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# Scale Space Geometry - PowerPoint PPT Presentation

Scale Space Geometry. Arjan Kuijper [email protected] 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

Arjan Kuijper

• 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

• 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

• Consider a simple 2D image.

• In this image, and its blurred versions we have

• Critical points L=0:

• Extrema (green)

• Minimum

• Maxima

• Isophotes L=0:

• 1-d curves, only intersecting in saddle points

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

• 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

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

• 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

• 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

• At a scale space saddle two manifolds intersect

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

• 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