Signal-to-noise comparison of deconvolution from wave-front sensing with traditional linear and speckle image reconstruction

It is well known that atmospheric turbulence severely degrades the performance of ground-based imaging systems. Techniques to overcome the effects of the atmosphere have been developing at a rapid pace over the past 10 years. These techniques can be grouped into two broad categories: predetection and postdetection techniques. A recent newcomer to the postdetection scene is deconvolution from wave-front sensing (DWFS). DWFS is a postdetection image-reconstruction technique that makes use of one feature of predetection techniques. A wave-front sensor (WFS) is used to record the wave-front phase distortion in the pupil of the telescope for each short-exposure image. The additional information provided by the WFS is used to estimate the system's point-spread function (PSF). The PSF is then used in conjunction with the ensemble of short-exposure images to obtain an estimate of the object intensity distribution through deconvolution. With the addition of DWFS to the suite of possible postdetection image-reconstruction techniques, it is natural to ask "How does DWFS compare with both traditional linear and speckle image-reconstruction techniques?" In the results we make a direct comparison based on a frequency-domain signal-to-noise-ratio performance metric. This metric is applied to each technique's image-reconstruction estimator. We find that DWFS nearly always results in improved performance over the estimators of traditional linear image reconstruction such as Wiener filtering. On the other hand, DWFS does not always outperform speckle-imaging techniques, and in cases that it does the improvement is small.     

Experimental measurements of estimator bias and the signal-to-noise ratio for deconvolution from wave-front sensing

Deconvolution from wave-front sensing (DWFS) has been proposed as a method for achieving high-resolution images of astronomical objects from ground-based telescopes. The technique consists of the simultaneous measurement of a short-exposure focal-plane speckled image, as well as the wave front, by use of a Shack-Hartmann sensor placed at the pupil plane. In early studies it was suspected that some problems would occur in poor seeing conditions; however, it was usually assumed that the technique would work well as long as the wave-front sensor subaperture spacing was less than r(0) (L/r(0) < 1). Atmosphere-induced phase errors in the pupil of a telescope imaging system produce both phase errors and magnitude errors in the effective short-exposure optical transfer function (OTF) of the system. Recently it has been shown that the commonly used estimator for this technique produces biased estimates of the magnitude errors. The significance of this bias problem is that one cannot properly estimate or correct for the frame-to-frame fluctuations in the magnitude of the OTF but can do so only for fluctuations in the phase. An auxiliary estimate must also be used to correct for the mean value of the magnitude error. The inability to compensate for the magnitude fluctuations results in a signal-to-noise ratio (SNR) that is less favorable for the technique than was previously thought. In some situations simpler techniques, such as the Knox-Thompson and bispectrum methods, which require only speckle gram data from the focal plane of the imaging system, can produce better results. We present experimental measurements based on observations of bright stars and the Jovian moon Ganymede that confirm previous theoretical predictions.     

Widening the effective field of view of adaptive-optics telescopes by deconvolution from wave-front sensing: average and signal-to-noise ratio performance

A fundamental problem of adaptive-optics systems is the very narrow corrected field of view that can be obtained because turbulence is extended in altitude throughout the atmosphere. The correctable field of view is of the order of 5-10 μrad at visible wavelengths and increases as the wavelength increases. Previous concepts to broaden the corrected field of view have been hardware oriented, requiring multiple wave-front sensor (WFS) measurements to control multiple deformable mirrors. We analyze the average and the signal-to-noise-ratio performance of an image measurement and postprocessing technique that uses simultaneous measurements of a short-exposure compensated image measured in an off-axis direction; an additional WFS measurement is taken in the off-axis direction. Results are presented for infinite-altitude WFS beacons driving both the WFS for the adaptive optics and the WFS looking in the off-axis direction, a variety of seeing and WFS light-level conditions, and off-axis angles from two to six times the isoplanatic angle. This technique improves the average effective transfer function out to a field angle of at least six times the isoplanatic angle while providing a higher signal-to-noise ratio in the spatial frequency domain.     

