- By
**ezra** - Follow User

- 86 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Scale Space Geometry' - ezra

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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

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.

- Equal intensities – isophotes, iso-intensity manifolds: L=c

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

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)

- Extrema (green)
- Isophotes L=0:
- 1-d curves, only intersecting in saddle points

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

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

Critical curves

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

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

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

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

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

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

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

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

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

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

How does it look like?

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

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

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

Scale space saddles

- At a scale space saddle two manifolds intersect

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

Manifolds in scale space

- Investigate structure through saddles.

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

Void scale space saddles

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

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

Download Presentation

Connecting to Server..