Global Geometry of Multichannel Sparse Blind Deconvolution on the - - PowerPoint PPT Presentation

Global Geometry

• f Multichannel Sparse Blind Deconvolution on the Sphere

Yanjun Li Yoram Bresler

CSL and Department of ECE, UIUC

Dec 4, 2018

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Multichannel Sparse Blind Deconvolution (MSBD)

Model: Given circular convolution: yi = xi ⊛ f, for i = 1, 2, . . . , N Solve for xi and f Assumptions: f ∈ Rn: invertible signal xi ∈ Rn: sparse filters Applications:

• pportunistic underwater acoustics

reflection seismology functional MRI super-resolution fluorescence microscopy Open problem: Guaranteed algorithm for unconstrained f

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Multichannel Sparse Blind Deconvolution (MSBD)

Model: Given circular convolution: yi = xi ⊛ f, for i = 1, 2, . . . , N Solve for xi and f Assumptions: f ∈ Rn: invertible signal xi ∈ Rn: sparse filters Applications:

• pportunistic underwater acoustics

reflection seismology functional MRI super-resolution fluorescence microscopy Open problem: Guaranteed algorithm for unconstrained f

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Multichannel Sparse Blind Deconvolution (MSBD)

Model: Given circular convolution: yi = xi ⊛ f, for i = 1, 2, . . . , N Solve for xi and f Assumptions: f ∈ Rn: invertible signal xi ∈ Rn: sparse filters Applications:

• pportunistic underwater acoustics

reflection seismology functional MRI super-resolution fluorescence microscopy Open problem: Guaranteed algorithm for unconstrained f

2

Multichannel Sparse Blind Deconvolution (MSBD)

Model: Given circular convolution: yi = xi ⊛ f, for i = 1, 2, . . . , N Solve for xi and f Assumptions: f ∈ Rn: invertible signal xi ∈ Rn: sparse filters Applications:

• pportunistic underwater acoustics

reflection seismology functional MRI super-resolution fluorescence microscopy Open problem: Guaranteed algorithm for unconstrained f

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Formulation

Solving for inverse filter Find the inverse h of f (P0) min

h∈Rn

1 N

N

• i=1

Cyih0, s.t. h = 0. Solution: scaled & shifted Smooth formulation

• min. “sparsity” ℓ1 norm ≈ max. “spiky” ℓ4 norm

(P1) min

h∈Rn − 1

4N

N

• i=1

CyiRh4

4,

s.t. h = 1.

Preconditioner R := (

1 θnN

N

i=1 C⊤ yiCyi)−1/2

Solution: signed & shifted

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Formulation

Solving for inverse filter Find the inverse h of f (P0) min

h∈Rn

1 N

N

• i=1

Cyih0, s.t. h = 0. Solution: scaled & shifted Smooth formulation

• min. “sparsity” ℓ1 norm ≈ max. “spiky” ℓ4 norm

(P1) min

h∈Rn − 1

4N

N

• i=1

CyiRh4

4,

s.t. h = 1.

Preconditioner R := (

1 θnN

N

i=1 C⊤ yiCyi)−1/2

Solution: signed & shifted

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Main Result

Theorem (Geometric Analysis [L. and Bresler, 2018])

If {xi}N

i=1 ⊂ Rn: Bernoulli-Rademacher

N polylog(n) Then w.h.p., local minima ⇐ ⇒ signed & shifted ground truth

• bjective function:
• near local minima: strongly convex
• near local maxima & saddle points: negative curvature (strict saddle points)

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Geometric Structure

• bjective

norm of gradient smallest curvature

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First-Order Algorithm

Optimize the sparsity promoting objective over the unit sphere Manifold gradient descent: h(t+1) = PSn−1

• h(t) − γ

∇L(h(t))

• gradient descent along the tangent space
• retraction (projection onto the sphere)

Time complexity (per iteration): O(Nn log n)

Theorem

If geometric properties random initialization h(0) ∼ Uniform(Sn−1) fixed step size Then manifold gradient descent converges to a local minimum (≈ signed & shifted ground truth) a.s. achieves accuracy ρ after T poly(n/ρ) steps

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Empirical Phase Transition

Random f ∈ Rn, Bernoulli-Rademacher xi ∈ Rn Noise level: 20 dB Iteration number T = 100, step size γ = 0.1 N vs. n N vs. θ N vs. κ (fix θ = 0.1) (fix n = 256) (fix n = 256, θ = 0.1)

64 128 192 256 64 128 192 256 n N 0.04 0.08 0.12 0.16 64 128 192 256 θ N 2 8 32 128 64 128 192 256 κ N

Empirical success:

• N nθ
• weak dependence on κ

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Empirical Convergence

Error Accuracy

50 100 0.5 1 t CfRh(t) − ej 50 100 0.2 0.4 0.6 0.8 1 t

Cf Rh(t)∞ Cf Rh(t)

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Application: SR Fluorescence Microscopy

Time resolved images fluorophores = ⇒ sparse random activation = ⇒ random

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Application: SR Fluorescence Microscopy

true image true kernel nonblind deconvolution miscalibrated kernel blind deconvolution estimated kernel

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Application: SR Fluorescence Microscopy

blurry image true kernel nonblind deconvolution miscalibrated kernel blind deconvolution estimated kernel

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Thank you!

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