Final Exam Study Guide CSE/EE 486 Fall 2007Lecture 2 Intensity Sufaces and GradientsImage visualized as surface. Terrain concepts.Gradient of functions in 1D and 2DNumerical derivatives. Taylor series. Finite differences.Image gradients.Functions of gradients: Magnitude. Orientation.!Lecture 3 Linear OperatorsLinear filters.Convolution vs Cross CorrelationHow to perform convolution. Border handling issues.Properties of convolution (e.g. commutative, associative, linear)Finite difference filters!Lecture 4 SmoothingGaussian image noise modelHow smoothing reduces noiseWhen is smoothing good? When is it bad?Box filter (Averaging filter)Gaussian smoothing filterSeparability. Cascading.Formula for sigma of cascaded Gaussians.!Lecture 5 Gradients and Edge DetectionSmoothing and differentiation (combine filters)Prewitt filter. Sobel Filter.Derivative of Gaussian filter.Edges are intensity discontinuities.Types of edges: step, ramp, ridgeRelationship between step/ramp edge and first derivativeFinding edges by thresholding on gradient magnitudeCanny criteria: good detection, good localization, low false positivesOptimal filter well approximated by deriv of Gaussian filterNon-maximum suppression for edge thinningHysteresis Thresholding definition!Lecture 6 Harris Corner DetectorsCorners have good localization in all directionsVisualization of small window shifting over uniform region, edge, and cornerHarris mathematics: sum of squared differences of shifted regionFirst order approx (Taylor series for 2D functions)Harris detector second moment matrixClassification via eigenvaluesCorner response measure RLecture 7 Correspondence MatchingThe Correspondence Problem.Dense correspondence vs sparse correspondenceCorrelation-based methodsSSD (sum of squared differences)Relation between SSD and correlationNCC (Normalized Cross Correlation)NCC response range is -1 to 1!Lecture 8 Introduction to StereoCan recover depth from two images (and why you can’t from just one)ParallaxAnaglyphs and red/cyan glasses. Basic idea of why it works.Julesz random dot experiment. What it is. What it proves.Geometry of a simple stereo system (right camera displaced along X axis)Stereo disparity. Equation relating depth, disparity, baseline and focal length.Meaning of epipolar constraint and epipolar line.Why are matches constrained to lie along an epipolar line?!Lecture 9 Stereo AlgorithmsConcept of Disparity space image (DSI)Lowest cost path through the DSIScanline consistency: Ordering constraint. Right and left occlusions.Cox and Hingorani solution to interscanline matching Dynamic programming Path can proceed in three directions (match, left occlude,right occlude) Costs for each choice of direction to proceed!Lecture 10 Image PyramidsCascaded Gaussians revisitedMultiresolution Pyramid Data StructureBasic operations: smooth, downsample, upsampleApplication: making thumbnail images for web pages subsampling leads to high-frequency artifacts (aliasing) smoothing before subsampling prevents aliasingScale space: different scales emphasize features of different sizes!Lecture 11 LoG Blob DetectionNumerical second derivatives (Taylor series)Finite difference Laplacian filterSecond derivative of Gaussian;Laplacian of Gaussian (LoG) filterFinding edges as Zero-crossings of LoG (or DoG)Approx LoG by difference of Gaussian (DoG) Leads to efficient implementation due to Gaussian separability, cascadabilityApplications of LoG: blob finding; compression.Lecture 12 Camera Projection (Extrinsics)Forward Projection ModelRelation between World Coords ; camera cords; image (film) cords; pixel coordsHomogeneous coordinatesPerspective ProjectionMatrix equations relating world to camera coordinate systemsExtrinsic Parameters Camera Offset Camera Rotation Relation between offset/rotation and rotation/translationLecture 13 Camera Projection (Intrinsics)Intrinsic Parameters relating film coordinates and pixel coordinates Focal length, scalex, scaley, offsetx, offsety Representation as a matrixGeometric Image mappings. Linear mappings can be written as matrices 2D planar transformations Translation, scale, rotation Euclidean, similarity, affine, and projective mappings Effects of each type of mapping What properties do they preserve (e.g. orientation, length, angles, parallelism, …)Lecture 14 Parameter EstimationFitting parameterized modelsDetermining how many point correspondences neededLeast squares line fittingBe able to solve simple least squares problems (sum of square error; take derivatives and set to zero; form matrix equation to solve)Algebraic distance vs orthogonal distanceImage Warping Forwards vs Backward Warping Bilinear InterpolationLecture 15 Robust Estimation and RANSACconcept of inliers vs outliersGeneral idea behind RANSAC procedure Sample minimal set of points; fit entity; count inliers supporting that entity; repeatLecture 16 Planar HomographiesHomographies Nonlinear transformation of 2D coords Linear transformation if you use homogeneous coordsProjection equations for points on a planar surface Frontal plane, calibrated camera: Similarity transformation Frontal plane, uncalibrated camera: Affine transformation Arbitrary plane, calibrated camera: Homography Arbitrary plane, uncalibrated camera: HomographyLecture 17 Stabilization and MosaicingVideo Stabilization Match all sequence frames to one reference frame (all must have some overlap) Frame to frame chaining of transformations (only need pairwise overlap)Ghosting (out of plane pixels don’t map correctly due to parallax)Images from rotating camera Images are related by homography, regardless of scene structure!!!!Panoramic mosaics (and Quicktime VR)Intensity/Color Blending: averaging; featheringLecture 18 Generalized Stereo: Epipolar GeometryEpipolar Geometry Concepts Epipoles Epipolar Lines Conjugate Epipolar linesLecture 19 The Essential/Fundamental MatrixBasic math behind the Essential matrix E Plane formed by camera centers and 3D scene point That plane intersects image planes in conjugate epipolar linesLonguet-Higgens equationFundamental Matrix F vs Essential matrix E Essential matrix – cameras calibrated Fundamental matrix – cameras can be uncalibratedHow to use F (or E) to map points in one image to lines in anotherHow to map points from im1 into lines in im2How to map points from im2 into lines in im1Lecture 20 The Eight-Point AlgorithmHow to