15-462: Computer Graphics

Total Page:16

File Type:pdf, Size:1020Kb

15-462: Computer Graphics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f(x,y) + - "#$%&'&()*+,-(&./0 !"#$%&'#()*+,#-.$+/0 1 2-.$3435*67*43(4$+ 1 2-.$3435*67*$38+ 1 2-.$3435*97*.$#8+ !! #$%&'(')*+,-.)'/01 2 3+*(.&&*)4+1+*+,-.)'/01*5'$%&'(')6 7+(.-1+* .&)4/-84*)4+9*'$%&9*.*(-:;+*/:*1-:<.(+=* )4+9*(.00/)*+>%&'(')&9*8+0+:.)+*)4+*%/'0)1* )4.)*(/$%:'1+*')? 2 #0*/:@+:*)/*8+0+:.)+*%/'0)1=*A+*0++@* .0/)4+:*</:$B !" #$%$&'(%)*+',-$()./0 1 !"#"$%&#'()%*+"&',-. .22'%+(3'+*$4$5)6)(7+(.+ 8'/'%$('+*./()/-.-0+*-%9'0+$/:+0-%2$*'0; 1 <.%+*-%9'0=+4$%$&'(%)*+',-$()./0+($>'+(3'+ 2.%& / ?+0@&A 1)2) 3@&A 4 ?+5@&A 1 <.%+BC+0-%2$*'0=+D'+3$9' / ?+0@.=&A 1)2)3@.=&A 4 ?+5@.=&A !" #$%$&'(%)*+',-$()./0 1 23'+!"#"$%&%#' 4.%+(3'0'+',-$()./0+$%'+ 0*$5$%0+(3$(+%$/6'+.7'%+$+*./()/-.-0+ 89.00):5;+)/4)/)('<+)/('%7$5= 1 >$%;)/6+(3'+9$%$&'('%0+.7'%+(3')%+'/()%'+ )/('%7$50+0&..(35;+6'/'%$('0+'7'%;+9.)/(+ ./+(3'+*-%7'+.%+0-%4$*'= !" #$%&'(')*+,-.)'/01 !"#$%&'#()*+,#-.$+/0 1 2#(#-+3(45*67*$48+ 1 2#(#-+3(45*/."+(+ !" #$%&'()*$+)#$,-. / 01'2$+( / 345-2&$6()*$+)'5+71()-6,)(5+*-'1( / 8-+.'162+&')9$$+,&6-21( : ;<.)=-+.'162+&')'$$+,&6-21(> : ;<-2)-+1)=-+.'162+&')'$$+,&6-21(> !" #$%&'()%*+,-).*&*//)0.,(-+12 c 3 4).(,56+1&()+&-$+&78,0(9+,-(6& :).9.-.;+&81+0&.,&<=&9/0+6.,5& :)/5)(91> 3 4).(,56+1&()+&1-/)+0&(1&(& b 1+?8+,*+&/7&-$)++&;+*-/)1@&+(*$& a 0+7.,.,5&(&;+)-+A> 3 B7-+,@&C+&D,/C&.,7/)9(-./,& ('/8-&-$+&;+)-.*+1@&18*$&(1&*/6/)@& -$(-&C+E0&6.D+&-/&.,-+):/6(-+&/;+)& -$+&C$/6+&-).(,56+> !" Barycentric Color Interpolation #$%&'%()'*%(+,)-&(.,',//(0.-%&)12 c 3 4$)'1.567)1&'8%+'&/'0/'&$.1' .-&)(6/7%&./-'.1'!"#$%&'(#)%* %++#,)'"(&-9 3 4$)'-%5)',/5)1':(/5'&$)';())<' 8/(0'!"#.- =$)%>+?'*),%@1)'&$)' b ,//(0.-%&)1'%()'8).A$&1'%11.A-)0' a &/'&$)'>)(&.,)19 3 B/.-&'a /-'&$)'&(.%-A7)'.1'&$)'/(.A.-' /:'&$)'-/-C/(&$/A/-%7',//(0.-%&)' 1+1&)59Goal: Assign a every point (x,y) or (x,y,z) to 3 4$)'>),&/(1':(/5'barycentric coordinatesa &/'b %-0':(/5' (α,β,γa). &/'c %()'&%<)-'%1'*%1.1'>),&/(19 !" Barycentric Coordinates !c !b !a source: http://en.wikipedia.org/wiki/File:Barycentric_coordinates_1.png Solution: !x = α!a + β!b + γ!c #$%&'%()'*%(+,)-&(.,',//(0.-%&)12 c Some cool properties: 3 4/.-&'p .1'.-1.0)'&$)'&(.%-56)'.7 %-0'/-6+'.7 8'9'! 9'": b 8'9'" 9'": a 8'9'# 9'" 3 ;7'/-)',/<=/-)-&'.1'>)(/:'p .1'/-'%-')05)? 3 ;7'&@/',/<=/-)-&1'%()'>)(/:'p .1'/-'%'A)(&)B? 3 C$)',//(0.-%&)1',%-'*)'D1)0'%1'@).5$&.-5' 7%,&/(1'7/('=(/=)(&.)1'/7'&$)'A)(&.,)1:'6.E)',/6/(? !" Barycentric Color Interpolation !c !b !x !a If: !x = α!a + β!b + γ!c Then: color(!x)=αcolor(!a)+βcolor(!b)+γcolor(!c) #$%&'()*%+','--%.+)$*(/ !"#$%&'#()*+,#-.$+/0 1 !'23+(/4'2*5('-*67*!#(8+/4#2 1 !'23+(/4'2*5('-*97*!#(8+/4#2 !" Transformations Translation,Translation, rotation,rotation, scalingscaling 2D2D 3D3D HomogeneousHomogeneous coordinatescoordinates TransformingTransforming normalsnormals ExamplesExamples Shirley Chapter 6 Uses of Transformations • Modeling – build complex models by positioning simple components – transform from object coordinates to world coordinates • Viewing – placing the virtual camera in the world – specifying transformation from world coordinates to camera coordinates • Animation – vary transformations over time to create motion CAMERA OBJECT WORLD Computer Graphics 15-462 3 Rigid Body Transformations Rotation angle and line about which to rotate Computer Graphics 15-462 4 Non-rigid Body Transformations