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UW-Madison CS 766 - Lecture Note

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Last LectureTodayFor dynamic ScenesFor dynamic scenesMore and BlendingTaking picturesWarped ImagesCylindrical panoramaInvariant Local FeaturesMore motivation… Corner detectorThe Basic IdeaMoravec corner detectorMoravec corner detectorMoravec corner detectorMoravec corner detectorMoravec corner detectorProblems of Moravec detectorHarris corner detectorHarris corner detectorHarris corner detectorHarris corner detectorHarris corner detectorHarris corner detectorHarris corner detectorHarris DetectorHarris Detector: WorkflowHarris Detector: WorkflowHarris Detector: WorkflowHarris Detector: WorkflowHarris Detector: WorkflowHarris Detector: Some PropertiesHarris Detector: Some PropertiesHarris Detector: Some PropertiesScale Invariant DetectionScale Invariant DetectionFeature selectionAdaptive Non-maximal SuppressionFeature descriptorsDescriptors Invariant to RotationDescriptor VectorMOPS descriptor vectorDetections at multiple scalesMulti-Scale Oriented PatchesFeature matchingFeature matchingWhat about outliers?Feature-space outlier rejectionFeature-space outlier rejectionFeature-space outliner rejectionMatching featuresRAndom SAmple ConsensusRAndom SAmple ConsensusLeast squares fitRANSAC for estimating homographyRANSACRANSAC in generalRANSAC algorithmHow to determine kExample: line fittingExample: line fittingModel fittingMeasure distancesCount inliersAnother trialThe best modelApplicationsAutomatic image stitchingAutomatic image stitchingAutomatic image stitchingAutomatic image stitchingAutomatic image stitchingCorrespondence ResultsObject Recognition ResultsObject Recognition ResultsObject Classification ResultsGeometry Estimation ResultsObject Tracking ResultsRobotics: Sony AiboLast LectureCreating virtual wide-angle cameraTodayMosaic for Dynamic ScenesFeature MatchingFor dynamic ScenesPoint Grey Ladybug2http://www.ptgrey.com/products/ladybug2/samples.aspFor dynamic sceneshttp://www1.cs.columbia.edu/CAVE/projects/cat_cam_360/cat_cam_360.phpMore and BlendingTaking picturesKaidan panoramic tripod headWarped ImagesCylindrical panorama1. Take pictures on a tripod (or handheld)2. Warp to cylindrical coordinate3. Compute pairwise alignments4. Fix up the end-to-end alignment5. Blending6. Crop the result and import into a viewerInvariant Local Features•Image content is transformed into local feature coordinates that are invariant to translation, rotation, scale, and other imaging parametersFeatures DescriptorsMore motivation… • Feature points are used for:–Image alignment (homography, fundamental matrix)–3D reconstruction–Motion tracking–Object recognition–Indexing and database retrieval–Robot navigation–… otherCorner detector• C.Harris, M.Stephens. “A Combined Corner and Edge Detector”. 1988The Basic Idea• We should easily recognize the point by looking through a small window• Shifting a window in any direction should give a large change in intensityMoravec corner detectorflatMoravec corner detectorflatMoravec corner detectorflat edgeMoravec corner detectorflat edgecornerisolated pointMoravec corner detectorChange of intensity for the shift [u,v]:[]2,(,) (,) ( , ) (,)xyEuv wxyIx uyvIxy=++−∑IntensityShifted intensityWindow functionFour shifts: (u,v) = (1,0), (1,1), (0,1), (-1, 1)Look for local maxima in min{E}Problems of Moravec detector• Noisy response due to a binary window function• Only a set of shifts at every 45 degree is considered• Only minimum of E is taken into accountÖHarris corner detector (1988) solves these problems.Harris corner detectorNoisy response due to a binary window function¾ Use a Gaussian functionHarris corner detectorOnly a set of shifts at every 45 degree is considered¾ Consider all small shifts by Taylor’s expansion∑∑∑===++=yxyxyxyyxxyxIyxIyxwCyxIyxwByxIyxwABvCuvAuvuE,,2,222),(),(),(),(),(),(),(2),(Harris corner detector[](,) ,uEuv uv Mv⎡⎤≅⎢⎥⎣⎦Equivalently, for small shifts [u,v] we have a bilinearapproximation:22,(, )xxyxyxy yIIIMwxyIII⎡⎤=⎢⎥⎢⎥⎣⎦∑, where M is a 2×2 matrix computed from image derivatives:Harris corner detectorOnly minimum of E is taken into account¾A new corner measurementHarris corner detector[](,) ,uEuv uv Mv⎡⎤≅⎢⎥⎣⎦Intensity change in shifting window: eigenvalue analysisλ1, λ2 – eigenvalues of Mdirection of the slowest changedirection of the fastest change(λmax)-1/2(λmin)-1/2Ellipse E(u,v) = constHarris corner detectorλ1λ2Cornerλ1and λ2are large,λ1 ~ λ2;E increases in all directionsλ1and λ2are small;E is almost constant in all directionsedge λ1>> λ2edge λ2>> λ1flatClassification of image points using eigenvalues of M:Harris corner detectorMeasure of corner response:MMRTracedet2121=+=λλλλHarris Detector• The Algorithm:–Find points with large corner response function R (R > threshold)–Take the points of local maxima of RHarris Detector: WorkflowHarris Detector: WorkflowCompute corner response RHarris Detector: WorkflowFind points with large corner response: R>thresholdHarris Detector: WorkflowTake only the points of local maxima of RHarris Detector: WorkflowHarris Detector: Some Properties• Rotation invarianceEllipse rotates but its shape (i.e. eigenvalues) remains the sameCorner response R is invariant to image rotationHarris Detector: Some Properties• Partial invariance to affine intensity change9 Only derivatives are used => invariance to intensity shift I → I + b9 Intensity scale: I → a IRx(image coordinate)thresholdRx(image coordinate)Harris Detector: Some Properties• But: non-invariant to image scale!All points will be classified as edgesCorner !Scale Invariant Detection• Consider regions (e.g. circles) of different sizes around a point• Regions of corresponding sizes will look the same in both imagesScale Invariant Detection• The problem: how do we choose corresponding circles independently in each image?• Choose the scale of the “best” cornerFeature selection• Distribute points evenly over the imageAdaptive Non-maximal Suppression• Desired: Fixed # of features per image– Want evenly distributed spatially…– Sort ponts by non-maximal suppression radius[Brown, Szeliski, Winder, CVPR’05]Feature descriptors• We know how to detect points• Next question: How to match them??Point descriptor should be:1. Invariant 2. DistinctiveDescriptors Invariant to Rotation• Find local orientationDominant direction of gradient• Extract image patches relative to this


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UW-Madison CS 766 - Lecture Note

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