Off-the-Grid Sparse Super-Resolution

E34ded36efe4b7abb12510d4e525fee8?s=47 Gabriel Peyré
February 19, 2017

Off-the-Grid Sparse Super-Resolution

E34ded36efe4b7abb12510d4e525fee8?s=128

Gabriel Peyré

February 19, 2017
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  1. Gabriel Peyré Off-the-Grid Sparse Super-resolution Joint work with Vincent Duval

    & Quentin Denoyelle É C O L E N O R M A L E S U P É R I E U R E Outline • What is the Lasso • Lasso with an orthogonal design • From projected gradient to proximal grad • Optimality conditions and subgradients (L • Coordinate descent algorithm … with some demos www.numerical-tours.com
  2. Sparse Super-resolution Neural spikes (1D) Astrophysics (2D) Seismic imaging (1.5D)

    A B Fig. 2. Experimental validation of the m of the system designed to create a slope of the maximum intensity projection of i TIRF illumination. (Scale bar: 5 μm.) The ev illumination angle θ of two selected bead sponding fitting theoretical model (con depth (respectively 10 and 89 nm). (D) Dep [Boulanger et al. 2014] Single-molecule fluorescence (3-D) Recover pointwise sources from noisy low-resolution observations.
  3. Sparse Super-resolution Neural spikes (1D) Astrophysics (2D) Seismic imaging (1.5D)

    A B Fig. 2. Experimental validation of the m of the system designed to create a slope of the maximum intensity projection of i TIRF illumination. (Scale bar: 5 μm.) The ev illumination angle θ of two selected bead sponding fitting theoretical model (con depth (respectively 10 and 89 nm). (D) Dep [Boulanger et al. 2014] Single-molecule fluorescence (3-D) Recover pointwise sources from noisy low-resolution observations. Practice: Scalable algorithms? Theory: Rayleigh limit?
  4. Overview • Sparse Spikes Super-resolution • Robust Support Recovery •

    Asymptotic Positive Measure Recovery • Off-the-Grid Optimization Algorithms • Application: Laplace Inversion for TIRF Imaging
  5. Discrete measure: ma,x = PN i=1 ai xi , a

    2 RN , x 2 TN Super-resolution of Measures ma,x m Radon measure m on T = ⇢ (R/Z)d Rd .
  6. Discrete measure: ma,x = PN i=1 ai xi , a

    2 RN , x 2 TN Super-resolution of Measures ma,x m Radon measure m on T = ⇢ (R/Z)d Rd . y = (m) + w Linear measurements: '(x) 2 H ' continuous. (m) def. = Z T '(x)dm(x) 2 H
  7. Discrete measure: ma,x = PN i=1 ai xi , a

    2 RN , x 2 TN Super-resolution of Measures ma,x m Radon measure m on T = ⇢ (R/Z)d Rd . '(x) = ˜ '(· x) Deconvolution: ! Signal-dependent recovery criteria. y = 2/fc y = 0.5/fc '(0) y = (m) + w Linear measurements: '(x) 2 H ' continuous. (m) def. = Z T '(x)dm(x) 2 H
  8. Discrete measure: ma,x = PN i=1 ai xi , a

    2 RN , x 2 TN Super-resolution of Measures ma,x m Radon measure m on T = ⇢ (R/Z)d Rd . '(x) = ˜ '(· x) Deconvolution: ! Signal-dependent recovery criteria. y = 2/fc y = 0.5/fc '(0) Fourier: '(x) = (ei`x)fc `= fc 2 C2fc+1 y = (m) + w Linear measurements: '(x) 2 H ' continuous. (m) def. = Z T '(x)dm(x) 2 H
  9. Discrete measure: ma,x = PN i=1 ai xi , a

    2 RN , x 2 TN Super-resolution of Measures ma,x m Radon measure m on T = ⇢ (R/Z)d Rd . '(x) = ˜ '(· x) Deconvolution: ! Signal-dependent recovery criteria. y = 2/fc y = 0.5/fc '(0) Fourier: '(x) = (ei`x)fc `= fc 2 C2fc+1 Laplace: '(x) = e x· 2 H def. = L2(R+) y = (m) + w Linear measurements: '(x) 2 H ' continuous. (m) def. = Z T '(x)dm(x) 2 H
  10. Grid-free Sparse Recovery Grid-free regularization: total variation of measures: |m|(T)

    = sup R ⌘dm : ⌘ 2 C(T), ||⌘||1 6 1 |ma,x |(T) = ||a||`1 ma,x |m|(T) = R |f| = ||f||L1 dm(x) = f(x)dx
  11. Grid-free Sparse Recovery Grid-free regularization: total variation of measures: |m|(T)

    = sup R ⌘dm : ⌘ 2 C(T), ||⌘||1 6 1 |ma,x |(T) = ||a||`1 ma,x |m|(T) = R |f| = ||f||L1 dm(x) = f(x)dx min m 1 2 || (m) y||2 + |m|(T) (P (y)) Sparse recovery: (P0(y)) min m {|m|(T) ; m = y} ! 0+
  12. Grid-free Sparse Recovery Grid-free regularization: total variation of measures: |m|(T)