Distance between points on object do not remain constant Computer Graphics 15-462 5 Basic 2D Transformations Scale Shear chalkboard Rotate Computer Graphics 15-462 6 Composition of Transformations • Created by stringing basic ones together, e.g. – “translate p to the origin, rotate, then translate back” can also be described as a rotation about p • Any sequence of linear transformations can be collapsed into a single matrix formed by multiplying the individual matrices together • Order matters! • Can apply a whole sequence of transformations at once chalkboard 0 1 2 3 Translate to the origin, rotate, then translate back. Computer Graphics 15-462 7 3D Transformations • 3-D transformations are very similar to the 2-D case • Scale • Shear • Rotation is a bit more complicated in 3-D – different rotation axes chalkboard Computer Graphics 15-462 8 Fixed Euler Angles for 3-D Rotations • Independent rotations about each coordinate axis – angle interpolation for animation generates bizarre motions – rotations are order-dependent, and there are no conventions about the order to use • Widely used anyway, because they're “simple” Computer Graphics 15-462 9 Euler Angles for 3-D Rotations Computer Graphics 15-462 10 Other representations of 3D orientation • Relative rather than fixed Euler angles • Axis/Angle: rotate by ! about axis V • Quaternions: – a generalization of complex numbers –3 give the rotation axis - magnitude is sin !/2 –1 gives cos !/2 (the amount of rotation) –unit magnitude - points on a 4-D unit sphere Computer Graphics 15-462 11 But what about translation? •Translation is not linear--how to represent as a matrix? chalkboard Computer Graphics 15-462 12 But what about translation? •Translation is not linear--how to represent as a matrix? chalkboard •Trick: add extra coordinate to each vector •This extra coordinate is the homogeneous coordinate, or w •When extra coordinate is used, vector is said to be represented in homogeneous coordinates •We call these matrices Homogeneous Transformations Computer Graphics 15-462 13 W? Where did that come from? • Practical answer: –W is a clever algebraic trick. –Don’t worry about it too much. The w value will be 1.0 for the time being (until we get to perspective viewing transformations) • More complete answer: –(x,y,w) coordinates form a 3D projective space. –All nonzero scalar multiples of (x,y,1) form an equivalence class of points that project to the same 2D Cartesian point (x,y). –For 3-D graphics, the 4D projective space point (x,y,z,w) maps to the 3D point (x,y,z) in the same way. Computer Graphics 15-462 14 Homogeneous 2D Transformations The basic 2D transformations become Translate: Scale: Rotate: !" %" !" %" 10tx sx 00 !cos" ( )sin( 0%" $" '" $" '" 01ty 0 sy 0 $"sin( cos( 0'" $" '" $" '" $" 0 0 1'" #"0 0 1&" #"0 0 1&" #" &" Now any sequence of translate/scale/rotate operations can be combined into a single homogeneous matrix by multiplication. 