Support Detection in Super-resolution

Support Detection in Super-resolution Carlos Fernandez-Granda (Stanford University) 7/2/2013 SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 1 / 25

Acknowledgements This work was supported by a Fundación Caja Madrid Fellowship SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 2 / 25

Super-resolution Aim : estimating ne-scale structure from low-resolution data SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 3 / 25

Super-resolution Aim : estimating ne-scale structure from low-resolution data Equivalently, extrapolating the high end of the spectrum SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 3 / 25

Applications Optics (diraction-limited systems), electronic imaging, signal processing, radar, spectroscopy, medical imaging, astronomy, geophysics, etc. SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 4 / 25

Applications Optics (diraction-limited systems), electronic imaging, signal processing, radar, spectroscopy, medical imaging, astronomy, geophysics, etc. Signals of interest are often modeled as superpositions of point sources Celestial bodies in astronomy Line spectra in speech analysis Molecules in uorescence microscopy SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 4 / 25

Mathematical model Signal : superposition of delta measures with support T x = j a j δ tj a j C, t j T [0, 1] SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 5 / 25

Mathematical model Signal : superposition of delta measures with support T x = j a j δ tj a j C, t j T [0, 1] Measurement process : low-pass ltering with cut-o frequency f c SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 5 / 25

Mathematical model Signal : superposition of delta measures with support T x = j a j δ tj a j C, t j T [0, 1] Measurement process : low-pass ltering with cut-o frequency f c Measurements : n = 2 f c + 1 noisy low-pass Fourier coecients y(k) = 1 0 = j e i 2πkt x (dt) + z(k) a j e i 2πkt j + z(k), k Z, k f c y = F n x + z SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 5 / 25

When is super-resolution well posed? Signal Spectrum 10 0 10 5 10 10 SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 6 / 25

When is super-resolution well posed? Signal Spectrum 10 0 10 5 10 10 10 0 10 5 10 10 SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 6 / 25

Minimum-separation condition Necessary condition : measurement operator must be well conditioned when acting upon signals of interest SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 7 / 25

Minimum-separation condition Necessary condition : measurement operator must be well conditioned when acting upon signals of interest Problem : even very sparse signals may be almost completely suppressed by low-pass ltering if they are highly clustered (sparsity is not enough) SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 7 / 25

Minimum-separation condition Necessary condition : measurement operator must be well conditioned when acting upon signals of interest Problem : even very sparse signals may be almost completely suppressed by low-pass ltering if they are highly clustered (sparsity is not enough) Additional conditions are necessary SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 7 / 25

Minimum-separation condition Necessary condition : measurement operator must be well conditioned when acting upon signals of interest Problem : even very sparse signals may be almost completely suppressed by low-pass ltering if they are highly clustered (sparsity is not enough) Additional conditions are necessary Minimum separation of the support T of a signal : (T ) = inf t t (t,t ) T : t t SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 7 / 25

Total-variation norm Continuous counterpart of the l 1 norm SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 8 / 25

Total-variation norm Continuous counterpart of the l 1 norm If x = j a jδ tj then x TV = j a j SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 8 / 25

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 SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 8 / 25

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]) SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 8 / 25

Recovery via convex programming In the absence of noise, i.e. if y = F n x, we solve min x x TV subject to F n x = y, over all nite complex measures x supported on [0, 1] SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 9 / 25

Recovery via convex programming In the absence of noise, i.e. if y = F n x, we solve min x x TV subject to F n 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 then recovery is exact (T ) 2 /f c SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 9 / 25

Estimation from noisy data We consider noise bounded in l 2 norm y = F n x + z, z 2 δ SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 10 / 25

Estimation from noisy data We consider noise bounded in l 2 norm y = F n x + z, z 2 δ Otherwise, the perturbation is arbitrary and can be adversarial SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 10 / 25

Estimation from noisy data We consider noise bounded in l 2 norm y = F n x + z, z 2 δ Otherwise, the perturbation is arbitrary and can be adversarial Estimation algorithm min x x TV subject to F n x y 2 δ, over all nite complex measures x supported on [0, 1] SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 10 / 25

Estimation from noisy data We consider noise bounded in l 2 norm y = F n x + z, z 2 δ Otherwise, the perturbation is arbitrary and can be adversarial Estimation algorithm min x x TV subject to F n x y 2 δ, over all nite complex measures x supported on [0, 1] The problem is innite-dimensional, but its dual is sdp-representable SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 10 / 25

Implementation Dual solution vector : Fourier coecients of low-pass polynomial that interpolates the sign of the primal solution SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 11 / 25

Implementation Dual solution vector : Fourier coecients of low-pass polynomial that interpolates the sign of the primal solution To estimate the support we SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 11 / 25

Implementation Dual solution vector : Fourier coecients of low-pass polynomial that interpolates the sign of the primal solution To estimate the support we 1 solve the sdp SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 11 / 25

Implementation Dual solution vector : Fourier coecients of low-pass polynomial that interpolates the sign of the primal solution To estimate the support we 1 solve the sdp 2 determine where the magnitude of the polynomial equals 1 SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 11 / 25

Example SNR : 14 db Measurements SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 12 / 25

Example SNR : 14 db Measurements Low res signal SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 12 / 25