    = sup R ⌘dm : ⌘ 2 C(T), ||⌘||1 6 1 |ma,x |(T) = ||a||`1 ma,x |m|(T) = R |f| = ||f||L1 dm(x) = f(x)dx [Fischer Jerome, 1974] If dim(Im( )) < +1, 9(a, x) 2 RN ⇥ TN with N 6 dim(Im( )) such that ma,x is a solution to P (y). Proposition: min m 1 2 || (m) y||2 + |m|(T) (P (y)) Sparse recovery: (P0(y)) min m {|m|(T) ; m = y} ! 0+
  13. Grid-free Sparse Recovery Grid-free regularization: total variation of measures: |m|(T)

    = sup R ⌘dm : ⌘ 2 C(T), ||⌘||1 6 1 |ma,x |(T) = ||a||`1 ma,x |m|(T) = R |f| = ||f||L1 dm(x) = f(x)dx [Fischer Jerome, 1974] If dim(Im( )) < +1, 9(a, x) 2 RN ⇥ TN with N 6 dim(Im( )) such that ma,x is a solution to P (y). Proposition: min m 1 2 || (m) y||2 + |m|(T) (P (y)) Sparse recovery: (P0(y)) min m {|m|(T) ; m = y} ! 0+ Other approaches: Greedy (MP/OMP/etc.) Prony (MUSIC/FRI/etc.) similar to Frank-Wolfe better/less general.
  14. (P0(y)) Robustness and Support-stability = 0.55/fc = 0.45/fc = 0.1/fc

    = 0.3/fc min m {|m|(T) ; m = y} Low-pass filter supp( ˆ ') = [ fc, fc]. When is m0 solution of P0( m0) ? = mini6=j |xi xj |
  15. (P0(y)) Robustness and Support-stability = 0.55/fc = 0.45/fc = 0.1/fc

    = 0.3/fc min m {|m|(T) ; m = y} Low-pass filter supp( ˆ ') = [ fc, fc]. When is m0 solution of P0( m0) ? Theorem: [Cand` es, Fernandez G.] > 1.26 fc ) m0 solves P0( m0). = mini6=j |xi xj |
  16. (P0(y)) Robustness and Support-stability = 0.55/fc = 0.45/fc = 0.1/fc

    = 0.3/fc min m {|m|(T) ; m = y} Low-pass filter supp( ˆ ') = [ fc, fc]. are solutions of P ( m0 + w)? How close to m0 When is m0 solution of P0( m0) ? Theorem: [Cand` es, Fernandez G.] > 1.26 fc ) m0 solves P0( m0). = mini6=j |xi xj | ! [Cand` es, Fernandez-G. 2012] ! [Fernandez-G.][de Castro 2012] Support approximation:
  17. (P0(y)) Robustness and Support-stability = 0.55/fc = 0.45/fc = 0.1/fc

    = 0.3/fc min m {|m|(T) ; m = y} Low-pass filter supp( ˆ ') = [ fc, fc]. are solutions of P ( m0 + w)? How close to m0 When is m0 solution of P0( m0) ? Theorem: [Cand` es, Fernandez G.] > 1.26 fc ) m0 solves P0( m0). = mini6=j |xi xj | ! [Cand` es, Fernandez-G. 2012] ! [Fernandez-G.][de Castro 2012] Support approximation: General kernels? Support recovery? No separation?
  18. Overview • Sparse Spikes Super-resolution • Robust Support Recovery •

    Asymptotic Positive Measure Recovery • Off-the-Grid Optimization Algorithms • Application: Laplace Inversion for TIRF Imaging
  19. = 1/fc = 0.6/fc Limit Certificate min m |m|(T) +

    1 2 || m y||2 P (y) Proposition: ⌘ ⌘ ⌘ def. = 1 ⇤(y m ) m solves (P (y)) , ⌘ 2 @|m |(T) , |⌘ | 6 1 and ⌘ (xi) = sign(ai)
  20. = 1/fc = 0.6/fc Limit Certificate min m |m|(T) +

    1 2 || m y||2 P (y) ⌘0 def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), ||⌘||1 6 1. Proposition: ⌘ ⌘ ⌘ def. = 1 ⇤(y m ) m solves (P (y)) , ⌘ 2 @|m |(T) , |⌘ | 6 1 and ⌘ (xi) = sign(ai) Theorem: m ! m0 = ma,x If ( , ||w||/ ) ! 0, then ⌘ ! ⌘0.
  21. −1 1 η 0 η V ⌘0 6= ⌘V −1

    1 η 0 η V ⌘0 = ⌘V Vanishing Derivative Pre-certificate Input measure: m0 = ma,x. ⌘0 def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), ||⌘||1 6 1. ⌘V def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), 8 i, ⌘0(xi) = 0.
  22. −1 1 η 0 η V ⌘0 6= ⌘V −1