Signal-to-noise ratio of phase sensing telescope interferometers

The paper described is the third part of a trilogy dealing with the principles, performance, and limitations of what the author named "telescope-interferometers" (TIs). The basic idea consists in transforming one telescope into a wavefront error (WFE) sensing device. This can be achieved in two different ways, namely, off-axis and phase-shifting TIs. In both cases the point-spread function measured in the focal plane of the telescope carries information about the transmitted WFE, which is retrieved by fast and simple algorithms suitable to an adaptive optics (AO) regime. The uncertainties of both types of TIs are evaluated in terms of noise and systematic errors. Numerical models are developed to establish the dependence of driving parameters such as useful spectral range, angular size of the observed star, or detector noise on the total WFE measurement error. The latter is found particularly sensitive to photon noise, which rapidly governs the achieved accuracy for telescope diameters higher than 10 m. A few practical examples are studied, showing that the TI method is applicable to AO systems for telescope diameters ranging from 10 to 50 m, depending on seeing conditions and magnitude of the observed stars. Also discussed is the case of a space-borne coronagraph, where the TI technique provides high sampling of the input WFE map.     

Biased estimators and object-spectrum estimation in the method of deconvolution from wave-front sensing

The method of deconvolution from wave-front sensing (DWFS), which is a method for improving the quality of astronomical images measured through atmospheric turbulence, uses simultaneous shortexposure measurements of both an image and the output of a wave-front sensor exposed to an image of the telescope pupil. The wave-front sensor measurements are used to reconstruct an estimate of the instantaneous generalized pupil function of the telescope, which is used to compute an estimate of the instantaneous optical transfer function (OTF). This estimate of the OTF is then used in a deconvolution procedure. We point out the existence and origin of an unnoticed bias in the estimator for the DWFS method. This bias leads to nonrandom errors in the estimated object spectrum beyond those expected to arise by virtue of low-pass filtering and noise, including the possibility of an overall system transfer function greater than unity at some spatial frequencies. An alternative measurement and postprocessing scheme to overco e this source of error is suggested.     

Ensemble-averaged Shack-Hartmann wave-front sensing for imaging through turbid media

A technique is described for ensemble-averaging the light wave emerging from a turbid medium, enabling the recovery of optical information that is otherwise lost in a speckle pattern. The technique recovers both an amplitude and a phase function for a wave that has been corrupted by severe scattering, without the use of holography. With the phase estimated, an ensemble-averaged field is constructed that can be backprojected to form an image of the object obscured by the scattering medium. Experimental results suggest that the technique can resolve two object points whose signals are unresolved on the exit surface of a diffuser.     

Wave-front sensing from defocused images by use of wave-front slopes

We describe a novel technique for deriving wave-front aberrations from two defocused intensity measurements. The intensity defines a probability density function, and the method is based on the evolution of the cumulative density function of the intensity with light propagation. In one dimension, the problem is easily solved with a histogram specification procedure, with a linear relationship between the wave-front slope and the difference in the abscissas of the histograms. In two dimensions, the method requires use of a Radon transform. Simulation results demonstrate that good reconstructions can be attained down to 100 photons in each detector. In addition, the method is insensitive to scintillation at the aperture.     

Signal-to-noise ratio development for quality engineering

Dr. Taguchi developed the concept of signal-to-noise (SN) ratio in quality engineering to evaluate the performance of a system. The objective is to develop systems which are robust against noise factors. The SN ratio indicates the degree of the predictable performance of a product or process in the presence of noise factors. Parameter design of the Taguchi method optimizes the SN ratio in the domain of control factors, so that performance could be made insensitive to the noise factors in order to improve product quality. If the domain of the control factors is a continuous space, the problem is a non-linear programming problem. Usually, in practice, there are only a few available levels for the control factors. Thus, experimental design methods can be useful for such problems. The SN ratio for four cases of dynamic characteristic problems is developed in this paper. This paper also gives the method to compute SN ratios for both equispaced and non-equispaced intervals for levels of signal factors. Two examples are presented to illustrate the method.     

Multiple-wave lateral shearing interferometry for wave-front sensing

Multiple-wave achromatic interferometric techniques are used to measure, with high accuracy and high transverse resolution, wave fronts of polychromatic light sources. The wave fronts to be measured are replicated by a diffraction grating into several copies interfering together, leading to an interference pattern. A CCD detector located in the vicinity of the grating records this interference pattern. Some of these wave-front sensors are able to resolve wave-front spatial frequencies 3 to 4 times higher than a conventional Shack-Hartmann technique using an equivalent CCD detector. Its dynamic is also much higher, 2 to 3 orders of magnitude.     

