Convergence of binomial-based derivative estimation for C 2 noisy discretized curves
|
|
- Abner Flowers
- 5 years ago
- Views:
Transcription
1 Journées RAIM 09, Lyon, octobre 2009 Convergence of binomial-based derivative estimation for C 2 noisy discretized curves H-A. Esbelin, R. Malgouyres, C. Cartade and S. Fourey Univ. Clermont 1, LAIC, France {esbelin,remy.malgouyres,colin}@laic.u-clermont1.fr GREYC, ENSICAEN Sebastien.Fourey@greyc.ensicaen.fr
2 1 Introduction Discrete Estimators from lines and planes 2 1D-2D Binomial Masks 3 Equations différentielles 4 Convolutions on surfaces First Order Partial Derivatives 5 Experiments and Results Normal Estimation Curvature Estimation
3 1 Introduction Discrete Estimators from lines and planes 2 1D-2D Binomial Masks 3 Equations différentielles 4 Convolutions on surfaces First Order Partial Derivatives 5 Experiments and Results Normal Estimation Curvature Estimation
4 Normal and Tangent Estimation DSS (Digital Straight Segments) based methods. [Debled-Rennesson, Klette, Kovalesky, Reveillès, Andrès, Lachaud.]
5 Same idea applied to surfaces... Approach based on digital plane recognition. [Sivignon, Dupont, Chassery]
6 Chen et al. (1985), Papier and Françon (1998) [Chen et al., 1985] Estimate the normal vector on each surfel by averaging the normals of their 4 e-neighbors. [Papier and Françon, 1998] Same averaging process but taking into account an arbitrarily large neighborhood (umbrella) with decreasing weights.
7 What does an umbrella look like?
8 1 Introduction Discrete Estimators from lines and planes 2 1D-2D Binomial Masks 3 Equations différentielles 4 Convolutions on surfaces First Order Partial Derivatives 5 Experiments and Results Normal Estimation Curvature Estimation
9
10 The Binomial Derivative Estimator Input : Discrete function Γ width hγ(i) = f (ih) + ɛ h (i) f is the sampled C 2 function, h the pixel s size and ɛ h the noise. ( ( 2m 1 u)(n) = 1 i=m ( ) ) 2m 1 2 2m 1 (u(n + i) u(n 1 + i)) m 1 + i i= m+1 Smoothed finite differences Binomial smoothing kernel
11 The Error Model Rounded case ] : ɛ h (i) 1 2h which is equivalent to Γ(i) = [ f (ih) h Floor case : 0 ɛ h (i) h which is equivalent to Γ(i) = f (ih) h Uniform Noise case : 0 ɛ h (i) Kh α with 0 < α 1 and K a positive constant. Note that the round case and the floor case are particular cases of uniform noise with α = 1.
12 Prior Results (DGCI 08) Theorem Suppose that f : R R is a C 3 function and f (3) is bounded, α ]0, 1], K R + and h R +. Suppose Γ : Z Z is such that hγ(i) f (hi) Kh α (uniform noise case). Then for m = h 2(α 3)/3, we have ( 2m 1 Γ)(n) f (nh) O(h 2α/3 ) Theorem Under the assumptions of Theorem 1, for some constant K and m sufficiently large, 2m 1 Γ(n) f (nh) h2 m 4 f (3) + K h α 1 m
13 More recent results (DGCI 09) Works for C 2 functions ; Uniform convergence for noisy parametrized curves. Theorem Suppose that f is a C 2 function and f (2) is bounded. Suppose Γ : Z Z is such that hγ(i) f (hi) Kh α (uniform noise case). Then if m = h (α 2)/1,01 we have ( 2m 1 Γ)(n) f (nh) O(h (0,51(α 0,01)/1,01 ). Theorem Under the assumptions of Theorem 3, for some constant K and m sufficiently large, 2m 1 Γ(n) f (nh) hm 0,51 f (2) + K h α 1 m
14 Convergence Result For Parametrized Discretizations Definition The binomial discrete tangent at M i is the real line going through M i directed by the vector ( m+1 (x i ), m+1 (y i )), when this vector is nonzero. Theorem Let g be a C 2 parametrization of a simple closed curve C. Suppose that for all i we have g(ih) hσ h (i) Kh α 1. Then for some constant K and m sufficiently large, 2m 1 Σ h (n) g (nh) hm 0,51 g (2) + K h α 1 m
15 Pixel-length Parametrization Definition A parametrization of a real curve γ is pixel-length if for all u et u such that γ x and γ y are monotonic between u and u, we have γ(u) γ(u ) 1 = u u. The idea is that for a curve with a pixel-length parametrization, the speed on the curve is the same as the speed of a discretization of the curve with each edgel taking the same time y x