    1 η 0 η V ⌘0 = ⌘V Vanishing Derivative Pre-certificate Input measure: m0 = ma,x. ⌘0 def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), ||⌘||1 6 1. ⌘V def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), 8 i, ⌘0(xi) = 0. ⌘V = P i ai h'(xi), '(·)i + bi h'0(xi), '(·)i ! 2N unknown (a, b), 2N equations
  23. −1 1 η 0 η V ⌘0 6= ⌘V −1

    1 η 0 η V ⌘0 = ⌘V Vanishing Derivative Pre-certificate Input measure: m0 = ma,x. ⌘0 def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), ||⌘||1 6 1. ⌘V def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), 8 i, ⌘0(xi) = 0. Theorem: ⌘V 2 ND(m0) =) ⌘V = ⌘0 () Non-degenerate certificate: ⌘ 2 ND(ma,x) : 8 t / 2 {x1, . . . , xN }, |⌘(t)| < 1 and 8 i, ⌘00(xi) 6= 0 ⌘V = P i ai h'(xi), '(·)i + bi h'0(xi), '(·)i ! 2N unknown (a, b), 2N equations
  24. Support Stability Theorem Theorem: the solution of P (y) for

    y = (m0) + w is for (||w||/ , ) = O(1), [Duval, Peyr´ e 2014] If ⌘V 2 ND(m0) for m0 = ma,x, then m = PN i=1 a? i x? i where ||(x, a) (x? , a?)|| = O(||w||) ||w|| Stable x
  25. Support Stability Theorem Theorem: the solution of P (y) for

    y = (m0) + w is for (||w||/ , ) = O(1), [Duval, Peyr´ e 2014] If ⌘V 2 ND(m0) for m0 = ma,x, then m = PN i=1 a? i x? i where ||(x, a) (x? , a?)|| = O(||w||) ||w|| Stable x ||w|| Unstable x
  26. When is Non-degenerate ? ⌘V ⌘V ⌘V ⌘V Input measure:

    '(0) . . . ˆ '(0) = 1[ fc,fc] m0 = ma, x, ! 0 = 1/fc [Cand` es, F. Granda] > 1.3
  27. When is Non-degenerate ? ⌘V ⌘V ⌘V ⌘V Input measure:

    '(0) . . . ˆ '(0) = 1[ fc,fc] ⌘V ⌘V ⌘V '(0) m0 = ma, x, ! 0 = 1/fc [Cand` es, F. Granda] > 1.3
  28. When is Non-degenerate ? ⌘V ⌘V ⌘V ⌘V Input measure:

    '(0) . . . ˆ '(0) = 1[ fc,fc] ⌘V ⌘V ⌘V '(0) m0 = ma, x, ! 0 = 1/fc [Cand` es, F. Granda] > 1.3 Valid for: Theorem: [Tang, Recht, 2013][Denoyelle 2017] 9C, ( > C ) =) (⌘V is non degenerate) '(x) = e (x ·)2/ 2 '(x) = 1 2+(x ·)2
  29. Overview • Sparse Spikes Super-resolution • Robust Support Recovery •

    Asymptotic Positive Measure Recovery • Off-the-Grid Optimization Algorithms • Application: Laplace Inversion for TIRF Imaging
  30. Super-resolution for Positive Measures Theorem: let and [de Castro et

    al. 2011] ! m0 is recovered when there is no noise. ⌘S(t) = 1 ⇢ QN i=1 sin(⇡(t xi))2 Input measure: m0 = ma,x where a 2 RN + . '(x) = (ei`x)fc `= fc for N 6 fc and ⇢ small enough, ⌘S 2 ND(m0). -1 1 ⌘S -1 1 ⌘S
  31. Super-resolution for Positive Measures Theorem: let and [de Castro et

    al. 2011] ! m0 is recovered when there is no noise. ⌘S(t) = 1 ⇢ QN i=1 sin(⇡(t xi))2 Input measure: m0 = ma,x where a 2 RN + . '(x) = (ei`x)fc `= fc for N 6 fc and ⇢ small enough, ⌘S 2 ND(m0). ! Extends to sampled Gaussian [Schiebinger et al 2015] -1 1 ⌘S -1 1 ⌘S
  32. Super-resolution for Positive Measures Theorem: let and [de Castro et

    al. 2011] ! m0 is recovered when there is no noise. ⌘S(t) = 1 ⇢ QN i=1 sin(⇡(t xi))2 Input measure: m0 = ma,x where a 2 RN + . [Morgenshtern, Cand` es, 2015] discrete `1 robustness. [Demanet, Nguyen, 2015] discrete `0 robustness. ! behavior as 8 i, xi ! 0 ? '(x) = (ei`x)fc `= fc for N 6 fc and ⇢ small enough, ⌘S 2 ND(m0). ! Extends to sampled Gaussian [Schiebinger et al 2015] -1 1 ⌘S -1 1 ⌘S
  33. Super-resolution for Positive Measures Theorem: let and [de Castro et

