SSVM07 Spatio-Temporal Scale-Spaces

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SSVM07 Spatio-Temporal Scale-Spaces

  1. 1. Spatio-Temporal Scale-Spaces Daniel Fagerström CVAP/CSC/KTH danielf@nada.kth.se
  2. 2. Motivation • Allow for differential geometric methods on moving images • Temporal causality • Recursive formulation based on current signal and earlier measurement rather than on previous signal – More realistic from a conceptual view point – Much more efficient from a computational view point
  3. 3. Earlier Approaches • Koenderink (88), Florack (97), Salden et al (98) – Requires convolution with past signal, non recursive • Lindeberg (02) – Recursive, but discrete in the spatio-temporal dimensions
  4. 4. Current Approach • ”Standard” axioms + Galelian similarity • Infinitesimal formulation • No time causal solution possible • But scale-space on space and memory exists
  5. 5. Galilean Similarity • Spatial and temporal translation, spatial rotation and spatio-temporal shear • Allows for measurement of relative motion µ ¶ µ ¶µ ¶ t0 ¿ 0 t = +a x0 v ¾R x x; v 2 Rn ; t 2 R; R 2 SO(n); ¾; ¿ 2 R+ and a 2 Rn+1
  6. 6. Point Measurement De¯nition © : L1 (Rn+1 ) ! C 1 (Rn+1 ), for any u; v 2 L1 (Rn+1 ) and ®; ¯ 2 R, is a point measurement operator if it ful¯lls: linearity ©(®u + ¯v) = ®©(u) + ¯©(v) gray level invariance k©ukL = kukL positivity u ¸ 0 ) ©u ¸ 0 1 1 point lims!0 ©s u = u
  7. 7. G Covariance De¯nition Given a Lie group g 2 G that acts on the base space as g 7! g ¢ x = Tg x, where Tg : (Rn+1 ). A family H 3 h 7! ©h of measurement operators ful¯lling, Tg ©h Tg ¡1 = ©g ¢h ; is called aG-covariant point measurement space. Where, G £ H 3 (g; h) 7! g ¢ h = ¾(g; h) = ¾g (h) 2 H; is a Lie group action on the set H.
  8. 8. G Scale-Space De¯nition A G scale-space is a minimal family of G-covariant point measure- ments that also is a semigroup of operators. I.e. that ful¯lls, ©h ©h = ©h ¢h : 1 2 1 2
  9. 9. Infinitesimal Criteria An infinitesimal condition for G scale- spaces: • Simplifies derivations as it liniarizes the problem • Allows for more general boundary conditions (which later will be seen to be necessary)
  10. 10. Infinitesimal Generators • The infinitesimal object that corresponds to a Lie groupLGis a g algebra G Lie = • The infinitesimal object that corresponds LH h to a Lie semigroup H is a Lie wedge = • Infinitesimal generators v 7! A A dT (e) : g ! L(§; §), g 3 = v = A(v), B = d©(e) : h ! L(§; §), h 3 w 7! Bw = B(w)
  11. 11. Pseudo Differential Operators The in¯ntesimal operators of transformation group and the semigroup of operators can be expressed in terms of pseudodi®erential operators, ªDO. ªDO's are de¯ned by Z Au(x) = (2¼)¡n eix¢» a(x; »)~(»)d»; u R P where u(») = e¡ix¢» u(x)dx and a(x; ») = j j· a® (x)» ® , and is called the ~ ® m symbol of A. The corresponding operator is denoted a(x; D). Translation invariant symbols are position independent, b(x; ») = b(»).
  12. 12. Infinitesimal Covariance The in¯nitesinal form of the covariance equation is, [Av ; Bw ] = BC(v;w) ; where g £ h 3 (v; w) 7! C(v; w) 2 h is the di®erential of ¾ with respect to both arguments. And an operator Bw full¯lling such a condition is called a covariant tensor operator. The operator C is denoted the covariance tensor and is a Lie algebra representation in the ¯rst argument and a Lie wedge representation in the second.
