Edward E. Fenimore

Coded aperture imaging with uniformly redundant arrays

Uniformly redundant arrays (URA) have autocorrelation functions with perfectly flat sidelobes. The URA combines the high-transmission characteristics of the random array with the flat sidelobe advantage of the nonredundant pinhole arrays. This gives the URA the capability to image low-intensity, low-contrast sources. Furthermore, whereas the inherent noise in random array imaging puts a limit on the obtainable SNR, the URA has no such limit. Computer simulations show that the URA with significant shot and background noise is vastly superior to random array techniques without noise. Implementation permits a detector which is smaller than its random array counterpart.

New family of binary arrays for coded aperture imaging

We introduce a new family of binary arrays for use in coded aperture imaging which are predicted to have properties and sensitivity (SNR) equal to that of the uniformly redundant array (URA). The new arrays, called MURAs (modified URAs), have decoding coefficients all of which are unimodular, resulting in a reconstructed image with noise terms completely independent of image-source structure. Although the new arrays are derived from quadratic residues, they do not belong to the cyclic difference set or set of pseudonoise sequences and consequently are constructible in configurations forbidden to those designs, thus providing the user with a wider selection of aperture patterns to match his particular needs. With the addition of MURAs to the family of binary arrays, all prime numbers can now be used for making optimal coded apertures, increasing the number of available square patterns by more than a factor of 3.

Coded aperture imaging: predicted performance of uniformly redundant arrays

Uniformly redundant arrays (URA) have autocorrelation functions with perfectly flat sidelobes. The URA combines the high-transmission characteristics of the random array with the flat sidelobe advantage of the nonredundant pinhole arrays. A general expression for the signal-to-noise ratio (SNR) has been developed for the URA as a function of the type of object being imaged and the design parameters of the aperture. The SNR expression is used to obtain an expression for the optimum aperture transmission. Currently, the only 2-D URAs known have a transmission of (1/2). This, however, is not a severe limitation because the use of the nonoptimum transmission of (1/2) never causes a reduction in the SNR of more than 30%. The predicted performance of the URA system is compared to the image obtainable from a single pinhole camera. Because the reconstructed image of the URA contains virtually uniform noise regardless of the original objects structure, the improvement over the single pinhole camera is much larger for the bright points than it is for the low intensity points. For a detector with high background noise, the URA will always give a much better image than the single pinhole camera regardless of the structure of the object. In the case of a detector with low background noise, the improvement of the URA relative to the single pinhole camera will have a lower limit of ~(2f)(-(1/2)), where f is the fraction of the field of view that is uniformly filled by the object.

Uniformly redundant arrays: digital reconstruction methods.

Several new digital reconstruction techniques for coded aperture imaging are developed which are especially applicable to uniformly redundant arrays (URAs). The techniques provide improved resolution without upsetting the artifact-free nature of URAs. Two new techniques are described; one which allows self-supporting URAs and one which avoids (or at least mitigates) a blur which has been associated with previous correlation analyses. Each of the methods and their resolution improvements are demonstrated with reconstructions of a laser-driven compression. Particular emphasis has been placed on the special sampling required of the encoded picture and the decoding function if artifacts are to be avoided. For large URAs, it is shown that another new digital technique, periodic decoding, is much faster. Periodic decoding does produce artifacts, but they usually are negligible.

Very Early Optical Afterglows of Gamma-Ray Bursts: Evidence for Relative Paucity of Detection

Very early observations with the Swift satellite of γ-ray burst (GRB) afterglows reveal that the optical component is not detected in a large number of cases. This is in contrast to the bright optical flashes previously discovered in some GRBs (e.g., GRB 990123 and GRB 021211). Comparisons of the X-ray afterglow flux to the optical afterglow flux and prompt γ-ray fluence is used to quantify the seemingly deficient optical, and in some cases X-ray, light at these early epochs. This comparison reveals that some of these bursts appear to have higher than normal γ-ray efficiencies. We discuss possible mechanisms and their feasibility for explaining the apparent lack of early optical emission. The mechanisms considered include, foreground extinction, circumburst absorption, Lyα blanketing and absorption due to high-redshift, low-density environments, rapid temporal decay, and intrinsic weakness of the reverse shock. Of these, foreground extinction, circumburst absorption, and high redshift provide the best explanations for most of the nondetections in our sample. There is tentative evidence of suppression of the strong reverse shock emission. This could be because of a Poynting flux-dominated flow or a pure nonrelativistic hydrodynamic reverse shock.

Tomographical imaging using uniformly redundant arrays

Recent work in coded aperture imaging has shown that the uniformly redundant array (URA) can image distant planar radioactive sources with no artifacts. This paper investigates the performance of two URA apertures when used in a close-up tomographic imaging system. It is shown that a URA based on m sequences is superior to one based on quadratic residues. The m-sequence array not only produces less noticeable defocus artifacts in tomographic imaging but is also more resilient to some described detrimental effects of close-up imaging. It is shown that, in spite of these close-up effects, the URA system retains tomographic depth resolution even as the source is moved close to the detector. The URAs based on m sequences provide better images than those obtained using random arrays. This compliments previous studies that have shown random arrays to have better tomographical properties than Fresnel zone plates and nonredundant arrays.