    al. 2011] ! m0 is recovered when there is no noise. ! noise robustness of support recovery ? ⌘S(t) = 1 ⇢ QN i=1 sin(⇡(t xi))2 Input measure: m0 = ma,x where a 2 RN + . [Morgenshtern, Cand` es, 2015] discrete `1 robustness. [Demanet, Nguyen, 2015] discrete `0 robustness. ! behavior as 8 i, xi ! 0 ? '(x) = (ei`x)fc `= fc for N 6 fc and ⇢ small enough, ⌘S 2 ND(m0). ! Extends to sampled Gaussian [Schiebinger et al 2015] -1 1 ⌘S -1 1 ⌘S
  34. Asymptotic of Vanishing Certificate 1 ⌘V Vanishing Derivative pre-certificate: ⌘V

    def. = argmin ⌘= ⇤p ||p|| m0 = ma, x where ! 0 s.t. 8 i, ⇢ ⌘( xi) = 1, ⌘0( xi) = 0. Valid only in 1-D, i.e. T = R or T = R/Z.
  35. Asymptotic of Vanishing Certificate 1 ⌘V 1 1 1 ⌘V

    ⌘V ⌘W Vanishing Derivative pre-certificate: ⌘V def. = argmin ⌘= ⇤p ||p|| m0 = ma, x where ! 0 s.t. 8 i, ⇢ ⌘( xi) = 1, ⌘0( xi) = 0. Valid only in 1-D, i.e. T = R or T = R/Z.
  36. Asymptotic of Vanishing Certificate 1 ⌘V 1 1 1 ⌘V

    ⌘V ⌘W s.t. ⇢ ⌘(0) = 1, ⌘0(0) = . . . = ⌘(2N 1)(0) = 0. Asymptotic pre-certificate: ⌘W def. = argmin ⌘= ⇤p ||p|| Vanishing Derivative pre-certificate: ⌘V def. = argmin ⌘= ⇤p ||p|| ! 0 m0 = ma, x where ! 0 s.t. 8 i, ⇢ ⌘( xi) = 1, ⌘0( xi) = 0. Valid only in 1-D, i.e. T = R or T = R/Z.
  37. Asymptotic Certificate 1 1 1 1 ⌘V = ⌘W ⌘W

    ⌘W ⌘W N = 1 N = 2 N = 3 N = 4 (2N 1)-Non degenerate: () ⌘W (2N)(0) 6= 0 ⇢ 8 t 6= 0, |⌘W (t)| < 1 ⌘W 2 NDN
  38. Asymptotic Certificate 1 1 1 1 ⌘V = ⌘W ⌘W

    ⌘W ⌘W N = 1 N = 2 N = 3 N = 4 (2N 1)-Non degenerate: () ⌘W (2N)(0) 6= 0 ⇢ 8 t 6= 0, |⌘W (t)| < 1 ⌘W 2 NDN Lemma: ! ⌘W govern stability as ! 0. If ⌘W 2 NDN , 9 0 > 0, 8 < 0, ⌘V 2 ND(m x,a)
  39. Asymptotic Robustness ||w || N = 2 ||w || N

    = 1 Theorem: the solution of P (y) for y = (m0) + w is for w , w 2N 1 , 2N 1 = O(1) PN i=1 a? i x? i If ⌘W 2 NDN , letting m0 = ma, x, then where ||(x, a) (x ? , a ?)|| = O ✓ ||w|| + 2N 1 ◆ [Denoyelle, D., P. 2015] ! signal/noise ⇠ 1/t2N 1 for super-resolution.
  40. Asymptotic Robustness ||w || N = 2 ||w || N

    = 1 Theorem: the solution of P (y) for y = (m0) + w is for w , w 2N 1 , 2N 1 = O(1) PN i=1 a? i x? i If ⌘W 2 NDN , letting m0 = ma, x, then where ||(x, a) (x ? , a ?)|| = O ✓ ||w|| + 2N 1 ◆ [Denoyelle, D., P. 2015] ! signal/noise ⇠ 1/t2N 1 for super-resolution. ! Extends to clusters: ⇠ t ⇠ t ⇠ t
  41. Asymptotic Robustness ||w || N = 2 ||w || N

    = 1 Theorem: the solution of P (y) for y = (m0) + w is for w , w 2N 1 , 2N 1 = O(1) PN i=1 a? i x? i If ⌘W 2 NDN , letting m0 = ma, x, then where ||(x, a) (x ? , a ?)|| = O ✓ ||w|| + 2N 1 ◆ [Denoyelle, D., P. 2015] ! signal/noise ⇠ 1/t2N 1 for super-resolution. ! Extends to clusters: ⇠ t ⇠ t ⇠ t [Poon, Peyr´ e 2017] signal/noise ⇠ 1/t4 ! Extends in dimension > 2 for N = 2 N = 2 N = 3 N = 4 N = 5 N = 2 N = 3 N = 4 N = N = 2 N = 2 N = 2 N = N = 1 N = 2 N = 2 N = Gaussian MEG-EEG
  42. When is Non-degenerate ? ⌘W Proposition: one has ⌘(2N) W

