The mathematical theory of compressed sensing (CS) asserts that one can acquire signals from measurements whose rate is much lower than the total bandwidth. Whereas the CS theory is now well developed, challenges concerning hardware implementations of CS-based acquisition devices-especially in optics-have only started being addressed. This paper presents an implementation of compressive sensing in fluorescence microscopy and its applications to biomedical imaging. Our CS microscope combines a dynamic structured wide-field illumination and a fast and sensitive single-point fluorescence detection to enable reconstructions of images of fluorescent beads, cells, and tissues with undersampling ratios (between the number of pixels and number of measurements) up to 32. We further demonstrate a hyperspectral mode and record images with 128 spectral channels and undersampling ratios up to 64, illustrating the potential benefits of CS acquisition for higher-dimensional signals, which typically exhibits extreme redundancy. Altogether, our results emphasize the interest of CS schemes for acquisition at a significantly reduced rate and point to some remaining challenges for CS fluorescence microscopy.biological imaging | compressed sensing | computational imaging | sparse signals F luorescence microscopy is a fundamental tool in basic and applied biomedical research. Because of its optical sensitivity and molecular specificity, fluorescence imaging is employed in an increasing number of applications which, in turn, are continuously driving the development of advanced microscopy systems that provide imaging data with ever higher spatio-temporal resolution and multiplexing capabilities. In fluorescence microscopy, one can schematically distinguish two kinds of imaging approaches, differing by their excitation and detection modalities (1). In wide-field (WF) microscopy, a large sample area is illuminated and the emitted light is recorded on a multidetector array, such as a CCD camera. In contrast, in raster scan (RS) microscopy, a point excitation is scanned through the sample and a point detector is used to detect the fluorescence signal at each position.While very distinct in their implementation and applications, these imaging modalities have in common that the acquisition is independent of the information content of the image. Rather, the number of measurements, either serial in RS or parallel in WF, is imposed by the Nyquist-Shannon theorem. This theorem states that the sampling frequency (namely the inverse of the image pixel size) must be twice the bandwidth of the signal, which is determined by the diffraction limit of the microscope lens equal to λ∕2NA (λ is the optical wavelength and NA the objective numerical aperture). Yet, most images, including those of biological interest, can be described by a number of parameters much lower than the total number of pixels. In everyday's world, a striking consequence of this compressibility is the ability of consumer cameras with several megapixel detectors to routinely reduce t...
In this paper, we prove that if R is an Archimedean reduced ring and satisfy ACC on annihilators, then Rrrxss is also an Archimedean reduced ring. More generally we prove that if R is a right Archimedean ring satisfying the ACC on annihilators and α is a rigid automorphism of R, then the skew power series ring Rrrx; αss is right Archimedean reduced ring. We also provide some examples to justify the assumptions we made to obtain the required result.
Following attempts at an analytic proof of the Pentagonal Number Theorem, we report on the discovery of a general principle leading to the unexpected cancellation of oscillating sums, of which n 2 ≤x (−1) n e 1 2
√x , making the Riemann Hypothesis estimate "trivial".
One of the most important methods of remote e‐voting is based on the ElGamal homomorphic cryptosystems. These systems are secure enough provided that the discrete logarithm problem is secure. However, one of their main challenges is the limitation on the number of voters. In this paper, a new scheme is suggested so as to increase the number of voters by designing parallel voting subprotocols. The required time for the tally computing process for 10 subprotocols decreases to less than 20% of the time in a similar scheme. The necessary memory space and the number of random integers do not show a significant change in comparison with those of similar voting protocols. Also, the communication complexity is of order O(Nt), which seems reasonable. The security comparisons show an acceptable rate of security in a reasonable length of keys for the proposed protocol. Simulations show that the number of covering voters in this protocol is 10 times more than that of a similar protocol.
We show that for every r ≥ 1, and all r distinct (sufficiently large) primes p1, ..., pr > p0(r), there exist infinitely many integers n such that 2n n is divisible by these primes to only low multiplicity. From a theorem of Kummer, an upper bound for the number of times that a prime pj can divide 2n n is 1 + log n/ log pj; and our theorem shows that we can find integers n where for j = 1, ..., r, pj divides 2n n with multiplicity at most o(log n). We connect this result to a famous conjecture by R. L. Graham on whether there are infinitely many integers n such that 2n n is coprime to 105.
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