  13. 13. Positivity and Gray Level Invariance • A positive translation invariant linear operator semigroup is generated by a negative definite symbol • For a gray level invariant semigroup that is generated by the generators B, the genrators B are conservative, i.e. B1=0
  14. 14. Infinitesimal G Scale-Space De¯nition A g scale-space wedge, is a minimal Lie wedge of negative de¯nite conservative operators h 3 w 7! Bw : L(§; §), that is a covariant tensor oper- ator, with respect of the Lie algebra action g 3 v 7! Av : L(§; §) and the Lie algebra and Lie wedge representation (v; w) 7! C(v; w). Theorem A G scale-space is generated by its corresponding g scale-space wedge, ½ @ s u = Bw u u(e; x) = f (x); where f 2 §.
  15. 15. Causal Affine Line Theorem A gl(1)(= f@x ; x@x g) temporally causal scale-space wedge is gener- ated by left sided Riemann Liouville fractional derivatives D ® with 0 < ® < 1, + where D§ (») = (¨i»)® = j» j® e¨i(»)®¼=2 : ®
  16. 16. Euclidean Similarity Theorem A Euclidean similarity es(2) = t(2) [ s(2) [ so(2) scale-space wedge is generated for any 0 < ® · 2 by the Riesz fractional derivative ¡(¡¢)®=2 (» j» j® ). t(2) = f@1 ; @2 g is the translation generator, s(2) = fx1 @1 + x2 @2 g is the scaling generator and so(2) = fx2 @1 ¡ x1 @2 g is the rotation generator.
  17. 17. No Time Causal Galilean Similarity Theorem A °s(2) scale-space wedge is generated by f@0 ; @0 @1 ; @1 g. 2 2 °s(2) = t(2) [ s(1) © s(1) [ °(1); where °(1) = fx0 @1 g is the Galilean boost that skewquot; space-time and s(1)©s(1) is a direct sum of the scaling generator in space and time respectively. Corollary It should be noted that the generated scale-space is symmetric booth in time and space and thus no time causal scale-spaces are possible with this axiomatization. And as °s(n); n ¸ 2 have °s(2) as a sub algebra, no time causal scale-spaces are possible for them either.
  18. 18. What to do? • A realistic temporal measurement system should not have access to the past signal, only the memory of the past signal • Let the memory have the same propeties as the past signal, i.e. it should be an affine half space • Define the time causal Galilean scale- space as a evolution equation on space and memory instead of on space and time
  19. 19. Galilean Similarity De¯nition Let °s(d + 1) = es(d) © gl(1) [ °(d). The d + 1-dimensional time causal Galilean scale-space R+ £ Rd £ R+ £ R £ Rd 3 (¾; v; ¿; t; x) 7! u(¾; v; ¿; t; x) 2 R, where ¾ is spatial scale, v is velocity, ¿ is memory (and temporal scale) is a °s(d + 1)-covariant, point measurement space in space-time (t; x) and a gl(1)-wedge in memory ¿ .
  20. 20. General Evolution Equation Theorem A d + 1-dimensional time causal Galilean scale-space (for d = 1; 2) is generated by the evolution equation, 8 < @tu = ¡v ¢ rxu + D¿ ¡u ®0 @¾ u = ¡(¡¢x)®=2u : u(0; 0; 0; t; x) = f(t; x); where 1 < ®0 · 2, 0 < ® · 2, ¾ is the spatial scale direction, v 2 Rd the velocity vector, @s = @t + v ¢ rx is the spatio-temporal direction, rx = (@1; : : : ; @d) is the spatial gradient and ¢x is the spatial Laplacian.
  21. 21. Heat Equation on Half-Space Theorem The equation, 8 < @tu = ¡v ¢ rxu + @2u ¿ @¾ u = ¢xu : u(0; 0; 0; t; x) = f(t; x); is unique in the family of evolution equations as it is the only one that has local generators.
  22. 22. Closed Form u(¾; v; ¿; t; x) = Á(¾; v; ¿; ¢; ¢) ¤ f (t; x); ¿ exp(¡ ¿ 2 ¡ (x¡tv)¢(x¡tv) ) p 4t Á(¾; v; ¿; t; x) = 4¾ 4¼t3=2 (4¼¾)d=2
  23. 23. Mixed Derivatives

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