Line Drawing and Clipping Week 1, Lecture 2

Size: px
Start display at page:

Download "Line Drawing and Clipping Week 1, Lecture 2"

Transcription

1 CS 43 Computer Graphcs I Lne Drawng and Clppng Week, Lecture 2 Davd Breen, Wllam Regl and Maxm Peysakhov Geometrc and Intellgent Computng Laboratory Department of Computer Scence Drexel Unversty Outlne Lne drawng Dgtal dfferental analyzer Bresenham s algorthm lne clppng Parametrc lne clppng 2 Scan-Converson Algorthms Scan-Converson: Computng pxel coordnates for deal lne on 2D raster grd Pxels best vsualzed as crcles/dots Why? Montor hardware 3 Dgtal Dfferental Analyzer (DDA) If slope s less then _x = (otherwze _y = ) Compute correspondng _y (_x) x k+ = x k + _x y k+ = y k + _y Issues: Avodng floatng pont ops, multplcatons, and real numbers y k+ y k _y _x x k x k+ 4 IBM Basc Idea: Only nteger arthmetc Incremental Consder the mplct equaton for a lne: 5 Gven: mplct lne equaton: Let: where r and q are ponts on the lne Then: Observe that all of these are ntegers and: for ponts above the lne for ponts below the lne Now.. 6

2 Suppose we just fnshed (assume <slope<) other cases symmetrc Whch pxel next? E or NE 7 Assume: = exact y value at y mdway between E and NE: Observe: If, then pck E Else pck NE If, t doesn t matter 8 Create modfed mplct functon (2x) Create a decson varable D to select, where D s the value of f at the mdpont: If then m s below the lne NE s the closest pxel If then m s above the lne E s the closest pxel 9 Note: because we multpled by 2x, D s now an nteger---whch s very good news How do we make ths ncremental?? Case I: When E s next What ncrement for computng an new D? Next mdpont s: Case II: When NE s next What ncrement for computng an new D? Next mdpont s: Hence, ncrement by: Hence, ncrement by: 2 2

3 How to get an ntal value for D? The Algorthm Suppose we start at: Intal mdpont s: Then: Assumptons: 3 Pre-computed:

4 Some ssues wth s Pxel densty vares based on slope straght lnes look darker, more pxels per CM Endpont order Lne from P to P2 should match P2 to P Always choose E when httng M, regardless of drecton 24 4

5 Some ssues wth s How to handle the lne when t hts the clp wndow? Vertcal ntersectons Could change lne slope Need to change nt cond. Horzontal ntersectons Agan, changes n the boundary condtons Can t just ntersect the lne w/ the box 25 Scssorng Clppng Performed durng scan converson of the lne (crcle, polygon) Compute the next pont (x,y) If x mn x x max and y mn y y max Then output the pont. Else do nothng Issues wth scssorng: Too slow Does more work then necessary We are not gong to use t. 26 The Lne Clppng Algorthm How to clp lnes to ft n wndows? easy to tell f whole lne falls w/n wndow harder to tell what part falls nsde Consder a straght lne And wndow: 27 Basc Idea: Frst, do easy test completely nsde or outsde the box? If no, we need a more complex test Note: we wll also need to fgure out how scan lne meets the box 28 The Easy Test: Compute 4-bt code based on endponts P and P 2 Lne s completely vsble ff both code values of endponts are,.e. If lne segments are completely outsde the wndow, then