16 Tangent Estimation for a general C 2 curve Theorem Let C be a simple closed curve C and M 0 C. Suppose that g is a regular C 2 parametrization of C and wlog M 0 = g(0). Let Σ h : Z Z 2 be a 4-connected discrete parametrized curve, lying in a tube of C of width Kh α. Suppose that Σ 2 h (0) M 0 Kh α (up to a translation on the parameters of Σ, this is alway possible). Then 2m 1 Σ h (0) T 0 K hm 0,51 + K h α 1 m
17 1 Introduction Discrete Estimators from lines and planes 2 1D-2D Binomial Masks 3 Equations différentielles 4 Convolutions on surfaces First Order Partial Derivatives 5 Experiments and Results Normal Estimation Curvature Estimation
18 Discrétisation d une EDO On discrétise l équation différentielle par x [a, b], y (x) = f (x, y(x)), y(x 0 ) = y 0 (D m u)(i) = hf (x i, u i ) m N, D m masque de dérivation
19 Exemples m = 2 u n+3 + u n+2 u n 1 u n 4h = u n+1 ou u n+2 m = 3 u n+4 + 2u n+3 2u n+1 u n 8h = u n+2
20 Calculs exacts suite récurrente linéaire d équation caractéristique (x 1)(x + 1) 2m 1 = 2 2m 1 hx m. une racine x 1 proche de 1 et 2m 1 racines x 2, x 3,..., x 2m proches de 1
21 Calculs exacts Équation y = y Développement limité de la solution numérique : Comme la n ième itérée est de la forme u n = a 1 x n 1 + a 2x n a 2mx n 2m x n 1 = (1 + x n + O( h 2 )) n = exp(x) + O(h) On choisit les conditions initiales pour que a 1 = 1 et a i = 0 pour i 1.
22 Calculs exacts
23 1 Introduction Discrete Estimators from lines and planes 2 1D-2D Binomial Masks 3 Equations différentielles 4 Convolutions on surfaces First Order Partial Derivatives 5 Experiments and Results Normal Estimation Curvature Estimation
24 Iterated classical smoothing masks Classical 2D averaging filter ( 16)
25 1D sampling, Parametrized Curves, Digital Surfaces
26 Adjacency Between Surfels e-adjacency v-adjacency e-neighborhood v-neighborhood
27 How To Define Convolutions on Surfaces? surfels
28 The Non-Planar Convolution Operator Let X Z 3 et Σ = δ(x). Furthermore, let E a vector space over R and two functions f and F : f : Σ E F : Σ Σ R Definition We define the operator Ψ, acting on f and F as follows : Ψ F (f ) : Σ E s f (s ) F(s, s ) (1) s Σ
29 Properties and Non Properties Non abelian : Ψ F (Ψ G (f )) Ψ G (Ψ F (f )) in general. Associative : Ψ ΨH (G)(f ) = Ψ H (Ψ G (f ))
30 The Iterated Convolution Operator X Z 3, Σ = δ(x), E is a vector space over R. f : Σ E, F : Σ Σ R. Definition Let n N, we define Ψ (n) F { Ψ (0) F Ψ (n) F (f ) = f (f ) = Ψ F(Ψ (n 1) F (f )) if n > 0. (2)
31 The Our Local Kernel
32 Classical 2D averaging filter ( 16) The local averaging mask H(s, s)
33 The Our Local Kernel
34 Over a plane with normal n = (1, 2, 3) ( surfels)
35 Over a plane with normal n = (1, 2, 3) ( surfels)
36 With Lauren-Papier method (breadth first search)
37 Over a plane with normal n = (1, 2, 3) ( surfels)
38 Over a hyperbolic paraboloid Response Breadth-first traversal
39 1 Introduction Discrete Estimators from lines and planes 2 1D-2D Binomial Masks 3 Equations différentielles 4 Convolutions on surfaces First Order Partial Derivatives 5 Experiments and Results Normal Estimation Curvature Estimation
40 We define two derivative masks D u (s, s) and D v (s, s) : D u (s, s) D v (s, s) Orientations and slice curves.
41 Partial derivative operators Using Ψ (n), H, D u and D v we define u and v : Le n Z, where (n) u (n) v = Ψ Du (Ψ (n) H (σ)) (3) = Ψ Dv (Ψ (n) H (σ)) (4) σ : Σ R 3 s (x,y,z), the center of the surfel s
42 Normal vectors estimation Let s Σ. We define Γ (n) (s), the estimated normal vector of Σ at the center of s (after n on-surface convolutions). Γ (n) (s) = (n) u (n) u (s) (n) v (s) (s) (n) v (s) (5)
43 Second order derivative operations, curvatures Using Ψ (n) H, u, and v (eventually twice ), we estimate the first and second order partial derivatives : S u, S v, 2 S u 2, 2 S v 2, and 2 S u v Then, using the coefficients of the first and second fundamental forms, we compute the Gaussian or mean curvatures.
