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Alternating minimization (AltMin) is a generic term for a widely popular approach in non-convex learning: often, it is possible to partition the variables into two (or more) sets, so that the problem is convex/tractable in one set if the other is held fixed (and vice versa). This allows for alternating between optimally updating one set of variables, and then the other. AltMin methods typically do not have associated global consistency guarantees; even though they are empirically observed to perform better than methods (e.g. based on convex optimization) that do have guarantees. In this thesis, we obtain rigorous performance guarantees for AltMin in three statistical learning settings: low rank matrix completion, phase retrieval and learning sparsely-used dictionaries. The overarching theme behind our results consists of two parts: (i) devising new initialization procedures (as opposed to doing so randomly, as is typical), and (ii) establishing exponential local convergence from this initialization. Our work shows that the pursuit of statistical guarantees can yield algorithmic improvements (initialization in our case) that perform better in practice. / text

La Synthèse d'Ouverture Optique permet d'obtenir la résolution d'un instrument de grand diamètre en faisant interférer les faisceaux issus de plusieurs sous-pupilles de diamètre inférieur. Néanmoins, la difficulté principale de cette méthode réside dans le cophasage de l'instrument, c'est-à-dire dans la mesure puis la correction des aberrations différentielles présentes entre les pupilles. Dans ce contexte, les techniques de type plan focal présentent un avantage certain. Ainsi, le phase retrieval et la diversité de phase (basés respectivement sur l'acquisition d'une ou d'au moins deux images dans des conditions de phase différentes) sont aujourd'hui couramment employés avec des instruments monolithiques; leur applicabilité au cophasage de systèmes multipupilles a également été démontrée en laboratoire. Toutefois, ces estimateurs présentent l'inconvénient d'être itératifs donc potentiellement coûteux en temps de calcul. Nous nous proposons dans ce manuscrit de développer de nouveaux estimateurs de cophasage qui soient analytiques, permettant ainsi une estimation en temps-réel des aberrations sur objet étendu. Pour cela, nous démontrons qu'en exprimant le critère à minimiser sous forme quadratique nous aboutissons à une estimée simple de la phase recherchée. Nous montrons également que nous pouvons parvenir à l'expression de ce nouveau critère en considérant une approximation affine de la fonction de transfert optique. Les performances obtenues en simulation révèlent qu'il est possible de fermer une boucle de cophasage à faible flux et par la même occasion de restaurer l'objet observé dans un but d'imagerie. La mise en oeuvre expérimentale de l'ensemble des algorithmes (itératifs et analytiques) dans le cadre de différents projets permet d'affirmer que l'approche plan focal peut désormais être utilisée pour cophaser des systèmes multipupilles complexes.

I present several new methods for the characterisation of ultrashort pulses using interferometry. A generalisation of the concatenation algorithm for spectral shearing interferometry enables interferograms taken at multiple shears to be combined. This improves the precision of the reconstructed phase in the presence of detector noise, and enables the relative phase between disjoint spectral components to be obtained without decreasing the spectral resolution. The algorithm is applied to experimental data from two different implementations of spectral shearing interferometry for ultrashort optical pulses. In one, the shears are acquired sequentially, and in the other they are acquired simultaneously. I develop a form of spatio-temporal ultrashort pulse characterisation which performs both spatial and spectral shearing interferometry simultaneously. It requires a similar geometrical setup to common implementations of spectral phase interferometry for direct electric-field reconstruction, but provides complete amplitude and phase characterisation in time and one spatial dimension. I develop the theory of lateral shearing interferometry for spectrally resolved wavefront sensing of extended ultraviolet and soft x-ray pulses generated using high-harmonic generation. A comprehensive set of wavefront measurements of harmonics 13-25 in Krypton show good agreement with theory, validating the technique. I propose and numerically demonstrate quantum-path interferometry mediated by a weak control field for high harmonic generation. This is a general technique for measuring the amplitude and relative phases of each contributing quantum path. The control field perturbatively modulates the phase of each path. The differing sensitivity of each path to the parameters of the control field allows their contributions to be distinguished from one another.

