Towards a Mathematical Theory of Super-resolution
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1 Towards a Mathematical Theory of Super-resolution Carlos Fernandez-Granda Information Theory Forum, Information Systems Laboratory, Stanford 10/18/2013
2 Acknowledgements This work was supported by a Fundación La Caixa Fellowship and a Fundación Caja Madrid Fellowship Collaborator : Emmanuel Candès (Department of Mathematics and of Statistics, Stanford)
3 Motivation : Limits of resolution in imaging The resolving power of lenses, however perfect, is limited (Lord Rayleigh) Diraction imposes a fundamental limit on the resolution of optical systems
4 Motivation Similar problems arise in electronic imaging, signal processing, radar, spectroscopy, medical imaging, astronomy, geophysics, etc.
5 Motivation Similar problems arise in electronic imaging, signal processing, radar, spectroscopy, medical imaging, astronomy, geophysics, etc. Signals of interest are often point sources : celestial bodies (astronomy), line spectra (signal processing), molecules (uorescence microscopy),...
6 Super-resolution Aim : estimating ne-scale structure from low-resolution data
7 Super-resolution Aim : estimating ne-scale structure from low-resolution data Equivalently, extrapolating the high end of the spectrum
8 Mathematical model Signal : superposition of Dirac measures with support T x = j a j δ tj a j C, t j T [0, 1]
9 Mathematical model Signal : superposition of Dirac measures with support T x = j a j δ tj a j C, t j T [0, 1] Measurements : low-pass ltering with cut-o frequency f c y = F c x y(k) = 1 0 (vector of low-pass Fourier coecients) e i 2πkt x (dt) = a j e i 2πkt j, k Z, k f c j
10 Equivalent problem : line-spectra estimation Swapping time and frequency Signal : superposition of sinusoids x(t) = j a j e i 2πf j t a j C, f j T [0, 1]
11 Equivalent problem : line-spectra estimation Swapping time and frequency Signal : superposition of sinusoids x(t) = j a j e i 2πf j t a j C, f j T [0, 1] Measurements : equispaced samples x(1), x(2), x(3),... x(n)
12 Equivalent problem : line-spectra estimation Swapping time and frequency Signal : superposition of sinusoids x(t) = j a j e i 2πf j t a j C, f j T [0, 1] Measurements : equispaced samples x(1), x(2), x(3),... x(n) Classical problem in signal processing
13 Can you nd the spikes?
14 Can you nd the spikes?
15 Prior art Based on Prony's method : MUSIC, ESPRIT, matrix pencil,...
16 Prior art Based on Prony's method : MUSIC, ESPRIT, matrix pencil,... An estimate of the number of spikes is needed
17 Prior art Based on Prony's method : MUSIC, ESPRIT, matrix pencil,... An estimate of the number of spikes is needed Do not apply to multiple dimensions (not even 2D)
18 Prior art Based on Prony's method : MUSIC, ESPRIT, matrix pencil,... An estimate of the number of spikes is needed Do not apply to multiple dimensions (not even 2D) Noise must be Gaussian and white
19 Prior art Based on Prony's method : MUSIC, ESPRIT, matrix pencil,... An estimate of the number of spikes is needed Do not apply to multiple dimensions (not even 2D) Noise must be Gaussian and white Incorporating additional structure is dicult (dierent point-spread functions, assumptions on the signal)
20 Prior art Based on Prony's method : MUSIC, ESPRIT, matrix pencil,... An estimate of the number of spikes is needed Do not apply to multiple dimensions (not even 2D) Noise must be Gaussian and white Incorporating additional structure is dicult (dierent point-spread functions, assumptions on the signal) This talk Non-parametric estimation
21 Prior art Based on Prony's method : MUSIC, ESPRIT, matrix pencil,... An estimate of the number of spikes is needed Do not apply to multiple dimensions (not even 2D) Noise must be Gaussian and white Incorporating additional structure is dicult (dierent point-spread functions, assumptions on the signal) This talk Non-parametric estimation Provably stable in the presence of noise
22 Prior art Based on Prony's method : MUSIC, ESPRIT, matrix pencil,... An estimate of the number of spikes is needed Do not apply to multiple dimensions (not even 2D) Noise must be Gaussian and white Incorporating additional structure is dicult (dierent point-spread functions, assumptions on the signal) This talk Non-parametric estimation Provably stable in the presence of noise Flexible variational framework based on convex programming
23 Outline of the talk Sparsity is not enough Theory Proof (sketch) Implementation via semidenite programming Robustness to noise