    (0) < 0. ! “locally” non-degenerate. (for convolutions)
  43. When is Non-degenerate ? ⌘W Proposition: one has ⌘(2N) W

    (0) < 0. ! “locally” non-degenerate. ˆ ' ⌘W ⌘W ⌘W N = 2 N = 3 N = 4 '(0) (for convolutions)
  44. Gaussian Deconvolution Gaussian convolution: Proposition: ⌘W (x) = e x2

    4 2 N 1 X k=0 (x/2 )2k k! In particular, ⌘W is non-degenerate. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.2 0 0.2 0.4 0.6 0.8 1 1 1 1 1 ! Gaussian deconvolution is support-stable. N = 1 N = 2 N = 3 N = 4 ⌘W ⌘W ⌘W ⌘W '(x) = e |x ·|2 2 2 (m) def. = Z '(x)dm(x) '(0)
  45. Overview • Sparse Spikes Super-resolution • Robust Support Recovery •

    Asymptotic Positive Measure Recovery • Off-the-Grid Optimization Algorithms • Application: Laplace Inversion for TIRF Imaging
  46. Algorithms min m |m|(T) + 1 2 || m y||2

    P (y) D (y) = sup || ⇤p||1 61 hp, yi 2 ||p||2 Primal Dual ! 1-dimensional ! 1-many constraints
  47. Algorithms min m |m|(T) + 1 2 || m y||2

    P (y) D (y) = sup || ⇤p||1 61 hp, yi 2 ||p||2 Primal Dual ! 1-dimensional ! 1-many constraints Algorithms: “-”: artifacts, slow. ! Lasso/Basis-Pursuit: discretize m. [Chen, Donoho, Saunders, 99] [Tibshirani, 96]
  48. Algorithms min m |m|(T) + 1 2 || m y||2

    P (y) D (y) = sup || ⇤p||1 61 hp, yi 2 ||p||2 Primal Dual ! 1-dimensional ! 1-many constraints Algorithms: “-”: only works for Fourier. “-”: artifacts, slow. ! SDP-represent D . [Cand` es, Fernandez-G. 2012] ! Lasso/Basis-Pursuit: discretize m. [Chen, Donoho, Saunders, 99] [Tibshirani, 96]
  49. Algorithms min m |m|(T) + 1 2 || m y||2

    P (y) D (y) = sup || ⇤p||1 61 hp, yi 2 ||p||2 Primal Dual ! 1-dimensional ! 1-many constraints Algorithms: “-”: only works for Fourier. “-”: artifacts, slow. ! SDP-represent D . [Cand` es, Fernandez-G. 2012] [Bredies,Pikkarainen 2010] ! Frank-Wolfe on P . ! Lasso/Basis-Pursuit: discretize m. [Chen, Donoho, Saunders, 99] [Tibshirani, 96]
  50. Algorithms min m |m|(T) + 1 2 || m y||2

    P (y) D (y) = sup || ⇤p||1 61 hp, yi 2 ||p||2 Primal Dual ! 1-dimensional ! 1-many constraints Algorithms: “-”: only works for Fourier. “-”: artifacts, slow. ! SDP-represent D . [Cand` es, Fernandez-G. 2012] [Bredies,Pikkarainen 2010] ! Frank-Wolfe on P . ! Lasso/Basis-Pursuit: discretize m. [Chen, Donoho, Saunders, 99] [Tibshirani, 96] Competitors: Prony’s methods (MUSIC, ESPRIT, FRI). “+”: always works when w = 0, less sensitive to sign. “-”: only for specific ' (e.g. Fourier), non trivial in 2-D.
  51. y zk Iterative Soft Thresholding Algorithm Computation grid z =

    (zk)K k=1 . Basis-pursuit / Lasso: ¯a def. = P k ak'(zk) ! Force m = ma,z in (P (y)). min a2RK 1 2 ||y ¯a||2 + ||a||1
  52. y zk Iterative Soft Thresholding Algorithm Computation grid z =

    (zk)K k=1 . Basis-pursuit / Lasso: Forward-Backward [Lions, Mercier 1979] (a.k.a. ISTA) ¯a def. = P k ak'(zk) ! Force m = ma,z in (P (y)). min a2RK 1 2 ||y ¯a||2 + ||a||1 S (a) a a(`+1) def. = S⌧ (a(`) ⌧ ¯⇤(¯a(`) y)) ⌧ < 2 ||¯||2
  53. y zk Iterative Soft Thresholding Algorithm Computation grid z =