3D transforms are modified similarly Computer Graphics 15-462 15 Rigid Body Transformations Rotation angle and line about which to rotate Computer Graphics 15-462 16 Rigid Body Transformations •A transformation matrix of the form xx xy tx yx yy ty 001 where the upper 2x2 submatrix is a rotation matrix and column 3 is a translation vector, is a rigid body transformation. •Any series of rotations and translations results in a rotation and translation of this form (and no change in the distance between vertices) Computer Graphics 15-462 17 Sequences of Transformations x' x' x' x' x' x' x' x' x' TRANSFORMED POINTS M PARAMETERS Often the same MATRICES transformations are applied M to many points Calculation time for the M matrices and combination is negligible compared to that of transforming the points x x x x x x x x x x UNTRANSFORMED Reduce the sequence to a POINTS single matrix, then transform Computer Graphics 15-462 18 Collapsing a Chain of Matrices. • Consider the composite function ABCD, i.e. p’ = ABCDp • Matrix multiplication isn’t commutative - the order is important • But matrix multiplication is associative, so can calculate from right to left or left to right: ABCD = (((AB) C) D) = (A (B (CD))). • Iteratively replace either the leading or the trailing pair by its product M ! D M ! A M ! CM or M ! MB both give the same result. M ! BM M ! MC M ! AM M ! MD Premultiply Postmultiply Computer Graphics 15-462 19 What is a Normal?– refresher Indication of outward facing direction for lighting and shading Order of definition of vertices in OpenGL Right hand rule Computer Graphics 15-462 20 Transforming Normals • It’s tempting to think of normal vectors as being like porcupine quills, so they would transform like points • Alas, it’s not so. • We need a different rule to transform normals. chalkboard Computer Graphics 15-462 21 Examples • Modeling with primitive shapes Computer Graphics 15-462 22 Announcements • Reading for Thursday: Shirley Ch: 2,6, & 7 • Project 1 due next Tuesday... • Style: 10% • Any questions? • Bulletin Board Mini-Demonstration.
Recommended publications
  • ARTG-2306-CRN-11966-Graphic Design I Computer Graphics
    Course Information Graphic Design 1: Computer Graphics ARTG 2306, CRN 11966, Section 001 FALL 2019 Class Hours: 1:30 pm - 4:20 pm, MW, Room LART 411 Instructor Contact Information Instructor: Nabil Gonzalez E-mail: nggonzalez@utep.edu Office: Office Hours: Instructor Information Nabil Gonzalez is your instructor for this course. She holds an Associate of Arts degree from El Paso Community College, a double BFA degree in Graphic Design and Printmaking from the University of Texas at El Paso and an MFA degree in Printmaking from the Rhode Island School of Design. As a studio artist, Nabil’s work has been focused on social and political views affecting the borderland as well as the exploration of identity, repetition and erasure. Her work has been shown throughout the United State, Mexico, Colombia and China. Her artist books and prints are included in museum collections in the United States. Course Description Graphic Design 1: Computer Graphics is an introduction to graphics, illustration, and page layout software on Macintosh computers. Students scan, generate, import, process, and combine images and text in black and white and in color. Industry standard desktop publishing software and imaging programs are used. The essential applications taught in this course are: Adobe Illustrator, Adobe Photoshop and Adobe InDesign. Course Prerequisite Information Course prerequisites include ARTF 1301, ARTF 1302, and ARTF 1304 each with a grade of “C” or better. Students are required to have a foundational understanding of the elements of design, the principles of composition, style, and content. Additionally, students must have developed fundamental drawing skills. These skills and knowledge sets are provided through the Department of Art’s Foundational Courses.
