Robust Sparse Analysis Regularization

4807c637e2e5e8a5c5e68b287e8492a9?s=47 Samuel Vaiter
November 09, 2011

Robust Sparse Analysis Regularization

Natimages'11, Institut Henri Poincaré, Paris, November 2011

4807c637e2e5e8a5c5e68b287e8492a9?s=128

Samuel Vaiter

November 09, 2011
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  1. Robust Sparse Analysis Regularization Samuel Vaiter Tuesday, November 8, 11

  2. Inverse Problems Tuesday, November 8, 11

  3. Inverse Problems Several problems Inpaiting Super-resolution Tuesday, November 8, 11

  4. Inverse Problems ill-posed Linear hypothesis One model y = x0

    + w Observations Operator Unknown signal Noise Several problems Inpaiting Super-resolution Tuesday, November 8, 11
  5. Inverse Problems ill-posed Linear hypothesis One model y = x0

    + w Observations Operator Unknown signal Noise Several problems Inpaiting Super-resolution Regularization x? 2 argmin x 2RN 1 2 || y x ||2 2 + J ( x ) Tuesday, November 8, 11
  6. Inverse Problems ill-posed Linear hypothesis One model y = x0

    + w Observations Operator Unknown signal Noise x? 2 argmin x = y J ( x ) Noiseless 0 Several problems Inpaiting Super-resolution Regularization x? 2 argmin x 2RN 1 2 || y x ||2 2 + J ( x ) Tuesday, November 8, 11
  7. space domain Synthesis Sparsity of Natural Images Tuesday, November 8,

    11
  8. space domain Synthesis Sparsity of Natural Images frequency-space domain Orthogonal

    Wavelets Tuesday, November 8, 11
  9. space domain Synthesis Sparsity of Natural Images frequency-space domain Orthogonal

    Wavelets Tuesday, November 8, 11
  10. Many almost null coe cients space domain Synthesis Sparsity of

    Natural Images frequency-space domain Orthogonal Wavelets Tuesday, November 8, 11
  11. Many almost null coe cients space domain Good approximation Sparsity

    of natural images in frequency-space domain J(x) = min ↵ || ↵ ||1 subject to x = D↵ Synthesis Sparsity of Natural Images frequency-space domain Orthogonal Wavelets Tuesday, November 8, 11
  12. An Other Sparsity Measure space domain Tuesday, November 8, 11

  13. An Other Sparsity Measure space domain analysis r Tuesday, November

    8, 11
  14. An Other Sparsity Measure space domain analysis r Tuesday, November

    8, 11
  15. non-null coe cients : discontinuities An Other Sparsity Measure space

    domain analysis r Tuesday, November 8, 11
  16. non-null coe cients : discontinuities Sparsity of cartoon images after

    gradient operator application J ( x ) = || D ⇤ x ||1 Here TV : D⇤ = r An Other Sparsity Measure space domain analysis r Tuesday, November 8, 11
  17. Sparse Regularizations Synthesis argmin ↵2RQ 1 2 ||y ↵||2 2

    + ||↵||1 = D x = D↵ Tuesday, November 8, 11
  18. Sparse Regularizations Synthesis argmin ↵2RQ 1 2 ||y ↵||2 2

    + ||↵||1 = D x = D↵ Analysis argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 Tuesday, November 8, 11
  19. Sparse Regularizations = 6= 0 D x ↵ Synthesis argmin

    ↵2RQ 1 2 ||y ↵||2 2 + ||↵||1 = D x = D↵ Analysis argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 Tuesday, November 8, 11
  20. Sparse Regularizations = 6= 0 D x ↵ = D⇤

    x ↵ Synthesis argmin ↵2RQ 1 2 ||y ↵||2 2 + ||↵||1 = D x = D↵ Analysis argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 Tuesday, November 8, 11
  21. Sparse approx. of x ? in D Sparse Regularizations =

