UMD CMSC 828 - Invariants (concluded); Lowe and Biederman

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Invariants (concluded); Lowe and BiedermanAnnouncementsKey PointsProjectiveLines: ParameterizationLine IntersectionIntersection of Parallel LinesInvariants of LinesInvariance in 3D to 2DNon-Invariance in 3D to 2DProof StrategyConstructing O1 … Ok-1SummaryLowe and BiedermanBackgroundViewpoint Invariant NAPsExamplesIssues with Non-Accidental PropertiesViewpoint InvarianceGeonsGeons for RecognitionEmpirical Support for GeonsEmpirical SupportEmpirical Support (2)Empirical Support (3): Degraded ObjectsPowerPoint PresentationSlide 27ConclusionsInvariants (concluded); Lowe and BiedermanAnnouncements•No class Thursday. Attend Rao lecture.•Double-check your paper assignments.Key Points•Rigid rotation is 3x3 orthonormal matrix.•3-D Translation is 3x4 matrix.•3-D Translation + Rotation is 3x4 matrix.•Scaled Orthographic Projection: Remove row three and allow scaling.•Planar Object, remove column 3.•Projective Transformations–Rigid Rotation of Planar Object Represented by 3x3 matrix.–When we write in homogeneous coordinates, projection implicit.–When we drop rigidity, 3x3 matrix is arbitrary.Projective1102,31,32,21,22,11,13,32,31,33,22,21,23,12,11,1yxtrrtrrtrryxtrrrtrrrtrrrzyxzyxRigid rotation and translation.Notation suggests that first two columns are orthonormal, and transformation has 6 degrees of freedom.11111wvwuwvuyxhgfedcbaProjective TransformationNotation suggests that transformation is unconstrained linear transformation. Points in homogenous coordinates are equivalent. Transformation has 8 degrees of freedom, because its scale is arbitrary.Lines: Parameterization•Equation for line: ax+by+c=0.•Parameterize line as l = (a,b,c)T.•p=(x,y,1)T is on line if <p,l>=0.Line Intersection•The intersection of l and l’ is l x l’ (where x denotes the cross product).•This follows from the fact that the cross product is orthogonal to both lines.Intersection of Parallel Lines•Suppose l and l’ are parallel. We can write l=(a,b,c), l’ = (a,b,c’). l x l’ = (c’-c)(b,-a,0). This equivalent to (b,-a,0).•This point corresponds to a line through the focal point that doesn’t intersect the image plane.•We can think of the real plane as points (a,b,c) where c isn’t equal to 0. When c = 0, we say these points lie on the ideal line at infinity.•Note that a projective transformation can map this to another line, the horizon, which we see.Invariants of Lines•Notice that affine transformations are the subgroup of projective transformations in which the last row is (0, 0, 1).•These map the line at infinity to itself.•So parallel lines are affine invariants, since they continue to intersect at infinity.Invariance in 3D to 2D•3D to 2D “Invariance” isn’t captured by mathematical definition of invariance because 3D to 2D transformations don’t form a group.–You can’t compose or invert them.•Definition: Let f be a function on images. We say f is an invariant iff for every Object O, if I1 and I2 are images of O, f(I1)=f(I2).•This means we can define f(O) as f(I) for I any image of O. O and I match only if f(O)=f(I).•f is a non-trivial invariant if there exist two image I1 and I2 such that f(I1)~=f(I2).Non-Invariance in 3D to 2D•Theorem: Assume valid objects are any 3D point sets of size k, for some k. Then there are no non-trivial invariants of the images of these objects under perspective projection.Proof Strategy•Let f be an invariant.•Suppose two objects, A and B have a common image. Then f(I)=f(J) if I and J are images of either A or B.•Given any O0, Ok, we construct a series of objects, O1, …, O(k-1), so that Oi and O(i+1) have a common image for all i, and Ok and j have a common image.•So for any pair of images, I, J, from any two objects, f(I) = f(J).Constructing O1 … Ok-1•Oi has its first i points identical to the first i points of Ok, and the remaining points identical to the remaining points of O0.•If two objects are identical except for one point, they produce the same image when viewed along a line joining those two points.–Along that line, those two points look the same.–The remaining points always look the same.Summary•Planar objects give rise to rich set of invariants.•3-D objects have no invariants.–We can deal with this by focusing on planar portions of objects.–Or special restricted classes of objects.–Or by relaxing notion of invariants.•However, invariants have become less popular in computer vision due to these limitations.Lowe and Biederman•Background•Viewpoint Invariant Non-Accidental Properties.–Lowe sees these as probabilistic.–Biederman drops this.–Primitive properties–Composing them into units/geons.•Use in Recognition.–Speed search.–Geons: analogy to speech. •Evidence for Value.–Computational speed.–Human psychology: parts; qualitative descriptions; view invariance.Background•Computational–2D approach to recognition.•Lowe is reacting to Marr.•Partly due to Lowe, recognition rarely involves reconstruction now. (But also 3D models more rare).–State of the art: –Little recognition of 3D objects, grouping implicit.–Speed, robustness a big concern.–2D recognition through search.•Psychology–Much more ambitious and specific than any prior theory of recognition (I believe).–P.O. widely studied, rarely related to other tasks.•Contrast.–CS must account for low-level processing.–Psych must account for categorization.Viewpoint Invariant NAPs•Non-Accidental Property–Happens rarely by chance–More frequently by scene structure.–p = property, c = chance, s = structure.)|()|()()|()()()|()|(cpPspPsPspPpPsPspPpsPLowe focuses on thisJepson and Richards consider this• Biederman downplays probabilistic inference.•Not concerned with background, feature detection.This is high due to viewpoint invariance.Examples(Copied from Lowe)Issues with Non-Accidental Properties•Is it “just” Bayesian inference?–Then why not model all information? •This may fit Lowe•Biederman relies more on certain inference.•See also Feldman, Jepson, Richards.Viewpoint Invariance•Match properties that are invariant to viewing conditions. –Parallelism, symmetry,


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