5. Conclusions and Future Work of This Thesis
5.2 Future Work
Although we have successfully applied ANN in two cases in optical metrology, the following issues related to the current work remain unsolved and worth further study:
(1) How many hidden nodes should one use in an ANN to reach the best performance for a specific application?
(2) Since the ANN is powerful for modeling a nonlinear system, it is possible to combine ANN with other solvers for nonlinear equations such as Landsweber iteration method to enhance the iteration performance.
(3) The initial guesses of the weighting parameters and biases may influence the learning speed and thereby the convergence. How to set up a better initial guess worth further study.
(4) In this thesis, we applied ANN to localize multiple bright spots in an optical microscopic image. It is also very interesting to develop an ANN to recognize a
complex object in an image. A related issue is how to distinguish the object from its environment (i.e., how to determine the boundary of the object?) Or how to derive a criterion to guide a computer to determine the boundary of an object automatically?
References
[1] E. M. Hempe, P. C. J., A. Podlipensky, "Measuring ultrashort laser pulses Intensity Autocorrelation", project for the "Elitestudiengang Physik Regensburg-Erlangen", (2006).
[2] D. J. Kane, R. Trebino, "Characterization of Arbitrary Femtosecond Pulses Using Frequency-Resolved Optical Gating", IEEE Journal of Quantum Electronic, Vol.29, No.2, P.571 (1993).
[3] T. Tanabe, H. Tanabe, Y. Teramura, and F. Kannari, "Spatiotemporal measurements based on spatial spectral interferometry for ultrashort optical pulses shaped by a Fourier pulse shaper", Opt. Soc, Vol.19, No.11, P.2795, (2002).
[4] M. R. Brady, "Subpixel Resolution Schemes for Multiphase Flows", Virginia Polytechnic Institute and State University, Master of Science, ( 2006).
[5] 蘇木春, 張孝德, "機器學習:類神經網路、模糊系統以及基因演算法則", 全華科 技圖書股份有限公司, (2006).
[6] V. V. Zharkova, V. Schetinin, "A Neural-Network Technique for Recognition of Filaments in Solar Images", Lecture Notes in Computer Science, Vol.2773, P.148, (2003).
[7] H. A. Rowley, S. Baluja, T. Kanade, "Neural Network-Based Face Detection", IEEE Trans. Pattern Analysis and Machine Intelligence, Vol.20, No.1, P.23, (1998)
[8] J. Korczak, P. Roger, "Stock timing using genetic algorithms", Applied Stochastic Models in Business and Industry, Vol.18, Issue.2, P.121, (2002).
[9] W. S. McCulloch,W. H. Pitts, "A logical calculus of the ideas immanent in nervous activity", Bulletin of Mathematical Biophysics, Vol. 5, P. 115, (1943)
[10] F. Rosenblatt, "The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain", Psychological Review, Vol.65, No.6, P.386, (1958).
[11] Marvin Minsky, Seymour Papert, "Perceptrons: An Introduction to Computational Geometry", MIT Press, (1969).
[12] T. Kohonen, "Self-organized formation of topologically correct feature maps", Biological Cybernetics, Vol.43, P.59, (1982).
[13] J. J. Hopfield, "Neural networks and physical systems with emergent collective computational abilities", Proc. Natl. Acad. Sci. USA, Vol.79, P.2554, (1982) . [14] D. E. Rumelhart, J. L. McClelland, "Parallel Distributed Processing", MIT Press,
(1986).
[15] G. A. Carpenter,S. Grossberg, D. B. Rosen, "Fuzzy ART: Fast stable learning and categorization of analog patterns by an adaptive resonance system", Neural Networks, Vol.4, Issue.6, P.759, (1991).
[16] L.-N. Hossein, M. Nasiruddin, T. Samad, "Effect of Initial Weights on Back-Propagation and It’s Variations", Systems, Man and Cybernetics, IEEE International Conference, Vol.1, P.218, (1989).
[17] J. K. Kruschke, J. R. Movellan, "Benefits of Gain: Speed Learning and Minimal Hidden Layers in Back-propagation Networks", IEEE Transactions on Systems, Man, and Cybernetics, Vol.21, No.1, P.273, (1991).
[18] Y. M. Pirez, D. Sarkar, "Back Propagation Algorithm with Controlled Oscillation of Weights", Neural Networks: IEEE International Conference, Vol.1, P.21, (1993).
[19] I. Ivanova, M. Kubat, "Initialization of neural networks by means of decision trees", Knowledge-Based Systems, Vol.8, No.6, P.333, (1995).
[20] M. Zhao, Y. Wu, X. Ding, "Speeding up the training process of the MFNN by optimizing the hidden layers' outputs", Neurocomputing, Vol.11, Issue.1, P.89, (1996).