44 1 Introduction Discrete Estimators from lines and planes 2 1D-2D Binomial Masks 3 Equations différentielles 4 Convolutions on surfaces First Order Partial Derivatives 5 Experiments and Results Normal Estimation Curvature Estimation
45 1 Introduction Discrete Estimators from lines and planes 2 1D-2D Binomial Masks 3 Equations différentielles 4 Convolutions on surfaces First Order Partial Derivatives 5 Experiments and Results Normal Estimation Curvature Estimation
46 On digitized spheres Average error (in degree) and standard deviation on spheres with increasing radii.
47 On digitized tori Average error (in degree) an standard deviation on tori with increasing large and small radii
48 1 Introduction Discrete Estimators from lines and planes 2 1D-2D Binomial Masks 3 Equations différentielles 4 Convolutions on surfaces First Order Partial Derivatives 5 Experiments and Results Normal Estimation Curvature Estimation
49 Gaussian Curvature on a Torus (35000 surfels)
50 Gaussian Curvature on a Torus Estimated and exact Gaussian curvatures on a torus. (Large radius is 80, small radius is 40.)
51 Gaussian Curvature on Bunny
52 Mean Curvature on Bunny
53 Questions
Binomial Convolutions and Derivatives Estimation from Noisy Discretizations
Binomial Convolutions and Derivatives Estimation from Noisy Discretizations Rémy Malgouyres 1, Florent Brunet 2,andSébastien Fourey 3 1 Univ. Clermont 1, LAIC, IUT Dépt Informatique, BP 86, F-63172 Aubière,
More informationMath 114 Spring 2013 Final Exam
1. Assume the acceleration of gravity is 10 m/sec downwards. A cannonball is fired at ground level. If the cannon ball rises to a height of 80 meters and travels a distance of 0 meters before it hits the
More information1 Introduction: connections and fiber bundles
[under construction] 1 Introduction: connections and fiber bundles Two main concepts of differential geometry are those of a covariant derivative and of a fiber bundle (in particular, a vector bundle).
More informationVoronoi-based geometry estimator for 3D digital surfaces
Voronoi-based geometry estimator for 3D digital surfaces Louis Cuel 1,2, Jacques-Olivier Lachaud 1, and Boris Thibert 2 1 Université de Savoie, Laboratoire LAMA Le bourget du lac, France 2 Université de
More informationImage processing and Computer Vision
1 / 1 Image processing and Computer Vision Continuous Optimization and applications to image processing Martin de La Gorce martin.de-la-gorce@enpc.fr February 2015 Optimization 2 / 1 We have a function
More informationDecouplings and applications
April 27, 2018 Let Ξ be a collection of frequency points ξ on some curved, compact manifold S of diameter 1 in R n (e.g. the unit sphere S n 1 ) Let B R = B(c, R) be a ball with radius R 1. Let also a
More informationImage Processing 1 (IP1) Bildverarbeitung 1
MIN-Fakultät Fachbereich Informatik Arbeitsbereich SAV/BV (KOGS) Image Processing 1 (IP1) Bildverarbeitung 1 Lecture 7 Spectral Image Processing and Convolution Winter Semester 2014/15 Slides: Prof. Bernd
More informationLinear Diffusion and Image Processing. Outline
Outline Linear Diffusion and Image Processing Fourier Transform Convolution Image Restoration: Linear Filtering Diffusion Processes for Noise Filtering linear scale space theory Gauss-Laplace pyramid for
More informationLaplacian Filters. Sobel Filters. Laplacian Filters. Laplacian Filters. Laplacian Filters. Laplacian Filters
Sobel Filters Note that smoothing the image before applying a Sobel filter typically gives better results. Even thresholding the Sobel filtered image cannot usually create precise, i.e., -pixel wide, edges.
