Fermat's spiral

About: Fermat's spiral is a research topic. Over the lifetime, 40 publications have been published within this topic receiving 643 citations.

Optimization of overlap uniformness for ptychography

Abstract: We demonstrate the advantages of imaging with ptychography scans that follow a Fermat spiral trajectory. This scan pattern provides a more uniform coverage and a higher overlap ratio with the same number of scan points over the same area than the presently used mesh and concentric [13] patterns. Under realistically imperfect measurement conditions, numerical simulations show that the quality of the reconstructed image is improved significantly with a Fermat spiral compared with a concentric scan pattern. The result is confirmed by the performance enhancement with experimental data, especially under low-overlap conditions. These results suggest that the Fermat spiral pattern increases the quality of the reconstructed image and tolerance to data with imperfections.

Connected fermat spirals for layered fabrication

Abstract: We develop a new kind of "space-filling" curves, connected Fermat spirals, and show their compelling properties as a tool path fill pattern for layered fabrication. Unlike classical space-filling curves such as the Peano or Hilbert curves, which constantly wind and bind to preserve locality, connected Fermat spirals are formed mostly by long, low-curvature paths. This geometric property, along with continuity, influences the quality and efficiency of layered fabrication. Given a connected 2D region, we first decompose it into a set of sub-regions, each of which can be filled with a single continuous Fermat spiral. We show that it is always possible to start and end a Fermat spiral fill at approximately the same location on the outer boundary of the filled region. This special property allows the Fermat spiral fills to be joined systematically along a graph traversal of the decomposed sub-regions. The result is a globally continuous curve. We demonstrate that printing 2D layers following tool paths as connected Fermat spirals leads to efficient and quality fabrication, compared to conventional fill patterns.

A new design and rationale for 3D orthogonally oversampled k-space trajectories.

Abstract: A novel center-out 3D trajectory for sampling magnetic resonance data is presented. The trajectory set is based on a single Fermat spiral waveform, which is substantially undersampled in the center of k-space. Multiple trajectories are combined in a “stacked cone” configuration to give very uniform sampling throughout a “hub,” which is very efficient in terms of gradient performance and uniform trajectory spacing. The fermat looped, orthogonally encoded trajectories (FLORET) design produces less gradient-efficient trajectories near the poles, so multiple orthogonal hub designs are shown. These multihub designs oversample k-space twice with orthogonal trajectories, which gives unique properties but also doubles the minimum scan time for critical sampling of k-space. The trajectory is shown to be much more efficient than the conventional stack of cones trajectory, and has nearly the same signal-to-noise ratio efficiency (but twice the minimum scan time) as a stack of spirals trajectory. As a center-out trajectory, it provides a shorter minimum echo time than stack of spirals, and its spherical k-space coverage can dramatically reduce Gibbs ringing. Magn Reson Med, 2011. © 2011 Wiley Periodicals, Inc.

2D array design based on Fermat spiral for ultrasound imaging

Abstract: The main challenge faced by 3D ultrasonic imaging with 2D array transducers is the large number of elements required to achieve an acceptable level of quality in the images. Therefore, the optimisation of the array layout, in order to reduce the number of active elements in the aperture, has been a research topic in the last years. Nowadays, array technology has made viable the production of 2D arrays with larger flexibility on elements size, shape and position, allowing to study other configurations dierent to the clasical matrix organization, such as circular, archimedes spiral or polygonal layout between others. In this work, the problem of designing an imaging system array with large apertures and a very limited number of active elements (Ne =128 and Ne= 256) using the Fermat spiral layout has been studied. As summary, a general discussion about the most interesting cases is presented.

Zone plates performing generalized Hankel transforms and their metrological applications

Abstract: A modified computer-generated Fresnel zone plane with the zone distribution determined by a Fermat spiral is proposed for performing generalized Hankel transforms. The special case of these elements realizing the generalized Hankel transform of first order is of special interest from the point of view of alignment applications. The doughnut shaped focal pattern, characteristic for the whole family of proposed zone plates exhibits then the smallest diameter. Its diameter is also smaller than other solutions proposed previously for obtaining focal spots with a black dip in the centre.