[21] S. C. Ng, S. H. Leung, A. Luk, "The Generalized Back-Propagation Algorithm with Convergence Analysis", Proceedings of the IEEE International Symposium on Circuits and Systems, Vol. 5, P.612, (1999).
[22] C. Dorrer, E. M. Kosik, I. A. Walmsley, "Direct space–time characterization of the electric fields of ultrashort optical pulses", Opt. Lett., Vol.27 No.7, P.548, (2002).
[23] A. M. Weiner, J. P. Heritage, and J. A. Salehi, "Encoding and decoding of femtosecond pulses", Opt. Lett., Vol.13, No.4, P.300, (1988).
[24] R. K. Shelton, L. S. Ma, H. C. Kapteyn, M. M. Murnane, J. L. Hall, Jun Ye "Active synchronization and carrier phase locking of two separate mode-locked femtosecond lasers", J. Modern Optics, Vol.49, Issue3, P.401, (2002).
[25] A. H. Zewail, "Femtochemistry, Recent progess in studies of dynamics and control of reactions and their transition states", J. Phys. Chem., Vol.100, P.12701, (1996).
[26] M. C. Chen, J. Y. Huang, L. J. Chen, "Coherent control multiphoton processes in semiconductor saturable Bragg reflector with freezing phase algorithm", Appl.
Phys. B, Vol.80, No.3, P.333, (2005).
[27] R.S. Judson, H. Rabitz, "Teaching lasers to control molecules", Phys. Rev. Lett., Vol.68, Issue 10, P.1500, (1992).
[28] N. Dudovich, B. Orion, Y. Silberberg, "Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy", Nature, Vol.418, P.512, (2002).
[29] T. Brixner, N. H. Damrauer, P. Niklaus, G. Gerber, "Photoselective adaptive femtosecond quantum control in the liquid phase", Nature, Vol.414, P.57 (2001).
[30] I. Pastirk, J. M. Dela Cruz, K.A. Walowicz, V. V. Lozovoy, M. Dantus, "Selective two-photon microscopy with shaped femtosecond pulses", Opt. Express, Vol.11, Issue.14, P.1695 (2003).
[31] D. Yelin, D. Meshulach, Y. Silberberg, "Adaptive femtosecond pulse compression",
Opt. Lett. Vol.22, P.1793, (1997).
[32] D. Meshulach, D. Yelin, Y. Silberberg, "Adaptive real-time femtosecond pulse shaping", J. Opt. Soc. Am. B, Vol.15, P.1615, (1998).
[33] T. Baumert, T. Brixner, V. Seyfried, M. Strehle, G. Gerber, "Femtosecond polarization pulse shaping", Appl. Phys. B, Vol.65, P.779 (1997).
[34] E. Zeek, R. Bartels, M.M.Murnane, H. C. Kapteyn, S. Backus, "Adaptive pulse compression for transform-limited 15-fs high-energy pulse generation", Opt. Lett., Vol.25, P.587, (2000).
[35] A. Efimov, M.D. Moores, N.M. Beach, J.L. Krause, D.H. Reitze, "Adaptive control of pulse phase in a chirped-pulse amplifier", Opt. Lett., Vol.23, P.1915, (1998).
[36] J. Kunde, B. Baumann, S. Arlt, F. Morier-Genoud, U. Siegner, U. Keller, "Adaptive feedback control of ultrafast semiconductor nonlinearities", Appl. Phys. Lett., Vol.77, P.924, (2000).
[37] A. YILDIZ, P. R. SELVIN, "Fluorescence Imaging with One Nanometer Accuracy:
Application to Molecular Motors", Acc. Chem. Res, Vol.38, No.7, P.574, (2005).
[38] M. J. Rust, M. Bates, X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)", Nature Methods Advance Online Publication, Vol.3, P.793, (2006).
[39] T. J. Gould, V. V. Verkhusha, S. T. Hess, "Imaging biological structures with fluorescence photoactivation localization microscopy", Nature Protocols, Vol.4, No.3, P.291, (2009).
[40] R. D. Kaufman, M. D., "Biophysical Mechanisms of Anesthetic Action: Historical Perspective and Review of Current Concepts", Anesthesiology, Vol.46, Issue.1, P.49, (1977).
[41] M. Smereka, I. Duleba, "CIRCULAR OBJECT DETECTION USING A MODIFIED HOUGH TRANSFORM", Int. J. Appl. Math. Comput. Sci., Vol.18, No.1, P.85, (2008).
[42] J. P. Fillard, H. M'timet, J. M. Lussert, M. Castagne , "Computer simulation of super-resolution point source image detection", Optical Engineering, Vol.32, No.11, P.2936, (1993).
[43] B. F. Alexander, K. C. Ng, Optical Engineering, Vol.30, No.9, P.1320, (1991).
[44] W.-K. Kuo, D.-T. Tang, "Laser spot location method for a laser pointer interaction application using a diffractive optical element", Optics & Laser Technology, Vol.39, P.1288, (2007).