24 Sparsity is not enough Theory Proof (sketch) Implementation via semidenite programming Robustness to noise
25 Compressed sensing vs super-resolution Estimation of sparse signals from undersampled measurements suggests connections to compressed sensing
26 Compressed sensing vs super-resolution Estimation of sparse signals from undersampled measurements suggests connections to compressed sensing Compressed sensing Super-resolution spectrum interpolation spectrum extrapolation
27 Compressed sensing vs super-resolution Compressed sensing Super-resolution spectrum interpolation spectrum extrapolation
28 Compressed sensing Compressed sensing theory establishes robust recovery of spikes from random Fourier measurements [Candès, Romberg & Tao 2004]
29 Compressed sensing Compressed sensing theory establishes robust recovery of spikes from random Fourier measurements [Candès, Romberg & Tao 2004] Crucial insight : measurement operator is well conditioned when acting upon sparse signals
30 Compressed sensing Compressed sensing theory establishes robust recovery of spikes from random Fourier measurements [Candès, Romberg & Tao 2004] Crucial insight : measurement operator is well conditioned when acting upon sparse signals Equivalently, the energy of all sparse signals is preserved by the randomized measurements (restricted isometry property)
31 Compressed sensing Compressed sensing theory establishes robust recovery of spikes from random Fourier measurements [Candès, Romberg & Tao 2004] Crucial insight : measurement operator is well conditioned when acting upon sparse signals Equivalently, the energy of all sparse signals is preserved by the randomized measurements (restricted isometry property) This is a necessary condition for stable estimation, but is it the case in super-resolution?
32 Simple experiment F x = y Discretize support to lie on a grid with N = 4096 points
33 Simple experiment F c x = y c Measure n low-pass DFT coecients, super-resolution factor (SRF) : N/n
34 Simple experiment F c = y c x Measure n low-pass DFT coecients, super-resolution factor (SRF) : N/n
35 Simple experiment F c,t = y c x T Restrict support of the signal to an interval of 48 contiguous points
36 Simple experiment x T = F c,t y c Compute singular values of resulting linear operator
37 Simple experiment SRF=2 SRF=4 SRF=8 SRF=16 1e 17 1e 13 1e 09 1e 05 1e SRF=2 SRF=4 SRF=8 SRF=16 For SRF = 4 measuring any unit-normed vector in a subspace of dimension 24 results in a signal with norm less than 10 7
38 Simple experiment SRF=2 SRF=4 SRF=8 SRF=16 1e 17 1e 13 1e 09 1e 05 1e SRF=2 SRF=4 SRF=8 SRF=16 For SRF = 4 measuring any unit-normed vector in a subspace of dimension 24 results in a signal with norm less than 10 7 At an SNR of 145 db, recovery is impossible by any method
39 Simple experiment SRF=2 SRF=4 SRF=8 SRF=16 1e 17 1e 13 1e 09 1e 05 1e SRF=2 SRF=4 SRF=8 SRF=16 For SRF = 4 measuring any unit-normed vector in a subspace of dimension 24 results in a signal with norm less than 10 7 At an SNR of 145 db, recovery is impossible by any method This phenomenon is characterized asymptotically by Slepian's seminal work on prolate spheroidal sequences
40 Conclusion Sparsity is not enough Signal Spectrum
41 Conclusion Sparsity is not enough Signal Spectrum Additional conditions are necessary to restrict our signal model
42 Sparsity is not enough Theory Proof (sketch) Implementation via semidenite programming Robustness to noise
43 Minimum separation To exclude highly-clustered signals from our model, we control the minimum separation of the support T = inf t t (t,t ) T : t t
44 Total-variation norm Continuous counterpart of the l 1 norm
45 Total-variation norm Continuous counterpart of the l 1 norm If x = j a jδ tj then x TV = j a j
46 Total-variation norm Continuous counterpart of the l 1 norm If x = j a jδ tj then x TV = j a j Not the total variation of a piecewise-constant function
47 Total-variation norm Continuous counterpart of the l 1 norm If x = a j jδ tj then x TV = a j j Not the total variation of a piecewise-constant function Formal denition : For a complex measure ν ν TV = sup ν (B j ), j=1 (supremum over all nite partitions B j of [0, 1])
48 Recovery via convex programming In the absence of noise, i.e. if y = F c x, we solve min x x TV subject to F c x = y, over all nite complex measures x supported on [0, 1]
49 Recovery via convex programming In the absence of noise, i.e. if y = F c x, we solve min x x TV subject to F c x = y, over all nite complex measures x supported on [0, 1] Theorem [Candès, F. 2012] If the minimum separation of the signal support T obeys 2 /f c := 2 λ c, then recovery is exact