基于哈特曼波前探测的流场层析重建系统仿真

基于哈特曼波前探测的流场层析重建技术结合了光学波前探测技术和计算机层析技术。重建系统由哈特曼传感器探测平行光束穿过流场后的投影波前,采用计算机层析技术重建流场物理量的空间分布。在介绍哈特曼流场层析重建原理的基础上,对流场重建的整个过程进行了计算机仿真,重建的RMS误差为0.0726。结果表明,该技术可以很好地实现流场的层析重建,在材料、流场研究等工程实际测量中具有良好的应用前景。     

Phase-diversity wave-front sensor for imaging systems

A phase-diversity wave-front sensor has been developed and tested at the Lockheed Palo Alto Research Labs (LPARL). The sensor consists of two CCD-array focal planes that record the best-focus image of an adaptive imaging system and an image that is defocused. This information is used to generate an object-independent function that is the input to a LPARL-developed neural network algorithm known as the General Regression Neural Network (GRNN). The GRNN algorithm calculates the wave-front errors that are present in the adaptive optics system. A control algorithm uses the calculated values to correct the errors in the optical system. Simulation studies and closed-loop experimental results are presented.     

Practical issues in wave-front sensing by use of phase diversity

We present the results of the phase-diversity algorithm applied to simulated and laboratory data. We show that the exact amount of defocus distance does not need to be known exactly for the phase-diversity algorithm on extended scene imaging. We determine, through computer simulation, the optimum diversity distance for various scene types. Using laboratory data, we compare the aberrations recovered with the phase-diversity algorithm and those measured with a Fizeau interferometer that uses a He-Ne laser. The two aberration sets agree with a Strehl ratio of over 0.9. The contrast of the recovered object is found to be ten times that of the raw image.     

Practical configuration for wave-front sensing with two-beam phase retrieval

A wave-front sensing approach based on two-beam phase retrieval is described. Light from an aberrated beam is split into two paths. A random phase and amplitude perturbation is applied to the beam in one path, and the interference patterns of the resultant two beams are measured in two planes along the axis of propagation. By modulation of one of the two beams, the intensity of each beam and the phase difference between the two beams are recovered in each plane. A rapidly convergent phase-retrieval algorithm is formulated by the method of sequential projections onto constraint sets. Examples are given illustrating the convergence properties of the approach.     

Evaluation of least-squares phase-diversity technique for space telescope wave-front sensing

Because of mechanical aspects of fabrication, launch, and operational environment, space telescope optics can suffer from unforeseen aberrations, detracting from their intended diffraction-limited performance goals. We give the results of simulation studies designed to explore how wave-front aberration information for such near-diffraction-limited telescopes can be estimated through a regularized, low-pass filtered version of the Gonsalves (least-squares) phase-diversity technique. We numerically simulate models of both monolithic and segmented space telescope mirrors; the segmented case is a simplified model of the proposed next generation space telescope. The simulation results quantify the accuracy of phase diversity as a wave-front sensing (WFS) technique in estimating the pupil phase map. The pupil phase is estimated from pairs of conventional and out-of-focus photon-limited point-source images. Image photon statistics are simulated for three different average light levels. Simulation results give an indication of the minimum light level required for reliable estimation of a large number of aberration parameters under the least-squares paradigm. For weak aberrations that average a 0.10lambda pupil rms, the average WFS estimation errors obtained here range from a worst case of 0.057lambda pupil rms to a best case of only 0.002lambda pupil rms, depending on the light level as well as on the types and degrees of freedom of the aberrations present.     

Target-in-the-loop beam control: basic considerations for analysis and wave-front sensing

Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related to maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive-index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing coherent outgoing-wave propagation, and the equation describing evolution of the mutual correlation function (MCF) for the backscattered wave (return wave). The resulting evolution equation for the MCF is further simplified by use of the smooth-refractive-index approximation. This approximation permits derivation of the transport equation for the return-wave brightness function, analyzed here by the method of characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wave-front sensors that perform sensing of speckle-averaged characteristics of the wave-front phase (TIL sensors). Analysis of the wave-front phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric-turbulence-related phase aberrations. We also show that wave-front sensing results depend on the extended target shape, surface roughness, and outgoing-beam intensity distribution on the target surface. For targets with smooth surfaces and nonflat shapes, the target-induced phase can contain aberrations. The presence of target-induced aberrations in the conjugated phase may result in a deterioration of adaptive system performance.     