    (zk)K k=1 . Basis-pursuit / Lasso: m0 ⇤y Forward-Backward [Lions, Mercier 1979] (a.k.a. ISTA) ¯a def. = P k ak'(zk) ! Force m = ma,z in (P (y)). ! Approximate Diracs by density ! post-processing. min a2RK 1 2 ||y ¯a||2 + ||a||1 S (a) a a(`+1) def. = S⌧ (a(`) ⌧ ¯⇤(¯a(`) y)) ⌧ < 2 ||¯||2 ! Slow for large K, ¯ ill-posed.
  54. Frank-Wolfe Based Methods Initialize: (a(0), x(0)) def. = (;, ;)

    min m 1 2 || m y||2 + |m|(T) , min a,x 1 2 || ma,x y||2 + |a|1 ` 0
  55. Frank-Wolfe Based Methods Initialize: (a(0), x(0)) def. = (;, ;)

    min m 1 2 || m y||2 + |m|(T) , min a,x 1 2 || ma,x y||2 + |a|1 ⌘(0) x? Grid generation: where ⌘(`) def. = 1 ⇤(y ma(`),x(`) ) x? def. = argmaxx |⌘(`)(x)| ` 0
  56. Frank-Wolfe Based Methods Initialize: (a(0), x(0)) def. = (;, ;)

    min m 1 2 || m y||2 + |m|(T) , min a,x 1 2 || ma,x y||2 + |a|1 Grid deformation: convex non-convex (a(`+1), x(`+1)) def. = argmin a,x 1 2 || ma,x y|| + ||a||1 BFGS initialized at ([a(`), 0], [x(`), x?]) ⌘(0) x? Grid generation: where ⌘(`) def. = 1 ⇤(y ma(`),x(`) ) x? def. = argmaxx |⌘(`)(x)| ` 0
  57. Frank-Wolfe Based Methods Initialize: (a(0), x(0)) def. = (;, ;)

    min m 1 2 || m y||2 + |m|(T) , min a,x 1 2 || ma,x y||2 + |a|1 Grid deformation: convex non-convex (a(`+1), x(`+1)) def. = argmin a,x 1 2 || ma,x y|| + ||a||1 BFGS initialized at ([a(`), 0], [x(`), x?]) ⌘(3) ⌘(0) x? Grid generation: where ⌘(`) def. = 1 ⇤(y ma(`),x(`) ) x? def. = argmaxx |⌘(`)(x)| ⌘(1) x? ⌘(2) x? ` 0
  58. Frank-Wolfe Based Methods Initialize: (a(0), x(0)) def. = (;, ;)

    ! Without moving x?: Frank-Wolfe, convergent. ! Non-convex update: still convergent, surprisingly e cient. min m 1 2 || m y||2 + |m|(T) , min a,x 1 2 || ma,x y||2 + |a|1 Grid deformation: convex non-convex (a(`+1), x(`+1)) def. = argmin a,x 1 2 || ma,x y|| + ||a||1 BFGS initialized at ([a(`), 0], [x(`), x?]) ⌘(3) ⌘(0) x? Grid generation: where ⌘(`) def. = 1 ⇤(y ma(`),x(`) ) x? def. = argmaxx |⌘(`)(x)| ⌘(1) x? ⌘(2) x? ` 0
  59. Overview • Sparse Spikes Super-resolution • Robust Support Recovery •

    Asymptotic Positive Measure Recovery • Off-the-Grid Optimization Algorithms • Application: Laplace Inversion for TIRF Imaging
  60. Laplace Transform Inversion Laplace transform: x = 2 x =

    20 t (m1) (m2) x m1 t m2 x [with E. Soubies] (m) def. = Z '(x)dm(x) '(x) = e x· '(x)
  61. Laplace Transform Inversion Laplace transform: x = 2 x =

