Blur resolved OCT: full-range interferometric synthetic aperture microscopy through dispersion encoding

We present a computational method for full-range interferometric synthetic aperture microscopy (ISAM) under dispersion encoding. With this, one can effectively double the depth range of optical coherence tomography (OCT), whilst dramatically enhancing the spatial resolution away from the focal plane. To this end, we propose a model-based iterative reconstruction (MBIR) method, where ISAM is directly considered in an optimization approach, and we make the discovery that sparsity promoting regularization effectively recovers the full-range signal. Within this work, we adopt an optimal nonuniform discrete fast Fourier transform (NUFFT) implementation of ISAM, which is both fast and numerically stable throughout iterations. We validate our method with several complex samples, scanned with a commercial SD-OCT system with no hardware modification. With this, we both demonstrate full-range ISAM imaging, and significantly outperform combinations of existing methods.Comment: 17 pages, 7 figures. The images have been compressed for arxiv - please follow DOI for full resolutio

Orientation of the opposition axis in mentally simulated grasping

Five normal subjects were tested in a simulated grasping task. A cylindrical container filled with water was placed on the center of a horizontal monitor screen. Subjects used a precision grip formed by the thumb and index finger of their right hand. After a preliminary run during which the container was present, it was replaced by an image of the upper surface of the cylinder appearing on the horizontal computer screen on which the real cylinder was placed during the preliminary run. In each trial the image was marked with two contact points which defined an opposition axis in various orientations with respect to the frontal plane. The subjects’ task consisted, once shown a stimulus, of judging as quickly as possible whether the previously experienced action of grasping the container full of water and pouring the water out would be easy, difficult or impossible with the fingers placed according to the opposition axis indicated on the circle. Response times were found to be longer for the grasps judged to be more difficult due to the orientation and position of the opposition axis. In a control experiment, three subjects actually performed the grasps with different orientations and positions of the opposition axis. The effects of these parameters on response time followed the same trends as during simulated movements. This result shows that simulated hand movements take into account the same biomechanical limitations as actually performed movements

Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts

We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Pad\'e approximation and constrained multivariate interpolation, and has applications in coding theory and security. This problem asks to find univariate polynomial relations between $m$ vectors of size $\sigma$; these relations should have small degree with respect to an input degree shift. For an arbitrary shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov normal form with a cost of $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ field operations, where $\omega$ is the exponent of matrix multiplication and the notation $\mathcal{O}\tilde{~}(\cdot)$ indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Pad\'e approximation and in the general interpolation case, compute non-normalized bases. Since for arbitrary shifts such bases may have size $\Theta(m^2 \sigma)$, the cost bound $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ was feasible only with restrictive assumptions on the shift that ensure small output sizes. The question of handling arbitrary shifts with the same complexity bound was left open. To obtain the target cost for any shift, we strengthen the properties of the output bases, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always $\mathcal{O}(m \sigma)$. Then, we design a divide-and-conquer scheme. We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms

A radix-independent error analysis of the Cornea-Harrison-Tang method

International audienceAssuming floating-point arithmetic with a fused multiply-add operation and rounding to nearest, the Cornea-Harrison-Tang method aims to evaluate expressions of the form $ab+cd$with high relative accuracy. In this paper we provide a rounding error analysis of this method,which unlike previous studiesis not restricted to binary floating-point arithmetic but holds for any radix $\beta$.We show first that an asymptotically optimal bound on the relative error of this method is$2u + O(u^2)$, where $u= \frac{1}{2}\beta^{1-p}$ is the unit roundoff in radix $\beta$ and precision $p$.Then we show that the possibility of removing the $O(u^2)$ term from this bound is governed bythe radix parity andthe tie-breaking strategy used for rounding: if $\beta$ is odd or rounding is \emph{to nearest even}, then the simpler bound $2u$ is obtained,while if $\beta$ is even and rounding is \emph{to nearest away}, then there exist floating-point inputs $a,b,c,d$ that lead to a relative error larger than $2u + \frac{2}{\beta} u^2 - 4u^3$.All these results hold provided underflows and overflows do not occurand under some mild assumptions on $p$ satisfied by IEEE 754-2008 formats

A (hopefully) friendly introduction to the complexity of polynomial matrix computations

This paper aims at a friendly introduction to the field of fast algorithms for polynomial matrices, and surveys the results of the ISSAC 2003 paper 'On the Complexity of Polynomial Matrix Computations' by Pascal Giorgi, Claude-Pierre Jeannerod, and Gilles Villard

Exploiting structure in floating-point arithmetic

Invited paper - MACIS 2015 (Sixth International Conference on Mathematical Aspects of Computer and Information Sciences)International audienceThe analysis of algorithms in IEEE floating-point arithmetic is most often carried out via repeated applications of the so-called standard model, which bounds the relative error of each basic operation by a common epsilon depending only on the format. While this approach has been eminently useful for establishing many accuracy and stability results, it fails to capture most of the low-level features that make floating-point arithmetic so highly structured. In this paper, we survey some of those properties and how to exploit them in rounding error analysis. In particular, we review some recent improvements of several classical, Wilkinson-style error bounds from linear algebra and complex arithmetic that all rely on such structure properties

Reply to our critics

Marc Jeannerod and I wrote a Précis of our 2003 book Ways of Seeing. The journal Dialogue asked Tim Schroeder, Alva Noë, Pierre Poirier and Martin Ratte to write a critical essay on our book. In this piece, we reply to our critics