Scale space geometry
Download
1 / 22

Scale Space Geometry - PowerPoint PPT Presentation


  • 86 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
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
Scale space geometry

Scale Space Geometry

Arjan Kuijper

arjan@itu.dk


Deep structure
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
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


Example image
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


What happens with these structures
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
Critical curves

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


Critical points
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
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
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 curves1
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
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 points1
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
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
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
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
How does it look like?

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


On creations
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
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
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
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
Void scale space saddles

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


Sources
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