R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Probl, pp.1217-1229, 1994.
DOI : 10.1088/0266-5611/10/6/003

T. Adali, P. Schreier, and L. Scharf, Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety, IEEE Transactions on Signal Processing, vol.59, issue.11, pp.5101-5125, 2011.
DOI : 10.1109/TSP.2011.2162954

L. G. Alexopoulos, G. R. Erickson, and F. Guilak, A method for quantifying cell size from differential interference contrast images: validation and application to osmotically stressed chondrocytes, Journal of Microscopy, vol.205, issue.2, pp.125-135, 2002.
DOI : 10.1046/j.0022-2720.2001.00976.x

R. D. Allen, G. B. David, and G. Nomarski, The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy, Zeitschrift Fur Wissenschaftliche Mikroskopie Und Mikroskopische Technik 69, pp.193-221, 1969.

H. Attouch, J. Bolte, and B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward???backward splitting, and regularized Gauss???Seidel methods, Mathematical Programming, vol.31, issue.1, pp.1-2, 2013.
DOI : 10.1137/0331048

URL : https://hal.archives-ouvertes.fr/hal-00790042

M. N. Avadhanulu and P. G. Kshirsagar, A Textbook of Engineering Physics, India: S. Chand and Company PVT. LTD, pp.94-236, 2008.

D. H. Ballard and C. M. Brown, Computer Vision, 1982.

J. Barzilai and J. M. Borwein, Two-Point Step Size Gradient Methods, IMA Journal of Numerical Analysis, vol.8, issue.1, pp.141-148, 1988.
DOI : 10.1093/imanum/8.1.141

L. Bautista, S. Rebegoldi, L. Blanc-féraud, M. Prato, L. Zanni et al., Phase estimation in differential-interference-contrast (DIC) microscopy, 2016 IEEE 13th International Symposium on Biomedical Imaging (ISBI), pp.136-139
DOI : 10.1109/ISBI.2016.7493229

URL : https://hal.archives-ouvertes.fr/hal-01349782

A. Beck and M. Teboulle, A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems, SIAM Journal on Imaging Sciences, vol.2, issue.1, pp.183-202, 2009.
DOI : 10.1137/080716542

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.231.3271

M. Bertero, P. Boccacci, G. Talenti, R. Zanella, and L. Zanni, A discrepancy principle for Poisson data, Inverse Probl. 26, p.105004, 2010.
DOI : 10.1088/0266-5611/26/10/105004

M. C. Bertilson, O. Von-hofsten, M. Lindblom, T. Wilhein, H. M. Hertz et al., Compact high-resolution differential interference contrast soft x-ray microscopy, Applied Physics Letters, vol.97, issue.6, pp.64104-064104, 2008.
DOI : 10.1116/1.1738671

S. Bonettini, A. Chiuso, and M. Prato, A Scaled Gradient Projection Method for Bayesian Learning in Dynamical Systems, SIAM Journal on Scientific Computing, vol.37, issue.3, pp.1297-1318, 2015.
DOI : 10.1137/140973529

S. Bonettini, R. Zanella, and L. Zanni, A scaled gradient projection method for constrained image deblurring, Inverse Probl. 25.1, p.15002, 2009.
DOI : 10.1088/0266-5611/25/1/015002

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.325.8771

S. Bonettini, I. Loris, F. Porta, M. Prato, and S. Rebegoldi, On the convergence of variable metric line?search based proximal-gradient method under the Kurdyka? Lojasiewicz inequality

S. Bonettini, G. Landi, E. L. Piccolomini, and L. Zanni, Scaling techniques for gradient projection-type methods in astronomical image deblurring, International Journal of Computer Mathematics, vol.3, issue.3, pp.9-29, 2013.
DOI : 10.1007/s10589-006-6446-0

S. Bonettini, I. Loris, F. Porta, and M. Prato, Variable Metric Inexact Line-Search-Based Methods for Nonsmooth Optimization, SIAM Journal on Optimization, vol.26, issue.2, pp.891-921, 2016.
DOI : 10.1137/15M1019325

URL : http://arxiv.org/abs/1506.00385

M. J. Booth, Adaptive optical microscopy: the ongoing quest for a perfect image, Light: Science & Applications, vol.8520, issue.4, pp.1-7, 2014.
DOI : 10.1364/OE.20.016532

E. Bostan, E. Froustey, B. Rappaz, E. Shaffer, D. Sage et al., Phase retrieval by using transport-of-intensity equation and differential interference contrast microscopy, 2014 IEEE International Conference on Image Processing (ICIP)
DOI : 10.1109/ICIP.2014.7025800