    [Show full text]
  • 4.3 Discovering Fractal Geometry in CAAD
    4.3 Discovering Fractal Geometry in CAAD Francisco Garcia, Angel Fernandez*, Javier Barrallo* Facultad de Informatica. Universidad de Deusto Bilbao. SPAIN E.T.S. de Arquitectura. Universidad del Pais Vasco. San Sebastian. SPAIN * Fractal geometry provides a powerful tool to explore the world of non-integer dimensions. Very short programs, easily comprehensible, can generate an extensive range of shapes and colors that can help us to understand the world we are living. This shapes are specially interesting in the simulation of plants, mountains, clouds and any kind of landscape, from deserts to rain-forests. The environment design, aleatory or conditioned, is one of the most important contributions of fractal geometry to CAAD. On a small scale, the design of fractal textures makes possible the simulation, in a very concise way, of wood, vegetation, water, minerals and a long list of materials very useful in photorealistic modeling. Introduction Fractal Geometry constitutes today one of the most fertile areas of investigation nowadays. Practically all the branches of scientific knowledge, like biology, mathematics, geology, chemistry, engineering, medicine, etc. have applied fractals to simulate and explain behaviors difficult to understand through traditional methods. Also in the world of computer aided design, fractal sets have shown up with strength, with numerous software applications using design tools based on fractal techniques. These techniques basically allow the effective and realistic reproduction of any kind of forms and textures that appear in nature: trees and plants, rocks and minerals, clouds, water, etc. For modern computer graphics, the access to these techniques, combined with ray tracing allow to create incredible landscapes and effects.
    [Show full text]
  • Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis
    Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues Meisam Sharify To cite this version: Meisam Sharify. Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues. Numerical Analysis [math.NA]. Ecole Polytechnique X, 2011. English. pastel-00643836 HAL Id: pastel-00643836 https://pastel.archives-ouvertes.fr/pastel-00643836 Submitted on 24 Nov 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Th`esepr´esent´eepour obtenir le titre de DOCTEUR DE L'ECOLE´ POLYTECHNIQUE Sp´ecialit´e: Math´ematiquesAppliqu´ees par Meisam Sharify Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues Jury Marianne Akian Pr´esident du jury St´ephaneGaubert Directeur Laurence Grammont Examinateur Laura Grigori Examinateur Andrei Sobolevski Rapporteur Fran¸coiseTisseur Rapporteur Paul Van Dooren Examinateur September 2011 Abstract Tropical algebra, which can be considered as a relatively new field in Mathemat- ics, emerged in several branches of science such as optimization, synchronization of production and transportation, discrete event systems, optimal control, oper- ations research, etc.
    [Show full text]
  • Volume Rendering
    Volume Rendering 1.1. Introduction Rapid advances in hardware have been transforming revolutionary approaches in computer graphics into reality. One typical example is the raster graphics that took place in the seventies, when hardware innovations enabled the transition from vector graphics to raster graphics. Another example which has a similar potential is currently shaping up in the field of volume graphics. This trend is rooted in the extensive research and development effort in scientific visualization in general and in volume visualization in particular. Visualization is the usage of computer-supported, interactive, visual representations of data to amplify cognition. Scientific visualization is the visualization of physically based data. Volume visualization is a method of extracting meaningful information from volumetric datasets through the use of interactive graphics and imaging, and is concerned with the representation, manipulation, and rendering of volumetric datasets. Its objective is to provide mechanisms for peering inside volumetric datasets and to enhance the visual understanding. Traditional 3D graphics is based on surface representation. Most common form is polygon-based surfaces for which affordable special-purpose rendering hardware have been developed in the recent years. Volume graphics has the potential to greatly advance the field of 3D graphics by offering a comprehensive alternative to conventional surface representation methods. The object of this thesis is to examine the existing methods for volume visualization and to find a way of efficiently rendering scientific data with commercially available hardware, like PC’s, without requiring dedicated systems. 1.2. Volume Rendering Our display screens are composed of a two-dimensional array of pixels each representing a unit area.
