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Johann Radon Institute for Computational and Applied Mathematics: 1/35 Signal- und Bildverarbeitung, Image Analysis and Processing Arjan Kuijper Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences Altenbergerstraße 56 A-4040 Linz, Austria

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Johann Radon Institute for Computational and Applied Mathematics: 2/35 Last Week Normal motion flow is equivalent to the mathematical morphological erosion or dilation with a ball. –The dilation and erosion operators are shown to be convolution operators with boolean operations on the operands. –Morphology with a quadratic structuring element links to Gaussian scale space –There exists a “pseudo-linear” equation linking them. The Mumford-Shah functional is designed to generate edges while denoising –Not unique –Complicated Active contours / snakes are defined as an energy minimizing splines that are supposed to converge to edges.

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Johann Radon Institute for Computational and Applied Mathematics: 3/35 Today Deep structure in Gaussian Scale Space –Critical points –Movement of critical points –Catastrophe points (singularity theory) Annihilations Creations –Scale space critical points –Iso-manifolds –Hierarchy –Topological segmentation

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Johann Radon Institute for Computational and Applied Mathematics: 4/35 Gaussian scale space Famous quote: “Gaussian scale space doesn’t work because it blurs everything away”

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Johann Radon Institute for Computational and Applied Mathematics: 5/35 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)

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Johann Radon Institute for Computational and Applied Mathematics: 6/35 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.

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Johann Radon Institute for Computational and Applied Mathematics: 7/35 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

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Johann Radon Institute for Computational and Applied Mathematics: 8/35 What happens with these structures? Causality: no creation of new level lines Outer scale: flat kernel –All level lines disappear –One extremum remains –Extrema and saddles (dis-)appear pair-wise View critical points in scale space: the critical curves.

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Johann Radon Institute for Computational and Applied Mathematics: 9/35 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 eigenvalues –Minimum: H has two positive eigenvalues –Saddle: H has a positive and a negative eigenvalue.

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Johann Radon Institute for Computational and Applied Mathematics: 10/35 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

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Johann Radon Institute for Computational and Applied Mathematics: 11/35 Diffusion equation We know that L t = L xx + L yy So we can construct polynomials (jets) 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

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Johann Radon Institute for Computational and Applied Mathematics: 12/35 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 saddle at (-x,0;- x 2 /2), or (-√-2t,0;t) –A catastrophe point at (0,0;0), an annihilation. What about the speed at such a catastrophe?

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Johann Radon Institute for Computational and Applied Mathematics: 13/35 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)

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Johann Radon Institute for Computational and Applied Mathematics: 14/35 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)

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Johann Radon Institute for Computational and Applied Mathematics: 15/35 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 : annihilations –det[M] > 0 : creations

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Johann Radon Institute for Computational and Applied Mathematics: 16/35 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.

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Johann Radon Institute for Computational and Applied Mathematics: 17/35 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

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Johann Radon Institute for Computational and Applied Mathematics: 18/35 How does it look like?

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Johann Radon Institute for Computational and Applied Mathematics: 19/35 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.

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Johann Radon Institute for Computational and Applied Mathematics: 20/35 Non-generic events …non-generic catastrophes are also of interest.

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Johann Radon Institute for Computational and Applied Mathematics: 21/35 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 saddle points. –Causality: no new level lines implies no extrema in scale space. –Visualize the intensity of the critical curves as a function of scale: the scale space saddles are the local extrema of these curves. Extrema (minima/maxima) branches in/de-crease monotonically.

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Johann Radon Institute for Computational and Applied Mathematics: 22/35 Example

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Johann Radon Institute for Computational and Applied Mathematics: 23/35 Scale space saddles At a scale space saddle two manifolds intersect

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Johann Radon Institute for Computational and Applied Mathematics: 24/35 Manifolds in scale space Investigate structure through saddles.

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Johann Radon Institute for Computational and Applied Mathematics: 25/35 Void scale space saddles

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Johann Radon Institute for Computational and Applied Mathematics: 26/35 Hierarchical Algorithm Initializing Build a scale space. Find the critical points at each scale level. Construct the critical branches. Find the catastrophe points. Construct and label the critical curves, including the one remaining extremum. Find the scale space saddles. Determining the manifolds Find for each annihilations extremum its critical iso-intensity manifold. Construct the dual manifolds.

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Johann Radon Institute for Computational and Applied Mathematics: 27/35 Hierarchical Algorithm Labeling Assign to each extremum the dual manifolds to which it belongs, sorted on intensity. Build a tree: Start with the remaining extremum at the coarsest scale as root. Trace to finer scale until at some value it is labeled to a dual manifold. Split into two branches, on the existing extremum, one the extremum having the critical manifold. Continue for all branches / extrema until all extrema are added to the tree.

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Johann Radon Institute for Computational and Applied Mathematics: 28/35 Consider the blobs

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Johann Radon Institute for Computational and Applied Mathematics: 29/35 Results

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Johann Radon Institute for Computational and Applied Mathematics: 30/35 Results

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Johann Radon Institute for Computational and Applied Mathematics: 31/35 The tree De 5 De 3 De 1 De 2 Ce 2 Ce 1 Ce 3 Ce 5 e 4 e 2 e 1 e 3 e 5 R demo

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Johann Radon Institute for Computational and Applied Mathematics: 32/35 A real example

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Johann Radon Institute for Computational and Applied Mathematics: 33/35 Find critical curves Pairs e6-s1, e1-s3, e3-s4, e2-s2

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Johann Radon Institute for Computational and Applied Mathematics: 34/35 Noise addition

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Johann Radon Institute for Computational and Applied Mathematics: 35/35 Conclusions A scale space approach justifies continuous calculations on discrete grids. Structure of the image is hidden in the deep structure of its scale space image. Essential keywords are –Critical curves –Singularities –Deep structure –Iso-manifolds

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