More informationEdge Detection. Image Processing - Computer Vision
Image Processing - Lesson 10 Edge Detection Image Processing - Computer Vision Low Level Edge detection masks Gradient Detectors Compass Detectors Second Derivative - Laplace detectors Edge Linking Image
More informationEfficient Inference in Fully Connected CRFs with Gaussian Edge Potentials
Efficient Inference in Fully Connected CRFs with Gaussian Edge Potentials Philipp Krähenbühl and Vladlen Koltun Stanford University Presenter: Yuan-Ting Hu 1 Conditional Random Field (CRF) E x I = φ u
More information1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is
1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order
More informationHistogram Processing
Histogram Processing The histogram of a digital image with gray levels in the range [0,L-] is a discrete function h ( r k ) = n k where r k n k = k th gray level = number of pixels in the image having
More informationITK Filters. Thresholding Edge Detection Gradients Second Order Derivatives Neighborhood Filters Smoothing Filters Distance Map Image Transforms
ITK Filters Thresholding Edge Detection Gradients Second Order Derivatives Neighborhood Filters Smoothing Filters Distance Map Image Transforms ITCS 6010:Biomedical Imaging and Visualization 1 ITK Filters:
More informationInverse problem and optimization
Inverse problem and optimization Laurent Condat, Nelly Pustelnik CNRS, Gipsa-lab CNRS, Laboratoire de Physique de l ENS de Lyon Decembre, 15th 2016 Inverse problem and optimization 2/36 Plan 1. Examples
More informationLow-level Image Processing
Low-level Image Processing In-Place Covariance Operators for Computer Vision Terry Caelli and Mark Ollila School of Computing, Curtin University of Technology, Perth, Western Australia, Box U 1987, Emaihtmc@cs.mu.oz.au
More informationEdge Detection. CS 650: Computer Vision
CS 650: Computer Vision Edges and Gradients Edge: local indication of an object transition Edge detection: local operators that find edges (usually involves convolution) Local intensity transitions are
More informationConvergence Properties of Curvature Scale Space Representations
Convergence Properties of Curvature Scale Space Representations Farzin Mokhtarian Department of Electronic and Electrical Engineering University of Surrey Guildford, Surrey GU2 5XH Email: F.Mokhtarian@ee.surrey.ac.uk
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationImage Filtering. Slides, adapted from. Steve Seitz and Rick Szeliski, U.Washington
Image Filtering Slides, adapted from Steve Seitz and Rick Szeliski, U.Washington The power of blur All is Vanity by Charles Allen Gillbert (1873-1929) Harmon LD & JuleszB (1973) The recognition of faces.
More information8. THE FARY-MILNOR THEOREM
Math 501 - Differential Geometry Herman Gluck Tuesday April 17, 2012 8. THE FARY-MILNOR THEOREM The curvature of a smooth curve in 3-space is 0 by definition, and its integral w.r.t. arc length, (s) ds,
More informationA Laplacian of Gaussian-based Approach for Spot Detection in Two-Dimensional Gel Electrophoresis Images
A Laplacian of Gaussian-based Approach for Spot Detection in Two-Dimensional Gel Electrophoresis Images Feng He 1, Bangshu Xiong 1, Chengli Sun 1, Xiaobin Xia 1 1 Key Laboratory of Nondestructive Test
More informationNonparametric Regression. Changliang Zou
Nonparametric Regression Institute of Statistics, Nankai University Email: nk.chlzou@gmail.com Smoothing parameter selection An overall measure of how well m h (x) performs in estimating m(x) over x (0,
More informationIntroduction to Computer Vision. 2D Linear Systems
Introduction to Computer Vision D Linear Systems Review: Linear Systems We define a system as a unit that converts an input function into an output function Independent variable System operator or Transfer
More informationConfidence and curvature estimation of curvilinear structures in 3-D
Confidence and curvature estimation of curvilinear structures in 3-D P. Bakker, L.J. van Vliet, P.W. Verbeek Pattern Recognition Group, Department of Applied Physics, Delft University of Technology, Lorentzweg
More informationA SHORT GALLERY OF CHARACTERISTIC FOLIATIONS
A SHORT GALLERY OF CHARACTERISTIC FOLIATIONS AUSTIN CHRISTIAN 1. Introduction The purpose of this note is to visualize some simple contact structures via their characteristic foliations. The emphasis is
More informationCurvature of Digital Curves
Curvature of Digital Curves Left: a symmetric curve (i.e., results should also be symmetric ). Right: high-curvature pixels should correspond to visual perception of corners. Page 1 March 2005 Categories
More informationTHE FOUR VERTEX THEOREM AND ITS CONVERSE in honor of Björn Dahlberg
Writhe - Four Vertex Theorem.14 June, 2006 THE FOUR VERTEX THEOREM AND ITS CONVERSE in honor of Björn Dahlberg The Four Vertex Theorem says that a simple closed curve in the plane, other than a circle,
More informationSlide a window along the input arc sequence S. Least-squares estimate. σ 2. σ Estimate 1. Statistically test the difference between θ 1 and θ 2
Corner Detection 2D Image Features Corners are important two dimensional features. Two dimensional image features are interesting local structures. They include junctions of dierent types Slide 3 They
More informationExercise: concepts from chapter 3
Reading:, Ch 3 1) The natural representation of a curve, c = c(s), satisfies the condition dc/ds = 1, where s is the natural parameter for the curve. a) Describe in words and a sketch what this condition
More informationCHAPTER 3. Gauss map. In this chapter we will study the Gauss map of surfaces in R 3.
