Microscopy has greatly advanced our understanding of biology. Although significant progress has recently been made in optical microscopy to break the diffraction-limit barrier, reliance of such techniques on fluorescent labeling technologies prohibits quantitative 3D imaging of the entire contents of cells. Cryoelectron microscopy can image pleomorphic structures at a resolution of 3-5 nm, but is only applicable to thin or sectioned specimens. Here, we report quantitative 3D imaging of a whole, unstained cell at a resolution of 50-60 nm by X-ray diffraction microscopy. We identified the 3D morphology and structure of cellular organelles including cell wall, vacuole, endoplasmic reticulum, mitochondria, granules, nucleus, and nucleolus inside a yeast spore cell. Furthermore, we observed a 3D structure protruding from the reconstructed yeast spore, suggesting the spore germination process. Using cryogenic technologies, a 3D resolution of 5-10 nm should be achievable by X-ray diffraction microscopy. This work hence paves a way for quantitative 3D imaging of a wide range of biological specimens at nanometer-scale resolutions that are too thick for electron microscopy.coherent diffractive imaging | equally sloped tomography | lensless imaging | iterative phase-retrieval algorithms | oversampling A pplication of the long penetration depth of X-rays to the imaging of large, unstained biological specimens has long been recognized as a possible solution to the thickness restrictions of electron microscopy. Indeed, using sizable protein crystals, X-ray crystallography is currently the primary methodology used for determining the 3D structure of protein molecules at near-atomic or atomic resolution. However, many biological specimens such as whole cells, cellular organelles, some viruses, and many important protein molecules are difficult or impossible to crystallize and hence their structures are not accessible by crystallography. Overcoming these limitations requires the employment of different techniques. One promising approach currently under rapid development is coherent diffraction microscopy (also termed coherent diffractive imaging or lensless imaging) in which the coherent diffraction pattern of a noncrystalline specimen or a nanocrystal is measured and then directly phased to obtain an image (1-28). The well-known phase problem is solved by using the oversampling method (29) in combination with the iterative algorithms (30-33). Since its first experimental demonstration in 1999 (1), coherent diffraction microscopy has been applied to imaging a wide range of materials science and biological specimens such as nanoparticles, nanocrystals, biomaterials, cells, cellular organelles, viruses by using synchrotron radiation (2-21), high harmonic generation (22-24), soft X-ray laser sources (23, 25), and free electron lasers (26-28). Until now, however, the radiation damage problem and the difficulty of acquiring high-quality 3D diffraction patterns from individual whole cells have prevented the successful high-resolution ...

The ability to determine the structure of matter in three dimensions has profoundly advanced our understanding of nature. Traditionally, the most widely used schemes for three-dimensional (3D) structure determination of an object are implemented by acquiring multiple measurements over various sample orientations, as in the case of crystallography and tomography, or by scanning a series of thin sections through the sample, as in confocal microscopy. Here we present a 3D imaging modality, termed ankylography (derived from the Greek words ankylos meaning 'curved' and graphein meaning 'writing'), which under certain circumstances enables complete 3D structure determination from a single exposure using a monochromatic incident beam. We demonstrate that when the diffraction pattern of a finite object is sampled at a sufficiently fine scale on the Ewald sphere, the 3D structure of the object is in principle determined by the 2D spherical pattern. We confirm the theoretical analysis by performing 3D numerical reconstructions of a sodium silicate glass structure at 2 A resolution, and a single poliovirus at 2-3 nm resolution, from 2D spherical diffraction patterns alone. Using diffraction data from a soft X-ray laser, we also provide a preliminary demonstration that ankylography is experimentally feasible by obtaining a 3D image of a test object from a single 2D diffraction pattern. With further development, this approach of obtaining complete 3D structure information from a single view could find broad applications in the physical and life sciences.

In compressed sensing, one takes n < N samples of an N-dimensional vector x 0 using an n × N matrix A, obtaining undersampled measurements y = Ax 0 . For random matrices with independent standard Gaussian entries, it is known that, when x 0 is k-sparse, there is a precisely determined phase transition: for a certain region in the (k/n,n/N)-phase diagram, convex optimization min || x || 1 subject to y = Ax, x ∈ X N typically finds the sparsest solution, whereas outside that region, it typically fails. It has been shown empirically that the same property-with the same phase transition location-holds for a wide range of non-Gaussian random matrix ensembles. We report extensive experiments showing that the Gaussian phase transition also describes numerous deterministic matrices, including Spikes and Sines, Spikes and Noiselets, Paley Frames, Delsarte-Goethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Namely, for each of these deterministic matrices in turn, for a typical k-sparse object, we observe that convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian random matrices. Our experiments considered coefficients constrained to X N for four different sets X ∈ {[0, 1], R + , R, C}, and the results establish our finding for each of the four associated phase transitions.sparse recovery | universality in random matrix theory equiangular tight frames | restricted isometry property | coherence C ompressed sensing aims to recover a sparse vector x 0 ∈ X N from indirect measurements y = Ax 0 ∈ X n with n < N, and therefore, the system of equations y = Ax 0 is underdetermined. Nevertheless, it has been shown that, under conditions on the sparsity of x 0 , by using a random measurement matrix A with Gaussian i.i.d entries and a nonlinear reconstruction technique based on convex optimization, one can, with high probability, exactly recover x 0 (1, 2). The cleanest expression of this phenomenon is visible in the large n; N asymptotic regime. We suppose that the object x 0 is k-sparse-has, at most, k nonzero entries-and consider the situation where k ∼ ρn and n ∼ δN. Fig. 1A depicts the phase diagram ðρ; δ; Þ ∈ ð0; 1Þ 2 and a curve ρ*ðδÞ separating a success phase from a failure phase. Namely, if ρ < ρ*ðδÞ, then with overwhelming probability for large N, convex optimization will recover x 0 exactly; however, if ρ > ρ*ðδÞ, then with overwhelming probability for large N convex optimization will fail. [Indeed, Fig. 1 depicts four curves ρ*ðδjXÞ of this kind for X ∈ f½0; 1; R + ; R; Cg-one for each of the different types of assumptions that we can make about the entries of x 0 ∈ X N (details below).]How special are Gaussian matrices to the above results? It was shown, first empirically in ref. 3 and recently, theoretically in ref. 4, that a wide range of random matrix ensembles exhibits precisely the same behavior, by which we mean the same phenomenon of separation into success and failure phases with the same phase boundary. Such universality, if exhib...