Simplification of 3D Meshes

Size: px
Start display at page:

Download "Simplification of 3D Meshes"

Transcription

1 Smplfcaton of 3D Meshes Addy Ngan /4/00 Outlne Motvaton Taxonomy of smplfcaton methods Hoppe et al, Mesh optmzaton Hoppe, Progressve meshes Smplfcaton of 3D Meshes 1

2 Motvaton Hgh detaled meshes becomng wdely avalable n computer vson, scentfc vsualzaton, terran data from satellte etc. Need to store, transmt, analyze, edt and dsplay them effcently. > 1M trangles! Goals Reduce number of polygons Faster renderng Less storage Smpler manpulaton Other qualtes General (non-manfold) Effcency Preserve attrbutes other than geometry Smplfcaton of 3D Meshes

3 Taxonomy of methods Manfold Smplfcaton Vertex decmaton Wavelet Edge collapse Non-manfold Smplfcaton Vertex clusterng Mesh Decmaton [Schroeder et al 9] Multple passes For each pass remove all vertces that matches certan crtera and retrangulate Crtera: dstance to plane/edge Advantage: smple algorthm, fast Dsadvantage: bg memory Smplfcaton of 3D Meshes 3

4 Re-Tlng [Turk 9] Steps: Generate random ponts on surface Iteratve repulson spread the ponts out unformly Add new set of ponts to the surface and mutual tessellate Remove old vertces one by one yeldng a new trangulaton Varant to put ponts adaptvely dependng on curvature Advantage : mantan topology Dsadvantage : complex algorthm, blur sharp features. Re-Tlng (con t( con t) Smplfcaton of 3D Meshes 4

5 3D Grd Method [Rossgnac[ Rossgnac-Borrel 93] Steps: Subdvde the boundng volume nto regular grd Merge all vertces wthn each cell together nto a new vertex Advantage: Form trangles very general, accordng to orgnal topology. fast Dsadvantage : low qualty, non-adaptve Optmzaton [Hoppe et al 93] Optmzaton based smplfcaton. Mnmze energy functon. Repeat sem-random changes to topology and optmze the geometry. Work on manfold Advantages: Hgh qualty, less senstve to nose Dsadvantages: slow Smplfcaton of 3D Meshes 5

6 Mesh Optmzaton Hugues Hoppe, Tony DeRose, Tom Duchamp, John McDonald, Werner Stuetzle, SIGGRAPH 93 Optmzaton Smlar problem to smplfcaton Startng wth sample data ponts from the surface and a ntal mesh, fnd a smpler mesh Mnmze energy functon that descrbes the concseness and accuracy of the mesh Smplfcaton of 3D Meshes 6

7 Mesh Representaton A mesh s represented by a par (K,V) K s a smplcal complex representng the connectvty of the vertces/edges/faces. V={v 1,,, v m } s a set of m vertex postons defnng the shape of the mesh n R 3 If mesh s not self-ntersectng, every pont can be represented by a barycentrc vector Energy Functon E(K,V) = E dst (K,V) + E rep (K) + E sprng (K,V) E dst (K,V) = E rep (K) = c rep m E sprng (K,V) = n = 1 d { j, k} K ( x, ϕ ( K )) j v v V k Wthout E sprng, spkes possble for regon wth no data ponts Smplfcaton of 3D Meshes 7

8 Smplfcaton of 3D Meshes 8 Mnmzng the Energy Functon Mnmzng the Energy Functon A bgger c rep sparser representaton. Outer loop optmze over K (Dscrete) Inner loop optmze over V for a fxed K (Contnuous) = Mnmze + For each x, dstance = New objectve functon : E(K,V,B)= Optmzaton for fxed K Optmzaton for fxed K = n K d 1 V )) (, ( ϕ x }, { k j K k j v v }, { V 1 ) ( mn k j K k j K n v v b x b + = V ) ( mn K b x b

9 Optmzaton over fxed K Two subproblems : For fxed vertces V, fnd optmal barycentrc coordnate vectors B by projecton For fxed B, fnd optmal V by solvng a lnear least squares problem Fnd optmal solutons to both subproblems, so E(K,V,B) must converge. Projecton Subproblem Fnd optmal B by projectng x onto the mesh. To accelerate, buld a spatal parttonng data structure so that for each pont only consder nearby subset of faces Assume a pont s projecton les n the neghborhood of ts projecton n prevous teraton (perform well n practce) Smplfcaton of 3D Meshes 9

10 Lnear Least Squares Subproblem Mnmze for x,y,z coordnates, v m-vector A (n+e) m matrx d (n+e)-vector For m mesh vertces, n data ponts, e edges Frst n rows of d contans the n data ponts Next e rows of d are zeroes v contans the m mesh vertces Av d Lnear Least Squares (con t( con t) Av d j j Frst n rows of A contans the barycentrc coordnates computed n projecton (at most 3 non-zero entres) Next e rows represent the sprng energy: each contans an κ and - κ entres n columns correspondng to ndces of edge s endponts (exactly non-zero entres) A s sparse. Use conjugate gradent method to solve n O(n+m) tme Smplfcaton of 3D Meshes 10

11 Optmzaton over K Take a legal move and accept f t gves lower energy. Termnate f a number of trals faled to gve a lower energy. Legal move : applcaton of one of these that leaves the topologcal type of K unchanged Three elementary transformatons: edge collapse/splt/swap Legal move Splt Always legal Collapse : f and only f For all vertces {k} adjacent to both {} and {j}, {,j,k} s a face If {} and {j} are both boundary vertces, {,j} s a boundary edge K has more than 4 vertces f {} and {j} both are not boundary vertces, or K has more than 3 vertces f ether {} or {j} are boundary vertces Swap f and only f {k,l} K Proof n Hoppe et al, TR Unversty of Washngton Smplfcaton of 3D Meshes 11

