scales Translations Rotations Shears Projective Projections Homographies or Collineations
2 Scale and Translation Scale u = s_x * x v = s_y * y Translation u = x + t_x v = y + t_y Rotation and Shears Rotation u = x*cos θ  y * sin θ v = x*sin θ + y * cos θ Shear u = x + Sh_x * y v = y + Sh_y * x 2
3 Shear Consider Sh_x only u = x + Sh_x*y; v = y; (Sh_y=) Homogenous Coords Translation is just an addition We can make it a function of multiplication by using homogenous coords p p Homogenous coords are equivalent up to a scale factor p p u v = tx ty x y wu wv w = tx ty wx wy w 3
4 Homogenous Coordinates Transform works on scale (w) * w w= Dividing by the scale w puts you in Cartesian coords Matrix Notation p Affine p u v = a b c d e f x y 4
5 Matrix notation of transforms Translation Scale p p p p u v = tx ty x y u v = Sx Sy x y Rotation p p Shear p p u v = cosθ sinθ sinθ cosθ x y u v = Shx Shy x y Concatenation You can concatenate several transforms into one matrix A = R S Sh T rotate scale shear translate 5
6 Affine Transformations Affine is a linear mapping plus a translation: A function f(x) is linear iff f(x+y) = f(x) + f(y) a*f(x) = f(a*x) A function T(x) is Affine if there exists a linear mapping L(x) and a constant c Such that: T() = L(x) + c (for all x) Affine Transforms Properties Preserves parallel lines Preserves equispaced points along lines equally spaced points on a line in the source space will produce equally spaced points on a line in the destination space (although the scale may be different) Preserves incident Points of intersection hold 6
7 Preserves Source Destination Source Destination 6 degrees of freedom u v = a b c d e f Y We can define an affine transform by specifying 3 point correspondences 3 (x,y) from the source space that map to 3 (u,v) in the destination space
8 Equations to solve for Affine Transform u = a*x + b*y + c u2 = a*x2 + b*y2 + c u3 = a*x3 + b*y3 + c v = d*x + e*y + f v2 = d*x2 + e*y2 + f v3 = d*x3 + e*y3 + f More commonly You can build an Affine transform by concatenating several transforms together translate to the origin Rotate by 2 degrees translate back from the origin scale by 5 GUI Interface tool that allows you to rotate, translate, scale, etc.. 8
9 Example Affine Transform Limitation Can map a triangle in source space To a triangle in destination space (or two parallelograms) 9
10 Affine Transform Limitation Can map a triangle in source space To a triangle in destination space (or two parallelograms) What about a rectangle to a general quadrilateral? Projective Transform Projective transform can transform general quadrilaterals between source and destination space Does not preserve parallel lines, or lengths Does not preserve equispaced points
11 Projective transform uses homogenous coords su sv s = a b c d e f g h x y u = u/s v = v/s Maps us back to Cartesian space Projective Transforms Very common in computer graphics Texture mapping a 3D polygon polygon has been projected onto a plane 3D polygon 3D perspective projection Texture map 2D polygon
12 Projective Transform Does not preserve length, equispacing Does Map lines to lines Preserve incidents Preserve cross ratio Cross Ratio Given 4 points on a line in source space and destination space D C B A A B C D Source Space Destination Space AC AD BC BD = A C A D B C B D Cross Ratio where Y is the Euclidean distance between point and Y 2
13 Solving for a Projective Transform 8 degrees of freedom su sv s = a b c d e f g h x y We need 4 point correspondences between source and destination image su = ax + by + c sv = dx + ey + f s = gx + hy + source destination Remember, with projective transform destination points are: dest_u = su/s dest_v = sv/s Solving for a Projective Transform su = ax + by + c sv = dx + ey + f s = gx + hy + u =su = ax + by + c s gx + hy + v =sv = dx + ey + f s gx + hy + (gx+hy+)u = ax + by + c (... ) gx + hy + gxu + hyu + u = ax + by + c u = ax + by + c gxu hyu 3