    [Show full text]
  • Animated Presentation of Static Infographics with Infomotion
    Eurographics Conference on Visualization (EuroVis) 2021 Volume 40 (2021), Number 3 R. Borgo, G. E. Marai, and T. von Landesberger (Guest Editors) Animated Presentation of Static Infographics with InfoMotion Yun Wang1, Yi Gao1;2, Ray Huang1, Weiwei Cui1, Haidong Zhang1, and Dongmei Zhang1 1Microsoft Research Asia 2Nanjing University (a) 5% (b) time Element Animation effect Meats, sweets slice spin link wipe dot zoom 35% icon zoom 10% Whole grains, title zoom OliVer oil pasta, beans, description wipe whole grain bread Mediterranean Diet 20% 30% 30% Vegetables and fruits Fish, seafood, poultry, Vegetables and fruits dairy food, eggs Figure 1: (a) An example infographic design. (b) The animated presentations for this infographic show the start time, duration, and animation effects applied to the visual elements. InfoMotion automatically generates animated presentations of static infographics. Abstract By displaying visual elements logically in temporal order, animated infographics can help readers better understand layers of information expressed in an infographic. While many techniques and tools target the quick generation of static infographics, few support animation designs. We propose InfoMotion that automatically generates animated presentations of static infographics. We first conduct a survey to explore the design space of animated infographics. Based on this survey, InfoMotion extracts graphical properties of an infographic to analyze the underlying information structures; then, animation effects are applied to the visual elements in the infographic in temporal order to present the infographic. The generated animations can be used in data videos or presentations. We demonstrate the utility of InfoMotion with two example applications, including mixed-initiative animation authoring and animation recommendation.
    [Show full text]
  • Two-Dimensional Rotational Kinematics Rigid Bodies
    Rigid Bodies A rigid body is an extended object in which the Two-Dimensional Rotational distance between any two points in the object is Kinematics constant in time. Springs or human bodies are non-rigid bodies. 8.01 W10D1 Rotation and Translation Recall: Translational Motion of of Rigid Body the Center of Mass Demonstration: Motion of a thrown baton • Total momentum of system of particles sys total pV= m cm • External force and acceleration of center of mass Translational motion: external force of gravity acts on center of mass sys totaldp totaldVcm total FAext==mm = cm Rotational Motion: object rotates about center of dt dt mass 1 Main Idea: Rotation of Rigid Two-Dimensional Rotation Body Torque produces angular acceleration about center of • Fixed axis rotation: mass Disc is rotating about axis τ total = I α passing through the cm cm cm center of the disc and is perpendicular to the I plane of the disc. cm is the moment of inertial about the center of mass • Plane of motion is fixed: α is the angular acceleration about center of mass cm For straight line motion, bicycle wheel rotates about fixed direction and center of mass is translating Rotational Kinematics Fixed Axis Rotation: Angular for Fixed Axis Rotation Velocity Angle variable θ A point like particle undergoing circular motion at a non-constant speed has SI unit: [rad] dθ ω ≡≡ω kkˆˆ (1)An angular velocity vector Angular velocity dt SI unit: −1 ⎣⎡rad⋅ s ⎦⎤ (2) an angular acceleration vector dθ Vector: ω ≡ Component dt dθ ω ≡ magnitude dt ω >+0, direction kˆ direction ω < 0, direction − kˆ 2 Fixed Axis Rotation: Angular Concept Question: Angular Acceleration Speed 2 ˆˆd θ Object A sits at the outer edge (rim) of a merry-go-round, and Angular acceleration: α ≡≡α kk2 object B sits halfway between the rim and the axis of rotation.
    [Show full text]
  • Information Graphics Design Challenges and Workflow Management Marco Giardina, University of Neuchâtel, Switzerland, Pablo Medi
    Online Journal of Communication and Media Technologies Volume: 3 – Issue: 1 – January - 2013 Information Graphics Design Challenges and Workflow Management Marco Giardina, University of Neuchâtel, Switzerland, Pablo Medina, Sensiel Research, Switzerland Abstract Infographics, though still in its infancy in the digital world, may offer an opportunity for media companies to enhance their business processes and value creation activities. This paper describes research about the influence of infographics production and dissemination on media companies’ workflow management. Drawing on infographics examples from New York Times print and online version, this contribution empirically explores the evolution from static to interactive multimedia infographics, the possibilities and design challenges of this journalistic emerging field and its impact on media companies’ activities in relation to technology changes and media-use patterns. Findings highlight some explorative ideas about the required workflow and journalism activities for a successful inception of infographics into online news dissemination practices of media companies. Conclusions suggest that delivering infographics represents a yet not fully tapped opportunity for media companies, but its successful inception on news production routines requires skilled professionals in audiovisual journalism and revised business models. Keywords: newspapers, visual communication, infographics, digital media technology © Online Journal of Communication and Media Technologies 108 Online Journal of Communication and Media Technologies Volume: 3 – Issue: 1 – January - 2013 During this time of unprecedented change in journalism, media practitioners and scholars find themselves mired in a new debate on the storytelling potential of data visualization narratives. News organization including the New York Times, Washington Post and The Guardian are at the fore of innovation and experimentation and regularly incorporate dynamic graphics into their journalism products (Segel, 2011).
