Galilean Differential Geometry Of Moving Images
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Galilean Differential Geometry Of Moving Images

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Presented at ECCV2004 (as a poster) and at Institut Mittag-Leffler 2003.

Presented at ECCV2004 (as a poster) and at Institut Mittag-Leffler 2003.

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Galilean Differential Geometry Of Moving Images Presentation Transcript

  • 1. Galilean Differential Geometry of Moving Images Daniel Fagerström CVAP/NADA/KTH [email_address]
  • 2. Differential Structure of Movies
    • How can we describe the local structure of an image sequence?
    • We will assume that a movie is a smooth function of 2+1 dimensional space-time
    • Looking for generic properties
  • 3. Approaches for Motion Analysis
    • Optical flow
    • Spatio-temporal texture
    • Spatio-temporal differential invariants
  • 4. Optical Flow
    • Geometry of the projected motion of particles in the observers field of view
    • Binding hypothesis needed to use it on image sequences
    • “ Top down”
    • Local formulation, but non-local, due to binding hypothesis
    • Undefined when particles appears and disappears, e.g. motion boundaries
    • Does not use image structure
  • 5. Spatio-Temporal Differential Invariants
    • Local geometry of spatio-temporal images
    • ” Bottom Up”, low level
    • No binding hypotheses, connection to the environment considered as a higher level problem
    • Well defined everywhere
    • Does use image structure, extension of low level vision for still images
  • 6. Overview
    • Galilean geometry
    • Moving frames
    • Image geometry
    • 1+1 dimensional Galilean differential invariants
    • 2+1 dimensional Galilean differential invariants
    • What is required for a more realistic movie model
  • 7. Galilean Geometry
    • Spatial and temporal translation a , spatial rotation R and spatio-temporal shear v
    Shear x t x t
  • 8. Galilean Geometry
    • Insensitive to constant relative motion for parallel projection, approximately otherwise
    • Simplest meaning full model
    • Assumed implicitly when one talk about optical flow invariants: div, rot, dev, i.e. first order flow
    • Shape properties from the environment can be derived from relative motion
    • (Newton physics describe Galilean invariants)
  • 9. Galilean Invariants
    • Planes of simultaneity (constant t ) are invariant and has Euclidean geometry: distances and angles are invariants
      • i.e. an image sequence
    • The temporal distance between planes of simultaneity is an invariant
  • 10. Galilean ON-System
    • An n+1 dimensional Galilean ON-system (e 1 ,e 2 ,  ,e 0 ) is s.t. (e 1 ,e 2 ,  ,e n ) is an Euclidean ON-system and ||e 0 || T =1
  • 11. Moving Frames
    • Galilean geometry has no metric
    • We will use Cartan's method of moving frames, that does not require a metric
    • Moving frame: e:M ! G ½ GL(n)
    • Attach a frame that is adapted to the local structure in each point
    • Differential geometry: the local change of the frame: de
  • 12. Moving Frames C(A) contains the differential geometric invariants expressed in the global frame i
  • 13. Image Geometry
    • Image space: E 2 ­ I - trivial fiber bundle with Euclidian base space and log intensity as fiber (Koenderink 02) z=f(x,y)
    • f is smooth
    • Image geometry
      • Global gray level transformations
      • Lightness gradients
  • 14. Gradient Gauge
    • For points where r f  0 we can choose an adapted ON-frame {  u ,  v } s.t. f u =0
    • All functions over  i u  j v f, i+j ¸ 1 becomes invariants w.r.t. rotation in space and translation in intensity
  • 15. Gray Level Invariants
  • 16. Hessian Gauge
    • For points where r f  0 we can choose an ON-frame {  p ,  q } s.t. f pq =0 and |f pp |>|f qq |
    • All functions over  i p  j q f, i+j ¸ 2 becomes invariants w.r.t. rotation in space, translation in intensity and addition of a linear light gradient (Koenderink 02)
  • 17. Galilean 1+1 D
    • Two cases:
      • Isophotes cut the spatial line, motion according to the constant brightness assumption
      • Isophotes are tangent to the spatial line (along curves), creation, annihilation
  • 18. Tangent Gauge
    • Let {  t ,  x } be a global Galilean ON-frame, for points where f x  0 we can define an adapted Galilean ON-frame {  s ,  x } s.t. f s =0.
  • 19. Isophote Invariants
  • 20. Hessian Gauge
    • Let {  t ,  x } be a global Galilean ON-frame, for points where f xx  0 we can define an adapted Galilean ON-frame {  r ,  x } s.t. f rx =0 .
  • 21. Hessian Invariants
  • 22. Galilean 2+1 D
    • General case
    • Also here are two different main cases
      • Isophote surfaces transversal to the spatial plane. Motion of isophote curves in the image
      • Isophote surfaces tangent to the plane. Creation, annihilation and saddle points
  • 23. Invariants in the General Case
    • a u , a v - acceleration
    •  u ,  v - divergence
    •  u ,  v - skew of the ”flow field”
    • - rotation of the plane in the temporal direction
    •  u ,  v - flow line curvature in the plane
  • 24. More Descriptive Invariants
    • D - rate of strain tensor for the spatio-temporal part of the frame field
    • a, curl D, div D, def D - are flow field invariants
    • a  ,  ,  ,  u ,  v - are not flow field invariants
  • 25. Tangent Gauge
    • Let {  t ,  x ,  y } be a global Galilean ON-frame, for points where ||{ f x ,f y }||  0 we can define an adapted Galilean ON-frame {  s ,  u ,  v } s.t. f s =f u =f su =0
    • Principal acceleration extrema
    • Direction of  u constant along  s – used in Guichard (98)
  • 26. Tangent Gauge
  • 27. Hessian Gauge
    • Let {  t ,  x ,  y } be a global Galilean ON-frame, we define an adapted Galilean ON-frame {  r ,  p ,  q } s.t. f pq = f rp = f rq =0 .
    • Also defined when the spatial tangent disappears, e.g. for creation and disappearance of structure
    •  r is the same vector field as when the optical flow constraint equation is solved for the spatial image gradient
  • 28. Hessian Gauge
  • 29. Real Image Sequences
    • Localized filters are not invariant w.r.t. Galilean shear, velocity adapted (  s – directed) filters are needed
    • More generic singular cases for imprecise measurements
  • 30. Conclusion
    • Theory about differential invariants for smooth Galilean spatio-temporal image sequences
    • Local operators
    • “ Bottom up”
    • Contains more information about the image sequence than optical flow
    • Extension of methods for still images