Temporal Scale-Spaces ScSp03

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Temporal Scale-Spaces ScSp03

  1. 1. Temporal Scale Spaces Daniel Fagerström CVAP/NADA/KTH [email_address]
  2. 2. Measurement of Temporal Signals <ul><li>Real time measurement </li></ul><ul><li>Temporal causality </li></ul><ul><li>Structure at different resolution </li></ul><ul><li>Well defined derivatives </li></ul>
  3. 3. Earlier Approaches <ul><li>Koenderink (88), Florack (97), ter Haar Romeny et. al. (01) – Logaritmic mapping of the past time-line, then ordinary scale space </li></ul><ul><li>Lindeberg & Fagerström (96) – Noncreation of local maxima with increasing scale </li></ul><ul><li>Salden (99) – Diffusion on the temporal time-line with absorbing or reflecting boundary condition </li></ul>
  4. 4. Current Approach <ul><li>Using axioms from Pauwels et. al. 95, replacing reflectional symmetry with temporal causality </li></ul>
  5. 5. Space vs. Time Space All directions simultaneously available Time The past can be memorized, the future can not Memory
  6. 6. Overview <ul><li>Motivation of axioms </li></ul><ul><li>Form of temporal scale space kernels </li></ul><ul><li>Time recursive realization </li></ul><ul><li>Numerical scheme </li></ul><ul><li>Comparison with earlier work </li></ul>
  7. 7. Principles <ul><li>Non committed representation </li></ul><ul><li>Embodiment of symmetries in the surroundings </li></ul><ul><li>Extended point </li></ul>
  8. 8. Measurement
  9. 9. Meaurement <ul><li>Choose well behaved sensor functions </li></ul><ul><li>Linearity </li></ul>
  10. 10. Examples <ul><li>Regular distribution </li></ul><ul><li>Dirac ”function” </li></ul><ul><li>Derivation </li></ul>
  11. 11. Causality <ul><li>Causality </li></ul><ul><li>Measurement time </li></ul>
  12. 12. Embodiment of Geometry <ul><li>The structure of the sensor system should correspond to regularities in the environment </li></ul><ul><li>Translation covariance : the measurement process is the same at each moment </li></ul><ul><li>Scaling covariance : all time spans are treated in the same way </li></ul>
  13. 13. Covariance <ul><li>Transformation groups </li></ul><ul><li>Corresponding action on a sensor </li></ul>
  14. 14. Orbits
  15. 15. Convolution <ul><li>Definition </li></ul><ul><li>Associativity </li></ul>
  16. 16. Cascade Property (Semi-Group)
  17. 17. Extended Point <ul><li>Unit area </li></ul><ul><li>Positivity </li></ul>
  18. 18. Temporal Scale Space Kernels <ul><li>Continuity </li></ul><ul><li>Positivity </li></ul><ul><li>Unit area </li></ul><ul><li>Causality </li></ul><ul><li>Dilatation covariance </li></ul><ul><li>Convolution semi-group </li></ul><ul><li>Pauwels et al used the same axioms, with (4) replaced by reflexion symmetry, to characterize spatial scale-spaces. </li></ul>
  19. 19. Characterization Laplace transform Semi-group and causality Continuity (Cauchy’s functional equation)
  20. 20. Characterization (Cont) Dilatation covariance Positivity (Bernstein)
  21. 21. Temporal Scale Space Kernels 0.3 0.5 0.8
  22. 22. Extremal Stable Density Functions <ul><li>Limit densities for sum of stochastic variables (Levy) </li></ul><ul><li>Infinite first and second moments </li></ul><ul><li>Increasing popularity in fysics, finance … </li></ul>
  23. 23. Explicit Form
  24. 24. Markov Property? <ul><li>The memory is the only access to the past </li></ul><ul><li>Convolution directly with the input signal is unrealistic </li></ul><ul><li>Need for an evaluation equation! </li></ul> t L(0,  )=0 L(t,  )=f(t)
  25. 25. Fractional Derivatives <ul><li>Linear </li></ul><ul><li>Equal to ordinary derivatives for integer order </li></ul><ul><li>Generalized Leibniz property </li></ul>
  26. 26. Evaluation Equations <ul><li>Signaling equation for  </li></ul><ul><li>Fractional Brownian motion </li></ul><ul><li>Diffusion for  </li></ul><ul><li>Scale space as state </li></ul><ul><li>Diffusion for  </li></ul>
  27. 27. Uniquenes of the Signaling Equation <ul><li>Locality: is a differential operator if 1/  is an integer </li></ul><ul><li>Positivity: have a positive Greens function if 1/    </li></ul><ul><li>Locality and positivity is only satisfied for  </li></ul>
  28. 28. Signaling Equation <ul><li>The temporal scale space is the only memory of earlier input </li></ul><ul><li>The memory diffuses over time </li></ul>
  29. 29. Numerical Scheme <ul><li>Implicit scheme </li></ul><ul><li>Second order stability in time and scale </li></ul><ul><li>4(add)+4(multiply)+2(divide) per mesh point </li></ul>t  3/2 -1/2
  30. 30. Example
  31. 31. Example
  32. 32. Comparison with Previous Work <ul><li>Florack (97) requires the measurement kernel to be a positive Schwartz test function, this rules out all stable density functions except Gaussians </li></ul><ul><li>Ter Haar Romeny et. al. (01) requires the measurement kernel to have finite first and second moment, only fulfilled for Gaussians, extremal (causal) stable density functions even has infinite first moment </li></ul>
  33. 33. Comparison with Earlier Work <ul><li>Koenderink (88) and Lindeberg & Fagerström (96) uses stronger requirements (than semi-group) on causality in the resolution domain </li></ul><ul><li>Salden (99) does not require translational covariance on the kernel only on the generator </li></ul>
  34. 34. Comparison with Earlier Work <ul><li>Lindeberg & Fagerström (96) and the current approach are the only ones that has a (known) time recursive formulation </li></ul><ul><li>Lindeberg & Fagerström (96) requires either time or scale to be discrete and lack scale covariance </li></ul><ul><li>The kernels in the current approach are less well localized than in the other approaches </li></ul>
  35. 35. Conclusion <ul><li>Causal scale invariant convolution semi group </li></ul><ul><li>Time recursive realization </li></ul><ul><li>Signaling equation </li></ul><ul><li>Efficient numerical scheme </li></ul>

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