EECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
|
|
|
- Frederica Lindsey
- 7 years ago
- Views:
Transcription
1 EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines
2 What is image processing? Image processing is the application of 2D signal processing methods to images Image representation and modeling Image enhancement Image restoration/image filtering Compression (computer vision)
3 Image enhancement Accentuate certain desired features for subsequent analysis or display. Contrast stretching / dynamic range adjustment Color histogram normalization Noise reduction Sharpening Edge detection Corner detection Image interpolation
4 What is interpolation? given a discrete space image g d [n,m] This corresponds to samples of some continuous space image g a (x, y) Compute values of the CS image g a (x, y) at (x,y) locations other than the sample locations. g a (x, y) g d [n,m] g d [n,m]
5 What is interpolation? Interpolation is critical for: Displaying Image zooming Warping Coding Motion estimation
6 Is perfect recovery always possible? No, (never actually), unless some assumptions on g a (x,y) are made There are uncountable infinite collection of functions g a (x, y) that agree with g d [n,m] at the sample points What are these assumptions? g a must be band limited Sampling rate must satisfy Nyquist theorem
7 g a must be band limited There exists such that
8 Sampling rate must satisfy Nyquist theorem v v v
9 Ideal Uniform Rectilinear Sampling 2D uniformly sampled grid
10 Ideal Uniform Rectilinear Sampling g a (x,y) g s (x,y)
11 g a (x,y,) How to recover g a?
12 How to recover g a?
13 Sinc interpolation We can recover g a (x, y) by interpolating the samples g d [n,m] using sinc functions
14 Interpolation Sampled g s MATLAB s interp2
15 What s the problem with sinc interpolation? Real world images need not be exactly bandlimited. Unbounded support Summations require infinitely many samples Computationally very expensive
16 Linear interpolation sinc interpolation formula Linear interpolation: h(x, y) = interpolation kernel Why this is a linear interpolation? Linear function of the samples g d [n,m]
17 Linear interpolation Consider alternative linear interpolation schemes that address issues with: Unbounded support Summations require infinitely many samples Computationally very expensive
18 Basic polynomial interpolation Here kernels that are piecewise polynomials Zero order or nearest neighbor interpolation Use the value at the nearest sample location Kernels are zero order polynomials
19 Nearest neighbor interpolation
20 Linear or bilinear interpolation tri function is a piecewise linear function of its (spatial) arguments Δ/2 Δ/2
21 Linear or bilinear interpolation For any point (x, y), we find the four nearest sample point fit a polynomial of the form Estimate alphas from system of 4 equations in 4 unknowns: Interpolating formula (x [0, 1] [0, 1]):
22 Linear or bilinear interpolation 0 1 If x [ 1, 1] [ 1, 1]: 1 1
23
24 Linear or bilinear interpolation
25 Linear interpolation by Delaunay triangulation Use a linear function within each triangle (three points determine a plane). MATLAB s griddata with the linear opt
26
27
28 What s the problem zero order or bilinear interpolation? neither the zero order or bilinear interpolator are differentiable. They are not continuous and smooth
29 Desirable properties of interpolators self consistency : Continuous and smooth (differentiable): Short spatial extent to minimize computation
30 Desirable properties of interpolators Frequency response approximately rect(). symmetric Shift invariant Minimum sidelobes to avoid ringing artifacts
31 Continuous and smooth Large sidelobes Non smooth Small sidelobes
32 Non smooth; non shift invariant
33 Cubic Polynomial interpolation interpolation kernel: Quadratic B splines
34
35 B spline interpolation Motivation: n order differentiable Short spatial extent to minimize computation
36 Background Splines are piecewise polynomials with pieces that are smoothly connected together The joining points of the polynomials are called knots
37 Background For a spline of degree n, each segment is a polynomial of degree n Do i need n+1 coefficient to describe each piece? Additional smoothness constraint imposes the continuity of the spline and its derivatives up to order (n 1) at the knots only one degree of freedom per segment!
38 B spline expansion Splines are uniquely characterized in terms of a B spline expansion integer shifts of the central B spline of degree n = central B spline = basis (B) spline
39 B splines Symmetrical bell shaped functions constructed from the (n+1) fold convolution
40 B splines
41 B splines n order B spline: Derived using:
42 B spline Using cubic B splines is a popular choice: Important: compactly supported!
43 Cubic B spline and its underlying components support
44 B spline expansion Each spline is unambiguously characterized by its sequence of B spline coefficients c(k)
45 Properties of B spline expansion Discrete signal representation! (even though the underlying model is a continuous representation). Easy to manipulate; E.g., derivatives:
46 B spline interpolation So far: B spline model of a given input signal s(x) s d [n] Interpolation problem: find coefficients c(k) such that spline function goes through the data points exactly That is, reconstruct signal using a spline representation!