50 Minimum-distance condition λ c /2 is the Rayleigh resolution limit (half-width of measurement lter)
51 Minimum-distance condition λ c /2 is the Rayleigh resolution limit (half-width of measurement lter) Numerical simulations show that TV-norm minimization fails if < λ c
52 Minimum-distance condition λ c /2 is the Rayleigh resolution limit (half-width of measurement lter) Numerical simulations show that TV-norm minimization fails if < λ c If < λ c /2 some signals are almost in the nullspace of the measurement operator (no method can achieve stable estimation)
53 Sparse recovery in overcomplete dictionaries If we discretize the support Sparse recovery via l 1 -norm minimization in an overcomplete Fourier dictionary
54 Sparse recovery in overcomplete dictionaries If we discretize the support Sparse recovery via l 1 -norm minimization in an overcomplete Fourier dictionary Previous theory based on dictionary incoherence is very weak, due to high column correlation
55 Sparse recovery in overcomplete dictionaries If we discretize the support Sparse recovery via l 1 -norm minimization in an overcomplete Fourier dictionary Previous theory based on dictionary incoherence is very weak, due to high column correlation If N = and n = 1000, how many spikes can we recover?
56 Sparse recovery in overcomplete dictionaries If we discretize the support Sparse recovery via l 1 -norm minimization in an overcomplete Fourier dictionary Previous theory based on dictionary incoherence is very weak, due to high column correlation If N = and n = 1000, how many spikes can we recover? Previous theory [Dossal 2005] : 3 spikes
57 Sparse recovery in overcomplete dictionaries If we discretize the support Sparse recovery via l 1 -norm minimization in an overcomplete Fourier dictionary Previous theory based on dictionary incoherence is very weak, due to high column correlation If N = and n = 1000, how many spikes can we recover? Previous theory [Dossal 2005] : 3 spikes Our result : n/4 = 250 spikes
58 Higher dimensions Signal : superposition of point sources (delta measures) in 2D Measurements : low-pass 2D Fourier coecients
59 Higher dimensions Signal : superposition of point sources (delta measures) in 2D Measurements : low-pass 2D Fourier coecients Theorem [Candès, F. 2012] TV-norm minimization yields exact recovery if 2.38 λ c
60 Higher dimensions Signal : superposition of point sources (delta measures) in 2D Measurements : low-pass 2D Fourier coecients Theorem [Candès, F. 2012] TV-norm minimization yields exact recovery if 2.38 λ c In dimension d, C d λ c, where C d only depends on d
61 Extensions Signal : piecewise-constant function Measurements : low-pass Fourier coecients
62 Extensions Signal : piecewise-constant function Measurements : low-pass Fourier coecients Corollary Solving min x (1) TV subject to F c x = y yields exact recovery if 2 λ c Similar result for cont. dierentiable piecewise-smooth functions
63 Sparsity is not enough Theory Proof (sketch) Implementation via semidenite programming Robustness to noise
64 Optimality conditions Consider the problem where f is convex Lemma min x f ( x) subject to A x = y, If there exists a subgradient g(x) of f at a feasible point x such that g(x) = A v for some v, then x is a solution
65 Optimality conditions Consider the problem where f is convex Lemma min x f ( x) subject to A x = y, If there exists a subgradient g(x) of f at a feasible point x such that g(x) = A v for some v, then x is a solution For any h such that Ah = 0 f (x + h) f (x) + g(x), h = f (x) + A v, h = f (x) + v, Ah = f (x) by denition of subgradient and convexity of f
66 Certicate of optimality q is a subgradient of the total-variation norm at x = j T a j e iφ j δ tj { q(t j ) = e iφ j, q(t) 1, t j T t [0, 1] \ T if (1)
67 Certicate of optimality q is a subgradient of the total-variation norm at x = j T a j e iφ j δ tj { q(t j ) = e iφ j, q(t) 1, t j T t [0, 1] \ T if (1) To certify optimality we also need q(t) = F c v = fc k= f c v k e i 2πkt
68 Certicate of optimality q is a subgradient of the total-variation norm at x = j T a j e iφ j δ tj { q(t j ) = e iφ j, q(t) 1, t j T t [0, 1] \ T if (1) To certify optimality we also need If (1) is strengthened to q(t) = F c v = fc k= f c v k e i 2πkt q(t) < 1, t [0, 1] \ T then x is the unique solution
69 Certicate of optimality
70 Construction of the certicate 1st idea : interpolation with a low-frequency fast-decaying kernel K q(t) = tj T α j K(t t j ),
71 Construction of the certicate 1st idea : interpolation with a low-frequency fast-decaying kernel K q(t) = tj T α j K(t t j ),