X-Ray Spectral Characteristics of Ginga Gamma-Ray Bursts

We have investigated the spectral characteristics of a sample of bright gamma-ray bursts detected with the gamma-ray burst sensors aboard the satellite Ginga. This instrument employed a proportional and scintillation counter to provide sensitivity to photons in the 2-400 keV region and as such provided a unique opportunity to characterize the largely unexplored X-ray properties of gamma-ray bursts. The photon spectra of the Ginga bursts are well described by a low-energy slope, a bend energy, and a high-energy slope. In the energy range where they can be compared, this result is consistent with burst spectral analyses obtained from the BATSE experiment aboard the Compton Gamma-Ray Observatory. However, below 20 keV we find evidence for a positive spectral number index in approximately 40% of our burst sample, with some evidence for a strong rolloff at lower energies in a few events. There is a correlation (Pearsons r = -0.62) between the low-energy slope and the bend energy. We find that the distribution of spectral bend energies extends below 10 keV. There has been some concern in cosmological models of gamma-ray bursts (GRBs) that the bend energy covers only a small dynamic range. Our result extends the observed dynamic range, and, since we observe bend energies down to the limit of our instrument, perhaps observations have not yet limited the range. The Ginga trigger range was virtually the same as that of BATSE, yet we find a different range of fit parameters. One possible explanation might be that GRBs have two break energies, one often in the 50-500 keV range and the other near 5 keV. Both BATSE and Ginga fit with only a single break energy, so BATSE tends to find breaks near the center of its energy range, and we tend to find breaks in our energy range. The observed ratio of energy emitted in the X-rays relative to the gamma rays can be much larger than a few percent and, in fact, is sometimes larger than unity. The average for our 22 bursts is 24%. We also investigated spectral evolution in two bursts. In these events we find strong evidence for spectral softening as well as a correlation between photon intensity and spectral hardness. We also find that the X-ray signal below 30 keV itself softens in both of these events. There is one example of a strong X-ray excess at low energy. In addition to providing further constraints on gamma-ray burst models, the description provided here of burst spectra down to 2 keV should prove useful to future planned efforts to detect bursts at X-ray energies.

Coded aperture imaging: the modulation transfer function for uniformly redundant arrays

Coded aperture imaging uses many pinholes to increase the SNR for intrinsically weak sources when the radiation can be neither reflected nor refracted. Effectively, the signal is multiplexed onto an image and then decoded, often by computer, to form a reconstructed image. We derive the modulation transfer function (MTF) of such a system employing uniformly redundant arrays (URA). We show that the MTF of a URA system is virtually the same as the MTF of an individual pinhole regardless of the shape or size of the pinhole. Thus, only the location of the pinholes is important for optimum multiplexing and decoding. The shape and size of the pinholes can then be selected based on other criteria. For example, one can generate self-supporting patterns, useful for energies typically encountered in the imaging of laser-driven compressions or in soft x-ray astronomy. Such patterns contain holes that are all the same size, easing the etching or plating fabrication efforts for the apertures. A new reconstruction method is introduced called delta decoding. It improves the resolution capabilities of a coded aperture system by mitigating a blur often introduced during the reconstruction step.

Fast delta Hadamard transform

In many fields (e.g., spectroscopy, imaging spectroscopy, photoacoustic imaging, coded aperture imaging) binary bit patterns known as m sequences are used to encode (by multiplexing) a series of measurements in order to obtain a larger throughput. The observed measurements must be decoded to obtain the desired spectrum (or image in the case of coded aperture imaging). Decoding in the past has used a technique called the fast Hadamard transform (FHT) whose chief advantage is that it can reduce the computational effort from N(2) multiplies to N log(2) N additions or subtractions. However, the FHT has the disadvantage that it does not readily allow one to sample more finely than the number of bits used in the m sequence. This can limit the obtainable resolution and cause confusion near the sample boundaries (phasing errors). We have developed both 1-D and 2-D methods (called fast delta Hadamard transforms, FDHT) which overcome both of the above limitations. Applications of the FDHT are discussed in the context of Hadamard spectroscopy and coded aperture imaging with uniformly redundant arrays. Special emphasis has been placed on how the FDHT can unite techniques used by both of these fields into the same mathematical basis.

Time-resolved and energy-resolved coded aperture images with URA tagging.

Coded aperture imaging with uniformly redundant arrays (URAs) is the standard technique for imaging above the limit of grazing incident x-ray telescopes. It is an ideal technique for high energy astrophysics because it has high throughput, excellent performance on point sources, and the ability to measure simultaneously signal and background. However, many sources of interest in high energy astrophysics are time variable or require detailed energy spectra. Until now, to obtain a single time (or energy) sample, the photons from the particular time (or energy) interval must be formed into an encoded pattern, then processed to obtain an image for that sample. Therefore, massive computations are required to cover the entire time and energy parameter space. We present a new method of coded aperture analysis called URA tagging, which provides time and/or energy resolved histories of sources with known positions without using a correlation operation. It can easily reduce the computation time by orders of magnitude ompared to the next fastest method, the fast delta Hadamard transform. URA tagging can also correct for improperly encoded images or motion blurred images. Whereas previous methods for quantifying performance have not taken into account the finite resolution or the quantized sampling, URA tagging provides a SNR equation that includes all such effects. URA tagging analysis explains why delta decoding has a somewhat poorer SNR than balanced correlation; naively, one expects the better angular resolution to yield a better SNR. In addition, we show that complementary URAs (exchanged opaque and transparent elements) have different properties, and those with an even number of transparent elements should be preferred.