The Global Positioning System (GPS) uses accurate, stable atomic clocks in satellites and on the ground to provide world-wide position and time determination. These clocks have gravitational and motional frequency shifts which are so large that, without carefully accounting for numerous relativistic effects, the system would not work. This paper discusses the conceptual basis, founded on special and general relativity, for navigation using GPS. Relativistic principles and effects which must be considered include the constancy of the speed of light, the equivalence principle, the Sagnac effect, time dilation, gravitational frequency shifts, and relativity of synchronization. Experimental tests of relativity obtained with a GPS receiver aboard the TOPEX/POSEIDON satellite will be discussed. Recently frequency jumps arising from satellite orbit adjustments have been identified as relativistic effects. These will be explained and some interesting applications of GPS will be discussed.

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Relativity in the Global Positioning System

Based on the coupling relationship among radial velocity, range walk, and Doppler frequency of the moving target's echoes, a novel method is proposed, i.e., Radon-Fourier transform (RFT), to realize the long-time coherent integration for radar target detection. The RFT realizes the echoes spatial-temporal decoupling via joint searching along range and velocity directions, as well as the successive coherent integration via the Doppler filter bank. Besides, four equivalent RFTs are obtained with respect to the different searching parameters. Furthermore, a generalized form of RFT, i.e., generalized Radon-Fourier transform (GRFT), is also defined for target detection with arbitrary parameterized motion. Due to the similarity between the RFT and the well-known moving target detection (MTD) method, this paper provides detailed comparisons between them on five aspects, i.e., coherent integration time, filter bank structure, blind speed response, detection performance, and computational complexity. It is shown that MTD is actually a special case of RFT and RFT is a kind of generalized Doppler filter bank processing for targets with across range unit (ARU) range walk. Finally, numerical experiments are provided to demonstrate the equivalence among four kinds of RFTs. Also, it is shown that the RFT may obtain the coherent integration gain in the different noisy background and the target's blind speed effect may be effectively suppressed. In the meantime, both the weak target detection performance and the radar coverage of high-speed targets may be significantly improved via RFT without change of the radar hardware system.

Radon-Fourier Transform for Radar Target Detection, I: Generalized Doppler Filter Bank

A method of and system for rendering a halftone image of a gray scale image by utilizing a pixel-by-pixel comparison of the gray scale image against a blue noise mask is disclosed in which the gray scale image is scanned on a pixel-by-pixel basis and compared on a pixel-by-pixel basis to an array of corresponding data points contained in a blue noise mask stored in a PROM or computer memory in order to produce the desired halftoned image. Both digital and optically implemented halftone methods are disclosed. Application specific modifications of the blue noise mask as well as its use for producing halftoned color images are also disclosed.

Method and apparatus for halftone rendering of a gray scale image using a blue noise mask

FastSPECT II is a recently commissioned 16-camera small-animal SPECT imager built with modular scintillation cameras and list-mode data-acquisition electronics. The instrument is housed in a lead-shielded enclosure and has exchangeable aperture assemblies and adjustable camera positions for selection of magnification, pinhole size, and field of view. The calibration of individual cameras and measurement of an overall system imaging matrix (1 mm/sup 3/ voxels) are supported via a five-axis motion-control system. Details of the system integration and results of characterization and performance measurements are presented along with first tomographic images. The dynamic imaging capabilities of the instrument are explored and discussed.

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FastSPECT II: a second-generation high-resolution dynamic SPECT imager

We need general relativity to understand extreme astrophysical realms. But the theory also turns out to be essential for the many mundane activities that nowadays rely on the precision of the GPS.

Relativity and the Global Positioning System

A "modular" scintillation camera is discussed as an alternative to using Anger cameras for gamma-ray imaging in nuclear medicine. Each module is an independent gamma camera and consists of a scintillation crystal, light pipe and mask plane, PMT's, and processing electronics. Groups of modules efficiently image radionuclide distributions by effectively utilizing crystal area. Performance of each module is maximized by using Monte-Carlo computer simulations to determine the optical design of the camera, optimizing the signal processing of the PMT signals using maximum-likelihood (ML) estimators, and incorporating digital lookup tables. Each event is completely processed in 2 ?sec, and FWHM of the PSF over the crystal area is expected to be 3 mm. Both one-dimensional and two-dimensional prototypes are tested for spatial and energy resolution.

A Modular Scintillation Camera for Use in Nuclear Medicine

The Wigner distribution function (WDF), a simultaneous coordinate and frequency representation of a signal, has properties useful in pattern recognition. Because the WDF is computationally demanding, its use is not usually appropriate in digital processing. Optical schemes have been developed to compute the WDF for one-dimensional (1 -D) signals, often using acousto-optic signal transducers. Some recent work has demonstrated the computation of two-dimensional (2-D) slices of the four-dimensional (4-D) WDF of a 2-D input transparency. In this latter case, the required 2-D Fourier transformation is performed by coherent optics. We demonstrate that computation of the WDF of real 2-D signals is susceptible to Radon transform solution. The 2-D operation is reduced to a series of 1 -D operations on the line-integral projections. The required projection data are produced optically, and the Fourier transformation is performed by efficient 1 -D processors (surface acoustic wave filters) by means of the chirp-transform algorithm. The resultant output gives 1 -D slices through the 4-D WDF nearly in real time, and the computation is not restricted to coherently illuminated transparencies. This approach may be useful in distinguishing patterns with known texture direction. The optical setup is easily modified to produce the cross-Wigner distribution function, a special case of the complex, or windowed, spectrogram.

Application Of The Radon Transform To Optical Production Of The Wigner Distribution Function

Application of the Radon transform to optical production of the Wigner distribution function

A hybrid system has been constructed to perform the complex Fourier transform of real 2-D data The system is based on the Radon transform; ie, operations are performed on 1-D projections of the data The projections are derived optically from transmissive or reflective objects, and the complex Fourier transform is performed with SAW filters via the chirp transform algorithm The real and imaginary parts of the 2-D transform are produced in two bipolar output channels

Two-dimensional complex Fourier transform via the Radon transform.

The authors explore the current use of modern image processing techniques to decipher ancient texts, and look toward future applications, including “discovering” original texts from overwritten documents.

Roger L. Easton

Papers

A Modular Scintillation Camera for Use in Nuclear Medicine

Application Of The Radon Transform To Optical Production Of The Wigner Distribution Function

Application of the Radon transform to optical production of the Wigner distribution function

Two-dimensional complex Fourier transform via the Radon transform.

Imaging the dead sea scrolls