Remote sensing of atmospheric turbulence and transverse winds from wave-front slope measurements from crossed optical paths

In the theory of atmospheric turbulence, the strength of the spatial variations of the index of refraction n is proportional to a parameter known as the atmospheric-structure constant. The atmosphericstructure constant is denoted C(2)(n)(z) and is a function of position along the optical path z. The characteristics of the temporal variations of the index of refraction are related to both C(2)(n)(z) and to the transverse wind velocity V(z). Current optical techniques for remotely sensing C(2)(n)(z) and V(z) rely primarily on the spatial or temporal cross-correlation properties of the intensity of the optical field. In the remote-sensing technique proposed here, we exploit the correlation properties of the wave-front slope measured from two point sources to obtain profiles of C(2)(n)(z) and V(z). The two sources are arranged to give crossed optical paths. The geometry of the crossed paths and the characteristics of the wave-front slope sensor determine the achievable resolution. The signal-to-noise ratio calculationsindicate the need for multiple measurements to obtain useful estimates of the desired quantities.     

Signal-to-noise ratio for temporal integrated drifting images: a model for perceived image sharpening

A formulation of signal-to-noise ratio is constructed that uses temporal integrated images from image sequences. Given a blurred image that drifts horizontally at various speeds and at various linear blurs, we prove that this formulation of the signal-to-noise ratio consistently increases with an increase in speed. This increase is shown to model the trends in the human vision system by which drifting blurred images are perceived with increased sharpness. The existing widely used objective quality techniques fail to model the perceptual increase in sharpness. This new formulation, along with other objective quality measures, is tested on several blurred drifting image sequences. The new formulation reflects the theoretically predicted increase in perceived sharpness.     

Multiscale gradient domain compression for astronomical high dynamic range imaging

Visualisation of high dynamic range images requires compression of the data to be properly displayed on media with more limited dynamic ranges. Astronomical images pose a difficult challenge for dynamic range compression algorithms, due to the nature of the imaged objects and to the lack of a reflectance illumination model based on spatial frequencies. As a result, most of the algorithms commonly used for daylight high dynamic range compression fail in achieving an optimal visualisation of astronomical targets. We propose an extended multiscale algorithm based on compression of the dynamic range in the gradient domain. Our algorithm effectively compresses the dynamic range, enhances local contrast and avoids noise amplification. This is achieved with a multiscale representation of the image and the use of luminance information. Our results show a significantly improved visualisation of astronomical images compared to the standard gradient domain compression, as well as more robustness to noise and better artefact suppression.     

Signal-to-noise ratio analysis to estimate ocean wave heights from X-band marine radar image time series

This work analyses the structure of the different contributions to the image spectrum derived by the three-dimensional Fourier decomposition of sea clutter time series measured by ordinary X-band marine radars. The goal of this investigation is to derive a method to estimate the significant wave height of the ocean wave fields imaged by the radar. The proposed method is an extension of a technique developed for the analysis of ocean wave fields by using synthetic aperture radar systems. The basic idea behind this method is that the significant wave height is linearly dependent on the square root of the signal-to-noise ratio, where the signal is assumed as the radar analysis estimation of the wave spectral energy and the noise is computed as the energy due to the sea surface roughness, which is closely related to the speckle of the radar image. The proposed method to estimate wave heights is validated using data sets of sea clutter images measured by a marine radar and significant wave heights derived from measurements taken by a buoy used as reference sensor.     

Pulse elongation and deconvolution filtering for medical ultrasonic imaging

Range sidelobe artifacts which are associated with pulse compression methods can be reduced with a new method composed of pulse elongation and deconvolution (PED). While pulse compression and PED yield similar signal-to-noise ratio (SNR) improvements, PED inherently minimizes the range sidelobe artifacts. The deconvolution is implemented as a stabilized inverse filter. With proper selection of the excitation waveform an exact inverse filter can be implemented. The excitation waveform is optimized in a minimum mean square error (MMSE) sense. An analytical expression for the power spectrum of the optimal pulse is presented and several techniques to numerically optimize the excitation pulse are shown. The effects of PED are demonstrated in computer simulations as well as ultrasonic images.