12 Evaluaton of a legal move Instead of re-computng the global energy after a legal move, only compute the change n energy for the local submesh For s K, defne star(s;k) = {s K : s nonempty subset of s } Edge Collapse To evaluate collapsng of an edge {,j}, take the submesh to be star({};k) star({j};k). Optmze over the new vertex h whle holdng all other constant Attempt optmzatons startng at v, v j, and ½(v +v j ). Accept the best one. Instead of checkng for self-ntersecton after collapse whch s expensve, use a threshold value for the maxmum dhedral angle of edges n star({h};k ) Smplfcaton of 3D Meshes 1

13 Edge splt/swap Edge splt Same procedure as collapse, usng the submesh to be star({,j}; K). Intal value of v h be ½(v +v j ). Edge swap- For a swap that replace an edge {,j} wth {k,l}, choose the best of the two optmzatons, one wth submesh star({k};k ) varyng vertex v k, another wth submesh star({l};k ) varyng vertex v l. Results Intal mesh (03 vertces) Sample Ponts (675 vertces) c rep =10-5 (487 vertces) c rep =10-4 (39 vertces) Smplfcaton of 3D Meshes 13

14 Results Results - Segmentaton Smplfcaton of 3D Meshes 14

15 Progressve Meshes Hugues Hoppe SIGGRAPH 96 Addtonal Goal Lossless,, contnuous-resoluton, progressve representaton More compact representaton Apart from geometry, also preserve other attrbutes (colors, normals,, materals, texture coordnates ) Smplfcaton of 3D Meshes 15

16 Mesh Representaton Edge collapse only s suffcent! n An ntal mesh Mˆ = M can be smplfed nto a coarser mesh M 0 by a sequence of edge collapse. (Mˆ = M The sequence s nvertble n ) ecoln ecol1 ecol n M M 0 Progressve Mesh (Mˆ = M n ) ecoln ecol1 ecol n M M 0 Smplfcaton of 3D Meshes 16

17 Energy Functon Old E(K,V) = E dst (K,V) + E rep (K) + E sprng (K,V) New E(M) = E dst (M) + E sprng (M) + E scalar (M) +E dscrete (M) where M = (K,V,D,S) E rep goes away! And so does the parameter c rep Preservng geometry E dst + E sprng Same as before, but only consder edge collapse n the outer loop. The possble legal collapses are placed n a prorty queue wth ts estmated energy change E For an edge collapse K K, E = E K -E K After each collapse, the cost of ts neghborhood edges n the prorty queue are updated. Smplfcaton of 3D Meshes 17

18 Preservng Scalar attrbutes For d scalar attrbutes, we could have added d dmenson to E dst. For effcency purpose, we separate t as a n dfferent term E (V) = (c ) x scalar scalar V ( ) = 1 b Frst solve for the vertex postons. Then usng the same b to fnd vertex attrbutes mnmzng E scalar by lnear least square Preservng Scalar Attrbutes 00x00 vertces 400 vertces Smplfcaton of 3D Meshes 18

19 Preservng dscontnuty We want to preserve dscontnuty because they often form notceable features sample an addtonal set of ponts X dsc from sharp edges of ntal mesh. Compute E dsc by projectng X dsc onto the correspondng sharp edges dsallow/penalze collapse of boundary and dscontnuty edges Wthout E dsc References Survey of Polygonal Surface Smplfcaton Algorthms, Paul Heckbert and Mchael Garland, tech. report, CS Dept.,Carnege Mellon U., to appear H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, Mesh Optmzaton, SIGGRAPH 93, Hugues Hoppe, Progressve Meshes, SIGGRAPH 96, Hugues Hoppe, Effcent mplementaton of progressve meshes, Computers & Graphcs, Vol., No. 1, 1998, pages Greg Turk, Re-Tlng Polygonal Surfaces, SIGGRAPH 9, Wllam J. Schroeder, Jonathan Zarge, and Wllam Lorensen, Decmaton of Trangle Meshes, SIGGRAPH 9, J. Rossgnac and P. Borrel, Mult-resoluton 3D approxmatons for renderng complex scenes Smplfcaton of 3D Meshes 19

Mesh Optimization. Abstract. 1 Introduction. Hugues Hoppe Tony DeRose Tom Duchamp y John McDonald z Werner Stuetzle z

Mesh Optimization. Abstract. 1 Introduction. Hugues Hoppe Tony DeRose Tom Duchamp y John McDonald z Werner Stuetzle z Mesh Optmzaton Hugues Hoppe Tony DeRose Tom Duchamp y John McDonald z Werner Stuetzle z Unversty of Washngton Seattle, WA 98195 Abstract We present a method for solvng the followng problem: Gven a set

More information

Interactive Rendering of Translucent Objects

Interactive Rendering of Translucent Objects Interactve Renderng of Translucent Objects Hendrk Lensch Mchael Goesele Phlppe Bekaert Jan Kautz Marcus Magnor Jochen Lang Hans-Peter Sedel 2003 Presented By: Mark Rubelmann Outlne Motvaton Background

More information

CS 534: Computer Vision Model Fitting

CS 534: Computer Vision Model Fitting CS 534: Computer Vson Model Fttng Sprng 004 Ahmed Elgammal Dept of Computer Scence CS 534 Model Fttng - 1 Outlnes Model fttng s mportant Least-squares fttng Maxmum lkelhood estmaton MAP estmaton Robust

More information

Subspace clustering. Clustering. Fundamental to all clustering techniques is the choice of distance measure between data points;

Subspace clustering. Clustering. Fundamental to all clustering techniques is the choice of distance measure between data points; Subspace clusterng Clusterng Fundamental to all clusterng technques s the choce of dstance measure between data ponts; D q ( ) ( ) 2 x x = x x, j k = 1 k jk Squared Eucldean dstance Assumpton: All features

More information

Modeling, Manipulating, and Visualizing Continuous Volumetric Data: A Novel Spline-based Approach

Modeling, Manipulating, and Visualizing Continuous Volumetric Data: A Novel Spline-based Approach Modelng, Manpulatng, and Vsualzng Contnuous Volumetrc Data: A Novel Splne-based Approach Jng Hua Center for Vsual Computng, Department of Computer Scence SUNY at Stony Brook Talk Outlne Introducton and

