Binarization of gray-scale hologram Project of ECE 533 Dec. 15, 2006 Fan Jiang1. Introduction Since Gabor’s first hologram, the meaning of this term have grown with increased use of the invention. This rapidly growing is also because the hologram can be generated by computers. Computer generated holograms (CGH) have been investigated intensively in recent years owing to their wide application range and their advantages in terms of flexibility, accuracy, size, weight and cost. Applications can be found, e.g., in optical information processing, in which CGH’s are used as filters to generate a required wave front, or in optical neural networks, in which they are used to accomplish complex synaptic interconnections. But, what is hologram? Hologram is a kind of mask that created by the diffraction wave of the object target and a reference beam. So in the hologram, information of the phase of the target is recorded. After reconstruction, the object can be recovered easily. In this project, the diffraction theory I used is Fraunhofer diffraction, which is just the Fourier Transform of the object. There is a rough process about making hologram and reconstruction in the figures below. Here the reconstruction is from the gray-scale hologram. One of the images is original image; while the other one is the conjugate image. Figure 1: Object (Target) Æ Hologram (gray-scale) Æ Reconstruction. One can generate a hologram with computer very easily, but after the calculation for a hologram, the picture is gray-scale. However, since the limitation of the lithography, we can only record binary information on the photo resist, which means, 0 10 20 30 40 50 6001020304050600 10 20 30 40 50 6001020304050600 10 20 30 40 50 600102030405060the intensity of the hologram can only contain 0 and 1. So we need to take care of the binarization process for the gray-scale hologram. There are a lot of papers about the binarization methods for hologram. And in this project, I will use the threshold as the key for the binarization and compare 3 different binarization processes. 2. 3 different methods for binarization The essential point for binarization is the threshold. If the intensity of the gray-scale is higher than threshold number, the binary hologram will have 1 for this point; otherwise, the number in the binary hologram will be 0. According to this idea, the production and the reproduction of binary holograms is simple. However, one has to take care of the binarization error because this error can cause enormous errors in the reconstruction with respect to the original object. So here, I have 3 different methods for binarization. They all have the threshold as the key part, but the threshold will be calculated in different way, or there will be post-processing in the binarization. I will show all the results of the binary hologram, and use the reconstruction result to be the comparison. 2.1 Approach 1: Simple threshold As said above, using a simple threshold can create a binary hologram. In the textbook of this course, we know how to choose the threshold is very important in the conversion. And all the optimal thresholds are depending on the histogram of the picture. It is said, in the histogram, we need to first clarify the object part and the background part. In the figure below, it’s the histogram of the hologram.Figure 2: Histogram of the gray-scale hologram (max. is 7.3916; min. is 0.11108) As seen in the figure, it is impossible to separate the object and the background. So in this case, we just use the median number of all the intensity data in the hologram as the threshold so that the binary hologram will have 50% black and 50% white. Run the Mathematica to find the median number, which is 1.228. And convert the gray-scale hologram to binary hologram. Here is the result. Figure 3: Binary hologram created using approach 1. 2.2 Approach 2: Random binarization According to “Gradual and random binarization of gray-scale holograms”, it is said most of the binarization error should be the random error. Hence, if we can binarize the n (n < total pixel number in the hologram) random pixels using their own threshold each time, and repeat this process until all the pixels are binarized, then we 2 4 6 82000400060008000100001200001020304050600102030405060can reduce this kind of random error. Using this idea, the algorithm is shown in the next page. Since I used different address for each pixel, and created n pseudorandom for the key to pick up the pixels, the pixels will be chosen randomly. And the threshold will be the median number of these n random pixels. So in each round, the chosen pixels cannot all be the same, hence the threshold for each round will be different. After all the pixels are binarized, the process ends. Gray-scale hologram Binarize n random pixelsAll the pixels are binarized? Yes No Binary hologram Figure 4: Algorithm of random binarization As a result, we should see that the histogram of the hologram gradually change to contain only the intensity of 1 and 0. (Figure 5)2 4 6 825005000750010000125001500017500 2 4 6 850001000015000200002500030000 1 2 3 45000100001500020000250003000035000 0.5 1 1.5 2 2.5 3100002000030000 0.5 1 1.5 2 2.5 310000200003000040000 0.25 0.5 0.75 1 1.25 1.5500100015002000 0.2 0.4 0.6 0.8 1 1.2500100015002000 0.2 0.4 0.6 0.8 1500100015002000 Figure 5: Histogram changes with the increasing rounds: After the 1st round; after the 10th round; after the 11th round; after the 20th round; after 30th round; after the 31st round; after the 50th round; after the last (61st) round. And the final result of the binary hologram is shown below.01020304050600102030405060 Figure 6: Binary hologram created using Approach 2. 2.3 Approach 3: High frequency binarization This binarization is created by our group. Here, we still use one threshold to convert the gray-scale hologram. But after this simple conversion, we have a post-processing. During this process, we picked up the maximum points under the threshold in the gray-scale hologram, and changed their intensity from 0 to 1 in the binary hologram. Similarly, we also picked up the minimum points above the threshold in the gray-scale hologram, and changed their intensity from 1 to 0 in the binary hologram. So far, all the peaks in the gray-scale hologram will have 1 in the binary hologram; while all the valleys in the gray-scale hologram will have 0 in the binary hologram. And these numbers are
View Full Document
Unlocking...