Many techniques in interferometry generate a two dimensional fringe-contour map o f the phase distribution in the form of Eqn. 2.22, which is rewritten in Eqn. 2.23 as it commonly appears when applied to two dimensional interfer- ograms.
g { x , y ) = a ( x , y ) + 6(z, y ) cos(<p(x, y ) ) (2.23) In this equation a ( x , y ) is the background intensity ( which may fluctuate in a real experiment ), b ( x , y ) is the amplitude and </>(x,y) is the phase.
The fringe analysis software operates over one or more binary coded im ages. These images contain quantised samples of the intensity values within an interferogram. Unless otherwise stated reference to phase images etc, are with respect to this binary coded form.
Whilst the generation o f this kind of fringe pattern permits a direct means of displaying a contour map, it has failings. The sign of the phase cannot be determined, i.e one cannot distinguish between hills and valleys in the map. Accuracy is limited by quantisation of the intensity, non-linearity of the detector, unwanted background variations in a ( x , y ) and other types of optical and electronic noise.
In order to be widely accepted the results of any experiment in Interferom etry, embodied in the interferogram, must be quickly and reliably extracted without expert knowledge. This would best be achieved by the automatic translation o f the interferogram into a numerical or pictorial map. The nu merical representation would then be easily compared or combined with other kinds o f measurement data.
The first obstacle to the extraction of measurement data from a fringe pattern ( represented by Eqn. 2.23 ) is the difficulty in uniting phase measure ments from neighbouring fringes. This is necessary to construct a continuous phase map of the measurement parameter across the complete field. The sec ond related impediment, is the problem of distinguishing between elevation and depression.
One method of fringe analysis which is still very popular, and intuitive, is fringe tracking. This is aimed at uniting data from adjacent fringes. The directional ambiguity under this regime is either resolved from shaping the
experimental arrangement so that fringes are generated in an incremental way ( by the application of tilt ) or by interactive means [37].
However, several, more generally applicable, automatic techniques for the solution of directional ambiguity have been evolved. The most prominent o f these are the Fourier Transform Technique ( FTT ) [6, 7] and Phase Stepping ( or Quasi Heterodyning ) [2, 3, 4, 5]. In the case of the FT T, a tilt or translation is used to solve the elevation/depression problem by yielding carrier fringes which are interpreted during an FFT process. In the case of Phase Stepping the solution is arrived at by combining several interferograms at different phases. Both techniques yield a ‘ phase fringe’ pattern which contains within it a coding of direction as well as displacement. These techniques are reviewed later in this chapter. This type o f fringe pattern is usually referred to as a wrapped phase map or, as it is derived from the tangent of phase, a ’tan’ fringe field.
The wrapped phase map contains a coding o f elevation and depression, as each point’s ‘ intensity’ ( when displayed as a grey scale image ), is a measure o f phase. For each fringe the intensity ranges from black at one extreme, representing a fringe phase o f 0, to intense white at the other representing a phase of 2n . Because the phase fringe field records a direct measurement o f phase, its analysis is known as Phase Unwrapping. Figure 2.5 shows an example of a computer generated wrapped phase map.
Such fringe fields are not usually analysed by fringe tracking. This tech nique has been replaced by fringe counting, which involves traversing the scan lines of the digitally processed field in search of the phase roll over points at the fringe edges. These are denoted by a sudden white/black or black/white transition in intensity between adjacent pixels. The aim is that by recording the positions o f such edges and the direction of phase roll over, by a suitable edge detection strategy, the phase may be summed in adjacent fringes to produce an unwrapped map. An example of this procedure is given by Nakadate [1 1] with respect to Speckle images.
In a Holographic system noise represents a problem to such a procedure, as it disrupts the fringe edges, upon which the technique relies. In a Speckle system the fringe field inherently consists of discrete points of information, presenting a much more serious problem. Therefore some other form of
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processing must also be applied. The problems of noise and Speckle may be greatly reduced by the use o f digital low pass filters, such as an average or median. These pass the relatively low spatial undulations of the fringes and filter out the rapid oscillations characteristic of noise.
However, phase unwrapping techniques have recently been developed which are immune to spike noise [13, 14, 15, 16]. Ghiglia describes the dis covery of a cellular automaton which can unwrap c o n s i s t e n t phase data in n
dimensions in a path-independent manner and which can automatically ac com m odate noise-induced ( pointlike ) inconsistencies. Although robust for
c o n s i s t e n t phase data the procedure does not address inconsistencies which are large in scale. That is, those which are larger than pixel sized spike noise and so may not be detected by a pixel level inspection. As described, the technique requires a large number of iterations to converge. Goldstein et al. [15] and Huntley [16] have described algorithms based on cuts which are immune to such noise and computationally efficient.
The algorithm described here is robust, immune to noise and computa tionally efficient. However, the advantage o f the MSTT algorithm over the former algorithms is that it is scalable. Utilizing small ‘ tiles’ it may unwrap a consistent phase map containing noise in a related manner to the methods o f Ghiglia, Goldstein and Huntley. By increasing the size of tiles, larger local features of inconsistency may be detected. Ghiglia describes these as ‘ natural or aliasing-induced path inconsistencies’ .
Natural path-inconsistencies can occur for a variety of reasons, because two independent objects overlap, for example. In satellite interferometry the tops of mountains and hills can be closer to the radar than their bases, so that in the interferograms, they appear to lean toward the radar. This is known as the layover effect. Shadows can cause large discontinuities, as can spaces between objects or their parts. Such spaces are especially problematic when they occur with a spatial frequency of the same order as the fringe spacing, as their size means that they may not be filtered out.
Aliasing-induced inconsistencies can be generated by the device recording the fringe pattern. This occurs when the device does not have a sufficient bandwidth to cope with that o f the fringe field. That is, the fringes are less than the size of a pixel in width.
In general, it is impossible to produce a totally unambiguous solution to a phase map with ‘ natural or aliasing induced path inconsistencies’ . How ever, the M STT technique incorporates strategies which minimise their ef fects where possible. In order to do so the strategy measures factors such as local fringe density, low modulation noise, fringe edge terminations and the extent to which the solution of neighbouring field areas agree.
The new M STT method has an inherently high tolerance o f errors in the interferogram, and operates on the ‘ phase fringe’ fields generated by either the Quasi-Heterodyne or FFT methods.
! - Scope Of Thesis
Figure 2.6: Relationship of Elements in Sector o f Fringe Analysis Considered The interaction of the various elements in a fully automated fringe anal ysis system are shown in Figure 2.6. The two systems, leading to the genera tion o f a wrapped phase map, allow some flexibility. It may be that analysis from a single interferogram is essential, in which case the Fourier Transform
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Technique may be employed, alternatively a need for greater accuracy [38] may promote the use of Quasi Heterodyning, which requires more than one interferogram.