More information

Angle-Independent 3D Reconstruction. Ji Zhang Mireille Boutin Daniel Aliaga

Angle-Independent 3D Reconstruction. Ji Zhang Mireille Boutin Daniel Aliaga Angle-Independent 3D Reconstructon J Zhang Mrelle Boutn Danel Alaga Goal: Structure from Moton To reconstruct the 3D geometry of a scene from a set of pctures (e.g. a move of the scene pont reconstructon

More information

Parameterization of Quadrilateral Meshes

Parameterization of Quadrilateral Meshes Parameterzaton of Quadrlateral Meshes L Lu 1, CaMng Zhang 1,, and Frank Cheng 3 1 School of Computer Scence and Technology, Shandong Unversty, Jnan, Chna Department of Computer Scence and Technology, Shandong

More information

The Greedy Method. Outline and Reading. Change Money Problem. Greedy Algorithms. Applications of the Greedy Strategy. The Greedy Method Technique

The Greedy Method. Outline and Reading. Change Money Problem. Greedy Algorithms. Applications of the Greedy Strategy. The Greedy Method Technique //00 :0 AM Outlne and Readng The Greedy Method The Greedy Method Technque (secton.) Fractonal Knapsack Problem (secton..) Task Schedulng (secton..) Mnmum Spannng Trees (secton.) Change Money Problem Greedy

More information

Feature Reduction and Selection

Feature Reduction and Selection Feature Reducton and Selecton Dr. Shuang LIANG School of Software Engneerng TongJ Unversty Fall, 2012 Today s Topcs Introducton Problems of Dmensonalty Feature Reducton Statstc methods Prncpal Components

More information

Some Tutorial about the Project. Computer Graphics

Some Tutorial about the Project. Computer Graphics Some Tutoral about the Project Lecture 6 Rastersaton, Antalasng, Texture Mappng, I have already covered all the topcs needed to fnsh the 1 st practcal Today, I wll brefly explan how to start workng on

More information

Computer Animation and Visualisation. Lecture 4. Rigging / Skinning

Computer Animation and Visualisation. Lecture 4. Rigging / Skinning Computer Anmaton and Vsualsaton Lecture 4. Rggng / Sknnng Taku Komura Overvew Sknnng / Rggng Background knowledge Lnear Blendng How to decde weghts? Example-based Method Anatomcal models Sknnng Assume

More information

A Comparison of Mesh Simplification Algorithms

A Comparison of Mesh Simplification Algorithms A Comparison of Mesh Simplification Algorithms Nicole Ortega Project Summary, Group 16 Browser Based Constructive Solid Geometry for Anatomical Models - Orthotics for cerebral palsy patients - Fusiform

More information

Outline. Type of Machine Learning. Examples of Application. Unsupervised Learning

Outline. Type of Machine Learning. Examples of Application. Unsupervised Learning Outlne Artfcal Intellgence and ts applcatons Lecture 8 Unsupervsed Learnng Professor Danel Yeung danyeung@eee.org Dr. Patrck Chan patrckchan@eee.org South Chna Unversty of Technology, Chna Introducton

More information

Range images. Range image registration. Examples of sampling patterns. Range images and range surfaces

Range images. Range image registration. Examples of sampling patterns. Range images and range surfaces Range mages For many structured lght scanners, the range data forms a hghly regular pattern known as a range mage. he samplng pattern s determned by the specfc scanner. Range mage regstraton 1 Examples

More information

Parallelism for Nested Loops with Non-uniform and Flow Dependences

Parallelism for Nested Loops with Non-uniform and Flow Dependences Parallelsm for Nested Loops wth Non-unform and Flow Dependences Sam-Jn Jeong Dept. of Informaton & Communcaton Engneerng, Cheonan Unversty, 5, Anseo-dong, Cheonan, Chungnam, 330-80, Korea. seong@cheonan.ac.kr

More information

Outline. Discriminative classifiers for image recognition. Where in the World? A nearest neighbor recognition example 4/14/2011. CS 376 Lecture 22 1

Outline. Discriminative classifiers for image recognition. Where in the World? A nearest neighbor recognition example 4/14/2011. CS 376 Lecture 22 1 4/14/011 Outlne Dscrmnatve classfers for mage recognton Wednesday, Aprl 13 Krsten Grauman UT-Austn Last tme: wndow-based generc obect detecton basc ppelne face detecton wth boostng as case study Today:

More information

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 15

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 15 CS434a/541a: Pattern Recognton Prof. Olga Veksler Lecture 15 Today New Topc: Unsupervsed Learnng Supervsed vs. unsupervsed learnng Unsupervsed learnng Net Tme: parametrc unsupervsed learnng Today: nonparametrc

More information

Unsupervised Learning

Unsupervised Learning Pattern Recognton Lecture 8 Outlne Introducton Unsupervsed Learnng Parametrc VS Non-Parametrc Approach Mxture of Denstes Maxmum-Lkelhood Estmates Clusterng Prof. Danel Yeung School of Computer Scence and

More information

CS246: Mining Massive Datasets Jure Leskovec, Stanford University

CS246: Mining Massive Datasets Jure Leskovec, Stanford University CS46: Mnng Massve Datasets Jure Leskovec, Stanford Unversty http://cs46.stanford.edu /19/013 Jure Leskovec, Stanford CS46: Mnng Massve Datasets, http://cs46.stanford.edu Perceptron: y = sgn( x Ho to fnd

More information

Unsupervised Learning and Clustering

Unsupervised Learning and Clustering Unsupervsed Learnng and Clusterng Supervsed vs. Unsupervsed Learnng Up to now we consdered supervsed learnng scenaro, where we are gven 1. samples 1,, n 2. class labels for all samples 1,, n Ths s also

More information

Machine Learning: Algorithms and Applications

Machine Learning: Algorithms and Applications 14/05/1 Machne Learnng: Algorthms and Applcatons Florano Zn Free Unversty of Bozen-Bolzano Faculty of Computer Scence Academc Year 011-01 Lecture 10: 14 May 01 Unsupervsed Learnng cont Sldes courtesy of

