EECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines


 Frederica Lindsey
 2 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
since by using a computer we are limited to the use of elementary arithmetic operations
> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations
More informationa = x 1 < x 2 < < x n = b, and a low degree polynomial is used to approximate f(x) on each subinterval. Example: piecewise linear approximation S(x)
Spline Background SPLINE INTERPOLATION Problem: high degree interpolating polynomials often have extra oscillations. Example: Runge function f(x) = 1 1+4x 2, x [ 1, 1]. 1 1/(1+4x 2 ) and P 8 (x) and P
More informationCSE168 Computer Graphics II, Rendering. Spring 2006 Matthias Zwicker
CSE168 Computer Graphics II, Rendering Spring 2006 Matthias Zwicker Last time Sampling and aliasing Aliasing Moire patterns Aliasing Sufficiently sampled Insufficiently sampled [R. Cook ] Fourier analysis
More informationPoint 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
More informationFourier Transform and Image Filtering. CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010
Fourier Transform and Image Filtering CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010 The Fourier Transform Fourier Transform Forward, mapping to frequency domain: Backward, inverse mapping to time domain:
More informationComputer 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: WireFrame Representation Object is represented as as a set of points
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 informationAliasing, 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
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 informationFinite 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
More informationComputer Graphics. Course SS 2007 Antialiasing. computer graphics & visualization
Computer Graphics Course SS 2007 Antialiasing How to avoid spatial aliasing caused by an undersampling of the signal, i.e. the sampling frequency is not high enough to cover all details Supersampling 
More informationHomework set 1: posted on website
Homework set 1: posted on website 1 Last time: Interpolation, fitting and smoothing Interpolated curve passes through all data points Fitted curve passes closest (by some criterion) to all data points
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More informationthe 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.
More informationparametric spline curves
parametric spline curves 1 curves used in many contexts fonts (2D) animation paths (3D) shape modeling (3D) different representation implicit curves parametric curves (mostly used) 2D and 3D curves are
More informationNatural 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
More informationCurve Fitting. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization.
Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization Subsections LeastSquares Regression Linear Regression General Linear LeastSquares Nonlinear
More informationCorrelation 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
More informationPiecewise 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 (computerassisted design), CAM (computerassisted manufacturing), and
More informationMoving Least Squares Approximation
Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the socalled moving least squares method. As we will see below, in this method the
More information3. 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
More informationjorge 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)
More information4.3 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
More informationINTERPOLATION. 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
More informationIntroduction 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
More informationPicture Perfect: The Mathematics of JPEG Compression
Picture Perfect: The Mathematics of JPEG Compression May 19, 2011 1 2 3 in 2D Sampling and the DCT in 2D 2D Compression Images Outline A typical color image might be 600 by 800 pixels. Images Outline A
More information222 The International Arab Journal of Information Technology, Vol. 1, No. 2, July 2004 particular pixels. High pass filter and low pass filter are gen
The International Arab Journal of Information Technology, Vol. 1, No. 2, July 2004 221 A New Approach for Contrast Enhancement Using Sigmoid Function Naglaa Hassan 1&2 and Norio Akamatsu 1 1 Department
More informationInterpolating and Approximating Implicit Surfaces from Polygon Soup
Interpolating and Approximating Implicit Surfaces from Polygon Soup 1. Briefly summarize the paper s contributions. Does it address a new problem? Does it present a new approach? Does it show new types
More informationConvolution. 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
More informationTime 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,
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 informationProblem definition: optical flow
Motion Estimation http://www.sandlotscience.com/distortions/breathing_objects.htm http://www.sandlotscience.com/ambiguous/barberpole.htm Why estimate motion? Lots of uses Track object behavior Correct
More informationA polynomial of degree n is a function of the form. f(x) = 3x 22x 4. Also, identify domain & range.
f(x) = 3x 22x 4. Also, identify domain & range. Identify polynomial functions. Recognize characteristics of graphs of polynomials. Determine end behavior. Use factoring to find zeros of polynomials.