D. Brandon and W. Kaplan, Microstructural Characterization of Materials, 2008.
DOI : 10.1002/9780470727133

E. Candès, X. Li, and M. Soltanolkotabi, Phase Retrieval via Wirtinger Flow: Theory and Algorithms, IEEE Transactions on Information Theory, vol.61, issue.4, pp.1985-2008, 2015.
DOI : 10.1109/TIT.2015.2399924

E. Candès and B. Recht, Exact Matrix Completion via Convex Optimization, Foundations of Computational Mathematics, vol.170, issue.1, pp.717-772, 2009.
DOI : 10.1017/CBO9780511814068

E. Candès, T. Strohmer, and V. Voroniski, PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming, Communications on Pure and Applied Mathematics, vol.38, issue.5, pp.1241-1274, 2012.
DOI : 10.1109/9.554402

M. Carlsson, On convexification/optimization of functionals including an l2-misfit term

P. Charbonnier, L. Blanc-féraud, G. Aubert, and M. Barlaud, Deterministic edge-preserving regularization in computed imaging, IEEE Transactions on Image Processing, vol.6, issue.2, pp.298-311, 1997.
DOI : 10.1109/83.551699

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.125.5878

X. Chen, B. Zheng, and H. Liu, Optical and Digital Microscopic Imaging Techniques and Applications in Pathology, Analytical Cellular Pathology, vol.34, issue.1-2, pp.5-18, 2011.
DOI : 10.1155/2011/150563

URL : http://doi.org/10.1155/2011/150563

C. Nice and J. K. Heath, A simple method allowing DIC imaging in conjunction with confocal microscopy, Journal of Microscopy, vol.2173, pp.265-274, 2005.

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, Quantitative DIC microscopy using a geometric phase shifter, Proc. SPIE 2984, Three-Dimensional Microscopy: Image Acquisition and Processing IV, pp.72-81, 1997.
DOI : 10.1117/12.271252

E. J. Cogswell and E. J. Sheppard, Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging, Journal of Microscopy, vol.18, issue.1
DOI : 10.1049/el:19820668

]. A. Cornelio, F. Porta, and M. Prato, A convergent least-squares regularized blind deconvolution approach, Applied Mathematics and Computation, vol.259, pp.173-186, 2015.
DOI : 10.1016/j.amc.2015.02.048

K. Creath, V Phase-Measurement Interferometry Techniques, In: Progress in Optics, vol.26, pp.349-393, 1988.
DOI : 10.1016/S0079-6638(08)70178-1

Y. H. Dai and Y. X. Yuan, Alternate minimization gradient method, IMA Journal of Numerical Analysis, vol.23, issue.3, pp.377-393, 2003.
DOI : 10.1093/imanum/23.3.377

R. De-asmundis, D. Di-serafino, F. Riccio, and G. Toraldo, On spectral properties of steepest descent methods, IMA Journal of Numerical Analysis, vol.33, issue.4, pp.1416-1435, 2013.
DOI : 10.1093/imanum/drs056

D. H. Brandwood, A Complex Gradient Operator and its Application in Adaptive Array Theory, IEEE Proceedings F -Communications, Radar and Signal Processing, pp.11-16, 1983.

T. Evgeniou, M. Pontil, and T. Poggio, Phase retrieval algorithms: a comparison, Applied Optics, vol.2115, pp.2758-2769, 1982.

R. Fletcher, A limited memory steepest descent method, Mathematical Programming, vol.24, issue.1-2, pp.1-2, 2012.
DOI : 10.1016/S0040-6090(02)01117-3

R. Fletcher, Practical methods of optimization. 2nd, 2000.
DOI : 10.1002/9781118723203

P. Frankel, G. Garrigos, and J. Peypouquet, Splitting Methods with Variable Metric for Kurdyka?????ojasiewicz Functions and General Convergence Rates, Journal of Optimization Theory and Applications, vol.122, issue.4, pp.874-900, 2015.
DOI : 10.1007/s00211-012-0475-7

M. M. Frigault, J. Lacoste, J. L. Swift, and C. M. Brown, Live-cell microscopy - tips and tools, Journal of Cell Science, vol.122, issue.6, pp.753-767, 2009.
DOI : 10.1242/jcs.033837

W. Galbraith and G. B. David, An aid to understanding differential interference contrast microscopy: computer simulation, Journal of Microscopy, vol.88, issue.2, pp.147-176, 1976.
DOI : 10.1111/j.1365-2818.1968.tb00616.x

R. W. Gerchberg and W. O. Saxton, A practical algorithm for the determination of phase from image and diffraction plane pictures, pp.237-246, 1972.