More information

Radial Basis Functions

Radial Basis Functions Radal Bass Functons Mesh Reconstructon Input: pont cloud Output: water-tght manfold mesh Explct Connectvty estmaton Implct Sgned dstance functon estmaton Image from: Reconstructon and Representaton of

More information

Loop Transformations for Parallelism & Locality. Review. Scalar Expansion. Scalar Expansion: Motivation

Loop Transformations for Parallelism & Locality. Review. Scalar Expansion. Scalar Expansion: Motivation Loop Transformatons for Parallelsm & Localty Last week Data dependences and loops Loop transformatons Parallelzaton Loop nterchange Today Scalar expanson for removng false dependences Loop nterchange Loop

More information

Active Contours/Snakes

Active Contours/Snakes Actve Contours/Snakes Erkut Erdem Acknowledgement: The sldes are adapted from the sldes prepared by K. Grauman of Unversty of Texas at Austn Fttng: Edges vs. boundares Edges useful sgnal to ndcate occludng

More information

Outline. Self-Organizing Maps (SOM) US Hebbian Learning, Cntd. The learning rule is Hebbian like:

Outline. Self-Organizing Maps (SOM) US Hebbian Learning, Cntd. The learning rule is Hebbian like: Self-Organzng Maps (SOM) Turgay İBRİKÇİ, PhD. Outlne Introducton Structures of SOM SOM Archtecture Neghborhoods SOM Algorthm Examples Summary 1 2 Unsupervsed Hebban Learnng US Hebban Learnng, Cntd 3 A

More information

Polyhedral Compilation Foundations

Polyhedral Compilation Foundations Polyhedral Complaton Foundatons Lous-Noël Pouchet pouchet@cse.oho-state.edu Dept. of Computer Scence and Engneerng, the Oho State Unversty Feb 8, 200 888., Class # Introducton: Polyhedral Complaton Foundatons

More information

LECTURE : MANIFOLD LEARNING

LECTURE : MANIFOLD LEARNING LECTURE : MANIFOLD LEARNING Rta Osadchy Some sldes are due to L.Saul, V. C. Raykar, N. Verma Topcs PCA MDS IsoMap LLE EgenMaps Done! Dmensonalty Reducton Data representaton Inputs are real-valued vectors

More information

Discriminative Dictionary Learning with Pairwise Constraints

Discriminative Dictionary Learning with Pairwise Constraints Dscrmnatve Dctonary Learnng wth Parwse Constrants Humn Guo Zhuoln Jang LARRY S. DAVIS UNIVERSITY OF MARYLAND Nov. 6 th, Outlne Introducton/motvaton Dctonary Learnng Dscrmnatve Dctonary Learnng wth Parwse

More information

6.854 Advanced Algorithms Petar Maymounkov Problem Set 11 (November 23, 2005) With: Benjamin Rossman, Oren Weimann, and Pouya Kheradpour

6.854 Advanced Algorithms Petar Maymounkov Problem Set 11 (November 23, 2005) With: Benjamin Rossman, Oren Weimann, and Pouya Kheradpour 6.854 Advanced Algorthms Petar Maymounkov Problem Set 11 (November 23, 2005) Wth: Benjamn Rossman, Oren Wemann, and Pouya Kheradpour Problem 1. We reduce vertex cover to MAX-SAT wth weghts, such that the

More information

AMath 483/583 Lecture 21 May 13, Notes: Notes: Jacobi iteration. Notes: Jacobi with OpenMP coarse grain

AMath 483/583 Lecture 21 May 13, Notes: Notes: Jacobi iteration. Notes: Jacobi with OpenMP coarse grain AMath 483/583 Lecture 21 May 13, 2011 Today: OpenMP and MPI versons of Jacob teraton Gauss-Sedel and SOR teratve methods Next week: More MPI Debuggng and totalvew GPU computng Read: Class notes and references

More information

Harmonic Coordinates for Character Articulation PIXAR

Harmonic Coordinates for Character Articulation PIXAR Harmonc Coordnates for Character Artculaton PIXAR Pushkar Josh Mark Meyer Tony DeRose Bran Green Tom Sanock We have a complex source mesh nsde of a smpler cage mesh We want vertex deformatons appled to

More information

Lecture #15 Lecture Notes

Lecture #15 Lecture Notes Lecture #15 Lecture Notes The ocean water column s very much a 3-D spatal entt and we need to represent that structure n an economcal way to deal wth t n calculatons. We wll dscuss one way to do so, emprcal

More information

Fitting: Deformable contours April 26 th, 2018

Fitting: Deformable contours April 26 th, 2018 4/6/08 Fttng: Deformable contours Aprl 6 th, 08 Yong Jae Lee UC Davs Recap so far: Groupng and Fttng Goal: move from array of pxel values (or flter outputs) to a collecton of regons, objects, and shapes.

More information

Face Recognition University at Buffalo CSE666 Lecture Slides Resources:

Face Recognition University at Buffalo CSE666 Lecture Slides Resources: Face Recognton Unversty at Buffalo CSE666 Lecture Sldes Resources: http://www.face-rec.org/algorthms/ Overvew of face recognton algorthms Correlaton - Pxel based correspondence between two face mages Structural

More information

An Iterative Solution Approach to Process Plant Layout using Mixed Integer Optimisation

An Iterative Solution Approach to Process Plant Layout using Mixed Integer Optimisation 17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 An Iteratve Soluton Approach to Process Plant Layout usng Mxed

More information

Hierarchical clustering for gene expression data analysis

Hierarchical clustering for gene expression data analysis Herarchcal clusterng for gene expresson data analyss Gorgo Valentn e-mal: valentn@ds.unm.t Clusterng of Mcroarray Data. Clusterng of gene expresson profles (rows) => dscovery of co-regulated and functonally

More information

High-Boost Mesh Filtering for 3-D Shape Enhancement

High-Boost Mesh Filtering for 3-D Shape Enhancement Hgh-Boost Mesh Flterng for 3-D Shape Enhancement Hrokazu Yagou Λ Alexander Belyaev y Damng We z Λ y z ; ; Shape Modelng Laboratory, Unversty of Azu, Azu-Wakamatsu 965-8580 Japan y Computer Graphcs Group,