More information1999, Denis Zorin. Image processing: image resizing
Image processing: image resizing How computer images work? Continuous real image square array of numbers (abstract pixels) display (physical pixels) perceived image Digitization (e.g. scanning) each physical
More informationCHAPTER 1 Splines and Bsplines an Introduction
CHAPTER 1 Splines and Bsplines 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
More informationComputer 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
More informationJUST 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
More informationLinear 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
More informationInfinite Algebra 1 supports the teaching of the Common Core State Standards listed below.
Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School
More informationTwoFrame Motion Estimation Based on Polynomial Expansion
TwoFrame Motion Estimation Based on Polynomial Expansion Gunnar Farnebäck Computer Vision Laboratory, Linköping University, SE581 83 Linköping, Sweden gf@isy.liu.se http://www.isy.liu.se/cvl/ Abstract.
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More informationLimit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f
More informationDepartment of Electronics and Communication Engineering 1
DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING III Year ECE / V Semester EC 6502 PRINCIPLES OF DIGITAL SIGNAL PROCESSING QUESTION BANK Department of
More informationObjectives. Materials
Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI84 Plus / TI83 Plus Graph paper Introduction One of the ways
More informationComputer Vision: Filtering
Computer Vision: Filtering Raquel Urtasun TTI Chicago Jan 10, 2013 Raquel Urtasun (TTIC) Computer Vision Jan 10, 2013 1 / 82 Today s lecture... Image formation Image Filtering Raquel Urtasun (TTIC) Computer
More information(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
More informationNEW 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
More informationGeometric Image Transformations
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
More informationDeinterlacing/interpolation of TV signals. Elham Shahinfard Advisors: Prof. M. Ahmadi Prof. M. SidAhmed
Deinterlacing/interpolation of TV signals Elham Shahinfard Advisors: Prof. M. Ahmadi Prof. M. SidAhmed Outline A Review of some terminologies Converting from NTSC to HDTV; What changes need to be considered?
More informationWe want to define smooth curves:  for defining paths of cameras or objects.  for defining 1D shapes of objects
lecture 10  cubic curves  cubic splines  bicubic surfaces We want to define smooth curves:  for defining paths of cameras or objects  for defining 1D shapes of objects We want to define smooth surfaces
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 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 informationLecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)
ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process
More informationAlgebra II Semester Exam Review Sheet
Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationMinimizing beamon time during radiotherapy treatment by totalvariation regularization
Minimizing beamon time during radiotherapy treatment by totalvariation regularization Xing Lab, Department of Radiation Oncology, Stanford. Zhu, L. & Xing, L. (2009) Search for IMRT inverse plans with
More informationRemote Sensing Image Processing
Remote Sensing Image Processing Preprocessing Geometric Correction Atmospheric correction Image enhancement Image classification Division of Spatial Information Science Graduate School Life and Environment
More informationSGN1158 Introduction to Signal Processing Test. Solutions
SGN1158 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:
More informationGeometric 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
More informationResearch Problems in Finite Element Theory: Analysis, Geometry, and Application
Research Problems in Finite Element Theory: Analysis, Geometry, and Application Andrew Gillette Department of Mathematics University of Arizona Research Tutorial Group Presentation Slides and more info
More informationNorth Carolina Math 1
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.
More informationWe can display an object on a monitor screen in three different computermodel 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
More informationHigh School Algebra 1 Common Core Standards & Learning Targets
High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities NQ.1. Use units as a way to understand problems
More informationApplication. Outline. 31 Polynomial Functions 32 Finding Rational Zeros of. Polynomial. 33 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 31 Polynomial Functions 32 Finding Rational Zeros of Polynomials 33 Approximating Real Zeros of Polynomials 34 Rational Functions Chapter 3 Group Activity:
More informationHøgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin email: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider twopoint
More informationApplications 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
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 informationIntroduction to polynomials
Worksheet 4.5 Polynomials Section 1 Introduction to polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n, (n N) where p 0, p 1,..., p n are constants and x os
More informationPSS 27.2 The Electric Field of a Continuous Distribution of Charge
Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight ProblemSolving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated.
