Faster approaches consider only one or a few spectral components. In this case the displacements retrieved may be subject to phase ambiguities that limit the actual unambiguous measurement range as described below. However there are numerous applications in which this limitation is not critical; provided that the unambiguous measurement range can be matched with practical requirements.2.1. Ambiguous Displacement Measurement from a Single Periodic PatternIn computer vision, the target displacements are retrieved through the processing of images captured by a static camera observing the moving object. The simplest way to apply phase computation to this task consists in associating some kind of periodic pattern to the target and thus to get periodically structured images for processing.
This is illustrated in Figure 1a in which the stripe set corresponds to the target image recorded in its initial position. Figure 1b shows the image recorded after a target displacement in the direction perpendicular to the stripes. The target displacement appears clearly through the stripe position and, as explained in Equation (2), it induces a phase shift ���� between the two stripe sets as represented in Figure 1c. The target displacement can then be determined by:��=����?P2��+kP(3)Figure 1.Correspondence between lateral position and phase of a sinusoidal pattern. (a) stripe set before displacement; (b) after displacement; (c) wrapped phase for both positions.
where P is the stripe period and k is an unknown integer standing for an entire number of stripe periods (We notice in Equation (3) that the vision system magnification does not need to be known since the actual period P of the target stripes serves as a dimensional reference, provided that the period is known or measured with sufficient accuracy.). Indeed due to the stripe periodicity, the displacement value is obtained modulo P since different positions distant from an entire number of periods produce indistinguishable images. This ambiguity is due to the definition domain ]?��, ��] of the inverse tangent function. It restricts the unambiguous measurement range to a single stripe period. The measurement range and measurement resolution are thus dependent on each other and the number of resolved positions is equal to K = 2��/�Ħ�, where �Ħ� is the resolution of the phase determination.
The only adjustment parameter is P that affects range and resolution in inverse proportions and thus does not affect the range-to-resolution ratio. The latter can only be improved through the phase GSK-3 computation performances. In practice, image digitizing, electronic noise and environmental disturbances form irreducible noise sources. As explained below, the use of a second stripe period is an alternative way to extend the measurement range without decreasing the resolution.2.2.