Example Measurements High res signal SNR : 14 db SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 12 / 25

Example Average localization error : 6.54 10 4 High res Signal Estimate SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 13 / 25

Support-detection accuracy Original signal, support T x = j a j δ tj a j C, t j T [0, 1] SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 14 / 25

Support-detection accuracy Original signal, support T x = j a j δ tj a j C, t j T [0, 1] Estimated signal, support ˆT ˆx = j â j δˆtj â j C, ˆtj ˆT [0, 1] SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 14 / 25

Support-detection accuracy Original signal, support T x = j a j δ tj a j C, t j T [0, 1] Estimated signal, support ˆT ˆx = j â j δˆtj â j C, ˆtj ˆT [0, 1] Main result : Theoretical guarantees on the quality of the estimate under the minimum-separation condition (T ) 2 /f c SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 14 / 25

Theorem [F. 2013] Spike detection (1) : a j {ˆtl T : ˆtl t j c/f c} â l C1 δ t j T, c := 0.1649 SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 15 / 25

Theorem [F. 2013] Support-detection accuracy (2) : {ˆtl T, t j T : ˆtl t j c/f c} â l (ˆtl ) 2 C 2 δ t j, c := 0.1649 f 2 c SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 16 / 25

Theorem [F. 2013] False positives (3) : {ˆtl T : ˆtl t j >c/f c t j T} â l C 3 δ, c := 0.1649 SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 17 / 25

Support-detection accuracy Corollary For any t i T, if a i > C 1 δ there exists ˆti ˆT such that ti ˆti 1 C 2 δ f c a i C 1 δ. SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 18 / 25

Support-detection accuracy Corollary For any t i T, if a i > C 1 δ there exists ˆti ˆT such that ti ˆti 1 C 2 δ f c a i C 1 δ. In previous work : SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 18 / 25

Support-detection accuracy Corollary For any t i T, if a i > C 1 δ there exists ˆti ˆT such that ti ˆti 1 C 2 δ f c a i C 1 δ. In previous work : The estimation errors of the dierent spikes cannot be decoupled [Candès, F. 2012] SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 18 / 25

Support-detection accuracy Corollary For any t i T, if a i > C 1 δ there exists ˆti ˆT such that ti ˆti 1 C 2 δ f c a i C 1 δ. In previous work : The estimation errors of the dierent spikes cannot be decoupled [Candès, F. 2012] The bounds depend on the amplitude of the estimate, not of the original signal [Azais et al 2013] SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 18 / 25

Consequence Robustness of the algorithm to high dynamic ranges Measurements High res signal SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 19 / 25

Consequence Robustness of the algorithm to high dynamic ranges High res Signal Estimate (Real) SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 19 / 25

Proof techniques Main proof technique : construction of interpolating polynomials SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 20 / 25

Proof techniques Main proof technique : construction of interpolating polynomials To establish the spike-detection bound a j â l C1 δ {ˆtl T : ˆtl t j c/f c} we construct a low-pass polynomial for each t j T t j T q tj (t) = f c k= f c i 2πkt b k e SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 20 / 25

Sketch of proof q tj satises q tj (t j ) = 1 and q tj (t l ) = 0 for t l T / {t j } SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 21 / 25

Sketch of proof High res Signal Spike at t j q tj satises q tj (t j ) = 1 and q tj (t l ) = 0 for t l T / {t j } q tj (t)x(dt) = a j [0,1] SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 21 / 25

Sketch of proof Estimate Estimate near t j We can establish (using the support-detection bound) q tj (t)ˆx(dt) = â l + Ω (δ) [0,1] {ˆtl T : ˆtl t j c/f c} SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 22 / 25

Sketch of proof By our assumption on the noise, F n x y 2 δ SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 23 / 25

Sketch of proof By our assumption on the noise, F n x y 2 δ Since ˆx is feasible F nˆx y 2 δ SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 23 / 25

Sketch of proof By our assumption on the noise, F n x y 2 δ Since ˆx is feasible F nˆx y 2 δ Consequently, by the triangle inequality F n (ˆx x) 2 2δ SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 23 / 25

Sketch of proof a j By our assumption on the noise, F n x y 2 δ Since ˆx is feasible F nˆx y 2 δ Consequently, by the triangle inequality F n (ˆx x) 2 2δ {ˆtl T : ˆtl t j c f c } â l = = q tj (t)x(dt) [0,1] f c k= f c [0,1] b k F n (x ˆx) k + Ω (δ) q tj (t)ˆx(dt) + Ω (δ) (Parseval) qtj F n (x ˆx) L2 2 + Ω (δ) (Cauchy-Schwarz) = Ω (δ) SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 23 / 25

Conclusion and future work Under a minimum-separation condition, super-resolution via convex programming achieves accurate support detection in the presence of noise even at high dynamic ranges Research directions : Lower bounds on support-detection accuracy Developing fast solvers to solve sdp formulation Super-resolution of images with sharp edges For more details, C. Fernandez-Granda. Support detection in super-resolution. Proceedings of SampTA 2013 E. J. Candès and C. Fernandez-Granda. Towards a mathematical theory of super-resolution. To appear in Comm. on Pure and Applied Math. SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 24 / 25

Thank you SampTA 2013 Support Detection in Super-resolution C. Fernandez-Granda 25 / 25