6 Otherwse, we clp the lnes: We know that there s a bt flp, w.o.l.g. assume ts (x,x ) Whch bt? Try `em all! suppose ts bt 4 Then x < WL and we know that x WL We need to fnd the pont: (x c,y c ) 3 Clearly: Usng smlar trangles Solvng for y c gves 32 Replace (x,y ) wth (x c,y c ) Re-compute values Contnue untl all ponts are nsde the clp wndow QED 33 Parametrc Lne Clppng Developed by Cyrus and Beck n 978 Used to clp 2D/3D lnes aganst convex polygon/polyhedron Lang and Barsky (984) algorthm effcent n clppng uprght 2D/3D clppng regons Frst we wll follow orgnal Cyrus-Beck development to ntroduce parametrc clppng Then we wll reduce Cyrus-Beck to more effcent Lang-Barsky case 34 The Cyrus-Beck Technque algorthm computes (x,y) ntersectons of the lne and clppng edge Cyrus-Beck fnds a value of parameter t for ntersectons of the lne and clppng edges Smple comparsons used to fnd actual ntersecton ponts Lang-Barsky optmzes t by examnng t values as they generated to reject some lne segments mmedately 35 Fndng the Intersecton Ponts Lne P(t) = P +t(p -P ) Pont on the edge P E N [P(t)-P ] = N [P + t(p-p )-P ] = N [P -P ] + N t[p-p ] = E E Let D = (P-P ) N [P - PE ] t = - N D Make sure. D, or P P 2. N D lnes are not parallel E 36 6

7 Fndng the Lne Segment Classfy pont as potentally enterng (PE) or leavng (PL) PE f crosses edge nto nsde half plane => angle P P and N greater 9 => N D < PL otherwse. Fnd T e = max(t e ) Fnd T l = mn(t l ) Dscard f T e > T l If T e < T e = If T l > T l = Use T e, T l to compute ntersecton coordnates (x e,y e ), (x l,y l ) 37 7

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017 U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that

More information

Review of Taylor Series. Read Section 1.2

Review of Taylor Series. Read Section 1.2 Revew of Taylor Seres Read Secton 1.2 1 Power Seres A power seres about c s an nfnte seres of the form k = 0 k a ( x c) = a + a ( x c) + a ( x c) + a ( x c) k 2 3 0 1 2 3 + In many cases, c = 0, and the

More information

Chapter Newton s Method

Chapter Newton s Method Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

Machine learning: Density estimation

Machine learning: Density estimation CS 70 Foundatons of AI Lecture 3 Machne learnng: ensty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square ata: ensty estmaton {.. n} x a vector of attrbute values Objectve: estmate the model of

More information

Economics 101. Lecture 4 - Equilibrium and Efficiency

Economics 101. Lecture 4 - Equilibrium and Efficiency Economcs 0 Lecture 4 - Equlbrum and Effcency Intro As dscussed n the prevous lecture, we wll now move from an envronment where we looed at consumers mang decsons n solaton to analyzng economes full of

More information

Mathematics Intersection of Lines

Mathematics Intersection of Lines a place of mnd F A C U L T Y O F E D U C A T I O N Department of Currculum and Pedagog Mathematcs Intersecton of Lnes Scence and Mathematcs Educaton Research Group Supported b UBC Teachng and Learnng Enhancement

More information

Ensemble Methods: Boosting

Ensemble Methods: Boosting Ensemble Methods: Boostng Ncholas Ruozz Unversty of Texas at Dallas Based on the sldes of Vbhav Gogate and Rob Schapre Last Tme Varance reducton va baggng Generate new tranng data sets by samplng wth replacement

More information

The path of ants Dragos Crisan, Andrei Petridean, 11 th grade. Colegiul National "Emil Racovita", Cluj-Napoca

The path of ants Dragos Crisan, Andrei Petridean, 11 th grade. Colegiul National Emil Racovita, Cluj-Napoca Ths artcle s wrtten by students. It may nclude omssons and mperfectons, whch were dentfed and reported as mnutely as possble by our revewers n the edtoral notes. The path of ants 07-08 Students names and

More information

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as: 1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors

More information

MTH 263 Practice Test #1 Spring 1999

MTH 263 Practice Test #1 Spring 1999 Pat Ross MTH 6 Practce Test # Sprng 999 Name. Fnd the area of the regon bounded by the graph r =acos (θ). Observe: Ths s a crcle of radus a, for r =acos (θ) r =a ³ x r r =ax x + y =ax x ax + y =0 x ax

More information

Lossy Compression. Compromise accuracy of reconstruction for increased compression.