CHAPTER 3 Gauss map In this chapter we will study the Gauss map of surfaces in R 3. 3.1. Surfaces in R 3 Let S R 3 be a submanifold of dimension 2. Let {U i, ϕ i } be a DS on S. For any p U i we have a
More informationCorner. Corners are the intersections of two edges of sufficiently different orientations.
2D Image Features Two dimensional image features are interesting local structures. They include junctions of different types like Y, T, X, and L. Much of the work on 2D features focuses on junction L,
More informationGEOMETRY HW (t, 0, e 1/t2 ), t > 0 1/t2, 0), t < 0. (0, 0, 0), t = 0
GEOMETRY HW CLAY SHONKWILER Consider the map.5.0 t, 0, e /t ), t > 0 αt) = t, e /t, 0), t < 0 0, 0, 0), t = 0 a) Prove that α is a differentiable curve. Proof. If we denote αt) = xt), yt), zt0), then it
More informationGEOMETRY HW Consider the parametrized surface (Enneper s surface)
GEOMETRY HW 4 CLAY SHONKWILER 3.3.5 Consider the parametrized surface (Enneper s surface φ(u, v (x u3 3 + uv2, v v3 3 + vu2, u 2 v 2 show that (a The coefficients of the first fundamental form are E G
More informationTHE DIFFERENTIAL GEOMETRY OF PARAMETRIC PRIMITIVES
THE DIFFERENTIAL GEOMETRY OF PARAMETRIC PRIMITIVES Ken Turkowski Media Technologies: Graphics Software Advanced Technology Group Apple Computer, Inc. (Draft Friday, May 18, 1990) Abstract: We derive the
More information(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.
MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1)
QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1) Each of the six questions is worth 10 points. 1) Let H be a (real or complex) Hilbert space. We say
More informationIntensity Transformations and Spatial Filtering: WHICH ONE LOOKS BETTER? Intensity Transformations and Spatial Filtering: WHICH ONE LOOKS BETTER?
: WHICH ONE LOOKS BETTER? 3.1 : WHICH ONE LOOKS BETTER? 3.2 1 Goal: Image enhancement seeks to improve the visual appearance of an image, or convert it to a form suited for analysis by a human or a machine.
More informationMAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.
MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant
More informationKernel-based density. Nuno Vasconcelos ECE Department, UCSD
Kernel-based density estimation Nuno Vasconcelos ECE Department, UCSD Announcement last week of classes we will have Cheetah Day (exact day TBA) what: 4 teams of 6 people each team will write a report
More informationEdge Detection. Introduction to Computer Vision. Useful Mathematics Funcs. The bad news
Edge Detection Introduction to Computer Vision CS / ECE 8B Thursday, April, 004 Edge detection (HO #5) Edge detection is a local area operator that seeks to find significant, meaningful changes in image
More informationComputer Vision & Digital Image Processing. Periodicity of the Fourier transform
Computer Vision & Digital Image Processing Fourier Transform Properties, the Laplacian, Convolution and Correlation Dr. D. J. Jackson Lecture 9- Periodicity of the Fourier transform The discrete Fourier
More informationReview Smoothing Spatial Filters Sharpening Spatial Filters. Spatial Filtering. Dr. Praveen Sankaran. Department of ECE NIT Calicut.
