A Sampled-Grating Model of Moire Patterns from Digital Imaging John Krumm and Steven A. Shafer CMU-RI-TR-89- 19 The Robotics Institute Carnegie Mellon University Pittsburgh, Pennsylvania 15213 July 1989 01989 Camegie Mellon Univeristy This research was supposred by the Defense Advanced Research Projects Agency, DoD. rhrough ARPA Order Number 4976, monitored by the AU Force Avionics Laboratory under Conhact F33615- 87€-1499 and by the Jet Propulsion Laboratory, California Institute of Technology, sponsored by the National Aeronautics and Space Administration under Contract 957989. Any opinions, findings, oonclusions or mmmendarions expressed in this publication are those of the authors and do not necessarily reflect the views of the United States Govemment a the let Propulsion Laboratory..- Con tents 1 Introduction 1 2 A Model of Moire Patterns from Crossed Gratings 2 3 Moire Patterns from Sampled Gratings 6 3.1 A Model of Multiple Stage Reconstmction and Sampling ............ 7 3.2 MoirePattems ................................... 11 4 Crossed-Gratings Models of Sampled Gratings 12 4.1 An Approximation With Two Ronchi Rulings ................... 13 4.2 An Approximation With Two Arbitrary Gratings ................. 15 5 Experimental Verification of Sampled-Grating Moire 16 5.1 Geometric Parameters ............................... 18 5.3 Moire Pattern Simulation ............................. 23 6 Crossed-Gratings Model of Sampled-Grating Moire 24 7 Conclusion 25 5.2 AmplitudeF'rediction ............................... 19 iiiList of Figures 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Crossed gratings making a moire pattern ..................... 1 Square wave transmittance profile ........................ 2 Ronchi ruling at 0" ................................ 4 Fourier transform of Ronchi ruling at 0" ..................... 4 Ronchi ruling at 15" ............................... 4 Fourier transform of Ronchi ruling at 15" .................... 4 Crossed Ronchi rulings at 0" and 15" ....................... 4 Fourier transform of crossed Ronchi rulings at o" and 15O ............ 4 Fourier transform of grating ........................... 8 Spectral orders after fist stage sampling ..................... 8 Spectral orders after second stage sampling. no additional aliasing ....... 8 Spectral orders after second stage sampling showing additional aliasing ..... 8 Block diagram illustrating multiple stage reconstruction and sampling ..... 9 Ronchi Ruling at 90" ............................... 14 Fourier transform of Ronchi ruling at 90" .................... 14 Crossed Ronchi rulings at 00 and 90" ....................... 14 Fourier transform of crossed Ronchi dings at 0" and 900 ............ 14 Crossed Ronchi rulings at W. 90" and 15O .................... 14 Fourier transform of crossed Ronchi rulings at 0". 900 and 15" 14 Method of mounting grating for experiment ................... 16 Sampled-Grating Moire Pattern .......................... 17 Predicted Moire Pattern .............................. 17 'Zhresholded Fourier Transform of Moire Pattern ................. 18 predicted Fourier Transform of Moire Pattern ................... 18 OTF's for monochromatic. incoherent light .................... 21 SpectralCurves .................................. 22 Fitted curve for PI ................................ 23 Slices of vertical Fourier transforms ....................... 24 Aliased moire Pattern simulated with three crossed gratings ........... 25 Aliased moire pattern simulated with two crossed gratings ............ 25 ......... iiiivAbstract Traditional “crossed-grating” moire, as well as newer ”sampled-grating” (scanning) moire. have proven to be effective methods of shape measurement. There is speculation that the moire patterns of a sampled gating, which are due to aliasing, can be modeled with crossed gratings. We compare the two and show that while crossed gratings can correctly predict the frequencies of a sampled grating, they cannot correctly predict the amplitudes. We go on to formulate a new model which accounts for multiple stages of sampling and transmis- sion, and show how neglecting multiple stages can lead to mistakes in moire analysis. We demonstrate our model with an experiment using a digital imaging system. V1 Introduction Moire patterns can be caused by crossed gratings or by the discrete sampling of a single grating. The crossed-gratings case can be seen in everyday smundings in layers of fences or screens. Figure 1 shows a contrived superposition of two crossed gratings that give rise to a moire pat- tern. If one of the crossed gratings is distorted, the moire pattern will be distorted, too. This phenomenon has been extensively exploited in metrology applications. One of the constituent gratings is projected onto a distorted object and observed through a second, flat grating. Since the resulting moire pattern is sensitive to small distortions in the object grating, accurate mea- surements of the object's shape can be derived. The object's shape is a function of the geometric layout of the original, undistorted gratings and the resuliing moire pattern. Post [Pos67] has shown that an analysis of the profiles of the crossed gratings can be used to sharpen the moire fringe patterns, making it easier to track the fringes and thus easier to determine the shape of the object. Therefore, both the geomeny and brightness profiles of the moire method are important to consider. Figure 1: Crossed gratings making a moire pattern Moire patterns can also be caused by aliasing in the discrete sampling of a single grating. The sampled-grating case can often be Seen on television when a relatively high-spatial-frequency grating, say a striped shirt, is shown. Idesawa et 01. [IYS771 have shown that the flat grating in traditional, crossed-grating, moire metrology can be replaced by a digital camera, still maintain- ing observable moire patterns using only a single, object grating. We will distinguish these two methods by calling the former "crossed-grating" moire and the later "sampled-grating" moire. The moire patterns from both methods appear similar, and it has been shown that sampled-grating moire can automate many time-consuming, subjective procedures of the traditional, crossed- grating paradigm [MSH88]. Idesawa et ul. assert that the camera can be modeled by a second grating, thus allowing the application of a