Lossy Compression. Compromise accuracy of reconstruction for increased compression. Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost

More information

Modeling curves. Graphs: y = ax+b, y = sin(x) Implicit ax + by + c = 0, x 2 +y 2 =r 2 Parametric:

Modeling curves. Graphs: y = ax+b, y = sin(x) Implicit ax + by + c = 0, x 2 +y 2 =r 2 Parametric: Modelng curves Types of Curves Graphs: y = ax+b, y = sn(x) Implct ax + by + c = 0, x 2 +y 2 =r 2 Parametrc: x = ax + bxt x = cos t y = ay + byt y = snt Parametrc are the most common mplct are also used,

More information

College of Computer & Information Science Fall 2009 Northeastern University 20 October 2009

College of Computer & Information Science Fall 2009 Northeastern University 20 October 2009 College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be

More information

Homework Assignment 3 Due in class, Thursday October 15

Homework Assignment 3 Due in class, Thursday October 15 Homework Assgnment 3 Due n class, Thursday October 15 SDS 383C Statstcal Modelng I 1 Rdge regresson and Lasso 1. Get the Prostrate cancer data from http://statweb.stanford.edu/~tbs/elemstatlearn/ datasets/prostate.data.

More information

Chapter 11: Simple Linear Regression and Correlation

Chapter 11: Simple Linear Regression and Correlation Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests

More information

CS 2750 Machine Learning. Lecture 5. Density estimation. CS 2750 Machine Learning. Announcements

CS 2750 Machine Learning. Lecture 5. Density estimation. CS 2750 Machine Learning. Announcements CS 750 Machne Learnng Lecture 5 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square CS 750 Machne Learnng Announcements Homework Due on Wednesday before the class Reports: hand n before

More information

A 2D Bounded Linear Program (H,c) 2D Linear Programming

A 2D Bounded Linear Program (H,c) 2D Linear Programming A 2D Bounded Lnear Program (H,c) h 3 v h 8 h 5 c h 4 h h 6 h 7 h 2 2D Lnear Programmng C s a polygonal regon, the ntersecton of n halfplanes. (H, c) s nfeasble, as C s empty. Feasble regon C s unbounded

More information

Affine transformations and convexity

Affine transformations and convexity Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

More information

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9 Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng

More information

18.1 Introduction and Recap

18.1 Introduction and Recap CS787: Advanced Algorthms Scrbe: Pryananda Shenoy and Shjn Kong Lecturer: Shuch Chawla Topc: Streamng Algorthmscontnued) Date: 0/26/2007 We contnue talng about streamng algorthms n ths lecture, ncludng

More information

Chapter 8 SCALAR QUANTIZATION

Chapter 8 SCALAR QUANTIZATION Outlne Chapter 8 SCALAR QUANTIZATION Yeuan-Kuen Lee [ CU, CSIE ] 8.1 Overvew 8. Introducton 8.4 Unform Quantzer 8.5 Adaptve Quantzaton 8.6 Nonunform Quantzaton 8.7 Entropy-Coded Quantzaton Ch 8 Scalar

More information

The Geometry of Logit and Probit

The Geometry of Logit and Probit The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.

More information

Physics 207: Lecture 20. Today s Agenda Homework for Monday

Physics 207: Lecture 20. Today s Agenda Homework for Monday Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems

More information

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve

More information

Linear Feature Engineering 11

Linear Feature Engineering 11 Lnear Feature Engneerng 11 2 Least-Squares 2.1 Smple least-squares Consder the followng dataset. We have a bunch of nputs x and correspondng outputs y. The partcular values n ths dataset are x y 0.23 0.19

More information

Turing Machines (intro)

Turing Machines (intro) CHAPTER 3 The Church-Turng Thess Contents Turng Machnes defntons, examples, Turng-recognzable and Turng-decdable languages Varants of Turng Machne Multtape Turng machnes, non-determnstc Turng Machnes,

More information

Digital Signal Processing

Digital Signal Processing Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over