More information

K-means and Hierarchical Clustering

K-means and Hierarchical Clustering Note to other teachers and users of these sldes. Andrew would be delghted f you found ths source materal useful n gvng your own lectures. Feel free to use these sldes verbatm, or to modfy them to ft your

More information

Solving two-person zero-sum game by Matlab

Solving two-person zero-sum game by Matlab Appled Mechancs and Materals Onlne: 2011-02-02 ISSN: 1662-7482, Vols. 50-51, pp 262-265 do:10.4028/www.scentfc.net/amm.50-51.262 2011 Trans Tech Publcatons, Swtzerland Solvng two-person zero-sum game by

More information

Interactive Multiresolution Surface Viewing

Interactive Multiresolution Surface Viewing Interactve Multresoluton Surface Vewng Andrew Certan z Jovan Popovć Tony DeRose Tom Duchamp Davd Salesn Werner Stuetzle y Department of Computer Scence and Engneerng Department of Mathematcs y Department

More information

Reading. 14. Subdivision curves. Recommended:

Reading. 14. Subdivision curves. Recommended: eadng ecommended: Stollntz, Deose, and Salesn. Wavelets for Computer Graphcs: heory and Applcatons, 996, secton 6.-6., A.5. 4. Subdvson curves Note: there s an error n Stollntz, et al., secton A.5. Equaton

More information

Programming in Fortran 90 : 2017/2018

Programming in Fortran 90 : 2017/2018 Programmng n Fortran 90 : 2017/2018 Programmng n Fortran 90 : 2017/2018 Exercse 1 : Evaluaton of functon dependng on nput Wrte a program who evaluate the functon f (x,y) for any two user specfed values

More information

Surface Mapping One. CS7GV3 Real-time Rendering

Surface Mapping One. CS7GV3 Real-time Rendering Surface Mappng One CS7GV3 Real-tme Renderng Textures Add complexty to scenes wthout addtonal geometry Textures store ths nformaton, can be any dmenson Many dfferent types: Dffuse most common Ambent, specular,

More information

Discretization of Functionally Based Heterogeneous Objects

Discretization of Functionally Based Heterogeneous Objects Dscretzaton of Functonally Based Heterogeneous Objects Elena Kartasheva Insttute for Mathematcal Modelng Russan Academy of Scence Moscow, Russa ekart@mamod.ru Oleg Fryaznov Insttute for Mathematcal Modelng

More information

Quality Improvement Algorithm for Tetrahedral Mesh Based on Optimal Delaunay Triangulation

Quality Improvement Algorithm for Tetrahedral Mesh Based on Optimal Delaunay Triangulation Intellgent Informaton Management, 013, 5, 191-195 Publshed Onlne November 013 (http://www.scrp.org/journal/m) http://dx.do.org/10.36/m.013.5601 Qualty Improvement Algorthm for Tetrahedral Mesh Based on

More information

Learning the Kernel Parameters in Kernel Minimum Distance Classifier

Learning the Kernel Parameters in Kernel Minimum Distance Classifier Learnng the Kernel Parameters n Kernel Mnmum Dstance Classfer Daoqang Zhang 1,, Songcan Chen and Zh-Hua Zhou 1* 1 Natonal Laboratory for Novel Software Technology Nanjng Unversty, Nanjng 193, Chna Department

More information

Chapter 6 Programmng the fnte element method Inow turn to the man subject of ths book: The mplementaton of the fnte element algorthm n computer programs. In order to make my dscusson as straghtforward

More information

Simple and Efficient Mesh Layout with Space-Filling Curves

Simple and Efficient Mesh Layout with Space-Filling Curves paper 2012/4/5 23:05 page 1 #1 Vol. [16], No. [1]: 1 15 Smple and Effcent Mesh Layout wth Space-Fllng Curves Huy T. Vo and Claudo T. Slva Polytechnc Insttute of New York Unversty Luz F. Schedegger Facebook,

More information

Robust Soft Shadow Mapping with Backprojection and Depth Peeling

Robust Soft Shadow Mapping with Backprojection and Depth Peeling paper 2008/3/20 15:47 page 19 #1 Vol. 13, No. 1: 19 29 Robust Soft Shadow Mappng wth Backprojecton and Depth Peelng Lous Bavol, Steven P. Callahan, and Claudo T. Slva Scentfc Computng and Imagng Insttute,

More information

SLAM Summer School 2006 Practical 2: SLAM using Monocular Vision

SLAM Summer School 2006 Practical 2: SLAM using Monocular Vision SLAM Summer School 2006 Practcal 2: SLAM usng Monocular Vson Javer Cvera, Unversty of Zaragoza Andrew J. Davson, Imperal College London J.M.M Montel, Unversty of Zaragoza. josemar@unzar.es, jcvera@unzar.es,

More information

Smoothing Spline ANOVA for variable screening

Smoothing Spline ANOVA for variable screening Smoothng Splne ANOVA for varable screenng a useful tool for metamodels tranng and mult-objectve optmzaton L. Rcco, E. Rgon, A. Turco Outlne RSM Introducton Possble couplng Test case MOO MOO wth Game Theory

More information

Simple and Efficient Mesh Layout with Space-Filling Curves

Simple and Efficient Mesh Layout with Space-Filling Curves paper 2011/11/7 11:21 page 1 #1 Vol. [VOL], No. [ISS]: 1 15 Smple and Effcent Mesh Layout wth Space-Fllng Curves Huy T. Vo and Claudo T. Slva Polytechnc Insttute of New York Unversty Luz F. Schedegger

More information

CGAL. Mesh Simplification. (Slides from Tom Funkhouser, Adam Finkelstein)

CGAL. Mesh Simplification. (Slides from Tom Funkhouser, Adam Finkelstein) CGAL Mesh Simplification (Slides from Tom Funkhouser, Adam Finkelstein) Siddhartha Chaudhuri http://www.cse.iitb.ac.in/~cs749 In a nutshell Problem: Meshes have too many polygons for storage, rendering,

More information

Level Set Extraction from Gridded 2D and 3D Data

Level Set Extraction from Gridded 2D and 3D Data Level Set Extraction from Gridded 2D and 3D Data David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International

More information

Classification / Regression Support Vector Machines

Classification / Regression Support Vector Machines Classfcaton / Regresson Support Vector Machnes Jeff Howbert Introducton to Machne Learnng Wnter 04 Topcs SVM classfers for lnearly separable classes SVM classfers for non-lnearly separable classes SVM

More information

Lecture 13: High-dimensional Images

Lecture 13: High-dimensional Images Lec : Hgh-dmensonal Images Grayscale Images Lecture : Hgh-dmensonal Images Math 90 Prof. Todd Wttman The Ctadel A grayscale mage s an nteger-valued D matrx. An 8-bt mage takes on values between 0 and 55.