47 Reconstruct signal using a spline representation s d [n]
48 B spline interpolation Relationship between coefficients and samples s d [n] = discrete B spline kernel Obtained by sampling the B spline of degree n expanded by a factor of m (m=1)
49 Cardinal B spline interpolation Cardinal splines
50 2D splines in images tensor product basis functions
51 B spline interpolation Conclusion: n order differentiable Short spatial extent to minimize computation
52 General image spatial transformations transform the spatial coordinates of an image f(x, y) so that (after being transformed), it better matches another image Ex: warping brain images into a standard coordinate system to facilitate automatic image analysis
53 image spatial transformations
54 General image spatial transformations Form of transformations: Shift only g(x, y) = f(x x 0, y y 0 ) Affine g(x, y) = f(ax + by, cx + dy) thin plate splines (describes smooth warpings using control points) General spatial transformation (e.g., morphing or warping an image) MATLAB s interp2 or griddata functions
55 Interpolation is critical!
56
57
Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics
Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group dimitri.vandeville@epfl.ch
Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points
Aliasing, Image Sampling and Reconstruction
Aliasing, Image Sampling and Reconstruction Recall: a pixel is a point It is NOT a box, disc or teeny wee light It has no dimension It occupies no area It can have a coordinate More than a point, it is
Finite Element Formulation for Plates - Handout 3 -
Finite Element Formulation for Plates - Handout 3 - Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the
Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition
Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition 1. Image Pre-Processing - Pixel Brightness Transformation - Geometric Transformation - Image Denoising 1 1. Image Pre-Processing
the points are called control points approximating curve
Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.
3. Interpolation. Closing the Gaps of Discretization... Beyond Polynomials
3. Interpolation Closing the Gaps of Discretization... Beyond Polynomials Closing the Gaps of Discretization... Beyond Polynomials, December 19, 2012 1 3.3. Polynomial Splines Idea of Polynomial Splines
Natural cubic splines
Natural cubic splines Arne Morten Kvarving Department of Mathematical Sciences Norwegian University of Science and Technology October 21 2008 Motivation We are given a large dataset, i.e. a function sampled
Correlation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs
Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in
Convolution. 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
Moving Least Squares Approximation
Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the so-called moving least squares method. As we will see below, in this method the
Piecewise Cubic Splines
280 CHAP. 5 CURVE FITTING Piecewise Cubic Splines The fitting of a polynomial curve to a set of data points has applications in CAD (computer-assisted design), CAM (computer-assisted manufacturing), and
INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).
INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x
4.3 Lagrange Approximation
206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average
jorge s. marques image processing
image processing images images: what are they? what is shown in this image? What is this? what is an image images describe the evolution of physical variables (intensity, color, reflectance, condutivity)
Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model
CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant
CHAPTER 1 Splines and B-splines an Introduction
CHAPTER 1 Splines and B-splines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the
(Refer Slide Time: 1:42)
Introduction to Computer Graphics Dr. Prem Kalra Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture - 10 Curves So today we are going to have a new topic. So far
Time series analysis Matlab tutorial. Joachim Gross
Time series analysis Matlab tutorial Joachim Gross Outline Terminology Sampling theorem Plotting Baseline correction Detrending Smoothing Filtering Decimation Remarks Focus on practical aspects, exercises,
Summary: Transformations. Lecture 14 Parameter Estimation Readings T&V Sec 5.1-5.3. Parameter Estimation: Fitting Geometric Models
Summary: Transformations Lecture 14 Parameter Estimation eadings T&V Sec 5.1-5.3 Euclidean similarity affine projective Parameter Estimation We will talk about estimating parameters of 1) Geometric models
JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:
Computer Graphics CS 543 Lecture 12 (Part 1) Curves. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)
Computer Graphics CS 54 Lecture 1 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines and flat surfaces Real world objects include
Objectives. Materials
Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the ways
1 Shapes of Cubic Functions
MA 1165 - Lecture 05 1 1/26/09 1 Shapes of Cubic Functions A cubic function (a.k.a. a third-degree polynomial function) is one that can be written in the form f(x) = ax 3 + bx 2 + cx + d. (1) Quadratic
Linear and quadratic Taylor polynomials for functions of several variables.
ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is
Two-Frame Motion Estimation Based on Polynomial Expansion
Two-Frame Motion Estimation Based on Polynomial Expansion Gunnar Farnebäck Computer Vision Laboratory, Linköping University, SE-581 83 Linköping, Sweden gf@isy.liu.se http://www.isy.liu.se/cvl/ Abstract.
Geometric Modelling & Curves
Geometric Modelling & Curves Geometric Modeling Creating symbolic models of the physical world has long been a goal of mathematicians, scientists, engineers, etc. Recently technology has advanced sufficiently
Digital Imaging and Multimedia. Filters. Ahmed Elgammal Dept. of Computer Science Rutgers University
Digital Imaging and Multimedia Filters Ahmed Elgammal Dept. of Computer Science Rutgers University Outlines What are Filters Linear Filters Convolution operation Properties of Linear Filters Application
SGN-1158 Introduction to Signal Processing Test. Solutions
SGN-1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:
Zeros of Polynomial Functions
Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction
Probability and Random Variables. Generation of random variables (r.v.)
Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly
Applications to Data Smoothing and Image Processing I
Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is
Robert Collins CSE598G. More on Mean-shift. R.Collins, CSE, PSU CSE598G Spring 2006
More on Mean-shift R.Collins, CSE, PSU Spring 2006 Recall: Kernel Density Estimation Given a set of data samples x i ; i=1...n Convolve with a kernel function H to generate a smooth function f(x) Equivalent
Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point
1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
CSE 167: Lecture 13: Bézier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011
CSE 167: Introduction to Computer Graphics Lecture 13: Bézier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011 Announcements Homework project #6 due Friday, Nov 18
Sampling Theorem Notes. Recall: That a time sampled signal is like taking a snap shot or picture of signal periodically.
Sampling Theorem We will show that a band limited signal can be reconstructed exactly from its discrete time samples. Recall: That a time sampled signal is like taking a snap shot or picture of signal
Serendipity Basis Functions for Any Degree in Any Dimension
Serendipity Basis Functions for Any Degree in Any Dimension Andrew Gillette Department of Mathematics University of Arizona joint work with Michael Floater (University of Oslo) http://math.arizona.edu/
8 Polynomials Worksheet
8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph
Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
it is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
PRACTICAL GUIDE TO DATA SMOOTHING AND FILTERING
PRACTICAL GUIDE TO DATA SMOOTHING AND FILTERING Ton van den Bogert October 3, 996 Summary: This guide presents an overview of filtering methods and the software which is available in the HPL.. What is
1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
Algebra 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
Finite Elements for 2 D Problems
Finite Elements for 2 D Problems General Formula for the Stiffness Matrix Displacements (u, v) in a plane element are interpolated from nodal displacements (ui, vi) using shape functions Ni as follows,
The continuous and discrete Fourier transforms
FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1
Computational Foundations of Cognitive Science
Computational Foundations of Cognitive Science Lecture 15: Convolutions and Kernels Frank Keller School of Informatics University of Edinburgh keller@inf.ed.ac.uk February 23, 2010 Frank Keller Computational
Trend and Seasonal Components
Chapter 2 Trend and Seasonal Components If the plot of a TS reveals an increase of the seasonal and noise fluctuations with the level of the process then some transformation may be necessary before doing
Sampling and Interpolation. Yao Wang Polytechnic University, Brooklyn, NY11201
Sampling and Interpolation Yao Wang Polytechnic University, Brooklyn, NY1121 http://eeweb.poly.edu/~yao Outline Basics of sampling and quantization A/D and D/A converters Sampling Nyquist sampling theorem
Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.
Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method
Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
Prentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
PCHS ALGEBRA PLACEMENT TEST
MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If
Mean Value Coordinates
Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its
A characterization of trace zero symmetric nonnegative 5x5 matrices
A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the
Lectures 6&7: Image Enhancement
Lectures 6&7: Image Enhancement Leena Ikonen Pattern Recognition (MVPR) Lappeenranta University of Technology (LUT) leena.ikonen@lut.fi http://www.it.lut.fi/ip/research/mvpr/ 1 Content Background Spatial
This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
Solving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
Building a Smooth Yield Curve. University of Chicago. Jeff Greco
Building a Smooth Yield Curve University of Chicago Jeff Greco email: jgreco@math.uchicago.edu Preliminaries As before, we will use continuously compounding Act/365 rates for both the zero coupon rates
1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
Video Coding with Cubic Spline Interpolation and Adaptive Motion Model Selection
Video Coding with Cubic Spline Interpolation and Adaptive Motion Model Selection Haricharan Lakshman, Heiko Schwarz and Thomas Wiegand Image Processing Department Fraunhofer Institute for Telecommunications
Basics of Polynomial Theory
3 Basics of Polynomial Theory 3.1 Polynomial Equations In geodesy and geoinformatics, most observations are related to unknowns parameters through equations of algebraic (polynomial) type. In cases where
Rolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
How To Understand The Nyquist Sampling Theorem
Nyquist Sampling Theorem By: Arnold Evia Table of Contents What is the Nyquist Sampling Theorem? Bandwidth Sampling Impulse Response Train Fourier Transform of Impulse Response Train Sampling in the Fourier
Image Interpolation by Pixel Level Data-Dependent Triangulation
Volume xx (200y), Number z, pp. 1 7 Image Interpolation by Pixel Level Data-Dependent Triangulation Dan Su, Philip Willis Department of Computer Science, University of Bath, Bath, BA2 7AY, U.K. mapds,
6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
The Convolution Operation
The Convolution Operation Convolution is a very natural mathematical operation which occurs in both discrete and continuous modes of various kinds. We often encounter it in the course of doing other operations
In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
Georgia Standards of Excellence Curriculum Map. Mathematics. GSE 8 th Grade
Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade
Modeling Curves and Surfaces
Modeling Curves and Surfaces Graphics I Modeling for Computer Graphics!? 1 How can we generate this kind of objects? Umm!? Mathematical Modeling! S Do not worry too much about your difficulties in mathematics,
Zeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
Microeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
FINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
Linear Filtering Part II
Linear Filtering Part II Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr Fourier theory Jean Baptiste Joseph Fourier had a crazy idea: Any periodic function can
POLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
DIEF, Department of Engineering Enzo Ferrari University of Modena e Reggio Emilia Italy Online Trajectory Planning for robotic systems
DIEF, Department of Engineering Enzo Ferrari University of Modena e Reggio Emilia Italy Online Trajectory Planning for robotic systems Luigi Biagiotti Luigi Biagiotti luigi.biagiotti@unimore.it Introduction
2 Integrating Both Sides
2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation
Matrices and Polynomials
APPENDIX 9 Matrices and Polynomials he Multiplication of Polynomials Let α(z) =α 0 +α 1 z+α 2 z 2 + α p z p and y(z) =y 0 +y 1 z+y 2 z 2 + y n z n be two polynomials of degrees p and n respectively. hen,
CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS
VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS MICHAEL DRMOTA, OMER GIMENEZ, AND MARC NOY Abstract. We show that the number of vertices of a given degree k in several kinds of series-parallel labelled
5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
2.5 Physically-based Animation
2.5 Physically-based Animation 320491: Advanced Graphics - Chapter 2 74 Physically-based animation Morphing allowed us to animate between two known states. Typically, only one state of an object is known.
Derive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
Polynomial Invariants
Polynomial Invariants Dylan Wilson October 9, 2014 (1) Today we will be interested in the following Question 1.1. What are all the possible polynomials in two variables f(x, y) such that f(x, y) = f(y,
Separable First Order Differential Equations
Separable First Order Differential Equations Form of Separable Equations which take the form = gx hy or These are differential equations = gxĥy, where gx is a continuous function of x and hy is a continuously
Essential 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
Design of FIR Filters
Design of FIR Filters Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 68 FIR as
DIGITAL IMAGE PROCESSING AND ANALYSIS
DIGITAL IMAGE PROCESSING AND ANALYSIS Human and Computer Vision Applications with CVIPtools SECOND EDITION SCOTT E UMBAUGH Uffi\ CRC Press Taylor &. Francis Group Boca Raton London New York CRC Press is
Differentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
Numerical integration of a function known only through data points
Numerical integration of a function known only through data points Suppose you are working on a project to determine the total amount of some quantity based on measurements of a rate. For example, you
Computational Optical Imaging - Optique Numerique. -- Deconvolution --
Computational Optical Imaging - Optique Numerique -- Deconvolution -- Winter 2014 Ivo Ihrke Deconvolution Ivo Ihrke Outline Deconvolution Theory example 1D deconvolution Fourier method Algebraic method
North Carolina Math 2
Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.