Moire topography is a well-established optical technique to measure the shape of three-dimensional surfaces, based on the geometric interference between an optical grid and its image deformed by an object surface. The technique produces fringes that represent contours of equal height, and from the recordings of several phase-shifted topograms surface height coordinates can be calculated. To perform these calculations, it is assumed that object height variation is small in comparison with the measurement setup dimensions, and this approximation leads to systematic errors in measurement accuracy. We present the mathematical description of the fringe formation process in projection moire topography, and on the basis of these equations we establish the relation between setup geometry and upper limits of the systematic measurement errors. We derive the equations that determine design specifications needed to reduce the effects of approximations to be below the measurement resolution of the setup. It is shown that setup geometry should be adapted to the gray-scale measurement resolution of the imaging system. We show that, using an iterative correction from one fringe order to the next, measurement accuracy can be maintained over the entire object depth.

Design considerations in projection phase-shift moiré topography based on theoretical analysis of fringe formation.

We analyze the sensitivity of a video-based moire technique used to measure vibrations. By use of modulated projected fringes, full-field distributions are mapped for both the vibration amplitude and the vibration phase. We also show methods for visual interpretation of the vibration fringe pattern, and we describe phase-stepping algorithms for full-field calculations of the distributions by use of digital image-processing equipment. Implementation of these methods makes it possible to measure vibration amplitude and phase over a large range of amplitudes, from < 1 μm to several millimeters. We show that the method bridges the traditional gap between holographic interferometry and moire methods.

Video-based vibration analysis using projected fringes.

Time-averaged moire fringes may be produced by projecting a grid onto a vibrating test item surface using a conventional (2 in. x 2 in.) slide-projector and recording with a 35-mm SLR camera. Quantitative analysis of the resultant fringes is generally performed with the help of equations developed by time-averaging the optical intensity of the projected grid throughout the vibration period of the test item surface. In the present paper the authors present a physical description for the production of time-averaged moire fringes involving the splitting of grid lines resulting from their behavior as simple harmonic oscillators. Superimposition of vibrating grid lines produces blurring of the photographic image; regions of zero-amplitude vibration (nodes) may be identified by the clarity of their grid lines, enabling the analysis of mode shapes from the photographs.

Infrastructure of time-averaged projection moire fringes in vibration analysis.

The probability of the moire effect in LCD displays is estimated as a function of angle based on the experimental data; a theoretical function (node spacing) is proposed basing on the distance between nodes. Both functions are close to each other. The connection between the probability of the moire effect and the Thomae's function is also found. The function proposed in this paper can be used in the minimization of the moire effect in visual displays, especially in autostereoscopic 3D displays.

Probability of the moiré effect in barrier and lenticular autostereoscopic 3D displays.

Surface contour from a low-frequency vibrating object using phase differences and the Fourier-transform method

This paper presents a qualitative and quantitative analysis of timeaveraged
projection moire vibration measurements for a plate and a cylindrical
shell. It is pointed out that although moire fringes are generally considered to
represent a constant contour difference or constant vibration level, in practice,
variations in local grid pitch values on the test-item surface result in wide
variations in measured displacement levels for given fringe numbers, which
must be taken into consideration for quantitative evaluation.

Avi Katz

Papers

Infrastructure of time-averaged projection moire fringes in vibration analysis.

Variation in displacement values along projection moire fringes in contour difference and vibration analysis of structures