A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

The phase of an image obtained with many magnetic resonance imaging techniques is related to some physical variable of interest. This phase needs to be unwrapped, which is complicated by the presence of noise and multiple objects of irregular shape. A new two-dimensional phase unwrapping algorithm is presented, along with simulation results.

An algorithm that suppresses translational motion artifacts in magnetic resonance imaging (MRI) by using post processing on a standard spin-warp image is presented. It is shown that translational motion causes an additional phase factor in the detected signal and that this phase error can be removed using an iterative algorithm of generalized projections. The method has been tested using computer simulations and it successfully removed most of the artifact. The algorithm converges even in the presence of severe noise.

The quality of magnetic resonance imaging systems has improved to the point that motion is a major limitation in many examinations. Translational motion in the imaging plane causes the phase of the data to be corrupted. An algorithm using computer post-processing is proposed to correct the phase of the data, and hence remove the artifact. This algorithm has superior convergence properties to an earlier algorithm, which is achieved by incorporating additional prior information specific to the situation. The algorithm is verified using a Shepp and Logan phantom with simulated motion in the imaging plane. It is shown that the algorithm can correct both periodic and random motion, and that the algorithm is not significantly degraded when noise is present.

Phase unwrapping refers to the determination of phase from modulo 2pi data, some of which may not be reliable. In 2D, this is equivalent to confining the support of the phase function to one or more arbitrarily shaped regions. A phase unwrapping algorithm is presented which works for 2D data known only within a set of nonconnected regions with possibly nonconvex boundaries. The algorithm includes the following steps: segmentation to identify connectivity, phase unwrapping within each segment using a Taylor series expansion, phase unwrapping between disconnected segments along an optimum path, and filling of phase information voids. The optimum path for intersegment unwrapping is determined by a minimum spanning tree algorithm. Although the algorithm is applicable to any 2D data, the main application addressed is magnetic resonance imaging (MRI) where phase maps are useful.