Minimum variance centroid thresholding

Nonlinear differential equations for the wavefront surface at arbitrary Hartmann-plane distances

In the Hartmann test, a wave aberration function W is estimated from the information of the spot diagram drawn in an observation plane. The distance from a reference plane to the observation plane, the Hartmann-plane distance, is typically chosen as z=f, where f is the radius of a reference sphere. The function W and the transversal aberrations {X,Y} calculated at the plane z=f are related by two well-known linear differential equations. Here, we propose two nonlinear differential equations to denote a more general relation between W and the transversal aberrations {U,V} calculated at any arbitrary Hartmann-plane distance z=r. We also show how to directly estimate the wavefront surface w from the information of {U,V}. The use of arbitrary r values could improve the reliability of the measurements of W, or w, when finding difficulties in adequate ray identification at z=f.

Extrapolation, interpolation, and identification of spots in Hartmann patterns

Based on Fourier analysis, we propose a simple method to extrapolate spots outside an aperture that bounds a Hartmann pattern, and interpolate spots in those regions where some spots are lost or damaged. It allows us to smoothen or remove the discontinuities of the wavefront slopes at the border of the aperture or in those inner regions where some spots are lost. The method changes the spot pattern by a fringe pattern. The extrapolated or interpolated spots are obtained from the intersection of the fringe maxima. The fringe pattern is also used to identify the spots, and it is particularly useful in highly distorted Hartmann patterns. Experimental results show how the method works.

Multi-image position detection

The exact measurement of positions is of fundamental importance in a multitude of image-sensor based optical measurement systems. We propose a new method for enhancing the accuracy of image-sensor based optical measurement systems by using a computer-generated hologram in front of the imaging system. Thereby, the measurement spot is replicated to a predefined pattern. Given enough light to correctly expose the sensor, the position detection accuracy can be considerably improved compared to the conventional one-spot approach. For the evaluation of the spot position we used center-of-gravity based averaging. We present simulated as well as experimental results showing an improvement by a factor of 3.6 to a positioning accuracy of better than three thousandths of a pixel for a standard industrial CCD sensor.

Pupil tracking with a Hartmann-Shack wavefront sensor

We present the theoretical background and experimental validation of a pupil tracking method based on measurement of the irradiance centroid of Hartmann-Shack aberrometric images. The experimental setup consists of a Hartmann-Shack (HS) sensor forming over the same camera the images of the eye's pupil and the aberrometric image. The calibration is made by comparing the controlled displacements induced to an artificial eye with the displacements estimated from the centroid of the pupil and of the HS focal plane. The pupil image is also used for validation of the method when operating with human eyes. The experimental results after calibration show a root mean square error of 10.45 mum for the artificial eye and 27, 10, and 6 mum rms for human eyes tested using Hartmann-Shack images, with signal-to-noise ratios of 6, 8, and 11, respectively. The performance of the method is similar to conventional commercial eye trackers. It avoids the need for using separate tracking devices and their associated synchronization problems. This technique can also be used to reprocess present and stored sets of Hartmann-Shack aberrometric images to estimate the ocular movements that occurred during the measurement runs, and, if convenient, to correct the measured aberrations from their influence.

Interferometric Shack-Hartmann wavefront sensor with an array of four-hole apertures

A modified Hartmann test based on the interference produced by a four-hole mask can be used to measure an unknown wavefront. To scan the wavefront, the interference pattern is measured for different positions of the mask. The position of the central fringe of the diamond-shaped interference pattern gives a measure of the local wavefront slopes. Using a set of four-hole apertures located behind an array of lenslets in such a way that each four-hole window is inside one lenslet area, a set of four-hole interference patterns can be obtained in the back focal plane of the lenslets without having to scan the wavefront. The central fringe area of each interference pattern is narrower than the area of the central maximum of the diffraction pattern of the lenslet, increasing the accuracy in the estimate of the lobe position as compared with the Shack-Hartmann wavefront sensor.

Estimation of centroid positions with a matched-filter algorithm: relevance for aberrometry of the eye

Most Shack-Hartmann based aberrometers use infrared light, for the comfort of the patients. A large amount of the light that is scattered from the retinal layers is recorded by the detector as background, from which it is not trivial to estimate the centroid of the Shack-Hartmann spot. For a centroiding algorithm, background light can lead to a systematic bias of the centroid positions towards the centre of the software window. We implement a matched filter algorithm for the estimation of the centroid positions of the Shack-Hartmann spots recorded by our aberrometer. We briefly present the performance of our algorithm, and recall the well-known robustness of the matched filter algorithm to background light. Using data collected on 5 human eyes, we parameterise a simple and fast centroiding algorithm and reduce the difference between the two algorithms down to a mean residual wavefront of 0.02 microm rms.