[45] S. B. Andersson, "Localization of a fluorescent source without numerical fitting", Optics Express, Vol.16, No.23, P.18714, (2008).
[46] X. Qu, D. Wu, L. Mets, N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy", PNAS (Biophysics), Vol.101 No.31, P.
11298 (2004).
[47] J. W. Cui, J. B. Tan, L. Ao, W. J. Kang, "Optimized algorithm of laser spot center location in strong noise", , 7th International Symposium on Measurement Technology and Intelligent Instruments, Journal of Physics, Conference Series 13, P.312, (2005).
[48] S. Arturus, "New Algorithm of protein spot center detection in two dimensional electrophoresis gel images", IX international PhD workshop, P.61, (2007).
[49] J. P. Fillard, "Subpixel Accuracy Location Estimation from Digital Signals", Optical Engineering, Vol.31 No.11, P.2465, (1992).
[50] J. Enderlein, "Tracking of fluorescent molecules diffusing within membranes", Appl. Phys. B, Vol.71, P.773, (2000).
[51] T. Basche, W. E. Moerner, M. Orrit, U. P. Wild, "Single-Molecule Optical Detection, Imaging and Spectroscopy", Weinheim, 1997.
Appendix I: Technical Details of BP ANN
To help understanding the implementation of a back-propagation artificial neuron network (BP ANN) reported in this thesis, some useful information is presented in this Appendix.
From the configuration of BP ANN shown in Fig. 2.9, the output from jth hidden node can be expressed as
1
The output of the kth output node becomes
1
The deviation of the real output Y from the desired output D can be found
2
We update the weighting parameters of BP ANN by using the gradient descent method
to minimize E, yielding ( 1) ( )
Case 1: the weighting parameters between the hidden and output layer are found to be
j
In Case 2: we can derive the weighting parameters between the input and hidden layer
as
Similar procedure can be employed to obtain Eq. (2-10) and Eq. (2-12) by simply
replacing
Since both of the derivatives are equal to one, therefore ( 1) ( )
During the training stage, Eq. (2-4) and Eq. (2-5) are used to obtain the output of BPANN. Eqs. (2-9) to (2-12) are then employed to update the weighting parameters of the BP ANN until the error E is small enough or the limit number of training loop is arrived. During the application stage of BP ANN, only Eq. (2-4) and Eq. (2-5) are used.
The detailed steps are summarized in the following:
1. Initialize the weighting parameters Ws and Ps of BP ANN with random numbers.
2. Input a training data set X and D and invoke Eq. (2-4) and Eq. (2-5) to derive Y.
3. Calculate the value of error with Eq. (2-6). If the value is small enough stop the
training process, otherwise continue to the next step.
4. Calculate the error for the output layer k.
5. Update the weighting parameters of the BP ANN for W and jk P with Eq. (2-9) k and Eq. (2-10).
6. Calculate the error for the hidden layer . j
7. Update the weighting parameters of the BP ANN for W and jk P with Eq. (2-11) k and Eq. (2-12).
8. Go to step 2 for next training data set or stop the training process if the limit number of training loop is arrived.
The Flow Chart of the Training Process of a BP ANN
Initialize
Stop DATA Input sample
Error Check
Update weight
Loop Check
Appendix II: The Code for Preparing the Data Used in This Thesis
To retrieve the phase information of a coherent optical pulse from its SHG spectral
Intensity, we follow the following procedure:
Step1: Generate a Gaussian spectral profile with the code of for w=1:64
A(w)=exp(-((w-32.5)/15)^2);
end
Step2: Prepare the polynomial form of phase with
mask=[1 1 1 1 1; 1 1 1 1 -1; 1 1 1 -1 1; 1 1 -1 1 1;
Phase(i)=paramater(1)*x^2+paramater(2)*x^3+paramater(3)*x^4 ...