More information

Loop Permutation. Loop Transformations for Parallelism & Locality. Legality of Loop Interchange. Loop Interchange (cont)

Loop Permutation. Loop Transformations for Parallelism & Locality. Legality of Loop Interchange. Loop Interchange (cont) Loop Transformatons for Parallelsm & Localty Prevously Data dependences and loops Loop transformatons Parallelzaton Loop nterchange Today Loop nterchange Loop transformatons and transformaton frameworks

More information

A Scalable Projective Bundle Adjustment Algorithm using the L Norm

A Scalable Projective Bundle Adjustment Algorithm using the L Norm Sxth Indan Conference on Computer Vson, Graphcs & Image Processng A Scalable Projectve Bundle Adjustment Algorthm usng the Norm Kaushk Mtra and Rama Chellappa Dept. of Electrcal and Computer Engneerng

More information

Collision Detection. Overview. Efficient Collision Detection. Collision Detection with Rays: Example. C = nm + (n choose 2)

Collision Detection. Overview. Efficient Collision Detection. Collision Detection with Rays: Example. C = nm + (n choose 2) Overvew Collson detecton wth Rays Collson detecton usng BSP trees Herarchcal Collson Detecton OBB tree, k-dop tree algorthms Multple object CD system Collson Detecton Fundamental to graphcs, VR applcatons

More information

What are the camera parameters? Where are the light sources? What is the mapping from radiance to pixel color? Want to solve for 3D geometry

What are the camera parameters? Where are the light sources? What is the mapping from radiance to pixel color? Want to solve for 3D geometry Today: Calbraton What are the camera parameters? Where are the lght sources? What s the mappng from radance to pel color? Why Calbrate? Want to solve for D geometry Alternatve approach Solve for D shape

More information

The Codesign Challenge

The Codesign Challenge ECE 4530 Codesgn Challenge Fall 2007 Hardware/Software Codesgn The Codesgn Challenge Objectves In the codesgn challenge, your task s to accelerate a gven software reference mplementaton as fast as possble.

More information

MULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION

MULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION MULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION Paulo Quntlano 1 & Antono Santa-Rosa 1 Federal Polce Department, Brasla, Brazl. E-mals: quntlano.pqs@dpf.gov.br and

More information

All-Pairs Shortest Paths. Approximate All-Pairs shortest paths Approximate distance oracles Spanners and Emulators. Uri Zwick Tel Aviv University

All-Pairs Shortest Paths. Approximate All-Pairs shortest paths Approximate distance oracles Spanners and Emulators. Uri Zwick Tel Aviv University Approxmate All-Pars shortest paths Approxmate dstance oracles Spanners and Emulators Ur Zwck Tel Avv Unversty Summer School on Shortest Paths (PATH05 DIKU, Unversty of Copenhagen All-Pars Shortest Paths

More information

S1 Note. Basis functions.

S1 Note. Basis functions. S1 Note. Bass functons. Contents Types of bass functons...1 The Fourer bass...2 B-splne bass...3 Power and type I error rates wth dfferent numbers of bass functons...4 Table S1. Smulaton results of type

More information

Cluster Analysis of Electrical Behavior

Cluster Analysis of Electrical Behavior Journal of Computer and Communcatons, 205, 3, 88-93 Publshed Onlne May 205 n ScRes. http://www.scrp.org/ournal/cc http://dx.do.org/0.4236/cc.205.350 Cluster Analyss of Electrcal Behavor Ln Lu Ln Lu, School

More information

Direction Fields over Point-Sampled Geometry

Direction Fields over Point-Sampled Geometry Drecton Felds over Pont-Sampled Geometry Marc Alexa Tobas Klug Carsten Stoll Interactve Graphcs Systems Group Department of Computer Scence Technsche Unverstät Darmstadt Fraunhoferstr. 5 64283 Darmstadt,

More information

Recognizing Faces. Outline

Recognizing Faces. Outline Recognzng Faces Drk Colbry Outlne Introducton and Motvaton Defnng a feature vector Prncpal Component Analyss Lnear Dscrmnate Analyss !"" #$""% http://www.nfotech.oulu.f/annual/2004 + &'()*) '+)* 2 ! &

More information

Fitting & Matching. Lecture 4 Prof. Bregler. Slides from: S. Lazebnik, S. Seitz, M. Pollefeys, A. Effros.

Fitting & Matching. Lecture 4 Prof. Bregler. Slides from: S. Lazebnik, S. Seitz, M. Pollefeys, A. Effros. Fttng & Matchng Lecture 4 Prof. Bregler Sldes from: S. Lazebnk, S. Setz, M. Pollefeys, A. Effros. How do we buld panorama? We need to match (algn) mages Matchng wth Features Detect feature ponts n both

More information

Monte Carlo 1: Integration

Monte Carlo 1: Integration Monte Carlo : Integraton Prevous lecture: Analytcal llumnaton formula Ths lecture: Monte Carlo Integraton Revew random varables and probablty Samplng from dstrbutons Samplng from shapes Numercal calculaton

More information

Scan Conversion & Shading

Scan Conversion & Shading Scan Converson & Shadng Thomas Funkhouser Prnceton Unversty C0S 426, Fall 1999 3D Renderng Ppelne (for drect llumnaton) 3D Prmtves 3D Modelng Coordnates Modelng Transformaton 3D World Coordnates Lghtng

More information

CSCI 104 Sorting Algorithms. Mark Redekopp David Kempe

CSCI 104 Sorting Algorithms. Mark Redekopp David Kempe CSCI 104 Sortng Algorthms Mark Redekopp Davd Kempe Algorthm Effcency SORTING 2 Sortng If we have an unordered lst, sequental search becomes our only choce If we wll perform a lot of searches t may be benefcal

More information

Barycentric Coordinates. From: Mean Value Coordinates for Closed Triangular Meshes by Ju et al.