14 8x8 System of Equations x y x u y u a u x y x u y u b u x 2 y 2 x 2 u 2 y 2 u 2 c u 2 x 3 y 3 x y x 3 u 3 x v y 3 u 3 y v d e = u 3 v x y x v y v f v x 2 y 2 x 2 v 2 y 2 v 2 g v 2 x 3 y 3 x 3 v 3 y 3 v 3 h v 3 Solve the system Matrix is in form: Ax = b Solve for x Gaussian elimination (LU decomposition) QR decomposition Entries of vector x are the coefficients for the projective transform 4
15 Properties of Transforms euclidean similarity affine projective Transformation translation rotation uniform scale nonuniform scale shear projection combination (w/ projection) Properties of Transforms euclidean similarity affine projective Invariant length angle ratio of lengths parallelism incident cross ratio 5
16 Transforming an Image origin Transforms and Images Coordinates x Y f(x,y) 6
17 Image Coords vs. Cartesian Coords Image Coords are generally defined by raster alignment Y [,Height], [,Width] Affine, projective transforms are converted into Cartesian coords Transforms and Images Coordinates (y value) origin x Y f(x,y) 7
18 Cartesian space to image space From Cartesian space to Image Space find (x_min, xmax) find(y_min, ymax) new size dimensions w = x_max x_min h = y_max y_min create newimage size (w, h) Translate transformed points, such that: T * (x,y) = (u,v) newimage( u + abs(x_min), v + abs(y_min) = I(x,y) Converting to an Image New Image Dimensions origin x Y 8
19 The transformed image origin New Image Dimensions x Y Creating the new image Forward Mapping Inverse Mapping Sampling 9
20 Mapping Pixels (,) (,N) 2 Forward Mapping (,) (,N) (,M) (M,N) (,M) (M,N) Source Image [u,v,s] T = A [x,y,] T Destination Image Transform Draw backs Forward Mapping Source pixels do not map directly to a single pixel in the destination space Possibility for holes in the destination image We can map the other direction 2
21 Reverse Mapping Reverse Mapping 2 2 (,) (,N) (,M) (M,N) black [x,y,s] T = A  [u,v,] T Inverse Mapping Advantages We assign an intensity to each pixel in the destination (no holes) Affine/projective transforms have inverses (not a problem) just reverse direction of the point correspondences We still don t have pixel to pixel mapping 2
22 Sampling the source How do we sample the source to determine the intensity for the destination? Sampling the source How do we sample the source to determine the intensity for the destination? 22
23 Mapping Source 2 x 2 pixels Option : Pick the pixel nearest to our center. Mapping Source 2 x 2 pixels Small change results in big difference Option : Pick the pixel nearest to our center. 23
24 Try different sampling Source 2 x 2 pixels 2 What if we assign an intensity to each vertex and then average? Pick the intensity which the vertex lies New Sample = Sampling Example Source 2 x 2 pixels 2 Move the destination slightly. Pick the intensity which the vertex lies New Sample = 24
25 No Difference Source 2 x 2 pixels Source 2 x 2 pixels Try different Sampling Source 2 x 2 pixels 2 3 What if we had more samples? 4 25
26 Try different Sampling Source 2 x 2 pixels 2 What if we sampled a larger area? 4 3 Try different Sampling Source 2 x 2 pixels How should we sample? 26
27 Common Sampling Approaches Nearest Neighbor Sample Take closest pixel value Bilinear Interpolation 2x2 (4) Samples Interpolate from these samples Slower BiCubic 4x4 (8) samples Construct a new sample using a nonlinear interpolation Slower Common Sampling Approaches Nearest Neighbor Bilinear Interpolation BiCubic 27
28 Common Sampling Approaches What do these approaches mean? How can evaluate what they are doing? Nearest Neighbor Bilinear Interpolation BiCubic Sampling and Signal Processing Proper sampling is a classic reconstruction problem Given a continuous signal f(x) How do you take discrete samples such that you can properly reconstruct the signal f(x) from the samples? f(x) x 28
Algebra and Linear Algebra
Vectors Coordinate frames 2D implicit curves 2D parametric curves 3D surfaces Algebra and Linear Algebra Algebra: numbers and operations on numbers 2 + 3 = 5 3 7 = 21 Linear algebra: tuples, triples,...
