Three-Dimensional Fourier Fringe Analysis
Three-Dimensional Fourier Fringe Analysis
Fringe projection techniques are considered to be an effective, reliable and robust optical non-contact method for measuring 3D surface form and shape. In these methods; a structured light patterned is projected onto the object surface. Due to the object surface shape, the projected pattern will be modified when viewed off axis. This pattern is captured by a CCD camera and stored using and a frame grabber. The image is then analysed by one of image processing algorithms. Finally, the phase is extracted and related to actual object height by use of some sort of phase-height relationship determined during calibration.
Many techniques are evolved in analysing the captured fringes patterns. Among all of this techniques Fourier transform fringe analysis (FTFA) technique by Takeda which was outstanding and provides good results. The original implantation of FTFA was one-dimensional, it only analysed a single line of the image at a time. However, it was not very long before the technique was extended to two-dimensions and, it has undergone considerable development over time.
Today Fourier transform fringe analysis (FTFA) is seen as a fast and flexible method of dealing with fringe patterns. But it is still 2-D, i.e. it deals with data on a video frame by video frame basis. The analysis of each frame is completely independent – no knowledge from the previous frames, or following frames, is available at the time of processing the current frame – this despite the fact that we know that there is continuity between frames because they describe the same surface either translated/rotated in the case of rigid body or possible slightly "stretched" in the case of a deformable body such as the surface of a human being.
In the case of a dynamically changing fringe pattern on a moving/deforming body, there is another way to view the data. Instead of seeing it as a series of disconnected 2-D images – we could consider the data as "stack" of images with time as the vertical axis as shown in Figure 1.
Figure 1: A 3-D Data Volume - A "Stack" Of 2-D Images Varying With Time As The Vertical Axis
The basic proposal of this research is to develop the necessary theoretical basis, tools and software to treat the entire time-stack with a single 3-D FTFA method. Such an approach would have a number of advantages;
- It would be significantly faster than the 2-D technique. Speed benefits would accrue from there only being a single data read operation along with a single filter etc.
- It is likely that the phase unwrapping operation would significantly benefit from the move to 3-D. Early FTFA systems employed very primitive phase unwrapping algorithms based on the rows of the image data, know as Schaffer’s method. In such an algorithm there is only one single path between any two points P1 and P2 – see Figure 2. If there is a phase discontinuity or ambiguity along this path then all of the data to the right of fault is lost. For this reason early unwrappers were notoriously unreliable and frequently failed. Basically in an 512 x 512 image the unwrapper would have to make 262144 "correct" decisions about the presence or a absence of a phase wrap, one error and potentially large areas of the data are lost.
Figure 2: A Simple Linear Unwrapping Path
This situation was significantly improved when 2-D unwrapping was developed using either tiles or confidence paths. The power of 2-D unwrapping to increase reliability stems principally from the fact that multiple paths now exist to link any two points in the image – as shown in Figure 3.
Figure 3: Multiple Unwrapping Paths For A 2-D
Just as the move to 2-D triggered a major growth in unwrapping algorithms - and an associated improvement in quality – so we envisage the move to 3-D will similarly lead a quantum improvement. While fringes will evolve, image-to-image in the 3-D stack the frame-sampling rate will be controlled to ensure that this evolution is gradual. Thus it should be possible to track the path of wrap not only "in-plane" but through the time dimension as well, image to image. For a limited case this has been done, but we do not believe it has been used with FTFA or successfully exploited for measuring dynamic surfaces.
So in short, 3-D FTFA has the potential to significantly enhance the speed and reliability of fringe analysis in the increasingly important class of dynamic problems.
Hussein Sudqi Hussein Abdul-Rahman.
