Camera Calibration class 9Camera calibrationSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Gold Standard algorithmSlide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Next class: More Single-View GeometryCamera Calibrationclass 9Multiple View GeometryComp 290-089Marc PollefeysCamera calibrationiixX ? PResectioningBasic equationsiiPXx iiPXx0Ap Basic equations0Ap minimal solutionOver-determined solution 5½ correspondences needed (say 6) P has 11 dof, 2 independent eq./pointsn 6 pointsApminimize subject to constraint 1p 1pˆ33pˆPDegenerate configurationsMore complicate than 2D case (see Ch.21)(i) Camera and points on a twisted cubic(ii) Points lie on plane or single line passing through projection centerLess obvious(i) Simple, as before(ii) Anisotropic scalingData normalization32Line correspondencesExtend DLT to linesilPT0PXl1Tii(back-project line)0PXl2Tii(2 independent eq.)Geometric errorGold Standard algorithmObjectiveGiven n≥6 3D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood Estimation of PAlgorithm(i) Linear solution: (a) Normalization: (b) DLT: (ii) Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error:(iii) Denormalization:iiUXX~iiTxx~UP~TP-1~ ~~Calibration example(i) Canny edge detection(ii) Straight line fitting to the detected edges(iii) Intersecting the lines to obtain the images cornerstypically precision <1/10 (HZ rule of thumb: 5n constraints for n unknownsErrors in the worldErrors in the image and in the worldiiXPxiXGeometric interpretation of algebraic error 2)xˆ,x(ˆiiiidw iiiiyxw PX1,ˆ,ˆˆP)depth(X;pˆˆ3iw)Xˆ,X(~)xˆ,x(ˆiiiiifddw then1pˆ if therefore,3note invariance to 2D and 3D similarities given proper normalizationEstimation of affine cameranote that in this case algebraic error = geometric errorGold Standard algorithmObjectiveGiven n≥4 3D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood Estimation of P(remember P3T=(0,0,0,1))Algorithm(i) Normalization:(ii) For each correspondence(iii) solution is(iv) Denormalization:iiUXX~iiTxx~UP~TP-1bpA88bAp88Restricted camera estimationMinimize geometric error impose constraint through parametrizationImage only 9 2n, otherwise 3n+9 5nFind best fit that satisfies•skew s is zero•pixels are square •principal point is known•complete camera matrix K is knownMinimize algebraic error assume map from param q P=K[R|-RC], i.e. p=g(q)minimize ||Ag(q)||Reduced measurement matrixOne only has to work with 12x12 matrix, not 2nx12pA~ApApApTTRestricted camera estimationInitialization •Use general DLT•Clamp values to desired values, e.g. s=0, x= yNote: can sometimes cause big jump in errorAlternative initialization•Use general DLT•Impose soft constraints•gradually increase weightsExterior orientationCalibrated camera, position and orientation unkown Pose estimation6 dof 3 points minimal (4 solutions in general)Covariance estimationML residual errorExample: n=197, =0.365, =0.37Covariance for estimated cameraCompute Jacobian at ML solution, then JJ1xTP(variance per parameter can be found on diagonal)2χ(chi-square distribution =distribution of sum of squares)cumulative-1short and long focal lengthRadial distortionCorrection of distortionChoice of the distortion function and centerComputing the parameters of the distortion function(i) Minimize with additional unknowns(ii) Straighten lines(iii) …Next class: More Single-View Geometry•Projective cameras and planes, lines, conics and quadrics.•Camera calibration and vanishing points, calibrating conic and the IAC**CPPQ TconeQCPP
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