ℓ 2 − ℓ 0 optimization for single molecule localization microscopy Laure Blanc-Féraud G. Aubert, A. Bechensteen, S. Rebegoldi*, E. Soubies** UCA, CNRS, INRIA - MORPHEME group * Informatiche e Matematiche, Università di Modena e Reggio Emilia, Modena, Italy ** BIG group, EPFL, Lausanne, Switzerland CMIPI Workshop — July 16-18, 2018
Outline of the talk I. Single molecule super-resolution microscopy: introduction II. ℓ 2 - ℓ 0 contrained optimization - continuous relaxation III. ℓ 2 - ℓ 0 contrained optimization - exact reformulation IV. Simulation results V. ℓ 2 - ℓ 0 penalized optimization - continuous exact relaxation CEL0 VI. Future work 2 / 36
I. Super-resolution: bypass the diffraction limit of light microscopy Conventional fluorescence microscopy limits ◮ physical diffraction limit of optical systems : Airy patch = PSF: Point Spread Function of the microscope ◮ overlapping patches limit at ≈ 200nm the distance between two molecules to be resolved (Rayleigh limit) 2D Super-resolution microscopy ◮ SIM Structured illumination microscopy [ Gustafsson, 2000 ] ◮ STED Stimulated emission Depletion [ Hell & al., 1994 ] ◮ SMLM Single Molecule Localization Microscopy : PALM Photo Activated Localisation Microscopy ([ Betzig & al 06 , Hess & al, 2006 ]) et STORM STochastic Optical Reconstruction Microscopy ([ Rust & al, 2006 ]) 3 / 36
I. Single Molecule Localization Microscopy: introduction ◮ Sequentially activate and image a small random set of fluorescent molecules, ◮ localize molecules ◮ assemble images Figure: PALM microscopy principle. From Zeiss tutorials [ http://zeiss-campus.magnet.fsu.edu/tutorials/index.html ] 4 / 36
I. Single Molecule Localization Microscopy: introduction ◮ Sequentially activate and image a small random set of fluorescent molecules, ◮ localize molecules ◮ assemble images Figure: PALM microscopy principle. From Zeiss tutorials [ http://zeiss-campus.magnet.fsu.edu/tutorials/index.html ] 4 / 36
I. Single Molecule Localization Microscopy: introduction ◮ Sequentially activate and image a small random set of fluorescent molecules, ◮ localize molecules ◮ assemble images Figure: PALM microscopy principle. From Zeiss tutorials [ http://zeiss-campus.magnet.fsu.edu/tutorials/index.html ] 4 / 36
I. Single Molecule Localization Microscopy: introduction ◮ Sequentially activate and image a small random set of fluorescent molecules, ◮ localize molecules ◮ assemble images Figure: PALM microscopy principle. From Zeiss tutorials [ http://zeiss-campus.magnet.fsu.edu/tutorials/index.html ] 4 / 36
I. Single Molecule Localization Microscopy: introduction ◮ Sequentially activate and image a small random set of fluorescent molecules, ◮ localize molecules ◮ assemble images Figure: PALM microscopy principle. From Zeiss tutorials [ http://zeiss-campus.magnet.fsu.edu/tutorials/index.html ] 4 / 36
I. Single Molecule Localization Microscopy: introduction ◮ Sequentially activate and image a small random set of fluorescent molecules, ◮ localize molecules ◮ assemble images Figure: PALM microscopy principle. From Zeiss tutorials [ http://zeiss-campus.magnet.fsu.edu/tutorials/index.html ] 4 / 36
I. Single Molecule Localization Microscopy: introduction ◮ Sequentially activate and image a small random set of fluorescent molecules, ◮ localize molecules ◮ assemble images Figure: PALM microscopy principle. From Zeiss tutorials [ http://zeiss-campus.magnet.fsu.edu/tutorials/index.html ] 4 / 36
I. Single Molecule Localization Microscopy: introduction ◮ Sequentially activate and image a small random set of fluorescent molecules, ◮ localize molecules ◮ assemble images Figure: PALM microscopy principle. From Zeiss tutorials [ http://zeiss-campus.magnet.fsu.edu/tutorials/index.html ] 4 / 36
I. Single Molecule Localization Microscopy: introduction ◮ Sequentially activate and image a small random set of fluorescent molecules, ◮ localize molecules ◮ assemble images Figure: PALM microscopy principle. From Zeiss tutorials [ http://zeiss-campus.magnet.fsu.edu/tutorials/index.html ] 4 / 36
I. Single Molecule Localization Microscopy: introduction 5 / 36
I. Single Molecule Localization Microscopy: introduction Limitations: number of acquisition needed to obtain the super-resolved image ◮ cost time and memory ◮ temporal resolution restricted (motion) → Increase molecule density ◮ Localization more difficult due to more overlapping Localization algorithms ◮ Challenge ISBI 2013 [ Sage & al, 2015 ] Challenge 2016 (bigwww.epfl.ch/smlm/challenge2016/index.html) ◮ PSF fitting, and derived methods for high density molecule localization (e.g. DAOSTORM, [ Holden & al 11 ] ). ◮ Deconvolution of measures and spike reconstruction : Gridless methods [ Denoyelle & al. ,2018 ] 6 / 36