More information

Integrals and Invariants of Euler-Lagrange Equations

Integrals and Invariants of Euler-Lagrange Equations Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,

More information

Least Squares Fitting of Data

Least Squares Fitting of Data Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2014. All Rghts Reserved. Created: July 15, 1999 Last Modfed: February 9, 2008 Contents 1 Lnear Fttng

More information

Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0

Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0 Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of

More information

12. The Hamilton-Jacobi Equation Michael Fowler

12. The Hamilton-Jacobi Equation Michael Fowler 1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and

More information

Chapter 3 Describing Data Using Numerical Measures

Chapter 3 Describing Data Using Numerical Measures Chapter 3 Student Lecture Notes 3-1 Chapter 3 Descrbng Data Usng Numercal Measures Fall 2006 Fundamentals of Busness Statstcs 1 Chapter Goals To establsh the usefulness of summary measures of data. The

More information

Spectral Graph Theory and its Applications September 16, Lecture 5

Spectral Graph Theory and its Applications September 16, Lecture 5 Spectral Graph Theory and ts Applcatons September 16, 2004 Lecturer: Danel A. Spelman Lecture 5 5.1 Introducton In ths lecture, we wll prove the followng theorem: Theorem 5.1.1. Let G be a planar graph

More information

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department

More information

Chapter 9: Statistical Inference and the Relationship between Two Variables

Chapter 9: Statistical Inference and the Relationship between Two Variables Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,

More information

Min Cut, Fast Cut, Polynomial Identities

Min Cut, Fast Cut, Polynomial Identities Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.

More information

PART 8. Partial Differential Equations PDEs

PART 8. Partial Differential Equations PDEs he Islamc Unverst of Gaza Facult of Engneerng Cvl Engneerng Department Numercal Analss ECIV 3306 PAR 8 Partal Dfferental Equatons PDEs Chapter 9; Fnte Dfference: Ellptc Equatons Assocate Prof. Mazen Abualtaef

More information

Section 8.1 Exercises

Section 8.1 Exercises Secton 8.1 Non-rght Trangles: Law of Snes and Cosnes 519 Secton 8.1 Exercses Solve for the unknown sdes and angles of the trangles shown. 10 70 50 1.. 18 40 110 45 5 6 3. 10 4. 75 15 5 6 90 70 65 5. 6.

More information

The Study of Teaching-learning-based Optimization Algorithm

The Study of Teaching-learning-based Optimization Algorithm Advanced Scence and Technology Letters Vol. (AST 06), pp.05- http://dx.do.org/0.57/astl.06. The Study of Teachng-learnng-based Optmzaton Algorthm u Sun, Yan fu, Lele Kong, Haolang Q,, Helongang Insttute

More information

Solutions to exam in SF1811 Optimization, Jan 14, 2015

Solutions to exam in SF1811 Optimization, Jan 14, 2015 Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable

More information

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)? Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2

More information

Double Layered Fuzzy Planar Graph

Double Layered Fuzzy Planar Graph Global Journal of Pure and Appled Mathematcs. ISSN 0973-768 Volume 3, Number 0 07), pp. 7365-7376 Research Inda Publcatons http://www.rpublcaton.com Double Layered Fuzzy Planar Graph J. Jon Arockaraj Assstant

More information

Unit 5: Quadratic Equations & Functions

Unit 5: Quadratic Equations & Functions Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton

More information

Some Inequalities Between Two Polygons Inscribed One in the Other

Some Inequalities Between Two Polygons Inscribed One in the Other BULLETIN of the Malaysan Mathematcal Scences Socety http://math.usm.my/bulletn Bull. Malays. Math. Sc. Soc. () 8() (005), 73 79 Some Inequaltes Between Two Polygons Inscrbed One n the Other Aurelo de Gennaro

More information

One Dimension Again. Chapter Fourteen

One Dimension Again. Chapter Fourteen hapter Fourteen One Dmenson Agan 4 Scalar Lne Integrals Now we agan consder the dea of the ntegral n one dmenson When we were ntroduced to the ntegral back n elementary school, we consdered only functons

