Scale Space Geometry Arjan Kuijper

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

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec What to look for Gaussian scale space is intensity-based.Gaussian scale space is intensity-based. Consider an n - dimensional image, i.e. a (n+1) dimensional Gaussian scale space (Gss) image.Consider an n - dimensional image, i.e. a (n+1) dimensional Gaussian scale space (Gss) image. Investigated intensity-related items.Investigated intensity-related items. “Things” with specialties w.r.t. intensity.“Things” with specialties w.r.t. intensity. Equal intensities – isophotes, iso-intensity manifolds: L=cEqual 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=0Critical intensities – maxima, minima, saddle points:  L=0  0 – dimensional points in the Gss image. Critical intensities – maxima, minima, saddle points,.....: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 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 What happens with these structures? Causality: no creation of new level lines Outer scale: flat kernel All level lines disappearAll level lines disappear All but one extrema disappearAll but one extrema disappear Example View critical points in scale space: the critical curves.

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

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec Critical points Let L(x,y) describe the image landscape. At critical points,  T L = (∂ x L,∂ y L) = (L x,L y ) = (0,0). To determine the type, consider de Hessian matrix H =  T L(x,y) = ((L xx, L xy ), (L xy, L yy )). Maximum: H has two negative eigenvaluesMaximum: H has two negative eigenvalues Minimum: H has two positive eigenvaluesMinimum: H has two positive eigenvalues Saddle: H has a positive and a negative eigenvalue.Saddle: H has a positive and a negative eigenvalue.

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec When things disappear Generically, det [H] = L xx L yy - L xy L xy <> = 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 Diffusion equation We know that L t = L xx + L yy So we can construct polynomials in scale space. Let’s make a Hessian with zero determinant: H=((6x,0),(0,2)) Thus L xx = 6x, L yy = 2, L xy = 0 And L t = 6x +2 Thus L = x 3 + 6xt + y 2 + 2t Consider the critical curves

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

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

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

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 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 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 L xx = 6x, L yy = 2 + f(x), L xy = 0 To obtain a path (√2t,0;t) require L t = -6x +2, so f(x) = -6x. Thus L = x 3 - 6xt + y 2 + 2t -6 x y 2

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

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

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

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

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

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