Spatial Filtering Dr. Praveen Sankaran Department of ECE NIT Calicut January 7, 203 Outline 2 Linear Nonlinear 3 Spatial Domain Refers to the image plane itself. Direct manipulation of image pixels. Figure:
More informationEfficient Inference in Fully Connected CRFs with Gaussian Edge Potentials
Efficient Inference in Fully Connected CRFs with Gaussian Edge Potentials by Phillip Krahenbuhl and Vladlen Koltun Presented by Adam Stambler Multi-class image segmentation Assign a class label to each
More informationLois de conservations scalaires hyperboliques stochastiques : existence, unicité et approximation numérique de la solution entropique
Lois de conservations scalaires hyperboliques stochastiques : existence, unicité et approximation numérique de la solution entropique Université Aix-Marseille Travail en collaboration avec C.Bauzet, V.Castel
More informationThe Yamabe invariant and surgery
The Yamabe invariant and surgery B. Ammann 1 M. Dahl 2 E. Humbert 3 1 Universität Regensburg Germany 2 Kungliga Tekniska Högskolan, Stockholm Sweden 3 Université François-Rabelais, Tours France Geometric
More informationParametric Curves. Calculus 2 Lia Vas
Calculus Lia Vas Parametric Curves In the past, we mostly worked with curves in the form y = f(x). However, this format does not encompass all the curves one encounters in applications. For example, consider
More informationPart IB Geometry. Theorems. Based on lectures by A. G. Kovalev Notes taken by Dexter Chua. Lent 2016
Part IB Geometry Theorems Based on lectures by A. G. Kovalev Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationDistance between multinomial and multivariate normal models
Chapter 9 Distance between multinomial and multivariate normal models SECTION 1 introduces Andrew Carter s recursive procedure for bounding the Le Cam distance between a multinomialmodeland its approximating
More informationIMAGE ENHANCEMENT II (CONVOLUTION)
MOTIVATION Recorded images often exhibit problems such as: blurry noisy Image enhancement aims to improve visual quality Cosmetic processing Usually empirical techniques, with ad hoc parameters ( whatever
More informationFast Local Laplacian Filters: Theory and Applications
Fast Local Laplacian Filters: Theory and Applications Mathieu Aubry (INRIA, ENPC), Sylvain Paris (Adobe), Sam Hasinoff (Google), Jan Kautz (UCL), and Frédo Durand (MIT) Input Unsharp Mask, not edge-aware
More informationA compactness theorem for Yamabe metrics
A compactness theorem for Yamabe metrics Heather acbeth November 6, 2012 A well-known corollary of Aubin s work on the Yamabe problem [Aub76a] is the fact that, in a conformal class other than the conformal
More informationMath 233. Practice Problems Chapter 15. i j k
Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed
More informationACIS ( , ) Total e e e e-10 4.
1 SUMMARY 1 G21.5-0.9 1 Summary Distance: 5 kpc ( Safi-Harb et al., 2001 ) Position of Central Source (J2000): ( 18 33 34.0, -10 34 15.3 ) X-ray size: 4.7 x4.6 Description: 1.1 Summary of Chandra Observations
More informationCS 534: Computer Vision Segmentation III Statistical Nonparametric Methods for Segmentation
CS 534: Computer Vision Segmentation III Statistical Nonparametric Methods for Segmentation Ahmed Elgammal Dept of Computer Science CS 534 Segmentation III- Nonparametric Methods - - 1 Outlines Density
More informationPattern Recognition and Machine Learning. Bishop Chapter 2: Probability Distributions
Pattern Recognition and Machine Learning Chapter 2: Probability Distributions Cécile Amblard Alex Kläser Jakob Verbeek October 11, 27 Probability Distributions: General Density Estimation: given a finite
More informationMorphological image processing
INF 4300 Digital Image Analysis Morphological image processing Fritz Albregtsen 09.11.2017 1 Today Gonzalez and Woods, Chapter 9 Except sections 9.5.7 (skeletons), 9.5.8 (pruning), 9.5.9 (reconstruction)
More informationReview Sheet for the Final
Review Sheet for the Final Math 6-4 4 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence
More informationPacking, Curvature, and Tangling
Packing, Curvature, and Tangling Osaka City University February 28, 2006 Gregory Buck and Jonathan Simon Department of Mathematics, St. Anselm College, Manchester, NH. Research supported by NSF Grant #DMS007747
More informationTopics. CS Advanced Computer Graphics. Differential Geometry Basics. James F. O Brien. Vector and Tensor Fields.
CS 94-3 Advanced Computer Graphics Differential Geometry Basics James F. O Brien Associate Professor U.C. Berkeley Topics Vector and Tensor Fields Divergence, curl, etc. Parametric Curves Tangents, curvature,
More informationk th -order Voronoi Diagrams
k th -order Voronoi Diagrams References: D.-T. Lee, On k-nearest neighbor Voronoi Diagrams in the plane, IEEE Transactions on Computers, Vol. 31, No. 6, pp. 478 487, 1982. B. Chazelle and H. Edelsbrunner,
More informationImage as a signal. Luc Brun. January 25, 2018
Image as a signal Luc Brun January 25, 2018 Introduction Smoothing Edge detection Fourier Transform 2 / 36 Different way to see an image A stochastic process, A random vector (I [0, 0], I [0, 1],..., I
More informationarxiv: v1 [math.dg] 30 Nov 2013
An Explicit Formula for the Spherical Curves with Constant Torsion arxiv:131.0140v1 [math.dg] 30 Nov 013 Demetre Kazaras University of Oregon 1 Introduction Ivan Sterling St. Mary s College of Maryland
More informationCoordinate Finite Type Rotational Surfaces in Euclidean Spaces
Filomat 28:10 (2014), 2131 2140 DOI 10.2298/FIL1410131B Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Coordinate Finite Type
More informationGaussian derivatives
Gaussian derivatives UCU Winter School 2017 James Pritts Czech Tecnical University January 16, 2017 1 Images taken from Noah Snavely s and Robert Collins s course notes Definition An image (grayscale)
More informationQuasi-invariant measures on the path space of a diffusion
Quasi-invariant measures on the path space of a diffusion Denis Bell 1 Department of Mathematics, University of North Florida, 4567 St. Johns Bluff Road South, Jacksonville, FL 32224, U. S. A. email: dbell@unf.edu,
More informationErkut Erdem. Hacettepe University February 24 th, Linear Diffusion 1. 2 Appendix - The Calculus of Variations 5.