More information2.1 COLOR AND GRAYSCALE LEVELS
2.1 COLOR AND GRAYSCALE LEVELS Various color and intensitylevel options can be made available to a user, depending on the capabilities and design objectives of a particular system. General purpose rasterscan
More informationAffine Transformations
Affine Transformations Reading Foley et al., Chapter 5.6 and Chapter 6 Supplemental David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, Second edition 2D geometry Pipeline 3D
More informationClass 11 Maths Chapter 10. Straight Lines
1 P a g e Class 11 Maths Chapter 10. Straight Lines Coordinate Geometry The branch of Mathematics in which geometrical problem are solved through algebra by using the coordinate system, is known as coordinate
More informationAffine Transformations. University of Texas at Austin CS384G  Computer Graphics Fall 2010 Don Fussell
Affine Transformations University of Texas at Austin CS384G  Computer Graphics Fall 2010 Don Fussell Logistics Required reading: Watt, Section 1.1. Further reading: Foley, et al, Chapter 5.15.5. David
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 informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationGeometric Transformations
CS3 INTRODUCTION TO COMPUTER GRAPHICS Geometric Transformations D and 3D CS3 INTRODUCTION TO COMPUTER GRAPHICS Grading Plan to be out Wednesdas one week after the due date CS3 INTRODUCTION TO COMPUTER
More informationHomographies and Panoramas. Slides from Steve Seitz, Rick Szeliski, Alexei Efros, Fredo Durand, and Kristin Grauman
Homographies and Panoramas Slides from Steve Seitz, Rick Szeliski, Alexei Efros, Fredo Durand, and Kristin Grauman cs129: Computational Photography James Hays, Brown, Fall 2012 Why Mosaic? Are you getting
More informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7 Linear AlgebraLab 2
Linear AlgebraLab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4) 2x + 3y + 6z = 10
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More informationRotation matrix. Fixed angle and Euler angle. Axis angle. Quaternion. Exponential map
3D orientation Rotation matrix Fixed angle and Euler angle Axis angle Quaternion Exponential map Joints and rotations Rotational DOFs are widely used in character animation 3 translational DOFs 48 rotational
More informationL 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:
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 informationMotivation. Goals. General Idea. Outline. (Nonuniform) Scale. Foundations of Computer Graphics
Foundations of Computer Graphics Basic 2D Transforms Motivation Many different coordinate systems in graphics World, model, body, arms, To relate them, we must transform between them Also, for modeling
More information2D Geometric Transformations
2D Geometric Transformations (Chapter 5 in FVD) 2D Geometric Transformations Question: How do we represent a geometric object in the plane? Answer: For now, assume that objects consist of points and lines.
More informationLinear transformations Affine transformations Transformations in 3D. Graphics 2011/2012, 4th quarter. Lecture 5: linear and affine transformations
Lecture 5 Linear and affine transformations Vector transformation: basic idea Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Multiplication of an n n matrix
More informationGeometric Transformation CS 211A
Geometric Transformation CS 211A What is transformation? Moving points (x,y) moves to (x+t, y+t) Can be in any dimension 2D Image warps 3D 3D Graphics and Vision Can also be considered as a movement to
More informationImage Warping and Morphing
Image Warping and Morphing Alexey Tikhonov 15463: Computational Photography Alexei Efros, CMU, Fall 2011 Women in Art video http://youtube.com/watch?v=nudion_hxs Image Warping in Biology D'Arcy Thompson
More information2. SPATIAL TRANSFORMATIONS
Digitales Video 1 2. SPATIAL TRANSFORMATIONS This chapter describes common spatial transformations derived for digital image warping applications in computer vision and computer graphics. A spatial transformation
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationLecture 2: Geometric Image Transformations
Lecture 2: Geometric Image Transformations Harvey Rhody Chester F. Carlson Center for Imaging Science Rochester Institute of Technology rhody@cis.rit.edu September 8, 2005 Abstract Geometric transformations
More informationgeometric transforms
geometric transforms 1 linear algebra review 2 matrices matrix and vector notation use column for vectors m 11 =[ ] M = [ m ij ] m 21 m 12 m 22 =[ ] v v 1 v = [ ] T v 1 v 2 2 3 matrix operations addition
More information0.1 Linear Transformations
.1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called
More informationMATH 304 Linear Algebra Lecture 24: Scalar product.
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent
More informationOrthogonal Matrices. u v = u v cos(θ) T (u) + T (v) = T (u + v). It s even easier to. If u and v are nonzero vectors then
Part 2. 1 Part 2. Orthogonal Matrices If u and v are nonzero vectors then u v = u v cos(θ) is 0 if and only if cos(θ) = 0, i.e., θ = 90. Hence, we say that two vectors u and v are perpendicular or orthogonal
More informationGeometric Transformations
Geometric Transformations Moving objects relative to a stationary coordinate system Common transformations: Translation Rotation Scaling Implemented using vectors and matrices Quick Review of Matrix Algebra
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 informationVector algebra Christian Miller CS Fall 2011
Vector algebra Christian Miller CS 354  Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority
More information13 Groups of Isometries
13 Groups of Isometries For any nonempty set X, the set S X of all onetoone mappings from X onto X is a group under the composition of mappings (Example 71(d)) In particular, if X happens to be the Euclidean
More informationModern Geometry Homework.
Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z
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 informationImage Processing and Computer Graphics. Texture Mapping. Matthias Teschner. Computer Science Department University of Freiburg
Image Processing and Computer Graphics Texture Mapping Matthias Teschner Computer Science Department University of Freiburg Motivation adding perpixel surface details without raising the geometric complexity
More informationLecture 2: 2D Fourier transforms and applications
Lecture 2: 2D Fourier transforms and applications B14 Image Analysis Michaelmas 2014 A. Zisserman Fourier transforms and spatial frequencies in 2D Definition and meaning The Convolution Theorem Applications
More informationTWODIMENSIONAL TRANSFORMATION
CHAPTER 2 TWODIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization
More informationMath 451: Euclidean and NonEuclidean Geometry MWF 3pm, Gasson 204 Homework 4 Solutions
Math 451: Euclidean and NonEuclidean Geometry MWF 3pm, Gasson 04 Homework 4 Solutions Exercises from 1.7 of the notes: 7.1, 7.6, 7.7, 7.10, 7.14, 7.16, 7.17 Exercise 7.1 (Isometries form a group). Show
More informationGeoGebra Geometry. Project Maths Development Team Page 1 of 24
GeoGebra Geometry Project Maths Development Team 2013 www.projectmaths.ie Page 1 of 24 Index Activity Topic Page 1 Introduction 4 2 To construct an equilateral triangle 5 3 To construct a parallelogram
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
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 information521493S 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 1618 in TS128 Question 2.1 Given two nonparallel,
More informationAQA Level 2 Certificate FURTHER MATHEMATICS
AQA Qualifications AQA Level 2 Certificate FURTHER MATHEMATICS Level 2 (8360) Our specification is published on our website (www.aqa.org.uk). We will let centres know in writing about any changes to the
More informationMath 4 Review Problems
Topics for Review #1 Functions function concept [section 1. of textbook] function representations: graph, table, f(x) formula domain and range Vertical Line Test (for whether a graph is a function) evaluating
More informationC4 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 informationLesson A  Natural Exponential Function and Natural Logarithm Functions
A Lesson A  Natural Exponential Function and Natural Logarithm Functions Natural Exponential Function In Lesson 2, we explored the world of logarithms in base 0. The natural logarithm has a base of e.
More informationLecture 3: Coordinate Systems and Transformations
Lecture 3: Coordinate Systems and Transformations Topics: 1. Coordinate systems and frames 2. Change of frames 3. Affine transformations 4. Rotation, translation, scaling, and shear 5. Rotation about an
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 informationContent. 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
More informationProblem Set 1 Solutions Math 109
Problem Set 1 Solutions Math 109 Exercise 1.6 Show that a regular tetrahedron has a total of twentyfour symmetries if reflections and products of reflections are allowed. Identify a symmetry which is
More informationBrightness and geometric transformations
Brightness and geometric transformations Václav Hlaváč Czech Technical University in Prague Center for Machine Perception (bridging groups of the) Czech Institute of Informatics, Robotics and Cybernetics
More informationIntroduction to Computer Graphics 8. Buffers and Mapping techniques (A)
Introduction to Computer Graphics 8. Buffers and Mapping techniques (A) National Chiao Tung Univ, Taiwan By: IChen Lin, Assistant Professor Textbook: Hearn and Baker, Computer Graphics, 3rd Ed., Prentice
More informationDefinition. Proof. Notation and Terminology
ÌÖ Ò ÓÖÑ Ø ÓÒ Ó Ø ÈÐ Ò A transformation of the Euclidean plane is a rule that assigns to each point in the plane another point in the plane. Transformations are also called mappings, or just maps. Examples
More informationSection 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
More informationEquilibrium of Concurrent Forces (Force Table)
Equilibrium of Concurrent Forces (Force Table) Objectives: Experimental objective Students will verify the conditions required (zero net force) for a system to be in equilibrium under the influence of
More informationA. 32 cu ft B. 49 cu ft C. 57 cu ft D. 1,145 cu ft. F. 96 sq in. G. 136 sq in. H. 192 sq in. J. 272 sq in. 5 in