I. Single Molecule Localization Microscopy: introduction Limitations: number of acquisition needed to obtain the super-resolved image ◮ cost time and memory ◮ temporal resolution restricted (motion) → Increase molecule density ◮ Localization more difficult due to more overlapping Localization algorithms ◮ Challenge ISBI 2013 [ Sage & al, 2015 ] Challenge 2016 (bigwww.epfl.ch/smlm/challenge2016/index.html) ◮ PSF fitting, and derived methods for high density molecule localization (e.g. DAOSTORM, [ Holden & al 11 ] ). ◮ Deconvolution of measures and spike reconstruction : Gridless methods [ Denoyelle & al. ,2018 ] 6 / 36
I. Single Molecule Localization Microscopy: introduction Deconvolution and reconstruction on a finer grid (e.g. FALCON, [ Min & al, 2014 ]) Image formation model PALM / STORM Y ∈ R M × M one acquisition. X ∈ R ML × ML an image where each pixel of Y is divided in L × L pixels. L=4 Reduction matrix ML ∈ R M × R ML Convolution matrix H ∈ R ML × R ML H( · ) M L ( · ) + η ∗ H(X) M L (H(X)) X PSF Y 7 / 36
I. Single Molecule Localization Microscopy: introduction Deconvolution and reconstruction on a finer grid (e.g. FALCON, [ Min & al, 2014 ]) Image formation model PALM / STORM Y ∈ R M × M one acquisition. X ∈ R ML × ML an image where each pixel of Y is divided in L × L pixels. L=4 Reduction matrix ML ∈ R M × R ML Convolution matrix H ∈ R ML × R ML H( · ) M L ( · ) + η ∗ H(X) M L (H(X)) X PSF Y 7 / 36
I. Single Molecule Localization Microscopy: introduction Deconvolution and reconstruction on a finer grid (e.g. FALCON, [ Min & al, 2014 ]) Image formation model PALM / STORM Y ∈ R M × M one acquisition. X ∈ R ML × ML an image where each pixel of Y is divided in L × L pixels. L=4 Reduction matrix ML ∈ R M × R ML Convolution matrix H ∈ R ML × R ML H( · ) M L ( · ) + η ∗ H(X) M L (H(X)) X PSF Y 7 / 36
I. Single Molecule Localization Microscopy: introduction Deconvolution and reconstruction on a finer grid (e.g. FALCON, [ Min & al, 2014 ]) Image formation model PALM / STORM Y ∈ R M × M one acquisition. X ∈ R ML × ML an image where each pixel of Y is divided in L × L pixels. L=4 Reduction matrix ML ∈ R M × R ML Convolution matrix H ∈ R ML × R ML H( · ) M L ( · ) + η ∗ H(X) M L (H(X)) X PSF Y 7 / 36
I. Single Molecule Localization Microscopy: introduction Deconvolution and reconstruction on a finer grid (e.g. FALCON, [ Min & al, 2014 ]) Image formation model PALM / STORM Y ∈ R M × M one acquisition. X ∈ R ML × ML an image where each pixel of Y is divided in L × L pixels. L=4 Reduction matrix ML ∈ R M × R ML Convolution matrix H ∈ R ML × R ML H( · ) M L ( · ) + η ∗ H(X) M L (H(X)) X PSF Y Model Y = AX + η, A = MLH 7 / 36
I. Single Molecule Localization Microscopy: introduction Deconvolution and reconstruction on a finer grid (e.g. FALCON, [ Min & al, 2014 ]) Image formation model PALM / STORM Y ∈ R M × M one acquisition. X ∈ R ML × ML an image where each pixel of Y is divided in L × L pixels. L=4 Reduction matrix ML ∈ R M × R ML Convolution matrix H ∈ R ML × R ML + η H( · ) M L ( · ) ∗ H(X) M L (H(X)) X PSF Y Problem ℓ 2 − ℓ 0 1 � Y − AX � 2 ˆ X ∈ arg min 2 , � X � 0 ≤ k 2 A = MLH ∈ R M × ML ◮ ◮ sparse solution modeled by using pseudo-norm- ℓ 0 : � x � 0 = ♯ � x i, i = 1 , . . . , N : x i � = 0 � 7 / 36
II. ℓ 2 - ℓ 0 optimization - continuous relaxation 1 2 � A x − d � 2 ˆ x = arg min 2 x ∈ R N , � x � 0 ≤ k � 1 2 � A x − d � 2 � ˆ x = arg min 2 + λ � x � 0 x ∈ R N ◮ non-convex, non-continuous and NP-hard optimization problem [ Natarajan 95 ] [ Davis & al 97 ]. ◮ Sparse Approximation in signal and image processing: intensive work Large literature on dedicated algorithms : ◮ ℓ 1 relaxation (Basis pursuit [ Chen & al, 1998 ], Compressive sensing [ Donoho & al 03 , Candes & Tao, 2005 ], LASSO [ Tibshirani, 1996 ], ...) ◮ Greedy algorithms (MP [ Mallat & al 93 ], ..., SBR [ Soussen & al 11 ], ...) ◮ Iterative Hard Thresholding (IHT) [ Blumensath & Davies 08 ], ◮ Non convex continuous relaxation (...MCP [ Zhang 10 ], ℓ p -norms 0 < p < 1 [ Chartrand 07 ], ...,[ Soubies & al, 2017 ],...) ◮ Reformulation ([ Yuan & Ghanem 16 ],...) 8 / 36
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