    20 t (m1) (m2) x m1 t m2 x [with E. Soubies] (m) def. = Z '(x)dm(x) '(x) = e x· '(x) Total internal reflection fluorescence microscopy (TIRFM) [Boulanger et al. 2014] varying the azimuth φ during the exposure time and can be modeled by the following expression: gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  θ − α Ω=cos θ  f À z Á dαdzdφ; where fðzÞ is the density of fluorophores in the medium con- volved by the emission point spread function and ρð · Þ represents slope of the glass slide recovered (Fig. 2D), the latter falling within the confidence interval deducted from the accuracy of the mea- surement of the different characteristic dimensions of the sample. Finally, from the dispersion of the estimated depth around the average slope (Fig. 2D), we can conclude that the localization precision obtained with this approach is higher than the corre- sponding precision given by estimating the location of the beads in the WF image stack as already mentioned (17). Estimating the 3D density of fluorophores convolved by the emission point spread function then would simply boil down to inverting the linear system. Some care has to be taken when inverting such system, as the inverse problem is at best badly con- ditioned. Nevertheless, constraints can be imposed to the solution such as positivity, and, in the case of time-lapse acquisitions, a multiframe regularization can be used in addition to the spatial and temporal regularization smoothness to solve the reconstruction problem. Moreover, to be effective, such a positivity constraint requires a correct knowledge of the background level. As a conse- quence, for each multiangle image stack, a background image is obtained by driving the beam out of the objective. Given that several convex constraints have to be satisfied at the same time, we propose to rely on a flavor of the PPXA algorithm (26) to estimate the tridimensional density of fluorophores (Fig. S4). More detailed information on how noise, object depth, and the required number of angles can be taken into account is discussed in SI Imaging Model and Reconstruction and Fig. S5. Finally, to take into account the variations of the medium index, we select an effective index within a predefined range by minimizing the reconstruction error at each pixel under a spatial smoothness constraint (Fig. S6). It is worth noting that the computation time for the reconstruction on 10 planes from a stack 512 × 512 images corresponding to 21 in- cidence angles ranges from 1 to 5 min depending on the number of iterations. Imaging in Vitro and in Vivo Actin Assembly. The proposed multi- angle TIRF image reconstruction approach was then tested on complex samples such as actin network architectures for which spatial resolution and dynamics remain an issue. We first chal- lenged the spatial organization of actin nucleation geometry A B C D Fig. 2. Experimental validation of the multiangle TIRF model. (A) Schema of the system designed to create a slope of fluorescent beads. (B) Overlay of the maximum intensity projection of image stack acquired with WF and TIRF illumination. (Scale bar: 5 μm.) The evolution of the intensity versus the illumination angle θ of two selected beads are plotted in C with the corre- sponding fitting theoretical model (continuous line) for their estimated depth (respectively 10 and 89 nm). (D) Depth of all of the beads estimated by fitting the theoretical TIRF model (in red) and the depth of the same beads estimated by fitting a Gaussian model in the WF image stack (in green). BIOPHYSICS AND COMPUTATIONAL BIOLOGY slope of the glass slide recovered (Fig. 2D), the latter falling within the confidence interval deducted from the accuracy of the mea- surement of the different characteristic dimensions of the sample. Finally, from the dispersion of the estimated depth around the average slope (Fig. 2D), we can conclude that the localization precision obtained with this approach is higher than the corre- sponding precision given by estimating the location of the beads in the WF image stack as already mentioned (17). Estimating the 3D density of fluorophores convolved by the emission point spread function then would simply boil down to inverting the linear system. Some care has to be taken when inverting such system, as the inverse problem is at best badly con- ditioned. Nevertheless, constraints can be imposed to the solution such as positivity, and, in the case of time-lapse acquisitions, a multiframe regularization can be used in addition to the spatial and temporal regularization smoothness to solve the reconstruction problem. Moreover, to be effective, such a positivity constraint requires a correct knowledge of the background level. As a conse- quence, for each multiangle image stack, a background image is obtained by driving the beam out of the objective. Given that several convex constraints have to be satisfied at the same time, we C D perimental validation of the multiangle TIRF model. (A) Schema em designed to create a slope of fluorescent beads. (B) Overlay mum intensity projection of image stack acquired with WF and ✓(t) y(t) ! multiple angles ✓(t). light depth x cell y(t) = m(t) ✓(t)
  62. Laplace Transform Inversion Laplace transform: x = 2 x =

    20 t (m1) (m2) x m1 t m2 x [with E. Soubies] N = 1 N = 3 ⌘W ⌘W ¯ x = 2 ¯ x = 20 Non-translation-invariant operator ¯ x x1 x2 ! ⌘W depends on ¯ x! Proposition: In particular, ⌘W is non-degenerate. ⌘W (x) = 1 ✓ x ¯ x x + ¯ x ◆2N (m) def. = Z '(x)dm(x) '(x) = e x· '(x) Total internal reflection fluorescence microscopy (TIRFM) [Boulanger et al. 2014] varying the azimuth φ during the exposure time and can be modeled by the following expression: gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  θ − α Ω=cos θ  f À z Á dαdzdφ; where fðzÞ is the density of fluorophores in the medium con- volved by the emission point spread function and ρð · Þ represents slope of the glass slide recovered (Fig. 2D), the latter falling within the confidence interval deducted from the accuracy of the mea- surement of the different characteristic dimensions of the sample. Finally, from the dispersion of the estimated depth around the average slope (Fig. 2D), we can conclude that the localization precision obtained with this approach is higher than the corre- sponding precision given by estimating the location of the beads in the WF image stack as already mentioned (17). Estimating the 3D density of fluorophores convolved by the emission point spread function then would simply boil down to inverting the linear system. Some care has to be taken when inverting such system, as the inverse problem is at best badly con- ditioned. Nevertheless, constraints can be imposed to the solution such as positivity, and, in the case of time-lapse acquisitions, a multiframe regularization can be used in addition to the spatial and temporal regularization smoothness to solve the reconstruction problem. Moreover, to be effective, such a positivity constraint requires a correct knowledge of the background level. As a conse- quence, for each multiangle image stack, a background image is obtained by driving the beam out of the objective. Given that several convex constraints have to be satisfied at the same time, we propose to rely on a flavor of the PPXA algorithm (26) to estimate the tridimensional density of fluorophores (Fig. S4). More detailed information on how noise, object depth, and the required number of angles can be taken into account is discussed in SI Imaging Model and Reconstruction and Fig. S5. Finally, to take into account the variations of the medium index, we select an effective index within a predefined range by minimizing the reconstruction error at each pixel under a spatial smoothness constraint (Fig. S6). It is worth noting that the computation time for the reconstruction on 10 planes from a stack 512 × 512 images corresponding to 21 in- cidence angles ranges from 1 to 5 min depending on the number of iterations. Imaging in Vitro and in Vivo Actin Assembly. The proposed multi- angle TIRF image reconstruction approach was then tested on complex samples such as actin network architectures for which spatial resolution and dynamics remain an issue. We first chal- lenged the spatial organization of actin nucleation geometry A B C D Fig. 2. Experimental validation of the multiangle TIRF model. (A) Schema of the system designed to create a slope of fluorescent beads. (B) Overlay of the maximum intensity projection of image stack acquired with WF and TIRF illumination. (Scale bar: 5 μm.) The evolution of the intensity versus the illumination angle θ of two selected beads are plotted in C with the corre- sponding fitting theoretical model (continuous line) for their estimated depth (respectively 10 and 89 nm). (D) Depth of all of the beads estimated by fitting the theoretical TIRF model (in red) and the depth of the same beads estimated by fitting a Gaussian model in the WF image stack (in green). BIOPHYSICS AND COMPUTATIONAL BIOLOGY slope of the glass slide recovered (Fig. 2D), the latter falling within the confidence interval deducted from the accuracy of the mea- surement of the different characteristic dimensions of the sample. Finally, from the dispersion of the estimated depth around the average slope (Fig. 2D), we can conclude that the localization precision obtained with this approach is higher than the corre- sponding precision given by estimating the location of the beads in the WF image stack as already mentioned (17). Estimating the 3D density of fluorophores convolved by the emission point spread function then would simply boil down to inverting the linear system. Some care has to be taken when inverting such system, as the inverse problem is at best badly con- ditioned. Nevertheless, constraints can be imposed to the solution such as positivity, and, in the case of time-lapse acquisitions, a multiframe regularization can be used in addition to the spatial and temporal regularization smoothness to solve the reconstruction problem. Moreover, to be effective, such a positivity constraint requires a correct knowledge of the background level. As a conse- quence, for each multiangle image stack, a background image is obtained by driving the beam out of the objective. Given that several convex constraints have to be satisfied at the same time, we C D perimental validation of the multiangle TIRF model. (A) Schema em designed to create a slope of fluorescent beads. (B) Overlay mum intensity projection of image stack acquired with WF and ✓(t) y(t) ! multiple angles ✓(t). light depth x cell y(t) = m(t) ✓(t)
  63. Spikes Detection Benchmark R = TP TP + FN P