LINEAR DIFFUSION Erkut Erdem Hacettepe University February 24 th, 2012 CONTENTS 1 Linear Diffusion 1 2 Appendix - The Calculus of Variations 5 References 6 1 LINEAR DIFFUSION The linear diffusion (heat)
More informationDigitally Continuous Multivalued Functions
Digitally Continuous Multivalued Functions Carmen Escribano, Antonio Giraldo, and María Asunción Sastre Departamento de Matemática Aplicada, Facultad de Informática Universidad Politécnica, Campus de Montegancedo
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More informationGaussian Processes. Le Song. Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012
Gaussian Processes Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 01 Pictorial view of embedding distribution Transform the entire distribution to expected features Feature space Feature
More information2016 EF Exam Texas A&M High School Students Contest Solutions October 22, 2016
6 EF Exam Texas A&M High School Students Contest Solutions October, 6. Assume that p and q are real numbers such that the polynomial x + is divisible by x + px + q. Find q. p Answer Solution (without knowledge
More informationLecture 7: Edge Detection
#1 Lecture 7: Edge Detection Saad J Bedros sbedros@umn.edu Review From Last Lecture Definition of an Edge First Order Derivative Approximation as Edge Detector #2 This Lecture Examples of Edge Detection
More informationMcGill University April 16, Advanced Calculus for Engineers
McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer
More informationMath S1201 Calculus 3 Chapters , 14.1
Math S1201 Calculus 3 Chapters 13.2 13.4, 14.1 Summer 2015 Instructor: Ilia Vovsha h@p://www.cs.columbia.edu/~vovsha/calc3 1 Outline CH 13.2 DerivaIves of Vector FuncIons Extension of definiion Tangent
More informationApril 13, We now extend the structure of the horseshoe to more general kinds of invariant. x (v) λ n v.
April 3, 005 - Hyperbolic Sets We now extend the structure of the horseshoe to more general kinds of invariant sets. Let r, and let f D r (M) where M is a Riemannian manifold. A compact f invariant set
More informationPart IB GEOMETRY (Lent 2016): Example Sheet 1
Part IB GEOMETRY (Lent 2016): Example Sheet 1 (a.g.kovalev@dpmms.cam.ac.uk) 1. Suppose that H is a hyperplane in Euclidean n-space R n defined by u x = c for some unit vector u and constant c. The reflection
More informationHyperbolic Geometry on Geometric Surfaces
Mathematics Seminar, 15 September 2010 Outline Introduction Hyperbolic geometry Abstract surfaces The hemisphere model as a geometric surface The Poincaré disk model as a geometric surface Conclusion Introduction
More informationSELECTED SAMPLE FINAL EXAM SOLUTIONS - MATH 5378, SPRING 2013
SELECTED SAMPLE FINAL EXAM SOLUTIONS - MATH 5378, SPRING 03 Problem (). This problem is perhaps too hard for an actual exam, but very instructional, and simpler problems using these ideas will be on the
More informationPractice Final Solutions
Practice Final Solutions Math 1, Fall 17 Problem 1. Find a parameterization for the given curve, including bounds on the parameter t. Part a) The ellipse in R whose major axis has endpoints, ) and 6, )
More informationChapter 3 Salient Feature Inference
Chapter 3 Salient Feature Inference he building block of our computational framework for inferring salient structure is the procedure that simultaneously interpolates smooth curves, or surfaces, or region
More informationUSAC Colloquium. Geometry of Bending Surfaces. Andrejs Treibergs. Wednesday, November 6, Figure: Bender. University of Utah
USAC Colloquium Geometry of Bending Surfaces Andrejs Treibergs University of Utah Wednesday, November 6, 2012 Figure: Bender 2. USAC Lecture: Geometry of Bending Surfaces The URL for these Beamer Slides:
More informationPart II : Connection
Indian Statistical Institute System Science and Informatics Unit Bengalore, India Bengalore, 19-22 October 2010 ESIEE University of Paris-Est France Part II : Connection - Set case - Function case Jean
More informationDierential Geometry Curves and surfaces Local properties Geometric foundations (critical for visual modeling and computing) Quantitative analysis Algo