More informationSignals and Sampling. Chapter 7. Motivation. Additional Reading. Motivation. Motivation
Chapter 7 Signals and Sampling 7.1 Sampling Theory 7.2 Image Sampling Interface Chapter 7 of Physically Based Rendering by Pharr&Humphreys 7.3 Stratified Sampling 7.4 LowDiscrepancy Sampling 7.5 BestCandidate
More informationImage Enhancement  Frequency Domain
Image Processing  Lesson 7 Image Enhancement  Frequency Domain Low Pass Filter High Pass Filter Band pass Filter Blurring Sharpening Image FFT Filtered FFT 1 Transform Transform Filtered Image filter
More informationDigital 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
More informationWhat is an Edge? Computer Vision Week 4. How to detect edges? What is an Edge? Edge Detection techniques. Edge Detection techniques.
What is an Edge? Computer Vision Week 4 Edge Detection Linear filtering; pyramids, wavelets Interest Operators surface normal discontinuity depth discontinuity surface color discontinuity illumination
More informationStability of time variant filters
Stable TV filters Stability of time variant filters Michael P. Lamoureux, Safa Ismail, and Gary F. Margrave ABSTRACT We report on a project to investigate the stability and boundedness properties of interpolated
More informationi = 0, 1,, N p(x i ) = f i,
Chapter 4 Curve Fitting We consider two commonly used methods for curve fitting, namely interpolation and least squares For interpolation, we use first polynomials then splines Then, we introduce least
More informationCHAPTER 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.
More informationCommon Core Algebra Critical Area 6: Systems of Equations and Inequalities and Linear Programming
Pacing: Weeks 3136 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
More informationImage Projection. Goal: Introduce the basic concepts and mathematics for image projection.
Image Projection Goal: Introduce the basic concepts and mathematics for image projection. Motivation: The mathematics of image projection allow us to answer two questions: Given a 3D scene, how does it
More informationSampling 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
More informationSolutions for 2015 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama
Solutions for 0 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama. For f(x = 6x 4(x +, find f(. Solution: The notation f( tells us to evaluate the
More informationComputational 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
More informationSerendipity 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/
More informationPolynomials Classwork
Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th degree polynomial function Def.: A polynomial function of degree n in the vaiable
More informationTrend 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
More information3.5 Spline interpolation
3.5 Spline interpolation Given a tabulated function f k = f(x k ), k = 0,... N, a spline is a polynomial between each pair of tabulated points, but one whose coefficients are determined slightly nonlocally.
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 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
More informationRobert Collins CSE598G. More on Meanshift. R.Collins, CSE, PSU CSE598G Spring 2006
More on Meanshift 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
More informationPowerTeaching i3: Algebra I Mathematics
PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and
More informationProbability 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
More informationDigital Filter Design (FIR) Using Frequency Sampling Method
Digital Filter Design (FIR) Using Frequency Sampling Method Amer Ali Ammar Dr. Mohamed. K. Julboub Dr.Ahmed. A. Elmghairbi Electric and Electronic Engineering Department, Faculty of Engineering Zawia University
More informationit 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
More information56 The Remainder and Factor Theorems
Use synthetic substitution to find f (4) and f ( 2) for each function. 1. f (x) = 2x 3 5x 2 x + 14 Divide the function by x 4. The remainder is 58. Therefore, f (4) = 58. Divide the function by x + 2.
More informationSYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation
SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline
More informationImage Interpolation by Pixel Level DataDependent Triangulation
Volume xx (200y), Number z, pp. 1 7 Image Interpolation by Pixel Level DataDependent Triangulation Dan Su, Philip Willis Department of Computer Science, University of Bath, Bath, BA2 7AY, U.K. mapds,
More information1.5. 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
More informationMean 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
More informationPolygon Scan Conversion & Shading
3D Rendering Pipeline (for direct illumination) Polygon Scan Conversion & Shading Greg Humphreys CS445: Intro Graphics University of Virginia, Fall 2004 3D Primitives 3D Modeling Coordinates Modeling Transformation
More informationSampling 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
More informationCommon Core State Standard I Can Statements 8 th Grade Mathematics. The Number System (NS)
CCSS Key: The Number System (NS) Expressions & Equations (EE) Functions (F) Geometry (G) Statistics & Probability (SP) Common Core State Standard I Can Statements 8 th Grade Mathematics 8.NS.1. Understand
More informationLagrange 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
More information