72 Construction of the certicate 1st idea : interpolation with a low-frequency fast-decaying kernel K q(t) = tj T α j K(t t j ),
73 Construction of the certicate 1st idea : interpolation with a low-frequency fast-decaying kernel K q(t) = tj T α j K(t t j ),
74 Construction of the certicate 1st idea : interpolation with a low-frequency fast-decaying kernel K q(t) = tj T α j K(t t j ),
75 Construction of the certicate Problem : magnitude of polynomial locally exceeds 1
76 Construction of the certicate Problem : magnitude of polynomial locally exceeds 1
77 Construction of the certicate Problem : magnitude of polynomial locally exceeds 1 Solution : add correction term and force q (t k ) = 0 for all t k T q(t) = tj T α j K(t t j ) + β j K (t t j )
78 Construction of the certicate Problem : magnitude of polynomial locally exceeds 1 Solution : add correction term and force q (t k ) = 0 for all t k T q(t) = tj T α j K(t t j ) + β j K (t t j )
79 Construction of the certicate Problem : magnitude of polynomial locally exceeds 1 Solution : add correction term and force q (t k ) = 0 for all t k T q(t) = tj T α j K(t t j ) + β j K (t t j )
80 Sparsity is not enough Theory Proof (sketch) Implementation via semidenite programming Robustness to noise
81 Practical implementation Primal problem : min x x TV subject to F c x = y, Innite-dimensional variable x (measure in [0, 1]) First option : Discretize domain and apply l 1 -norm minimization
82 Practical implementation Primal problem : min x x TV subject to F c x = y, Innite-dimensional variable x (measure in [0, 1]) First option : Discretize domain and apply l 1 -norm minimization Dual problem : max Re [y u] subject to F u c 1, n := 2f c + 1 u C n Finite-dimensional variable u, but innite-dimensional constraint F u = c u k e i 2πkt k f c Second option : Recast dual problem as semidenite program
83 Lemma : Semidenite representation The Fejér-Riesz Theorem and the semidenite representation of polynomial sums of squares imply that F c u 1 is equivalent to There exists a Hermitian matrix Q C n n such that [ ] Q u u 0, 1 n j Q i,i+j = i=1 { 1, j = 0, 0, j = 1, 2,..., n 1.
84 Using the dual solution We can solve the dual problem, but how do we retrieve a primal solution?
85 Using the dual solution We can solve the dual problem, but how do we retrieve a primal solution? Dual solution vector : Fourier coecients of low-pass polynomial that interpolates the sign of the primal solution (follows from strong duality)
86 Using the dual solution We can solve the dual problem, but how do we retrieve a primal solution? Dual solution vector : Fourier coecients of low-pass polynomial that interpolates the sign of the primal solution (follows from strong duality) Idea : Use the polynomial to locate the support of the signal and then estimate the amplitudes by least squares
87 Super-resolution via semidenite programming Measurements High res signal
88 Super-resolution via semidenite programming Solve semidenite program to obtain dual solution
89 Super-resolution via semidenite programming Locate points at which corresponding polynomial has unit magnitude
90 Super-resolution via semidenite programming High res Signal Estimate 3. Estimate amplitudes via least squares
91 Sparsity is not enough Theory Proof (sketch) Implementation via semidenite programming Robustness to noise
92 Estimation from noisy data Without noise, we achieve perfect precision, i.e. innite resolution y = F c x
93 Estimation from noisy data Without noise, we achieve perfect precision, i.e. innite resolution In practice, there is always noise y = F c x y = F c x + z Understanding the performance in a noisy setting is crucial for applications
94 Estimation from noisy data Without noise, we achieve perfect precision, i.e. innite resolution In practice, there is always noise y = F c x y = F c x + z Understanding the performance in a noisy setting is crucial for applications Metrics : 1. Approximation error at a higher resolution 2. Support-detection error
95 Super-resolution factor : spatial viewpoint Super-resolution factor SRF = λ c λ f
96 Super-resolution factor : spectral viewpoint Super-resolution factor SRF = f f c
97 Approximation at a higher resolution Resolution at scale λ is quantied by convolution with kernel φ λ of width λ At the resolution of the measurements φ λc (x est x) δ L 1
98 Approximation at a higher resolution Resolution at scale λ is quantied by convolution with kernel φ λ of width λ At the resolution of the measurements φ λc (x est x) δ L 1 How does the estimate degrade at a higher resolution?