J. C. Gilbert and J. Nocedal, Global Convergence Properties of Conjugate Gradient Methods for Optimization, SIAM Journal on Optimization, vol.2, issue.1, pp.21-42, 1992.
DOI : 10.1137/0802003

URL : https://hal.archives-ouvertes.fr/inria-00075291

D. Goldstein, Polarized Light. 3rd, 2011.

T. J. Holmes, Signal-processing characteristics of differential-interference-contrast microscopy 2: Noise considerations in signal recovery, Applied Optics, vol.27, issue.7, pp.1302-1308, 1988.
DOI : 10.1364/AO.27.001302

T. J. Holmes and W. J. Levy, Signal-processing characteristics of differential-interferencecontrast microscopy, Applied Optics, vol.2618, pp.3929-3939, 1987.
DOI : 10.1364/ao.26.003929

A. B. Johnson and L. J. Lewis, Molecular Biology of the Cell. 4th, New York: Garland Science, 2002.

F. Kagalwala and T. Kanade, Computational Model of Image Formation Process in DIC Microscopy, pp.193-207, 1998.

Z. Kam, Microscopic differential interference contrast image processing by line integration (LID) and deconvolution, In: Bioimaging, vol.64, pp.166-176, 1998.
DOI : 10.1002/1361-6374(199812)6:4<166::aid-bio2>3.3.co;2-p

H. Kamiokaa, T. Honjoa, and T. Takano-yamamoto, A three-dimensional distribution of osteocyte processes revealed by the combination of confocal laser scanning microscopy and differential interference contrast microscopy, Bone, vol.28, issue.2, pp.145-149, 2001.
DOI : 10.1016/S8756-3282(00)00421-X

C. T. Kelley, Iterative Methods for Optimization, Philadelphia: SIAM, pp.43-44, 1999.
DOI : 10.1137/1.9781611970920

S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard, Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging, Optics Letters, vol.35, issue.3, pp.447-449, 2010.
DOI : 10.1364/OL.35.000447.m001

S. Krantz and H. R. Parks, A primer of Real Analytic Functions, Birkhäuser, 2002.

K. Kurdyka, On gradients of functions definable in o-minimal structures, Annales de l???institut Fourier, vol.48, issue.3, pp.769-783, 1998.
DOI : 10.5802/aif.1638

W. Lang, Nomarski differential interference contrast microscopy II. Formation of the interference image, Zeiss Inf, pp.12-16, 1969.

H. Lantéri, M. Roche, and C. Aime, Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms, Inverse Problems, vol.18, issue.5, pp.1397-1419, 2002.
DOI : 10.1088/0266-5611/18/5/313

S. ?ojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les Équations aux Dérivées Partielles. Paris: Éditions du Centre National de la Recherche Scientifique, pp.87-89, 1963.

C. J. Sheppard, N. I. Smith, M. R. Arnison, K. G. Larkin, and C. J. , Linear phase imaging using differential interference contrast microscopy, Journal of Microscopy, vol.2141, pp.7-12, 2004.

N. I. Smith, P. W. Fekete, M. R. Arnison, C. J. Cogswell, and K. G. Larkin, Using the Hilbert transform for 3D visualization ofdifferential interference contrast microscope images, Journal of Microscopy, vol.1991, pp.79-84, 2000.

D. Malacara, Phase-Shifting Interferometry, Revista Mexicana de Física, vol.1, pp.36-42, 1990.
DOI : 10.1201/9781420027273.ch7

S. B. Mehta and R. Oldenbourg, Image simulation for biological microscopy: microlith, Biomedical Optics Express, vol.5, issue.6, pp.1822-1838, 2014.
DOI : 10.1364/BOE.5.001822.m002

URL : http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4052850

S. B. Mehta and C. J. Sheppard, Partially coherent image formation in differential interference contrast (DIC) microscope, Optics Express, vol.16, issue.24, pp.19462-19479, 2008.
DOI : 10.1364/OE.16.019462.m004

S. B. Mehta and C. J. Sheppard, Sample-less calibration of the differential interference contrast microscope, Applied Optics, vol.49, issue.15, pp.2954-2968, 2010.
DOI : 10.1364/AO.49.002954.m005

E. B. Van-munster, L. J. Van, J. A. Vliet, and . Aten, Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope, Journal of Microscopy, vol.188, issue.2, pp.149-157, 1997.
DOI : 10.1046/j.1365-2818.1997.2570815.x