Barycentric Coordinates. From: Mean Value Coordinates for Closed Triangular Meshes by Ju et al. Barycentrc Coordnates From: Mean Value Coordnates for Closed Trangular Meshes by Ju et al. Motvaton Data nterpolaton from the vertces of a boundary polygon to ts nteror Boundary value problems Shadng Space

More information

Robust Soft Shadow Mapping with Depth Peeling

Robust Soft Shadow Mapping with Depth Peeling 1 Robust Soft Shadow Mappng wth Depth Peelng Lous Bavol, Steven P. Callahan, Cláudo T. Slva UUSCI-2006-028 Scentfc Computng and Imagng Insttute Unversty of Utah Salt Lake Cty, UT 84112 USA August 11, 2006

More information

SIGGRAPH Interactive Image Cutout. Interactive Graph Cut. Interactive Graph Cut. Interactive Graph Cut. Hard Constraints. Lazy Snapping.

SIGGRAPH Interactive Image Cutout. Interactive Graph Cut. Interactive Graph Cut. Interactive Graph Cut. Hard Constraints. Lazy Snapping. SIGGRAPH 004 Interactve Image Cutout Lazy Snappng Yn L Jan Sun Ch-Keung Tang Heung-Yeung Shum Mcrosoft Research Asa Hong Kong Unversty Separate an object from ts background Compose the object on another

More information

Mesh Simplification. Mesh Simplification. Mesh Simplification Goals. Mesh Simplification Motivation. Vertex Clustering. Mesh Simplification Overview

Mesh Simplification. Mesh Simplification. Mesh Simplification Goals. Mesh Simplification Motivation. Vertex Clustering. Mesh Simplification Overview Mesh Simplification Mesh Simplification Adam Finkelstein Princeton University COS 56, Fall 008 Slides from: Funkhouser Division, Viewpoint, Cohen Mesh Simplification Motivation Interactive visualization

More information

Multicriteria Decision Making

Multicriteria Decision Making Multcrtera Decson Makng Andrés Ramos (Andres.Ramos@comllas.edu) Pedro Sánchez (Pedro.Sanchez@comllas.edu) Sonja Wogrn (Sonja.Wogrn@comllas.edu) Contents 1. Basc concepts 2. Contnuous methods 3. Dscrete

More information

A Fast Content-Based Multimedia Retrieval Technique Using Compressed Data

A Fast Content-Based Multimedia Retrieval Technique Using Compressed Data A Fast Content-Based Multmeda Retreval Technque Usng Compressed Data Borko Furht and Pornvt Saksobhavvat NSF Multmeda Laboratory Florda Atlantc Unversty, Boca Raton, Florda 3343 ABSTRACT In ths paper,

More information

Scan Conversion & Shading

Scan Conversion & Shading 1 3D Renderng Ppelne (for drect llumnaton) 2 Scan Converson & Shadng Adam Fnkelsten Prnceton Unversty C0S 426, Fall 2001 3DPrmtves 3D Modelng Coordnates Modelng Transformaton 3D World Coordnates Lghtng

More information

Two Dimensional Projective Point Matching

Two Dimensional Projective Point Matching Two Dmensonal Projectve Pont Matchng Jason Denton & J. Ross Beverdge Colorado State Unversty Computer Scence Department Ft. Collns, CO 80523 denton@cs.colostate.edu Abstract Pont matchng s the task of

More information

Mesh Editing in ROI with Dual Laplacian

Mesh Editing in ROI with Dual Laplacian Mesh Edtng n ROI wth Dual Laplacan Luo Qong, Lu Bo, Ma Zhan-guo, Zhang Hong-bn College of Computer Scence, Beng Unversty of Technology, Chna lqngng@sohu.com, lubo@but.edu.cn,mzgsy@63.com,zhb@publc.bta.net.cn

More information

Discussion. History and Outline. Smoothness of Indirect Lighting. Irradiance Caching. Irradiance Calculation. Advanced Computer Graphics (Fall 2009)

Discussion. History and Outline. Smoothness of Indirect Lighting. Irradiance Caching. Irradiance Calculation. Advanced Computer Graphics (Fall 2009) Advanced Computer Graphcs (Fall 2009 CS 29, Renderng Lecture 6: Recent Advances n Monte Carlo Offlne Renderng Rav Ramamoorth http://nst.eecs.berkeley.edu/~cs29-13/fa09 Dscusson Problems dfferent over years.

More information

CS229 Class Project: Fusion arc treatment planning strategy by adaptive learning cost function based beam selection Ho Jin Kim

CS229 Class Project: Fusion arc treatment planning strategy by adaptive learning cost function based beam selection Ho Jin Kim CS229 Class Project: Fuson arc treatment plannng strategy by adaptve learnng cost functon based beam selecton Ho Jn Km (eekhj87@stanford.edu) Abstract Current therapeutc modaltes n radaton therapy such

More information

Progressive Simplicial Complexes

Progressive Simplicial Complexes Progressve Smplcal Complexes Jovan Popovć Carnege Mellon Unversty Hugues Hoppe Mcrosoft Research ABSTRACT In ths paper, we ntroduce the progressve smplcal complex (PSC) representaton, a new format for

More information

Geometric Error Estimation

Geometric Error Estimation Geometrc Error Estmaton Houman Borouchak Project-team GAMMA 3 UTT Troyes, France Emal: houman.borouchak@utt.fr Patrck Laug Project-team GAMMA 3 INRIA Pars - Rocquencourt, France Emal: patrck.laug@nra.fr