Hardware accelerated optical alignment of lasers using beam-specific matched filters

Accurate automated alignment of laser beams in the National Ignition Facility (NIF) is essential for achieving extreme temperature and pressure required for inertial confinement fusion. The alignment achieved by the integrated control systems relies on algorithms processing video images to determine the position of the laser beam images in real time. Alignment images that exhibit wide variations in beam quality require a matched-filter algorithm for position detection. One challenge in designing a matched-filter-based algorithm is to construct a filter template that is resilient to variations in imaging conditions while guaranteeing accurate position determination. A second challenge is to process images for thousands of templates in under a second, as may be required in future high-energy laser systems. This paper describes the development of a new analytical template that captures key recurring features present in the beam image to accurately estimate the beam position under good image quality conditions. Depending on the features present in a particular beam, the analytical template allows us to create a highly tailored template containing only those selected features. The second objective is achieved by exploiting the parallelism inherent in the algorithm to accelerate processing using parallel hardware that provides significant performance improvement over conventional processors. In particular, a Xilinx Virtex II Pro field programmable gate array (FPGA) hardware implementation processing 32 templates provided a speed increase of about 253 times over an optimized software implementation running on a 2.2 GHz AMD Opteron core.

Modified Shack-Hartmann wavefront sensor using an array of superresolution pupil filters

A conventional Shack-Hartmann wavefront sensor works with an array of lenslets which produces an array of focal spots in the back focal plane of the lenses. The displacements of the focal spots from a reference position give a measure of the mean local wavefront slopes. To determine the positions of the focal spots, centroiding algorithms have to be used. In this work, the use of superresolution pupil filters to reduce the size of the focal spots is analyzed, as well as its effect on the variance of the centroid estimate, seeking for an enhancing of the sensor accuracy.

Best linear unbiased axial localization in three-dimensional fluorescent bead tracking with subnanometer resolution using off-focus images

A three-dimensional particle tracking technique, based on microscope off-focus images, was introduced in Z. Zhang and C.-H. Menq, Appl. Opt.47, 2361 (2008) and applied to bright-field imaging. This paper presents two major improvements to the axial localization algorithm of the 3D particle tracking technique. First, it extends the algorithm to measure fluorescent particles in the presence of photobleaching and excitation variation. Second, it enhances the measurement resolution by achieving the best linear unbiased estimation of the particle's axial position. Similarly to the original algorithm, a radius vector is first converted from the off-focus 2D image of the particle, and the axial position is estimated by comparing the radius vector with an object-specific model, calibrated automatically prior to each experiment. Although it was an intensity-based method, by normalizing the radius vectors the improved algorithm becomes a shape-based method, thus invariant to image intensity change and robust to photobleaching. Moreover, when considering the noise variance of each point in the radius vector and their correlations, the best linear unbiased estimation based on a linearized model is achieved. It is shown that variance equalization and correlation-weighted optimization greatly reduce the estimation variance and lead to near-uniform localization resolution over the entire measurement range. Estimation resolution is theoretically analyzed and validated by experiments. Theoretical analysis enables the prediction of measurement resolution based on calibration data. Finally, experimental results are presented to illustrate the performance of the measurement method in terms of measurement precision and range, as well as its robustness to intensity variation.

Three-dimensional particle tracking with subnanometer resolution using off-focus images

A three-dimensional (3D) particle tracking algorithm based on microscope off-focus images is presented in this paper. Subnanometer resolution in all three axes at 400 Hz sampling rate is achieved using a complementary metal-oxide-semiconductor (CMOS) camera. At each sampling, the lateral position of the spherical particle is first estimated by the centroid method. The axial position is then estimated by comparing the radius vector, which is converted from the off-focus two-dimensional image of the particle with no information loss, with an object-specific model, calibrated automatically prior to each experiment. Estimation bias and variance of the 3D tracking algorithm are characterized through analytical analysis. It leads to an analytical model, enabling prediction of the measurement performance based on calibration data. Finally, experimental results are presented to illustrate the performance of the measurement method in terms of precision and range. The validity of the theoretical analysis is also experimentally confirmed.

Improvement of Shack-Hartmann wave-front sensor measurement for extreme adaptive optics

The development of high-performance adaptive optics systems requires the optimization of wave-front sensors (WFSs) working in the high-order correction regime. We propose a new method to improve the wave-front slope estimation of a Shack-Hartmann WFS in such a regime. Based on a detailed analysis of the different errors in the slope estimation with a classical centroid and with the new method, the gain in terms of wave-front-sensing accuracy in both the detector and the photon noise regimes is stressed. This improvement is proposed without major system disruption.

Influence of thresholding on centroid statistics: full analytical description

The centroid method is a common procedure for subpixel location that is applied to a large number of optical sensors. In practice, it is always accompanied by thresholding algorithms used to eliminate undesirable background that may decrease precision. We present a full analytical description of the interaction between centroiding and thresholding applied over an intensity distribution corrupted by additive Gaussian noise. An in depth analysis of the most outstanding statistical properties of this relation (mean and variance) is also presented by means of simulated and experimental data. This work provides fundamental concepts to the designers of sensors that are based on centroid measurements to allow them to use thresholding correctly before centroid computation.