    [Show full text]
  • 1.2 Rules for Translations
    1.2. Rules for Translations www.ck12.org 1.2 Rules for Translations Here you will learn the different notation used for translations. The figure below shows a pattern of a floor tile. Write the mapping rule for the translation of the two blue floor tiles. Watch This First watch this video to learn about writing rules for translations. MEDIA Click image to the left for more content. CK-12 FoundationChapter10RulesforTranslationsA Then watch this video to see some examples. MEDIA Click image to the left for more content. CK-12 FoundationChapter10RulesforTranslationsB 18 www.ck12.org Chapter 1. Unit 1: Transformations, Congruence and Similarity Guidance In geometry, a transformation is an operation that moves, flips, or changes a shape (called the preimage) to create a new shape (called the image). A translation is a type of transformation that moves each point in a figure the same distance in the same direction. Translations are often referred to as slides. You can describe a translation using words like "moved up 3 and over 5 to the left" or with notation. There are two types of notation to know. T x y 1. One notation looks like (3, 5). This notation tells you to add 3 to the values and add 5 to the values. 2. The second notation is a mapping rule of the form (x,y) → (x−7,y+5). This notation tells you that the x and y coordinates are translated to x − 7 and y + 5. The mapping rule notation is the most common. Example A Sarah describes a translation as point P moving from P(−2,2) to P(1,−1).
    [Show full text]
  • Inviwo — a Visualization System with Usage Abstraction Levels
    IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL X, NO. Y, MAY 2019 1 Inviwo — A Visualization System with Usage Abstraction Levels Daniel Jonsson,¨ Peter Steneteg, Erik Sunden,´ Rickard Englund, Sathish Kottravel, Martin Falk, Member, IEEE, Anders Ynnerman, Ingrid Hotz, and Timo Ropinski Member, IEEE, Abstract—The complexity of today’s visualization applications demands specific visualization systems tailored for the development of these applications. Frequently, such systems utilize levels of abstraction to improve the application development process, for instance by providing a data flow network editor. Unfortunately, these abstractions result in several issues, which need to be circumvented through an abstraction-centered system design. Often, a high level of abstraction hides low level details, which makes it difficult to directly access the underlying computing platform, which would be important to achieve an optimal performance. Therefore, we propose a layer structure developed for modern and sustainable visualization systems allowing developers to interact with all contained abstraction levels. We refer to this interaction capabilities as usage abstraction levels, since we target application developers with various levels of experience. We formulate the requirements for such a system, derive the desired architecture, and present how the concepts have been exemplary realized within the Inviwo visualization system. Furthermore, we address several specific challenges that arise during the realization of such a layered architecture, such as communication between different computing platforms, performance centered encapsulation, as well as layer-independent development by supporting cross layer documentation and debugging capabilities. Index Terms—Visualization systems, data visualization, visual analytics, data analysis, computer graphics, image processing. F 1 INTRODUCTION The field of visualization is maturing, and a shift can be employing different layers of abstraction.