7.5 The student will a) describe volume and surface area of cylinders; b) solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c) describe how changing
More informationScanline Polygon Fill Algorithm
Scanline Polygon Fill Algorithm Look at individual scan lines Compute intersection points with polygon edges Fill between alternate pairs of intersection points Scanline Polygon Fill Algorithm 1. Set up
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
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 information0017 Understanding and Using Vector and Transformational Geometries
Tom Coleman INTD 301 Final Project Dr. Johannes Vector geometry: 0017 Understanding and Using Vector and Transformational Geometries 3D Cartesian coordinate representation:  A vector v is written as
More informationCS 4204 Computer Graphics
CS 4204 Computer Graphics 2D and 3D Transformations Doug Bowman Adapted from notes by Yong Cao Virginia Tech 1 Transformations What are they? changing something to something else via rules mathematics:
More informationChapter 14. Transformations
Chapter 14 Transformations Questions For Thought 1. When you take a picture, how does the real world image become a reduced celluloid or digital image? 2. How are maps of the Earth made to scale? 3. How
More informationElimination Methods. intersecting two circles computing solutions at infinity. transformation to half angles. the Sylvester matrix
Elimination Methods 1 Projective Coordinates intersecting two circles computing solutions at infinity 2 Molecular Configurations transformation to half angles 3 Resultants the Sylvester matrix 4 Cascading
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 informationProjective Geometry: A Short Introduction. Lecture Notes Edmond Boyer
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Contents 1 Introduction 2 11 Objective 2 12 Historical Background 3 13 Bibliography 4 2 Projective Spaces 5 21 Definitions 5 22 Properties
More informationHigher Geometry Problems
Higher Geometry Problems ( Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement
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 informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More informationIntroduction to 2D and 3D Computer Graphics Mastering 2D & 3D Computer Graphics Pipelines
Introduction to 2D and 3D Computer Graphics Mastering 2D & 3D Computer Graphics Pipelines CS447 31 Mastering 2D & 3D Graphics Overview of 2D & 3D Pipelines What are pipelines? What are the fundamental
More information2D Geometrical Transformations. Foley & Van Dam, Chapter 5
2D Geometrical Transformations Fole & Van Dam, Chapter 5 2D Geometrical Transformations Translation Scaling Rotation Shear Matri notation Compositions Homogeneous coordinates 2D Geometrical Transformations
More informationGeometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v
12.4 Cross Product Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! u x v is perpendicular to u and v The length of u x v is uv u v sin The
More informationINTRODUCTION TO RENDERING TECHNIQUES
INTRODUCTION TO RENDERING TECHNIQUES 22 Mar. 212 Yanir Kleiman What is 3D Graphics? Why 3D? Draw one frame at a time Model only once X 24 frames per second Color / texture only once 15, frames for a feature
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 informationSummary: Transformations. Lecture 14 Parameter Estimation Readings T&V Sec 5.15.3. Parameter Estimation: Fitting Geometric Models
Summary: Transformations Lecture 14 Parameter Estimation eadings T&V Sec 5.15.3 Euclidean similarity affine projective Parameter Estimation We will talk about estimating parameters of 1) Geometric models
More informationGeometric Objects and Transformations
Geometric Objects and ransformations How to represent basic geometric types, such as points and vectors? How to convert between various represenations? (through a linear transformation!) How to establish
More informationMCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 15 January. Elimination Methods
Elimination Methods In this lecture we define projective coordinates, give another application, explain the Sylvester resultant method, illustrate how to cascade resultants to compute discriminant, and
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationAntialiasing. CS 319 Advanced Topics in Computer Graphics John C. Hart
Antialiasing CS 319 Advanced Topics in Computer Graphics John C. Hart Aliasing Aliasing occurs when signals are sampled too infrequently, giving the illusion of a lower frequency signal alias noun (c.