A number of topics related to Fourier imaging are investigated. Relationships between the magnitude of errors in the amplitude and phase of the Fourier transform of images and the mean square error in reconstructed images are derived. The differing effects of amplitude and phase errors are evaluated, and "equivalent" amplitude and phase errors are derived. A model of the probability density function of the Fourier amplitudes of images is derived. The fundamental basis of phase dominance is studied and quantitated. Inconsistencies in published counter-examples of phase dominance are highlighted. The key characteristics of natural images that lead to their observed power spectral behaviour with spatial frequency are determined.

During the last years we have seen the birth of free-electron lasers, a new type of light source ten billion times brighter than syncrotrons and able to produce pulses only a few femtoseconds long. One of the main motivations for building these multi-million dollar machines was the prospect of imaging biological samples such as proteins and viruses in 3D without the need for crystallization or staining. This thesis contains some of the first biological results from free-electron lasers. These results include 2D images, both of whole cells and of the giant mimivirus and also con- tains a 3D density map of the mimivirus assembled from diffraction patterns from many virus particles. These are important proof-of-concept experiments but they also mark the point where free-electron lasers start to produce biologically relevant results. The most noteworthy of these results is the unexpectedly non-uniform density distribution of the internals of the mimivirus. We also present Hawk, the only open-source software toolkit for analysing single particle diffraction data. The Uppsala-developed program suite supports a wide range fo algorithms and takes advantage of Graphics Processing Units which makes it very computationally efficient. Last, the problem introduced by structural variability in samples is discussed. This includes a description of the problem and how it can be overcome, and also how it could be turned into an advantage that allows us to image samples in all of their conformational states.

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No description available.

X-ray computed tomography is a nondestructive method for 3D imaging of inner structure of objects; it is, however, restricted by absorption properties of materials. With phase contrast imaging, observation of samples with low absorption or with parts with similar absorption becomes possible. For a long time, phase contrast imaging has been restricted to synchrotron radiation sources or special techniques due to requirement of spatial coherence of radiation. Along with recent development of X-ray micro and nanofocus tubes and X-ray detectors a phase contrast imaging becomes available also with laboratory sources. In this work an overview of phase contrast imaging methods is given, with an emphassis on propagation based method. Examples of this method and application of phase retrieval algorithm on samples from RIGAKU Nano3DX and GE phoenix v|tome|x L240 stations within Laboratory of X-ray micro and nanotomography CEITEC BUT are shown.

Tato práce se zaměřuje na pokročilou zobrazovací technologii, rentgenovou počítačovou tomografii (CT). Tato nedestruktivní technika je využívána pro výzkum různých biomateiálů ve tkáňovém inženýrství a materiálové vědě obecně (skafoldy, polymery, keramické materiály, kompozity aj.). Vizualizace a kvantifikace ve 3D jsou výhodné v rámci multidisciplinárního přístupu, který je často v těchto odvětvích uplatňován. Záměr této práce lze rozdělit do dvou oblastí. Prvním tématem je optimalizace měřicí procedury různých měkkých materiálů pomocí CT s laboratorními rentgenovými zdroji. To zahrnuje převážně zobrazování ve fázovém kontrastu, konkrétně metodu volného šíření záření (VŠZ). Tato práce teoreticky popisuje VŠZ a demonstruje tento jev na řadě experimentů. Následné nezbytné zpracování dat získaných VŠZ je implementováno a vyhodnoceno na základě míry zlepšení obrazových dat. Druhé téma ukazuje konkrétní aplikace CT v materiálovém inženýrství. Několik studií s různými CT zařízeními ukazuje příklady možných aplikací a obrazového zpracování. Příklady korelace CT dat s jinými doplňkovými technikami ukazují, jak může být CT aplikována v multioborovém přístupu ke komplexnímu řešení vědeckých problémů.