    [Show full text]
  • THESIS MULTIDIMENSIONAL SCALING: INFINITE METRIC MEASURE SPACES Submitted by Lara Kassab Department of Mathematics in Partial Fu
    THESIS MULTIDIMENSIONAL SCALING: INFINITE METRIC MEASURE SPACES Submitted by Lara Kassab Department of Mathematics In partial fulfillment of the requirements For the Degree of Master of Science Colorado State University Fort Collins, Colorado Spring 2019 Master’s Committee: Advisor: Henry Adams Michael Kirby Bailey Fosdick Copyright by Lara Kassab 2019 All Rights Reserved ABSTRACT MULTIDIMENSIONAL SCALING: INFINITE METRIC MEASURE SPACES Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces, along with its optimality properties and goodness of fit. This allows us to study the MDS embeddings of the geodesic circle S1 into Rm for all m, and to ask questions about the MDS embeddings of the geodesic n-spheres Sn into Rm. Furthermore, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space X, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of X? Convergence is understood when each metric space in the sequence has the same finite number of points, or when each metric space has a finite number of points tending to infinity. We are also interested in notions of convergence when each metric space in the sequence has an arbitrary (possibly infinite) number of points. ii ACKNOWLEDGEMENTS We would like to thank Henry Adams, Mark Blumstein, Bailey Fosdick, Michael Kirby, Henry Kvinge, Facundo Mémoli, Louis Scharf, the students in Michael Kirby’s Spring 2018 class, and the Pattern Analysis Laboratory at Colorado State University for their helpful conversations and support throughout this project.
    [Show full text]
  • Multidisciplinary Design Project Engineering Dictionary Version 0.0.2
    Multidisciplinary Design Project Engineering Dictionary Version 0.0.2 February 15, 2006 . DRAFT Cambridge-MIT Institute Multidisciplinary Design Project This Dictionary/Glossary of Engineering terms has been compiled to compliment the work developed as part of the Multi-disciplinary Design Project (MDP), which is a programme to develop teaching material and kits to aid the running of mechtronics projects in Universities and Schools. The project is being carried out with support from the Cambridge-MIT Institute undergraduate teaching programe. For more information about the project please visit the MDP website at http://www-mdp.eng.cam.ac.uk or contact Dr. Peter Long Prof. Alex Slocum Cambridge University Engineering Department Massachusetts Institute of Technology Trumpington Street, 77 Massachusetts Ave. Cambridge. Cambridge MA 02139-4307 CB2 1PZ. USA e-mail: pjgl2@eng.cam.ac.uk e-mail: slocum@mit.edu tel: +44 (0) 1223 332779 tel: +1 617 253 0012 For information about the CMI initiative please see Cambridge-MIT Institute website :- http://www.cambridge-mit.org CMI CMI, University of Cambridge Massachusetts Institute of Technology 10 Miller’s Yard, 77 Massachusetts Ave. Mill Lane, Cambridge MA 02139-4307 Cambridge. CB2 1RQ. USA tel: +44 (0) 1223 327207 tel. +1 617 253 7732 fax: +44 (0) 1223 765891 fax. +1 617 258 8539 . DRAFT 2 CMI-MDP Programme 1 Introduction This dictionary/glossary has not been developed as a definative work but as a useful reference book for engi- neering students to search when looking for the meaning of a word/phrase. It has been compiled from a number of existing glossaries together with a number of local additions.
    [Show full text]
  • An Online System for Classifying Computer Graphics Images from Natural Photographs
    An Online System for Classifying Computer Graphics Images from Natural Photographs Tian-Tsong Ng and Shih-Fu Chang fttng,sfchangg@ee.columbia.edu Department of Electrical Engineering Columbia University New York, USA ABSTRACT We describe an online system for classifying computer generated images and camera-captured photographic images, as part of our effort in building a complete passive-blind system for image tampering detection (project website at http: //www.ee.columbia.edu/trustfoto). Users are able to submit any image from a local or an online source to the system and get classification results with confidence scores. Our system has implemented three different algorithms from the state of the art based on the geometry, the wavelet, and the cartoon features. We describe the important algorithmic issues involved for achieving satisfactory performances in both speed and accuracy as well as the capability to handle diverse types of input images. We studied the effects of image size reduction on classification accuracy and speed, and found different size reduction methods worked best for different classification methods. In addition, we incorporated machine learning techniques, such as fusion and subclass-based bagging, in order to counter the effect of performance degradation caused by image size reduction. With all these improvements, we are able to speed up the classification speed by more than two times while keeping the classification accuracy almost intact at about 82%. Keywords: Computer graphics, photograph, classification, online system, geometry features, classifier fusion 1. INTRODUCTION The level of photorealism capable by today’s computer graphics makes distinguishing photographic and computer graphic images difficult.
    [Show full text]