+paramater(4)*x^5+paramater(5)*x^6;
end
Step3: Add the phase some Legendre polynomials and yield the associated SHG spectral intensity with
PhaseAdd=[
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
6.2832 5.6943 5.1244 4.5735 4.0416 3.5287 3.0347 2.5598 2.1039 1.667 1.249 0.85011 0.47017 0.10923 -0.23271 -0.55566 -0.8596
-1.1446 -1.4105 -1.6575 -1.8854 -2.0944 -2.2844 -2.4553 -2.6073 -2.7403 -2.8543 -2.9493 -3.0252 -3.0822 -3.1202 -3.1392 -3.1392 -3.1202 -3.0822 -3.0252 -2.9493 -2.8543 -2.7403 -2.6073 -2.4553 -2.2844 -2.0944 -1.8854 -1.6575 -1.4105 -1.1446 -0.8596 -0.55566 -0.23271 0.10923 0.47017 0.85011 1.249 1.667 2.1039 2.5598 3.0347 3.5287 4.0416 4.5735 5.1244 5.6943 6.2832;
-6.2832 -5.1334 -4.0755 -3.1067 -2.2237 -1.4237 -0.70356 -0.060332 0.50902 1.0075 1.4382 1.804 2.108 2.3532 2.5426 2.6792 2.7661 2.8063 2.8027 2.7584 2.6765 2.5598 2.4115 2.2346 2.032 1.8068 1.562 1.3006 1.0257 0.74015 0.4471 0.14954
-0.14954 -0.4471 -0.74015 -1.0257 -1.3006 -1.562 -1.8068 -2.032 -2.2346 -2.4115 -2.5598 -2.6765 -2.7584 -2.8027 -2.8063 -2.7661 -2.6792 -2.5426 -2.3532 -2.108 -1.804 -1.4382 -1.0075 -0.50902 0.060332 0.70356 1.4237 2.2237 3.1067 4.0755 5.1334 6.2832;
6.2832 4.4275 2.8361 1.4888 0.36615 -0.55052 -1.2793 -1.8377 -2.2425 -2.5096 -2.6546 -2.6921 -2.6363 -2.5005 -2.2974 -2.0392 -1.7372 -1.4022 -1.0441 -0.67247 -0.29589 0.07757 0.44053 0.78629 1.1088 1.4028 1.6634 1.8868 2.0695 2.2089 2.3029 2.3503 2.3503 2.3029 2.2089 2.0695 1.8868 1.6634 1.4028 1.1088 0.78629
0.44053 0.07757 -0.29589 -0.67247 -1.0441 -1.4022 -1.7372 -2.0392 -2.2974 -2.5005 -2.6363 -2.6921 -2.6546 -2.5096 -2.2425 -1.8377 -1.2793 -0.55052 0.36615 1.4888 2.8361 4.4275 6.2832;
-6.2832 -3.6098 -1.5204 0.06077 1.2036 1.9726 2.427 2.6211 2.6041 2.4206 2.1108 1.7104 1.2512 0.7608 0.26336 -0.22069 -0.6742
-1.0832 -1.4367 -1.7264 -1.9467 -2.0944 -2.1684 -2.1696 -2.1008 -1.9665 -1.7724 -1.5257 -1.2344 -0.90767 -0.55507 -0.18678 0.18678 0.55507 0.90767 1.2344 1.5257 1.7724 1.9665 2.1008 2.1696 2.1684 2.0944 1.9467 1.7264 1.4367 1.0832 0.6742 0.22069
-0.26336 -0.7608 -1.2512 -1.7104 -2.1108 -2.4206 -2.6041 -2.6211 -2.427 -1.9726 -1.2036 -0.06077 1.5204 3.6098 6.2832;
6.2832 2.7183 0.24698 -1.3414 -2.2315 -2.5836 -2.5359 -2.2061 -1.693 -1.0785 -0.42916 0.20267 0.7769 1.2645 1.6463 1.9113 2.0559 2.0826 1.9989 1.8164 1.5495 1.2153 0.8318 0.41806 -0.0070076
-0.42505 -0.81884 -1.1727 -1.4731 -1.7087 -1.8706 -1.9531 -1.9531 -1.8706 -1.7087 -1.4731 -1.1727 -0.81884 -0.42505 -0.0070076 0.41806
0.8318 1.2153 1.5495 1.8164 1.9989 2.0826 2.0559 1.9113 1.6463 1.2645 0.7769 0.20267 -0.42916 -1.0785 -1.693 -2.2061
-2.5359 -2.5836 -2.2315 -1.3414 0.24698 2.7183 6.2832 ];
SHGI=zeros(1,127*6);
for phasecount=1:6
PhaseMod=Phase+PhaseAdd(phasecount,:);
SHGI(127*phasecount-126:127*phasecount)=SHG(A,PhaseMod);
end
Finally, the training data for ANN are prepared with a pair of input SHGI(1,127*6) and output Phase(1,64).
To localize the central position of an image of light spot with 1111 pixels, Step1: Generate an image of light spot with random central position by
Size=11;
X0=rand(1,1)*Size;
Y0=rand(1,1)*Size;
Step2: Prepare the image of 11*11 pixels with a 2D Gaussian profile
for X=1:Size
for Y=1:Size
G(X,Y)=exp(-((X-X0)^2+(Y-Y0)^2)/3^2)+exp(-((X-X1)^2+(Y-Y1)^2)/3^2);
end end
Step3: Add random noise to the image
noiselevel = (a value from 0 to 0.5) for X=1:Size
for Y=1:Size
Gr(X,Y)=G(X,Y)+rand(1,1)* noiselevel- noiselevel/2;
end end
Finally, the training data for ANN are prepared with a pair of input Gr(11,11) and output X0,Y0.