Improved bounds for the CF algorithm

International audienceWe consider the problem of isolating the real roots of a square-free polynomial with integer coefficients using the classic variant of the continued fraction algorithm (CF), introduced by Akritas. %% We compute a lower bound on the positive real roots of univariate polynomials using exponential search. This allows us to derive a worst case bound of \sOB( d^4\tau^2) for isolating the real roots of a polynomial with integer coefficients using the {\em classic variant of CF}, where $d$ is the degree of the polynomial and $\tau$ the maximum bitsize of its coefficients. This improves the previous bound of Sharma by a factor of $d^3$ and matches the bound derived by Mehlhorn and Ray for another variant of CF which is combined with subdivision; it also matches the worst case bound of the classical subdivision-based solvers \func{sturm}, \func{descartes}, and \func{bernstein}

Applications of the golden angle in cardiovascular MRI

The use of radial trajectories has been seen as a potential solution to highly efficient cardiovascular magnetic resonance imaging (MRI). By acquiring a broad range of spatial frequencies per repetition time, the acquisition is time-efficient and robust against motion. Of particular interest is the golden angle profile order, which promises a near-uniform k-space coverage for an arbitrary number of readouts, enabling flexible data resorting, which is critical for efficient cardiovascular MRI. In Study I the use of 2D golden angle profile ordering is explored for imaging pulmonary embolisms. The insensitivity to motion and flow is used to reduce the artifacts that otherwise degrade images of the pulmonary vasculature when imaging with thin slices. It was found that the proposed technique could improve the image quality. Another source of artifacts arises when gradients are rapidly switched, and local induction of eddy currents may perturb spin equilibrium. In Study II, we propose a generalized golden angle profile orderings in 3D which reduces eddy-current artifacts. We demonstrate the efficacy of our generalization through numerical simulations, phantom imaging and imaging of a healthy volunteer. In Study III an improved 2D golden angle profile ordering was explored which resulted in a higher degree of k-space uniformity after physiological binning. This novel profile ordering was used in combination with a phase-contrast readout to enable quantification of myocardial tissue velocity and transmitral blood flow velocity, which are essential parameters for diastolic function assessment. When compared to echocardiography, it was found that MRI could accurately quantify myocardial tissue velocity, whereas transmitral blood flow velocity was underestimated. Study IV explored a further development of Study III by proposing a 3D version of the improved profile ordering. This novel ordering was used to acquire whole-heart functional images during free-breathing in less than one minute. Together, these results indicate that golden-angle-based imaging has the potential to improve cardiovascular MRI in several areas

High-dimensional quantum information processing with linear optics

Quantum information processing (QIP) is an interdisciplinary field concerned with the development of computers and information processing systems that utilize quantum mechanical properties of nature to carry out their function. QIP systems have become vastly more practical since the turn of the century. Today, QIP applications span imaging, cryptographic security, computation, and simulation (quantum systems that mimic other quantum systems). Many important strategies improve quantum versions of classical information system hardware, such as single photon detectors and quantum repeaters. Another more abstract strategy engineers high-dimensional quantum state spaces, so that each successful event carries more information than traditional two-level systems allow. Photonic states in particular bring the added advantages of weak environmental coupling and data transmission near the speed of light, allowing for simpler control and lower system design complexity. In this dissertation, numerous novel, scalable designs for practical high-dimensional linear-optical QIP systems are presented. First, a correlated photon imaging scheme using orbital angular momentum (OAM) states to detect rotational symmetries in objects using measurements, as well as building images out of those interactions is reported. Then, a statistical detection method using chains of OAM superpositions distributed according to the Fibonacci sequence is established and expanded upon. It is shown that the approach gives rise to schemes for sorting, detecting, and generating the recursively defined high-dimensional states on which some quantum cryptographic protocols depend. Finally, an ongoing study based on a generalization of the standard optical multiport for applications in quantum computation and simulation is reported upon. The architecture allows photons to reverse momentum inside the device. This in turn enables realistic implementation of controllable linear-optical scattering vertices for carrying out quantum walks on arbitrary graph structures, a powerful tool for any quantum computer. It is shown that the novel architecture provides new, efficient capabilities for the optical quantum simulation of Hamiltonians and topologically protected states. Further, these simulations use exponentially fewer resources than feedforward techniques, scale linearly to higher-dimensional systems, and use only linear optics, thus offering a concrete experimentally achievable implementation of graphical models of discrete-time quantum systems

On Continued Fraction Expansion of Real Roots of Polynomial Systems, Complexity and Condition Numbers

International audienceWe elaborate on a correspondence between the coeffcients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions, that use only integer arithmetic (in contrast to Bernstein basis) and are feasible over unbounded regions. Then, we study an algorithm to split this representation and we obtain a subdivision scheme for the domain of multivariate polynomial functions. This implies a new algorithm for real root isolation, MCF, that generalizes the Continued Fraction (CF) algorithm of univariate polynomials. A partial extension of Vincent's Theorem for multivariate polynomials is presented, which allows us to prove the termination of the algorithm. Bounding functions, projection and preconditioning are employed to speed up the scheme. The resulting isolation boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. Finally, we present new complexity bounds for a simplified version of the algorithm in the bit complexity model, and also bounds in the real RAM model for a family of subdivision algorithms in terms of the real condition number of the system. Examples computed with our C++ implementation illustrate the practical aspects of our method

Domino tiling, gene recognition and mice

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999.Includes bibliographical references (p. 186-192).by Lior Samuel Pachter.Ph.D

Doctor of Philosophy

dissertationThis dissertation presents original research that improves the ability of magnetic resonance imaging (MRI) to measure temperature in aqueous tissue using the proton resonance frequency (PRF) shift and T1 measurements in fat tissue in order to monitor focused ultrasound (FUS) treatments. The inherent errors involved in measuring the longitudinal relaxation time T1 using the variable flip angle method with a two-dimensional (2D) acquisition are presented. The edges of the slice profile can contribute a significant amount of signal for large flip angles at steady state, which causes significant errors in the T1 estimate. Only a narrow range of flip angle combinations provided accurate T1 estimates. Respiration motion causes phase artifacts, which lead to errors when measuring temperature changes using the PRF method. A respiration correction method for 3D imaging temperature of the breast is presented. Free induction decay (FID) navigators were used to measure and correct phase offsets induced by respiration. The precision of PRF temperature measurements within the breast was improved by an average factor of 2.1 with final temperature precision of approximately 1 °C. Locating the position of the ultrasound focus in MR coordinates of an ultrasound transducer with multiple degrees of freedom can be difficult. A rapid method for predicting the position using 3 tracker coils with a special MRI pulse iv sequence is presented. The Euclidean transformation of the coil's current positions to their calibration positions was used to predict the current focus position. The focus position was predicted to within approximately 2.1 mm in less than 1 s. MRI typically has tradeoffs between imaging field of view and spatial and temporal resolution. A method for acquiring a large field of view with high spatial and temporal resolution is presented. This method used a multiecho pseudo-golden angle stack of stars imaging sequence to acquire the large field of view with high spatial resolution and k-space weighted image contrast (KWIC) to increase the temporal resolution. The pseudo-golden angle allowed for removal of artifacts introduced by the KWIC reconstruction algorithm. The multiple echoes allowed for high readout bandwidth to reduce blurring due to off resonance and chemical shift as well as provide separate water/fat images, estimates of the initial signal magnitude M(0), T2 * time constant, and combination of echo phases. The combined echo phases provided significant improvement to the PRF temperature precision, and ranged from ~0.3-1.0 °C within human breast. M(0) and T2 * values can possibly be used as a measure of temperature in fat