99 Approximation at a higher resolution Resolution at scale λ is quantied by convolution with kernel φ λ of width λ At the resolution of the measurements φ λc (x est x) δ L 1 How does the estimate degrade at a higher resolution? Theorem [Candès, F. 2012] If 2 /f c then the solution ˆx to min x x TV subject to F c x y 2 δ, satises φλf (ˆx x) L1 SRF2 δ
100 Example SNR : 25 db Measurements High res signal
101 Example SNR : 25 db High res Signal Estimate
102 Example SNR : 25 db Noisy measurements Low res signal High res signal Estimate
103 Support detection Original signal, support T x = j a j δ tj a j C, t j T [0, 1]
104 Support detection Original signal, support T x = j a j δ tj a j C, t j T [0, 1] Estimated signal, support T ˆx = j â j δˆtj â j C, ˆtj T [0, 1]
105 Support detection Original signal, support T x = j a j δ tj a j C, t j T [0, 1] Estimated signal, support T ˆx = j â j δˆtj â j C, ˆtj T [0, 1] How accurately can we detect the support at a certain noise level δ?
106 Support-detection accuracy Theorem [F. 2013] For any t i T, if a i > C 1 δ there exists ˆti T such that ti ˆti 1 C 2 δ f c a i C 1 δ
107 Support-detection accuracy Theorem [F. 2013] For any t i T, if a i > C 1 δ there exists ˆti T such that ti ˆti 1 C 2 δ f c a i C 1 δ The support-detection accuracy is not aected by aliasing (no dependence on the amplitude of the signal at other locations)
108 Consequence Robustness of the algorithm to high dynamic ranges Measurements High res signal
109 Consequence Robustness of the algorithm to high dynamic ranges High res Signal Estimate (Real)
110 Conclusion and future work To obtain theoretical guarantees for super-resolution in realistic settings, we need conditions that avoid clustered supports
111 Conclusion and future work To obtain theoretical guarantees for super-resolution in realistic settings, we need conditions that avoid clustered supports Under a minimum-separation condition, convex programming achieves exact recovery
112 Conclusion and future work To obtain theoretical guarantees for super-resolution in realistic settings, we need conditions that avoid clustered supports Under a minimum-separation condition, convex programming achieves exact recovery The optimization problem can be recast as a tractable semidenite program
113 Conclusion and future work To obtain theoretical guarantees for super-resolution in realistic settings, we need conditions that avoid clustered supports Under a minimum-separation condition, convex programming achieves exact recovery The optimization problem can be recast as a tractable semidenite program The method is provably robust to noise
114 Conclusion and future work To obtain theoretical guarantees for super-resolution in realistic settings, we need conditions that avoid clustered supports Under a minimum-separation condition, convex programming achieves exact recovery The optimization problem can be recast as a tractable semidenite program The method is provably robust to noise Research directions : Super-resolution of images with sharp edges Developing fast solvers to solve sdp formulation Extending results to other overcomplete dictionaries
115 For more details Towards a mathematical theory of super-resolution. E. J. Candès and C. Fernandez-Granda. Comm. on Pure and Applied Math. Super-resolution from noisy data. E. J. Candès and C. Fernandez-Granda. Journal of Fourier Analysis and Applications Support detection in super-resolution. C. Fernandez-Granda. Proceedings of SampTA 2013 Prolate spheroidal wave functions, Fourier analysis, and uncertainty V - The discrete case. D. Slepian. Bell System Technical Journal, 57 : , 1978
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