The generalized analytical quadrature filter from a set of interferograms with arbitrary phase shifts is obtained. Both symmetrical and non symmetrical algorithms for any order are reported. The analytic expression is obtained through the convolution of a set of two-frame algorithms and expressed in terms of the combinatorial theory. Finally, the solution is applied to obtain several generalized tunable quadrature filters.

The main purpose of this paper is to present a method to design tunable quadrature filters in phase shifting interferometry. From a general tunable two-frame algorithm introduced, a set of individual filters corresponding to each quadrature conditions of the filter is obtained. Then, through a convolution algorithm of this set of filters the desired symmetric quadrature filter is recovered. Finally, the method is applied to obtain several tunable filters, like four and five-frame algorithms.

In this paper we present several eight-frame algorithms for their use in phase shifting profilometry and their application for the analysis of semi-fossilized materials. All algorithms are obtained from a set of two-frame algorithms and designed to compensate common errors such as phase shift detuning and bias errors.

There has been long historical interest in digital preservation of morphological features of biological materials, especially because often the preservation of sensitive molecules is critical for evolutionary studies. To this end, we successfully applied the recent advances of the fringe projection profilometry technique, in conjunction with white light and a new phase algorithm, to digitalize the shape of a fossil rodent hemimandible. We were able to generate a cloud of points in an array of data that allowed us to plot a three-dimensional (3-D) digital restoration of the entire fossil sample. The maximum resolution of this system is given by the diffraction limit (in the order of microns), and we show that this enhanced system can be used with objects in a range of 1-30 mm, minimizing the systematic errors induced by small vibrations or light fluctuations and, consequently, improving the signal-to-noise ratio of the recovered cloud data. This is a useful tool to preserve 3-D images of fossils and other biological objects for which rather detailed morphological information is required, like in research studies for biologists and paleontologists or, as in the present case, when researchers need a morphological replica because the sample will be destroyed for ancient DNA extraction.

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