    = TP TP + FP ||w|| FW+BFGS ISTA ISTA+Postprocessing R P R P
  64. Palm-TIRF Hybridisation x x y z x x y z

    Shot #1 ... Shot #2 BLASSO ... Kernel: '(x, y, z) = k(||(x, y) ·||)e z· fuse Gaussian Laplace Gaussian Laplace (joint work with Emmanuel Soubi` es)
  65. Conclusion varying the azimuth φ during the ex modeled by

    the following expression: gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  θ Ω= A B C D Fig. 2. Experimental validation of the mult of the system designed to create a slope of of the maximum intensity projection of imag TIRF illumination. (Scale bar: 5 μm.) The evolu illumination angle θ of two selected beads a sponding fitting theoretical model (continu depth (respectively 10 and 89 nm). (D) Depth o fitting the theoretical TIRF model (in red) an estimated by fitting a Gaussian model in the [Boulanger et al. 2014] Super-resolution should be o↵-the-grid! Theory: `2 errors meaningless. 6= compressed sensing. Super-resolution , signal/noise vs t
  66. Conclusion varying the azimuth φ during the ex modeled by

    the following expression: gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  θ Ω= A B C D Fig. 2. Experimental validation of the mult of the system designed to create a slope of of the maximum intensity projection of imag TIRF illumination. (Scale bar: 5 μm.) The evolu illumination angle θ of two selected beads a sponding fitting theoretical model (continu depth (respectively 10 and 89 nm). (D) Depth o fitting the theoretical TIRF model (in red) an estimated by fitting a Gaussian model in the [Boulanger et al. 2014] Super-resolution should be o↵-the-grid! Theory: `2 errors meaningless. 6= compressed sensing. Practice: Adaptive grid refinement. Non-convex step crucial. Surprisingly e cient. Super-resolution , signal/noise vs t
  67. Conclusion Open problem: other regularizations (e.g. piecewise constant) ? see

    [Chambolle, Duval, Peyr´ e, Poon 2016] for TV denoising. varying the azimuth φ during the ex modeled by the following expression: gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  θ Ω= A B C D Fig. 2. Experimental validation of the mult of the system designed to create a slope of of the maximum intensity projection of imag TIRF illumination. (Scale bar: 5 μm.) The evolu illumination angle θ of two selected beads a sponding fitting theoretical model (continu depth (respectively 10 and 89 nm). (D) Depth o fitting the theoretical TIRF model (in red) an estimated by fitting a Gaussian model in the [Boulanger et al. 2014] Super-resolution should be o↵-the-grid! Theory: `2 errors meaningless. 6= compressed sensing. Practice: Adaptive grid refinement. Non-convex step crucial. Surprisingly e cient. Super-resolution , signal/noise vs t