More information

CIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M

CIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M CIS56: achne Learnng Lecture 3 (Sept 6, 003) Preparaton help: Xaoyng Huang Lnear Regresson Lnear regresson can be represented by a functonal form: f(; θ) = θ 0 0 +θ + + θ = θ = 0 ote: 0 s a dummy attrbute

More information

Here is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)

Here is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y) Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,

More information

MATH 829: Introduction to Data Mining and Analysis The EM algorithm (part 2)

MATH 829: Introduction to Data Mining and Analysis The EM algorithm (part 2) 1/16 MATH 829: Introducton to Data Mnng and Analyss The EM algorthm (part 2) Domnque Gullot Departments of Mathematcal Scences Unversty of Delaware Aprl 20, 2016 Recall 2/16 We are gven ndependent observatons

More information

Lecture Notes on Linear Regression

Lecture Notes on Linear Regression Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume

More information

Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane.

Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane. Queens College, CUNY, Department of Computer Scence Numercal Methods CSCI 361 / 761 Sprng 2018 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2018 16 Lecture 16a May 3, 2018 Numercal soluton of systems

More information

Lecture 21: Numerical methods for pricing American type derivatives

Lecture 21: Numerical methods for pricing American type derivatives Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)

More information

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011 Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected

More information

Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:

Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law: CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and

More information

Formulas for the Determinant

Formulas for the Determinant page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use

More information

Calculation of time complexity (3%)

Calculation of time complexity (3%) Problem 1. (30%) Calculaton of tme complexty (3%) Gven n ctes, usng exhaust search to see every result takes O(n!). Calculaton of tme needed to solve the problem (2%) 40 ctes:40! dfferent tours 40 add

More information

Statistical Mechanics and Combinatorics : Lecture III

Statistical Mechanics and Combinatorics : Lecture III Statstcal Mechancs and Combnatorcs : Lecture III Dmer Model Dmer defntons Defnton A dmer coverng (perfect matchng) of a fnte graph s a set of edges whch covers every vertex exactly once, e every vertex

More information

Error Bars in both X and Y

Error Bars in both X and Y Error Bars n both X and Y Wrong ways to ft a lne : 1. y(x) a x +b (σ x 0). x(y) c y + d (σ y 0) 3. splt dfference between 1 and. Example: Prmordal He abundance: Extrapolate ft lne to [ O / H ] 0. [ He

More information

Supplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso

Supplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed

More information

Generalized Linear Methods

Generalized Linear Methods Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set

More information

Lecture 5 Decoding Binary BCH Codes

Lecture 5 Decoding Binary BCH Codes Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture

More information

CALCULUS CLASSROOM CAPSULES

CALCULUS CLASSROOM CAPSULES CALCULUS CLASSROOM CAPSULES SESSION S86 Dr. Sham Alfred Rartan Valley Communty College salfred@rartanval.edu 38th AMATYC Annual Conference Jacksonvlle, Florda November 8-, 202 2 Calculus Classroom Capsules

More information

Lecture 2 Solution of Nonlinear Equations ( Root Finding Problems )

Lecture 2 Solution of Nonlinear Equations ( Root Finding Problems ) Lecture Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton o Methods Analytcal Solutons Graphcal Methods Numercal Methods Bracketng Methods Open Methods Convergence Notatons Root Fndng

More information

Suggested solutions for the exam in SF2863 Systems Engineering. June 12,

Suggested solutions for the exam in SF2863 Systems Engineering. June 12, Suggested solutons for the exam n SF2863 Systems Engneerng. June 12, 2012 14.00 19.00 Examner: Per Enqvst, phone: 790 62 98 1. We can thnk of the farm as a Jackson network. The strawberry feld s modelled

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

More information

ˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m)

ˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m) 7.. = = 3 = 4 = 5. The electrc feld s constant everywhere between the plates. Ths s ndcated by the electrc feld vectors, whch are all the same length and n the same drecton. 7.5. Model: The dstances to