    6= 0 D x ↵ = D⇤ x ↵ Synthesis argmin ↵2RQ 1 2 ||y ↵||2 2 + ||↵||1 = D x = D↵ Analysis argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 Tuesday, November 8, 11
  22. Correlation of x ? and D sparse Sparse approx. of

    x ? in D Sparse Regularizations = 6= 0 D x ↵ = D⇤ x ↵ Synthesis argmin ↵2RQ 1 2 ||y ↵||2 2 + ||↵||1 = D x = D↵ Analysis argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 Tuesday, November 8, 11
  23. Signal Model of Synthesis Sparsity Tuesday, November 8, 11

  24. Signal Model of Synthesis Sparsity d1 d2 Tuesday, November 8,

    11
  25. Signal Model of Synthesis Sparsity d1 d2 Tuesday, November 8,

    11
  26. ||↵||1 = 1 Signal Model of Synthesis Sparsity d1 d2

    Tuesday, November 8, 11
  27. ||↵||1 = 1 Signal Model of Synthesis Sparsity d1 d2

    y = ↵ Tuesday, November 8, 11
  28. ||↵||1 = 1 Signal Model of Synthesis Sparsity d1 d2

    y = ↵ Tuesday, November 8, 11
  29. ||↵||1 = 1 Signal Model of Synthesis Sparsity d1 d2

    y = ↵ ↵? Tuesday, November 8, 11
  30. ||↵||1 = 1 sparsest solution Signal Model of Synthesis Sparsity

    d1 d2 y = ↵ ↵? Tuesday, November 8, 11
  31. Analysis Counterpart Tuesday, November 8, 11

  32. Analysis Counterpart d1 d2 d3 Tuesday, November 8, 11

  33. Analysis Counterpart d1 d2 d3 Tuesday, November 8, 11

  34. Analysis Counterpart d1 d2 d3 || D ⇤ x ||1

    = 1 Tuesday, November 8, 11
  35. Analysis Counterpart d1 d2 d3 || D ⇤ x ||1

    = 1 y = x Tuesday, November 8, 11
  36. Analysis Counterpart d1 d2 d3 || D ⇤ x ||1

    = 1 y = x Tuesday, November 8, 11
  37. Analysis Counterpart d1 d2 d3 || D ⇤ x ||1

    = 1 y = x x ? Tuesday, November 8, 11
  38. Behaviour of Solutions x?( , , D ) = argmin

    x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 Tuesday, November 8, 11
  39. piecewise a ne y 7! x ? Observations Behaviour of

    Solutions x?( , , D ) = argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 Tuesday, November 8, 11
  40. piecewise a ne y 7! x ? Observations Behaviour of

    Solutions x?( , , D ) = argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 P depends on sign(D ⇤ x ? (y)) and AP is a linear operator If [y, ¯ y] ⇢ P x ?(¯ y ) = AP x ?( y ) u y ¯ y P Tuesday, November 8, 11
  41. piecewise a ne y 7! x ? Observations 7! x

    ? piecewise a ne Scaling Behaviour of Solutions x?( , , D ) = argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 P depends on sign(D ⇤ x ? (y)) and AP is a linear operator If [y, ¯ y] ⇢ P x ?(¯ y ) = AP x ?( y ) u y ¯ y P Tuesday, November 8, 11
  42. piecewise a ne y 7! x ? Observations 7! x

    ? piecewise a ne Scaling D 7! x ? open question Dictionary Behaviour of Solutions x?( , , D ) = argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 P depends on sign(D ⇤ x ? (y)) and AP is a linear operator If [y, ¯ y] ⇢ P x ?(¯ y ) = AP x ?( y ) u y ¯ y P Tuesday, November 8, 11
  43. there exists a unique solution x ? such that ||x?

    x0 ||2 = O(||w||2) supp( D ⇤ x ?) ✓ supp( D ⇤ x0) then, for every x0 such that supp(D ⇤ x0) = I, If RC(I) < 1 and > ||w||2CI, RC ( I ) : function of the support Robustness of Estimators D ⇤ x0 D ⇤ x ? x?( , , D ) = argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 y = x0 + w Tuesday, November 8, 11
  44. Joint work with — Gabriel Peyr´ e (CEREMADE, Dauphine) —

    Charles Dossal (IMB, Bordeaux I) — Jalal Fadili (GREYC, ENSICAEN) Any questions ? Robust Sparse Analysis Regularization arXiv:1109.6222 Thanks Tuesday, November 8, 11