CAPÍTULO 3. APROXIMACIONES AL EMPRENDIMIENTO
3.1 MARCO TEÓRICO
CHAPTER 4: DISPLACEMENT ANALYSIS
4.1 INTRODUCTION 7 1
4.2 AUTOMATING OPTICAL ANALYSIS OF YOUN G ' S FRINGES 72
4.2. 1 Young's Fringes 72
4.2.2 Hardware Arrangement for Young's Fringe Analysis 75
4.2.3 Software Processing of Young's Fringes 75
4.2.4 Performance of Fringe Analysis System 80
4.3 DIGITAL MOTION ANALYSIS 83
4.3. 1 Digital Images 83
4.3.2 Digital Motion Analysis: Literature & Methods 86
4.3.3 Maximum Cross Correlation 87
4.3.4 Extensions on MCC 89
4.3.5 MCC Algorithm 94
4.4 IMAGE REGISTRATION 95
4.4. 1 Degrees of Freedom 95
4.4.2 Camera and Tripod Arrangement 98
4.4.3 Removing Image Misalignment 1 02
4.5 BACKGROUND TO FULL-FIELD DISTANCE MEASUREMENT 108
4.1 INTRODUCTION
This chapter considers the challenges involved in estimating the two-dimensional displacement field representing the motion of a large deforming object. Displacement vectors are obtained from time l apsed image pairs using two methods; one optical and the other digital. The conventional optical filtering techniques have been discussed in section 2.2. Point-by-point filtering has been employed in this research, as opposed to the full-field method, on two accounts; ( 1 ) the displacement estimates from Young's fringes are more accurate than from full-field isothetics; and (2) the analysis equipment is minimal and easy to arrange.
In order to obtain a ful l vector field describing the object's motion however, a complete grid of points across the negative must be analysed and this is time consuming and tedious if done by hand. Most researchers therefore employ some method of at least partially automating the process of fringe analysis. The hardware and software developed for automation in this work is described in section 4.2. This system is robust for use with the lower visibility fringes and variable halo profiles generated in sunlight speckle photography. An example of the resulting displacement field is given in the context of the material in section 4.4.
Optical metrologists are increasingly employing digital image proce sing in their work. Previously its main application was the automation of diffraction fringe analysis but more recently, techniques are being adopted from the image processing community which make it possible to directly examine digital images of the deforming object. An introduction is given in section 4.3 to digital motion analysis techniques about which a huge body of literature exists. The technique of maximum cross correlation is i l lustrated as being the most appropriate to the needs of this work. Full-field results from this technique are also shown in section 4.4.
Once the displacement vectors have been obtained from a time lapsed pair of images there remain two further tasks.
In many laboratory applications of speckle photography, it is possible to make double exposures of a deforming object without moving the camera. Generally when photographing large objects in sunlight however, the time l apse between exposures may be in the order of days or longer time periods and weather conditions prevent the camera from remaining in position, loaded with the piece of film on which the first exposure has been recorded. In addition it would be common to have a number of photographic
sites so the camera would be moved from place to place. Use of single, rather than double, exposures is thus a practical necessity during the recording stage. It is also very advantageous during the filtering stage because it allows the pairing up of any two speckle negatives with arbitrary time lapses, whereas use of a double exposure l imits the analysis to the two exposures recorded on the film. A pair of single exposed negatives can , in addition, be given a displacement with respect to one another so that even very small object movements can yield diffraction fringes, that is, the datum can be changed. Consideration must therefore be given to the alignment of the pair of exposures. The process of image registration is explained in section 4.4, and a description is given of the novel equipment and methods developed both for the field when the images are recorded and for the laboratory when they are analysed.
Finally, in order to relate the displacement vectors obtained from the negative pair to the physical object movement, it is necessary to have an estimate of photographic magnification. A deforming object l ike a glacier is unlikely to lie normal to the optical axis and besides wil l be highly three dimensional, so the photographic magnification will be different at every point. The full-field potential of sunlight speckle photography can only be realised if the distance measurement is full-field also. Section 4.5 describes the initial investigations into ful l-field object distance measurement.
4.2 AUTOMATING OPTICAL ANALYSIS OF YOUNG'S
FRINGES
4.2.1 Young 's Fringes
Figure 4- 1: Typical sunlight diffraction fringes generated from a pair of time lapsed
The equations describing optical filtering of a single exposed negative have been presented in section 3.4. Now if a double exposed negative (or pair of single exposed negatives) with displacement d =
[d1 d2]
is placed in the optical filtering arrangement, the autocorrelation halo is seen to be modulated by cosine-squared fringes (figure 4- 1 )with
? 1tud =
h
o(u), V. cos---'AL
(4- 1 )
where
h(
u ) is the intensity distribution at the Fourier plane; composed as shown In figure 2-9. This is given in many places including Franc;on ( 1 979) and Asundi &Chiang ( 1 982b). V is the fringe visibility which determines the amplitude of the fringes as defined in eq(3-2). The fringe spacing is proportional to the magnitude of d as given in eq(2-3) and the fringes lie at right angles to the direction of displacement. There is 1 800 direction ambiguity inherent in analysis of Young's fringes, as shown in figure 4-2, and one needs