More information

Maximal Margin Classifier

Maximal Margin Classifier CS81B/Stat41B: Advanced Topcs n Learnng & Decson Makng Mamal Margn Classfer Lecturer: Mchael Jordan Scrbes: Jana van Greunen Corrected verson - /1/004 1 References/Recommended Readng 1.1 Webstes www.kernel-machnes.org

More information

EN40: Dynamics and Vibrations. Homework 4: Work, Energy and Linear Momentum Due Friday March 1 st

EN40: Dynamics and Vibrations. Homework 4: Work, Energy and Linear Momentum Due Friday March 1 st EN40: Dynamcs and bratons Homework 4: Work, Energy and Lnear Momentum Due Frday March 1 st School of Engneerng Brown Unversty 1. The fgure (from ths publcaton) shows the energy per unt area requred to

More information

Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0

Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0 Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

(2mn, m 2 n 2, m 2 + n 2 )

(2mn, m 2 n 2, m 2 + n 2 ) MATH 16T Homewk Solutons 1. Recall that a natural number n N s a perfect square f n = m f some m N. a) Let n = p α even f = 1,,..., k. be the prme factzaton of some n. Prove that n s a perfect square f

More information

PHYS 705: Classical Mechanics. Calculus of Variations II

PHYS 705: Classical Mechanics. Calculus of Variations II 1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary

More information

VQ widely used in coding speech, image, and video

VQ widely used in coding speech, image, and video at Scalar quantzers are specal cases of vector quantzers (VQ): they are constraned to look at one sample at a tme (memoryless) VQ does not have such constrant better RD perfomance expected Source codng

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 5 Lecture 0 Canoncal Transformatons (Chapter 9) What We Dd Last Tme Hamlton s Prncple n the Hamltonan formalsm Dervaton was smple δi δ p H(, p, t) = 0 Adonal end-pont constrants δ t ( )

More information

Tutorial 2. COMP4134 Biometrics Authentication. February 9, Jun Xu, Teaching Asistant

Tutorial 2. COMP4134 Biometrics Authentication. February 9, Jun Xu, Teaching Asistant Tutoral 2 COMP434 ometrcs uthentcaton Jun Xu, Teachng sstant csjunxu@comp.polyu.edu.hk February 9, 207 Table of Contents Problems Problem : nswer the questons Problem 2: Power law functon Problem 3: Convoluton

More information

Celestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg

Celestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg PHY 454 - celestal-mechancs - J. Hedberg - 207 Celestal Mechancs. Basc Orbts. Why crcles? 2. Tycho Brahe 3. Kepler 4. 3 laws of orbtng bodes 2. Newtonan Mechancs 3. Newton's Laws. Law of Gravtaton 2. The

More information

1 GSW Iterative Techniques for y = Ax

1 GSW Iterative Techniques for y = Ax 1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn

More information

MLE and Bayesian Estimation. Jie Tang Department of Computer Science & Technology Tsinghua University 2012

MLE and Bayesian Estimation. Jie Tang Department of Computer Science & Technology Tsinghua University 2012 MLE and Bayesan Estmaton Je Tang Department of Computer Scence & Technology Tsnghua Unversty 01 1 Lnear Regresson? As the frst step, we need to decde how we re gong to represent the functon f. One example:

More information

Report on Image warping

Report on Image warping Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.

More information

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of

More information

The Second Anti-Mathima on Game Theory

The Second Anti-Mathima on Game Theory The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player

More information

: Numerical Analysis Topic 2: Solution of Nonlinear Equations Lectures 5-11:

: Numerical Analysis Topic 2: Solution of Nonlinear Equations Lectures 5-11: 764: Numercal Analyss Topc : Soluton o Nonlnear Equatons Lectures 5-: UIN Malang Read Chapters 5 and 6 o the tetbook 764_Topc Lecture 5 Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton

More information

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of

More information

11. Dynamics in Rotating Frames of Reference

11. Dynamics in Rotating Frames of Reference Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons

More information

Chapter 3. r r. Position, Velocity, and Acceleration Revisited

Chapter 3. r r. Position, Velocity, and Acceleration Revisited Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector

More information

Grover s Algorithm + Quantum Zeno Effect + Vaidman

Grover s Algorithm + Quantum Zeno Effect + Vaidman Grover s Algorthm + Quantum Zeno Effect + Vadman CS 294-2 Bomb 10/12/04 Fall 2004 Lecture 11 Grover s algorthm Recall that Grover s algorthm for searchng over a space of sze wors as follows: consder the

More information

Statistics MINITAB - Lab 2

Statistics MINITAB - Lab 2 Statstcs 20080 MINITAB - Lab 2 1. Smple Lnear Regresson In smple lnear regresson we attempt to model a lnear relatonshp between two varables wth a straght lne and make statstcal nferences concernng that

More information

ANOVA. The Observations y ij

ANOVA. The Observations y ij ANOVA Stands for ANalyss Of VArance But t s a test of dfferences n means The dea: The Observatons y j Treatment group = 1 = 2 = k y 11 y 21 y k,1 y 12 y 22 y k,2 y 1, n1 y 2, n2 y k, nk means: m 1 m 2

More information

Kernel Methods and SVMs Extension

Kernel Methods and SVMs Extension Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general

More information

= 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W]

= 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W] Secton 1.3: Acceleraton Tutoral 1 Practce, page 24 1. Gven: 0 m/s; 15.0 m/s [S]; t 12.5 s Requred: Analyss: a av v t v f v t a v av f v t 15.0 m/s [S] 0 m/s 12.5 s 15.0 m/s [S] 12.5 s 1.20 m/s 2 [S] Statement:

More information

The Second Eigenvalue of Planar Graphs

The Second Eigenvalue of Planar Graphs Spectral Graph Theory Lecture 20 The Second Egenvalue of Planar Graphs Danel A. Spelman November 11, 2015 Dsclamer These notes are not necessarly an accurate representaton of what happened n class. The

More information

Single Variable Optimization

Single Variable Optimization 8/4/07 Course Instructor Dr. Raymond C. Rump Oce: A 337 Phone: (95) 747 6958 E Mal: rcrump@utep.edu Topc 8b Sngle Varable Optmzaton EE 4386/530 Computatonal Methods n EE Outlne Mathematcal Prelmnares Sngle

More information

Spatial Statistics and Analysis Methods (for GEOG 104 class).

Spatial Statistics and Analysis Methods (for GEOG 104 class). Spatal Statstcs and Analyss Methods (for GEOG 104 class). Provded by Dr. An L, San Dego State Unversty. 1 Ponts Types of spatal data Pont pattern analyss (PPA; such as nearest neghbor dstance, quadrat

More information

MA 323 Geometric Modelling Course Notes: Day 13 Bezier Curves & Bernstein Polynomials

MA 323 Geometric Modelling Course Notes: Day 13 Bezier Curves & Bernstein Polynomials MA 323 Geometrc Modellng Course Notes: Day 13 Bezer Curves & Bernsten Polynomals Davd L. Fnn Over the past few days, we have looked at de Casteljau s algorthm for generatng a polynomal curve, and we have

More information

A New Algorithm for Finding a Fuzzy Optimal. Solution for Fuzzy Transportation Problems

A New Algorithm for Finding a Fuzzy Optimal. Solution for Fuzzy Transportation Problems Appled Mathematcal Scences, Vol. 4, 200, no. 2, 79-90 A New Algorthm for Fndng a Fuzzy Optmal Soluton for Fuzzy Transportaton Problems P. Pandan and G. Nataraan Department of Mathematcs, School of Scence

More information

Multilayer Perceptron (MLP)

Multilayer Perceptron (MLP) Multlayer Perceptron (MLP) Seungjn Cho Department of Computer Scence and Engneerng Pohang Unversty of Scence and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjn@postech.ac.kr 1 / 20 Outlne

More information