Dierential Geometry Curves and surfaces Local properties Geometric foundations (critical for visual modeling and computing) Quantitative analysis Algorithm development Shape control and interrogation Curves
More informationThe Convolution of a Paraboloid and a Parametrized Surface
Journal for Geometry and Graphics Volume 7 (2003), No. 2, 57 7. The Convolution of a Paraboloid and a Parametrized Surface Martin Peternell, Friedrich Manhart Institute of Geometry, Vienna University of
More informationBiharmonic tori in Euclidean spheres
Biharmonic tori in Euclidean spheres Cezar Oniciuc Alexandru Ioan Cuza University of Iaşi Geometry Day 2017 University of Patras June 10 Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras,
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More informationSatellite image deconvolution using complex wavelet packets
Satellite image deconvolution using complex wavelet packets André Jalobeanu, Laure Blanc-Féraud, Josiane Zerubia ARIANA research group INRIA Sophia Antipolis, France CNRS / INRIA / UNSA www.inria.fr/ariana
More informationarxiv: v2 [gr-qc] 9 Sep 2010
arxiv:1006.2933v2 [gr-qc] 9 Sep 2010 Global geometry of T 2 symmetric spacetimes with weak regularity Philippe G. LeFloch a and Jacques Smulevici b a Laboratoire Jacques-Louis Lions & Centre National de
More informationMATH H53 : Final exam
MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out
More informationCreated by T. Madas SURFACE INTEGRALS. Created by T. Madas
SURFACE INTEGRALS Question 1 Find the area of the plane with equation x + 3y + 6z = 60, 0 x 4, 0 y 6. 8 Question A surface has Cartesian equation y z x + + = 1. 4 5 Determine the area of the surface which
More informationChapter 16. Local Operations
Chapter 16 Local Operations g[x, y] =O{f[x ± x, y ± y]} In many common image processing operations, the output pixel is a weighted combination of the gray values of pixels in the neighborhood of the input
More informationSolutions: Problem Set 4 Math 201B, Winter 2007
Solutions: Problem Set 4 Math 2B, Winter 27 Problem. (a Define f : by { x /2 if < x
More informationImage Reconstruction And Poisson s equation
Chapter 1, p. 1/58 Image Reconstruction And Poisson s equation School of Engineering Sciences Parallel s for Large-Scale Problems I Chapter 1, p. 2/58 Outline 1 2 3 4 Chapter 1, p. 3/58 Question What have
More informationMath Exam IV - Fall 2011
Math 233 - Exam IV - Fall 2011 December 15, 2011 - Renato Feres NAME: STUDENT ID NUMBER: General instructions: This exam has 16 questions, each worth the same amount. Check that no pages are missing and
More informationFraunhofer Institute for Computer Graphics Research Interactive Graphics Systems Group, TU Darmstadt Fraunhoferstrasse 5, Darmstadt, Germany
Scale Space and PDE methods in image analysis and processing Arjan Kuijper Fraunhofer Institute for Computer Graphics Research Interactive Graphics Systems Group, TU Darmstadt Fraunhoferstrasse 5, 64283
More informationPractice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009.
Practice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009. Solutions (1) Let Γ be a discrete group acting on a manifold M. (a) Define what it means for Γ to act freely. Solution: Γ acts
More informationChaotic Modeling and Simulation (CMSIM) : , Geodesics Revisited. Pavel Pokorny
Chaotic Modeling and Simulation (CMSIM) : 28 298, 22 Geodesics Revisited Pavel Pokorny Prague Institute of Chemical Technology, Prague, Czech Republic (E-mail: pavel.pokorny@vscht.cz) Abstract. Metric
More informationWelcome to Copenhagen!
Welcome to Copenhagen! Schedule: Monday Tuesday Wednesday Thursday Friday 8 Registration and welcome 9 Crash course on Crash course on Introduction to Differential and Differential and Information Geometry
More informationProperties of the Pseudosphere
Properties of the Pseudosphere Simon Rubinstein Salzedo May 25, 2004 Euclidean, Hyperbolic, and Elliptic Geometries Euclid began his study of geometry with five axioms. They are. Exactly one line may drawn
More informationThe Frequency Domain. Many slides borrowed from Steve Seitz
The Frequency Domain Many slides borrowed from Steve Seitz Somewhere in Cinque Terre, May 2005 15-463: Computational Photography Alexei Efros, CMU, Spring 2010 Salvador Dali Gala Contemplating the Mediterranean
More information