More information

Multi-Resolution Geometric Fusion

Multi-Resolution Geometric Fusion Internatonal Conference on Recent Advances n 3-D Dgtal Imagng and Modellng, Ottawa, Canada May 12 15, 1997 Mult-Resoluton Geometrc Fuson Adran Hlton and John Illngworth Centre for Vson, Speech and Sgnal

More information

Monte Carlo 1: Integration

Monte Carlo 1: Integration Monte Carlo : Integraton Prevous lecture: Analytcal llumnaton formula Ths lecture: Monte Carlo Integraton Revew random varables and probablty Samplng from dstrbutons Samplng from shapes Numercal calculaton

More information

High resolution 3D Tau-p transform by matching pursuit Weiping Cao* and Warren S. Ross, Shearwater GeoServices

High resolution 3D Tau-p transform by matching pursuit Weiping Cao* and Warren S. Ross, Shearwater GeoServices Hgh resoluton 3D Tau-p transform by matchng pursut Wepng Cao* and Warren S. Ross, Shearwater GeoServces Summary The 3D Tau-p transform s of vtal sgnfcance for processng sesmc data acqured wth modern wde

More information

Multi-stable Perception. Necker Cube

Multi-stable Perception. Necker Cube Mult-stable Percepton Necker Cube Spnnng dancer lluson, Nobuuk Kaahara Fttng and Algnment Computer Vson Szelsk 6.1 James Has Acknowledgment: Man sldes from Derek Hoem, Lana Lazebnk, and Grauman&Lebe 2008

More information

A Robust Method for Estimating the Fundamental Matrix

A Robust Method for Estimating the Fundamental Matrix Proc. VIIth Dgtal Image Computng: Technques and Applcatons, Sun C., Talbot H., Ourseln S. and Adraansen T. (Eds.), 0- Dec. 003, Sydney A Robust Method for Estmatng the Fundamental Matrx C.L. Feng and Y.S.

More information

Kent State University CS 4/ Design and Analysis of Algorithms. Dept. of Math & Computer Science LECT-16. Dynamic Programming

Kent State University CS 4/ Design and Analysis of Algorithms. Dept. of Math & Computer Science LECT-16. Dynamic Programming CS 4/560 Desgn and Analyss of Algorthms Kent State Unversty Dept. of Math & Computer Scence LECT-6 Dynamc Programmng 2 Dynamc Programmng Dynamc Programmng, lke the dvde-and-conquer method, solves problems

More information

Smooth Approximation to Surface Meshes of Arbitrary Topology with Locally Blended Radial Basis Functions

Smooth Approximation to Surface Meshes of Arbitrary Topology with Locally Blended Radial Basis Functions 587 Smooth Approxmaton to Surface eshes of Arbtrary Topology wth Locally Blended Radal Bass Functons ngyong Pang 1,, Weyn a 1, Zhgeng Pan and Fuyan Zhang 1 Cty Unversty of Hong Kong, mewma@ctyu.edu.hk

More information

Graph-based Clustering

Graph-based Clustering Graphbased Clusterng Transform the data nto a graph representaton ertces are the data ponts to be clustered Edges are eghted based on smlarty beteen data ponts Graph parttonng Þ Each connected component

More information

Discussion. History and Outline. Smoothness of Indirect Lighting. Irradiance Calculation. Irradiance Caching. Advanced Computer Graphics (Fall 2009)

Discussion. History and Outline. Smoothness of Indirect Lighting. Irradiance Calculation. Irradiance Caching. Advanced Computer Graphics (Fall 2009) Advanced Computer Graphcs (Fall 2009 CS 283, Lecture 13: Recent Advances n Monte Carlo Offlne Renderng Rav Ramamoorth http://nst.eecs.berkeley.edu/~cs283/fa10 Dscusson Problems dfferent over years. Intally,

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Decson surface s a hyperplane (lne n 2D) n feature space (smlar to the Perceptron) Arguably, the most mportant recent dscovery n machne learnng In a nutshell: map the data to a predetermned

More information

3D vector computer graphics

3D vector computer graphics 3D vector computer graphcs Paolo Varagnolo: freelance engneer Padova Aprl 2016 Prvate Practce ----------------------------------- 1. Introducton Vector 3D model representaton n computer graphcs requres

More information

Fast Computation of Shortest Path for Visiting Segments in the Plane

Fast Computation of Shortest Path for Visiting Segments in the Plane Send Orders for Reprnts to reprnts@benthamscence.ae 4 The Open Cybernetcs & Systemcs Journal, 04, 8, 4-9 Open Access Fast Computaton of Shortest Path for Vstng Segments n the Plane Ljuan Wang,, Bo Jang

More information

Exercises (Part 4) Introduction to R UCLA/CCPR. John Fox, February 2005

Exercises (Part 4) Introduction to R UCLA/CCPR. John Fox, February 2005 Exercses (Part 4) Introducton to R UCLA/CCPR John Fox, February 2005 1. A challengng problem: Iterated weghted least squares (IWLS) s a standard method of fttng generalzed lnear models to data. As descrbed

More information

Multiresolution Modeling for Real-Time Facial Animation

Multiresolution Modeling for Real-Time Facial Animation Multresoluton Modelng for Real-me Facal Anmaton Soo-Kyun Km, Jong-In Cho and Chang-Hun Km Department of Computer Scence and Engneerng, Korea Unersty {ncesk, cho, chkm}@cgr.korea.ac.kr Fgure 1. Same expresson

More information

Support Vector Machines

Support Vector Machines /9/207 MIST.6060 Busness Intellgence and Data Mnng What are Support Vector Machnes? Support Vector Machnes Support Vector Machnes (SVMs) are supervsed learnng technques that analyze data and recognze patterns.

More information

Steps for Computing the Dissimilarity, Entropy, Herfindahl-Hirschman and. Accessibility (Gravity with Competition) Indices

Steps for Computing the Dissimilarity, Entropy, Herfindahl-Hirschman and. Accessibility (Gravity with Competition) Indices Steps for Computng the Dssmlarty, Entropy, Herfndahl-Hrschman and Accessblty (Gravty wth Competton) Indces I. Dssmlarty Index Measurement: The followng formula can be used to measure the evenness between

More information