D. B. Murphy, Fundamentals of Light Microscopy and Electronic Imaging, 2001.
DOI : 10.1002/9781118382905

J. Nocedal and S. J. Wright, Numerical optimization. 2nd, 2006.

T. Ono, R. Okamoto, and S. Takeuchi, An entanglement-enhanced microscope, Nature Communications, vol.64, pp.1-7, 2013.
DOI : 10.1103/PhysRevA.65.050303

URL : http://arxiv.org/abs/1401.8075

F. J. Pedrotti and L. S. Pedrotti, Introduction to Optics, 1992.

L. S. Pedrotti, Basic Physical Optics, pp.117-167, 2008.
DOI : 10.1117/3.784938.ch4

M. Prato, R. Cavicchioli, L. Zanni, P. Boccacci, and M. Bertero, Efficient deconvolution methods for astronomical imaging: algorithms and IDL-GPU codes, Astronomy & Astrophysics, vol.539, pp.539-133, 2012.
DOI : 10.1051/0004-6361/201118681

URL : http://arxiv.org/abs/1210.2258

C. Preza, Rotational-diversity phase estimation from differential-interference-contrast microscopy images, Journal of the Optical Society of America A, vol.17, issue.3, pp.415-424, 2000.
DOI : 10.1364/JOSAA.17.000415

C. Preza, S. V. King, and C. J. , Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images, Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, 2006.
DOI : 10.1117/12.661550

C. Preza, D. L. Snyder, and J. Conchello, Image reconstruction for three-dimensional transmitted-light DIC microscopy " . In: Three-Dimensional Microscopy: Image Acquisition and Processing IV, Proc. SPIE, 1997.
DOI : 10.1117/12.271264

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.442

C. Preza, D. L. Snyder, and J. Conchello, Theoretical development and experimental evaluation of imaging models for differential-interference-contrast microscopy, Journal of the Optical Society of America A, vol.16, issue.9, pp.2185-2199, 1999.
DOI : 10.1364/JOSAA.16.002185

L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, vol.60, issue.1-4, pp.1-4, 1992.
DOI : 10.1016/0167-2789(92)90242-F

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao et al., Phase Retrieval with Application to Optical Imaging, IEEE Signal Processing Magazine, vol.323, pp.87-109, 2015.
DOI : 10.1109/msp.2014.2352673

URL : http://bib-pubdb1.desy.de//record/220020/files/1402.7350v1.pdf

M. Shribak and S. Inoué, Orientation-independent differential interference contrast microscopy, Applied Optics, vol.453, pp.460-469, 2006.
DOI : 10.1017/s1431927606063434

W. Sun, N. Fang, and B. G. Trewyn, Endocytosis of a single mesoporous silica nanoparticle into a human lung cancer cell observed by differential interference contrast microscopy, Analytical and Bioanalytical Chemistry, vol.8, issue.3, p.2119, 2008.
DOI : 10.1007/s00216-008-2162-1

W. Wirtinger, Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen, Mathematische Annalen, vol.9720, pp.357-376, 1926.
DOI : 10.1007/bf01447872

G. Wang, W. Sun, Y. Luo, and N. Fang, Resolving Rotational Motions of Nano-objects in Engineered Environments and Live Cells with Gold Nanorods and Differential Interference Contrast Microscopy, Journal of the American Chemical Society, vol.132, issue.46, pp.46-16417, 2010.
DOI : 10.1021/ja106506k

L. Wang and H. Wu, Biomedical Optics: Principles and Imaging, 2007.
DOI : 10.1002/9780470177013

R. Wayne, Light and Video Microscopy, 2014.

S. M. Wilson and A. Bacic, Preparation of plant cells for transmission electron microscopy to optimize immunogold labeling of carbohydrate and protein epitopes, Nature Protocols, vol.37, issue.9, pp.1716-1727, 2012.
DOI : 10.1242/jcs.080085

Y. Xu and W. Yin, A Block Coordinate Descent Method for Regularized Multiconvex Optimization with Applications to Nonnegative Tensor Factorization and Completion, SIAM Journal on Imaging Sciences, vol.6, issue.3, pp.1758-1789, 2013.
DOI : 10.1137/120887795

B. Zhou, L. Gao, and Y. H. Dai, Gradient Methods with Adaptive Step-Sizes, Computational Optimization and Applications, vol.14, issue.1, pp.69-86, 2006.
DOI : 10.6028/jres.049.044