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Scale Space Geometry

Scale Space Geometry. Arjan Kuijper arjan@itu.dk. Deep structure. The challenge is to understand the image really on all the levels simultaneously, and not as an unrelated set of derived images at different levels of blurring. Jan Koenderink (1984). What to look for.

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Scale Space Geometry

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  1. Scale Space Geometry Arjan Kuijper arjan@itu.dk

  2. Deep structure • The challenge is to understand the imagereally on all the levels simultaneously,and not as an unrelated set of derived imagesat different levels of blurring. • Jan Koenderink (1984) Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  3. What to look for • Gaussian scale space is intensity-based. • Consider an n - dimensional image, i.e. a (n+1) dimensional Gaussian scale space (Gss) image. • Investigated intensity-related items. • “Things” with specialties w.r.t. intensity. • Equal intensities – isophotes, iso-intensity manifolds: L=c • n - dimensional iso-manifolds in the Gss image • (n-1) - dimensional manifolds in the image. • Critical intensities – maxima, minima, saddle points: L=0 • 0 – dimensional points in the Gss image. • Critical intensities – maxima, minima, saddle points, .....: • 0 – dimensional critical points in the blurred image, • 1 – dimensional critical curves in the Gss image. Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  4. Example image • Consider a simple 2D image. • In this image, and its blurred versions we have • Critical points L=0: • Extrema (green) • Minimum • Maxima • Saddles (Red) • Isophotes L=0: • 1-d curves, only intersecting in saddle points Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  5. What happens with these structures? • Causality: no creation of new level lines • Outer scale: flat kernel • All level lines disappear • All but one extrema disappear • Example • View critical points in scale space: the criticalcurves. Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  6. Critical curves Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  7. Critical points • Let L(x,y) describe the image landscape. • At critical points, TL = (∂xL,∂yL) = (Lx,Ly) = (0,0). • To determine the type, consider de Hessian matrix • H = TL(x,y) = ((Lxx , Lxy), (Lxy , Lyy)). • Maximum: H has two negative eigenvalues • Minimum: H has two positive eigenvalues • Saddle: H has a positive and a negative eigenvalue. Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  8. When things disappear • Generically, det [H] = Lxx Lyy - Lxy Lxy <> = 0, there is no eigenvalue equal to 0.This yields an over-determined system. • In scale space there is an extra parameter, so an extra possibility: det [H] = 0. • So, what happens if det [H] = 0? -> Consider the scale space image Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  9. Diffusion equation • We know that Lt = Lxx + LyySo we can construct polynomials in scale space. • Let’s make a Hessian with zero determinant: • H=((6x,0),(0,2)) • Thus Lxx = 6x, Lyy = 2, Lxy = 0And Lt = 6x +2 • Thus L = x3 + 6xt + y2 + 2t • Consider the critical curves Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  10. Critical Curves • L = x3 + 6xt + y2 + 2t • Lx = 3x2 + 6t, Ly = 2y • For (x,y;t) we have • A minimum at (x,0;-x2/2), or (√-2t,0;t) • A saddle at (-x,0;- x2/2), or (-√-2t,0;t) • A catastrophe point at (0,0;0), an annihilation. • What about the speed at such a catastrophe? Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  11. Speed of critical points • Higher order derivatives: -L = H x + DL t • x = -H-1(L + DL t) • Obviously goes wrong at catastrophe points, since then det(H)=0. • The velocity becomes infinite: ∂t (√-2t,0;t)= (-1/√-2t,0;1) Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  12. Speed of critical points • Reparametrize t = det(H) t : x = -H-1(L + DL det(H) t) • Perfectly defined at catastrophe points • The velocity becomes 0: -H-1(DL det(H) t -> v = (1,0, t) Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  13. To detect catastrophes • Do the same trick for the determinant: • -L = H x + DL t-det(H) = det(H) x + Ddet(H) t • Set M = ((H, DL), (det(H), Ddet(H)) • Then if at catastrophes • det[M] < 0 : annihilations • det[M] > 0 : creations Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  14. Creations • Obviously, critical points can also be created. • This does not violate the causality principle. • That only excluded new level lines to be created. • At creations level lines split, think of a camel with two humps. Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  15. To create a creation • Let’s again make a Hessian with zero determinant: • H=((6x,0),(0,2+f(x))) • With f(0)=0. • Thus Lxx = 6x, Lyy = 2 + f(x), Lxy = 0 • To obtain a path (√2t,0;t) require Lt = -6x +2, so f(x) = -6x. • Thus L = x3 - 6xt + y2 + 2t -6 x y2 Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  16. How does it look like? Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  17. On creations • For creations the y-direction is needed: • Creations only occur if D>1. • Creations can be understood when they are regarded as perturbations of non-generic catastrophes. • At non-generic catastrophes the Hessian is “more” degenerated: there are more zero eigenvalues and/or they are “more” zero. Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  18. Critical points in scale space • L = 0DL = 0 • Scale space critical points are always spatial saddle points. • Scale space critical points are always saddle points. • Causality: no new level lines implies no extrema in scale space. Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  19. Scale space saddles • At a scale space saddle two manifolds intersect Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  20. Manifolds in scale space • Investigate structure through saddles. Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  21. Void scale space saddles Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

  22. Sources • Local Morse theory for solutions to the heat equation and Gaussian blurringJ. DamonJournal of differential equations 115 (2): 386-401, 1995 • The topological structure of scale-space images L. M. J. Florack, A. KuijperJournal of Mathematical Imaging and Vision 12 (1):65-79, 2000. • The deep structure of Gaussian scale space images Arjan Kuijper • Superficial and deep structure in linear diffusion scale space:Isophotes, critical points and separatricesLewis Griffin and A. Colchester.Image and Vision Computing 13 (7): 543-557, 1995 Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

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