OSU CS 559 - Distribution Tracking By Active Contours , Motivations and Generalizations

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Distribution Tracking By Active Contours Distribution Tracking By Active Contours, Motivations and GeneralizationsAli TorkamaniOutlineOutliney Motivationy Calculus of Variationsy Statistical Distancey Early works on Snakes and Active Contour Modelsy Active Contour For Distribution Trackingy Generalizations of Active Contour Modelsy ReferencesOutlineOutliney Motivationy Calculus of Variationsy Statistical Distancey Early works on Snakes and Active Contour Modelsy Active Contour For Distribution Trackingy Generalizations of Active Contour Modelsy ReferencesMotivationMotivationy Back to boundary detectiony This time using perceptual grouping.y Non-parametricW’ l ki f f ifi hyWe’re not looking for a contour of a specific shape.y Just a good contour.Sometimes edge detectors find the boundary pretty And Sometimes edge detectors are not that well.good.Canny: Threshold=0.3Canny: Threshold=0.7Canny: Threshold=0.05Canny: Threshold=0.1SobelCanny: Threshold=0.3Canny: Threshold=0.7Canny: Threshold=0.05Canny: Threshold=0.1SobelImproving Boundary DetectionImproving Boundary Detectiony Integrate information over distance.y Use Gestalt cuesy SmoothnessClyClosurey Humans integrate contour informationProblem: Finding The Right PathProblem: Finding The Right PathCalculus of variationsCalculus of variationsy Motivationy Calculus of Variationsy Statistical Distancey Early works on Snakes and Active Contour Modelsy Active Contour For Distribution Trackingy Generalizations of Active Contour Modelsy ReferencesCalculus of VariationsCalculus of Variationsy Functional Analysisy Finding the function y=y(x) that Maximizes :dft∫′2)(dxxyyft∫′1),,(Calculus of VariationsCalculus of Variationsy Functional AnalysisFd h f () h M yFinding the function y=y(x) that Maximizes :dft∫′2)(dxxyyft∫′1),,(y can be any any function of y, y’ and x, e.g.:),,( xyyf′)()()()(),,(42xxyxyxyxyxyyf+′+−′=′y but y is an explicit function of x, like )(),,(xxyyyfxxxy+=)sin()(ExampleExampley What’s the function of the shortest path between two point’s l? plane? y Answer: a straight line!y find y such that Minimizes the arc length:2*)(1minarg2dxxyyty′+=∫2)(1)(),,(1xyyfxyyfty′+=′=′EulerLagrange EquationEuler-Lagrange Equation),,(][2′=∫dxxyyfyIt),,(][1∂∂∫fdfdxxyyfyIt0=′∂∂−∂∂yfdxdyfStraight LineStraight Linedxxyyt′+=∫)(1minarg2*2fdfxyyfxyyfyyty∂∂′+=′=′∫0)(1)(),,()(g21yfydxy=∂∂=′∂−∂00xyxydfdxyxyyf=′′=′=∂′+′=′∂∂0)()()(1)(2baxxyxyxyxyxydxydx+==′′=′+′+=′+=′∂)(0)(0)(1))(1()(1222y)(Calculus of variationsCalculus of variationsy Finding the optimal functional:y By Green’s Theorem:Calculus of variationsCalculus of variationsy Euler-LagrangeSt ti ti l Di tStatistical Distancey Motivationy Calculus of Variationsy Statistical Distancey Early works on Snakes and Active Contour Modelsy Active Contour For Distribution Trackingy Generalizations of Active Contour Modelsy ReferencesStatistical MeasuresStatistical Measuresy Kullback-Leibler Measurey Bhattacharyya CoefficientEarly works on Snakes and active tcontoursy Motivationy Calculus of Variationsy Statistical Distancey Early works on Snakes and Active Contour Modelsy Active Contour For Distribution Trackingy Generalizations of Active Contour Modelsy ReferencesSnakesSnakesy A very flexible non-parametric contour modely Desired Curve=Optimal Path, n points: p(1),…,p(n)p,pp(),,p()y Cost Functiony Smoothness, Discrete Curvature: if you go from p(j-1) to p(j) to p(j+1) how much does direction change?yChange of direction of gradient from p(j) to p(j+1)Change of direction of gradient from p(j) to p(j+1))],()([*),(11,...,1+++=∑iijiinppppfpgppdEnβα• d(p(j),p(j+1)) is distance between consecutive path points 1 or sqrt(2).•( (j)) t th f di t 1=i•g(p(j)) measures strength of gradient • f measures smoothness, curvature• Finally α and β are parameters [1]SnakesSnakesn)],()([*),(111,...,1++=+=∑iijiinippppfpgppdEnβα10max1)(2<<∇∇=+=ρρiiiiiiIIllpg“Dynamic Programming for Detecting, Tracking, and Matching Deformable Contours”, by Geiger, Gupta, Costa, and Vlontzos, IEEE Trans. PAMI 17(3)294-302 1995 ) 302, 1995.)Snakes: More General ApproachSnakes: More General Approachy Posistion of the snake:))(),(()( sysxsv=y Energy Functional:Snakes: Cont’dSnakes: Cont’d…y Internal Energy:y First and Second Order Derivatives: Tangent and Curvaturey α(s) and β(s): Weights Controlling Importance, Continuity y Image Energy:Snakes: Image EnergySnakes: Image Energyy Line Functional: Image Intensity itselfy Dependence on w_liney Attracted to lighter or darker borderyEd EyEdge EnergyyOr MarrHildrethEdge Detector Based Edge EnergyyOr Marr-HildrethEdge Detector Based Edge Energy:Snakes: Termination FunctionalSnakes: Termination Functionaly In Order to find Termination of line segments and Cornersy Smoothing Image by a Gaussian Filter:y Computing The Gradient Angle:y Curvature in Contours of Smoothed Image:Ati C t F Ditibti T kigActive Contour For Distribution Trackingy Motivationy Calculus of Variationsy Statistical Distancey Early works on Snakes and Active Contour Modelsy Active Contour For Distribution Trackingy Generalizations of Active Contour Modelsy ReferencesAti C t Ditibti T kigActive Contour, Distribution Trackingy Photometric Variablesy Colory TextureyIntensityyIntensityy Cumulative Density Funstion inside a region:∫∫∫−=ωωθωdxdxxZzzF))(()|(y θ is the heaviside function, z is the photometric variable, Z(x) is the video frame.Probability Density FunctionProbability Density Functiony PDFy q(z): The distribution of interesty KL DistanceBh h MyBhattacharyya MeasureKL FlowKL Flowy The goal is to minimizey Gradient Descent:y By substituting:KL Flow: Cont’dKL Flow: Cont’dy Finally:y Similarly For Bhattacharyya:Some ResultsSome Resultsy Tracking by KL Flow:ComparisonComparisony Geodesic Active Region (Kimmel)Generalizations of Active Contour MdlModelsy Motivationy Early works on Snakes and Active Contour Modelsy Calculus of Variationsy Statistical Distancey Active Contour For Distribution Trackingy Generalizations of Active Contour Modelsy ReferencesOther GeneralizationsOther Generalizationsy Background Mismatch (Freedman)y Geometric Active Contours(Yezzi)y Active Contour with Occlusion Handling (Yilmaz)y Very Logically simple occlusion modelsy Highly discriminate color distribution (Black and white)yNo dense ScenarioyNo dense


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OSU CS 559 - Distribution Tracking By Active Contours , Motivations and Generalizations

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