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- 1. Object Recognition with Pictorial Structures Pedro F. Felzenszwalb University of Chicago pﬀ@cs.uchicago.edu Joint work with Daniel P. Huttenlocher
- 2. Pictorial structuresPart-based representation: • Each part models local visual properties. • “Springs” model spatial relationships. • Joint estimation of part locations. – No hard detection of parts or features. – No initialization parameters. 1
- 3. • Model is represented by a graph G = (V, E). – V = {v1, . . . , vn} are the parts. – (vi, vj ) ∈ E indicates a connection between parts.• mi(li) is the cost of placing part i at location li.• dij (li, lj ) is a deformation cost.• Optimal location for object is given by L∗ = (l1, . . . , ln), ∗ ∗ n L∗ = argmin mi(li) + dij (li, lj ) L i=1 (vi,vj )∈E 2
- 4. Eﬃcient minimization n L∗ = argmin mi(li) + dij (li, lj ) L i=1 (vi,vj )∈E• n parts and h locations gives hn conﬁgurations.• If graph is a tree we can use dynamic programming. – O(nh2), much better but still slow.• If dij (li, lj ) = ||Tij (li) − Tji(lj )||2 can use DT. – O(nh), as good as matching each part separately!! 3
- 5. Distance transform Given a set of points on a grid P ⊆ G,the quadratic distance transform of P is, DP (q) = min ||q − p||2 p∈P P DP 4
- 6. Generalized distance transformGiven a function f : G → R, Df (q) = min ||q − p||2 + f (p) p∈G – for each location q, ﬁnd nearby location p with f (p) small. – equals DT of points P if f is an indicator function. 0 if p ∈ P f (p) = . ∞ otherwise 5
- 7. 1D case: Df (q) = minp∈G (q − p)2 + f (p)For each p, Df (q) is below the parabola rooted at (p, f (p)).Df (q) is deﬁned by the lower envelope of h parabolas. 1 f ( ) 2 f ( ) § h 1 f ( ) 0 f ( ) § . . . . . . . . . . . . . 0 1 2 h 1 6
- 8. There is a simple geometric algorithm that computes Df (p) inO(h) time for the 1D case. – similar to Graham’s scan convex hull algorithm. – about 20 lines of C code.The 2D case is “separable”, it can be solved by sequential 1Dtransformations along rows and columns of the grid.See Distance Transforms of Sampled Functions, Felzen-szwalb and Huttenlocher. 7
- 9. Simple face model• Locations are positions in the image grid.• Match cost mi(li) for placing part i at li.• Central part v1 - the nose.• Each part has an ideal position pi relative to nose. – Let T1i(l1) = l1 + pi, n n E(l1, . . . , ln) = mi(li) + ||li − T1i(l1)||2 i=1 i=2 8
- 10. Eﬃcient minimization n nL∗ = argmin mi(li) + ||li − T1i(l1)||2 L i=1 i=2 nL∗ = argmin m1(l1) + mi(li) + ||li − T1i(l1)||2 L i=2 n ∗l1 = argmin m1(l1) + min(mi(li) + ||li − T1i(l1)||2) l1 i=2 li n ∗l1 = argmin m1(l1) + Dmi (T1i(l1)) l1 i=2 9
- 11. Matching results 10
- 12. Matching results 11
- 13. Summary• Generic framework for part-based modeling.• Global minimization for deformable objects can be fast.• Soft detection avoids unnecessary early decisions.• Partial occlusion is handled automatically. 12

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