More informationThe Gradient and Level Sets
The Gradient and Level Sets. Let f(x, y) = x + y. (a) Find the gradient f. Solution. f(x, y) = x, y. (b) Pick your favorite positive number k, and let C be the curve f(x, y) = k. Draw the curve on the
More informationacute angle acute triangle Cartesian coordinate system concave polygon congruent figures
acute angle acute triangle Cartesian coordinate system concave polygon congruent figures convex polygon coordinate grid coordinates dilatation equilateral triangle horizontal axis intersecting lines isosceles
More informationPreAlgebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems
Academic Content Standards Grade Eight Ohio PreAlgebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
More informationWe can use more sectors (i.e., decrease the sector s angle θ) to get a better approximation:
Section 1.4 Areas of Polar Curves In this section we will find a formula for determining the area of regions bounded by polar curves. To do this, wee again make use of the idea of approximating a region
More informationCAMI Education linked to CAPS: Mathematics
 1  TOPIC 1.1 Whole numbers _CAPS Curriculum TERM 1 CONTENT Properties of numbers Describe the real number system by recognizing, defining and distinguishing properties of: Natural numbers Whole numbers
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationPolygon Scan Conversion and ZBuffering
Polygon Scan Conversion and ZBuffering Rasterization Rasterization takes shapes like triangles and determines which pixels to fill. 2 Filling Polygons First approach:. Polygon ScanConversion Rasterize
More informationPerimeter and Area of Geometric Figures on the Coordinate Plane
Perimeter and Area of Geometric Figures on the Coordinate Plane There are more than 00 national flags in the world. One of the largest is the flag of Brazil flown in Three Powers Plaza in Brasilia. This
More informationLattice Point Geometry: Pick s Theorem and Minkowski s Theorem. Senior Exercise in Mathematics. Jennifer Garbett Kenyon College
Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem Senior Exercise in Mathematics Jennifer Garbett Kenyon College November 18, 010 Contents 1 Introduction 1 Primitive Lattice Triangles 5.1
More informationGeoGebra. 10 lessons. Gerrit Stols
GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter
More informationTessellations. A tessellation is created when a shape is repeated over and over again to cover the plane without any overlaps or gaps.
Tessellations Katherine Sheu A tessellation is created when a shape is repeated over and over again to cover the plane without any overlaps or gaps. 1. The picture below can be extended to a tessellation
More informationLINE INTEGRALS OF VECTOR FUNCTIONS: GREEN S THEOREM. Contents. 2. Green s Theorem 3
LINE INTEGRALS OF VETOR FUNTIONS: GREEN S THEOREM ontents 1. A differential criterion for conservative vector fields 1 2. Green s Theorem 3 1. A differential criterion for conservative vector fields We
More informationTrigonometric Identities and Conditional Equations C
Trigonometric Identities and Conditional Equations C TRIGONOMETRIC functions are widely used in solving realworld problems and in the development of mathematics. Whatever their use, it is often of value
More information2 Topics in 3D Geometry
2 Topics in 3D Geometry In two dimensional space, we can graph curves and lines. In three dimensional space, there is so much extra space that we can graph planes and surfaces in addition to lines and
More informationCentroid and Moment of Inertia
Centroid and Moment of Inertia This program will find the centroid and area moments of inertia for a composite area with respect to a given coordinate system. Then the program will calculate the moments
More informationWarm up Factoring Remember: a 2 b 2 = (a b)(a + b)
Warm up Factoring Remember: a 2 b 2 = (a b)(a + b) 1. x 2 16 2. x 2 + 10x + 25 3. 81y 2 4 4. 3x 3 15x 2 + 18x 5. 16h 4 81 6. x 2 + 2xh + h 2 7. x 2 +3x 4 8. x 2 3x + 4 9. 2x 2 11x + 5 10. x 4 + x 2 20
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
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 information6. ISOMETRIES isometry central isometry translation Theorem 1: Proof:
6. ISOMETRIES 6.1. Isometries Fundamental to the theory of symmetry are the concepts of distance and angle. So we work within R n, considered as an innerproduct space. This is the usual n dimensional
More informationSection 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.
. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and
More informationEquilibrium. To determine the mass of unknown objects by utilizing the known force requirements of an equilibrium
Equilibrium Object To determine the mass of unknown objects by utilizing the known force requirements of an equilibrium situation. 2 Apparatus orce table, masses, mass pans, metal loop, pulleys, strings,
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.8 INTRODUCTION TO LINEAR TRANSFORMATIONS LINEAR TRANSFORMATIONS A transformation (or function or mapping) T from R n to R m is a rule that assigns to each vector
More informationRealtime 3D Computer Graphics Virtual Reality
Realtime 3D Computer Graphics Virtual Realit Viewing and projection Classical and General Viewing Transformation Pipeline CPU Pol. DL Pixel Per Vertex Texture Raster Frag FB object ee clip normalized device
More information