Sparse reconstruction in digital holography

Sparse reconstruction in digital holography Loïc Denis Observatoire de Lyon

joint work with Corinne Fournier, Ferréol Soulez and Eric Thiébaut

Loïc Denis

Sparse reconstruction in digital holography

Context Digital holography: 2 possible meanings 1

Generate numerically a hologram for 3D visualization:

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2

Record a hologram on a digital camera and reconstruct it numerically:

Applications: - metrology (particles, surface deformation) - (fast) imaging of semi-transparent objects Loïc Denis


Context (cont) Digital holography: 3D+t particle imaging Particle tracking is an essential technique in experimental fluid dynamics. Holography has long been used to instantaneously measure particle sizes and 3D locations:

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Context (cont) Digital holography: 3D+t particle imaging Particle tracking is an essential technique in experimental fluid dynamics. Holography has long been used to instantaneously measure particle sizes and 3D locations. With digital holography time-resolved measurements can be done:

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What is measured in particle holography Direct model Diffraction can be modeled as a convolution with a (complexed valued) chirp kernel (Fresnel’s approximation). A spherical opaque particle creates concentric fringes on the hologram:

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What is measured in particle holography Direct model Diffraction can be modeled as a convolution with a (complexed valued) chirp kernel (Fresnel’s approximation). A spherical opaque particle creates concentric fringes on the hologram. Inter-particle interferences can be neglected → linear direct model: Direct model: d = c 1 − H · ϑ + , with d the data, c an offset, H the dictionnary of diffraction patterns, ϑ the unknown particle distribution. Loïc Denis


What is measured in particle holography Direct model Diffraction can be modeled as a convolution with a (complexed valued) chirp kernel (Fresnel’s approximation). A spherical opaque particle creates concentric fringes on the hologram . Inter-particle interferences can be neglected → linear direct model: Direct model: ¯ · ϑ + , d¯ = H with proper centering.

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The detection/reconstruction problems 1- Detection problem From a particle hologram, detect all particles (3D location + size)

2- Reconstruction problem From a hologram, reconstruct the 2D or 3D transmittance distrib.

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Detection: a CLEAN/matching-pursuit approach

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Detection: a CLEAN/matching-pursuit approach → Ferréol Soulez’s PhD thesis 1- Global detection The model is shift-invariant → correlationbased detection of the “best matching” diffraction pattern. 2- Local optimization The underlying “dictionary” should ideally be continuous → local (LevenbergMarquardt) optimization. 3- Residuals update Subtract the fitted diffraction pattern, and re-iterate steps 1-3 to detect another particle. Loïc Denis


Detection: a CLEAN/matching-pursuit approach (cont) Results: 1- 3D+t reconstruction. 2- Out-of-field particle detection:

→ accuracy estimation with Cramèr-Rao bounds (cf. Corinne’s talk) Loïc Denis


Reconstruction: minimization of sparsity-promoting `1 norm

Principle: `1 norm minimization is widely used to get a solution that contains many zeros (i.e., a sparse solution). The reconstruction problem is recast into a (non-smooth) minimization problem: ˆ sparse = arg min ϑ ϑ

1 kHϑ ¯ 2

− d¯ k22 + τ kϑk1

Using a weighted `1 norm is useful to account for non-normalized dictionaries.

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Reconstruction: minimization of sparsity-promoting `1 norm Results:

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Conclusion

1- Hologram reconstruction as a linear inverse problem 2- CLEAN/matching-pursuit approach to particle detection gave unprecedented results 3- Global optimization with sparsity-promoting `1 norm improves on classical (diffraction-based) reconstruction

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Ferreol Soulez, Loic Denis, Corinne Fournier, Eric Thiebaut, and Charles Goepfert. Inverse-problem approach for particle digital holography: accurate location based on local optimization. Journal of the Optical Society of America A, 24(4):1164–1171, April 2007. Ferreol Soulez, Loic Denis, Eric Thiebaut, Corinne Fournier, and Charles Goepfert. Inverse problem approach in particle digital holography: out-of-field particle detection made possible. Journal of the Optical Society of America A, 24(12):3708–3716, December 2007. L Denis, D A Lorenz, and D Trede. Greedy solution of ill-posed problems: error bounds and exact inversion. Inverse Problems, 25(11):115017, 2009. Loic Denis, Dirk Lorenz, Eric Thiebaut, Corinne Fournier, and Dennis Trede. Inline hologram reconstruction with sparsity constraints. Optics Letters, 34(22):3475–3477, November 2009.

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