116.891Computer Vision and ApplicationsProf. Trevor. DarrellLecture 11: Model-based vision• Hypothesize and test• Interpretation Trees• Alignment• Pose Clustering• Geometric HashingReadings: F&P Ch 18.1-18.52Last timeProjective SFM– Projective spaces– Cross ratio– Factorization algorithm– Euclidean upgrade3Projective transformationsA projectivity is an invertible mapping h from P2to itself such that three points x1,x2,x3lie on the same line if and only if h(x1),h(x2),h(x3) do.Definition:A mapping h:P2→P2is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2reprented by a vector x it is true that h(x)=HxTheorem:Definition: Projective transformation=321333231232221131211321'''xxxhhhhhhhhhxxxxx' H=or8DOFprojectivity=collineation=projective transformation=homography[F&P, www.cs.unc.edu/~marc/mvg]4The value of this cross ratio is independent of the intersecting line or plane:[F&P]5Two-frame reconstruction(i) Compute F from correspondences(ii) Compute camera matrices from F(iii) Compute 3D point for each pair of corresponding pointscomputation of Fuse x‘iFxi=0 equations, linear in coeff. F8 points (linear), 7 points (non-linear), 8+ (least-squares)(more on this next class)computation of camera matricestriangulationcompute intersection of two backprojected raysPossible choice: ]e'|F][[e'P' 0]|[IP×==[www.cs.unc.edu/~marc/mvg]6Perspective factorizationAll equations can be collected for all i aswhere,with:PMm==ΛΛΛ=mnnPPPPmmmm...,...212211 m are known, but Λi,P and M are unknown…Observe that PM is a product of a 3mx4 matrix and a 4xnmatrix, i.e. it is a rank 4 matrix[www.cs.unc.edu/~marc/mvg][][ ]()imiiimimiiimmmλ,...,λ,λdiagM,...,M,M,,...,,212121=Λ==Mm mjmijiijij,...,1,,...,1,Mmλ === P27Iterative perspective factorizationWhen Λiare unknown the following algorithm can be used:1. Set λij=1 (affine approximation).2. Factorize PM and obtain an estimate of P and M. If s5is sufficiently small then STOP.3. Use m, P and M to estimate Λifrom the camera equations (linearly) miΛi=PiM4. Goto 2.In general the algorithm minimizes the proximity measureP(Λ,P,M)=s5Structure and motion recovered up to an arbitrary projective transformation[www.cs.unc.edu/~marc/mvg]8Given a camera with known intrinsic parameters, we can take the calibration matrix to be the identity and write the perspective projection equation in some Euclidean world coordinate system asfor any non-zero scale factor λ. If Miand Pjdenote the shape and motion parameters measured in some Euclidean coordinate system, there must exist a 4 ×4 matrix Q such thatEuclidean upgrade[F&P]9Today: “Model-based Vision”Still feature and geometry-based, but now with moving objects rather than cameras…Topics:– Hypothesize and test– Interpretation Trees– Alignment– Pose Clustering– Invariances– Geometric Hashing10Approach•Given– CAD Models (with features)– Detected features in an image• Hypothesize and test recognition…– Guess – Render – Compare11Hypothesize and Test Recognition• Hypothesize object identity and correspondence– Recover pose– Render object in camera– Compare to image• Issues– where do the hypotheses come from?– How do we compare to image (verification)?12Features?• Pointsbut also,•Lines• Conics• Other fitted curves• Regions (particularly the center of a region, etc.)313How to generate hypotheses?• Brute force– Construct a correspondence for all object features to every correctly sized subset of image points– Expensive search, which is also redundant.– L objects with N features– M features in image–O(LMN) !• Add geometric constraints to prune search, leading to interpretation tree search• Try subsets of features (frame groups)… 14Interpretation Trees• Tree of possible model-image feature assignments• Depth-first search• Prune when unary (binary, …) constraint violated– length–area– orientation(a,1)(b,2)……15Interpretation Trees[ A.M. Wallace. 1988. ]“Wild cards” handle spurious image features16Adding constraints• Correspondences between image features and model features are not independent.• A small number of good correspondences yields a reliable pose estimation --- the others must be consistent with this.• Generate hypotheses using small numbers of correspondences (e.g. triples of points for a calibrated perspective camera, etc., etc.)17Pose consistency / Alignment• Given known camera type in some unknown configuration (pose)– Hypothesize configuration from set of initial features– Backproject –Test• “Frame group” -- set of sufficient correspondences to estimate configuration, e.g.,– 3 points– intersection of 2 or 3 line segments, and 1 point18Alignment419 20Pose clustering• Each model leads to many correct sets of correspondences, each of which has the same pose• Vote on pose, in an accumulator array (per object)21PoseClustering2223Pose clusteringProblems– Clutter may lead to more votes than the target!– Difficult to pick the right bin sizeConfidence-weighted clustering– See where model frame group is reliable (visible!)– Downweight / discount votes from frame groups at poses where that frame group is unreliable…24pick feature pairdark regions show reliable views of those features525 2627 2829Detecting 0.1% inliers among 99.9% outliers?• Example: David Lowe’s SIFT-based Recognition system• Goal: recognize clusters of just 3 consistent features among 3000 feature match hypotheses• Approach– Vote for each potential match according to model ID and pose– Insert into multiple bins to allow for error in similarity approximation– Using a hash table instead of an array avoids need to form empty bins or predict array size[Lowe]30Lowe’s Model verification step• Examine all clusters with at least 3 features• Perform least-squares affine fit to model. • Discard outliers and perform top-down check for additional features.• Evaluate probability that match is correct– Use Bayesian model, with probability that features would arise by chance if object was not present– Takes account of object size in image, textured regions, model feature count in database, accuracy of fit (Lowe, CVPR 01)[Lowe]631Solution for affine parameters• Affine