Face Recognition Using Face Unit Radial Basis Function Networks

Introduction

• This page is home to my research for the CS 198 course I've completed at Harvey Mudd College during the spring semester of 2000. The course, entitled Advanced Problems in Computer Science, consists of independent student directed research into a topic of interest. Prof. Robert Keller was gracious enough to be my faculty advisor for my research. This research is a continuation of my final project for CS152 - Neural Networks which I took during the fall semester of 1999. To get an understanding of what I'm researching, please read my neural network project web page from fall 1999. For that project I developed a system for human face recognition, based on a radial basis function (RBF) neural network with Gabor wavelet image preprocessing. This system was based on papers published by Dr. A. J. Howell describing approaches to human face regcognition using RBF networks. I succeeded in implementing the system in C++, however system performance was not very good. I deceided to continue work on this system by enrolling in CS 198, hoping to achieive results as good as those published by Dr. A. J. Howell.

Original Research Goals

• Before getting approval to register for the course, I wrote down some ideas I had for improving and extending my existing face_net system. Below is a list of things I set out to accomplish during the semester.
• Work on improving the approximation of the neuron weight vector. I believed that the existing method using a Singular Value Decomposition (SVD) matrix pseudoinverse was not working correctly. I wanted to try a non-SVD pseudoinverse methods, possibly using Gaussian or Gauss-Jordan elimination.
• Work on improving the algorithm for determining of neuron widths. The algorithms I has tested did not apear to be working well with the neuron widths calculated by the SVD pseudoinverse.
• Create a network of these face unit networks, whose output will show who the net thinks the image is of, instead of the two yes and no outputs for each face unit.

Research Achievements

• Determined that the neuron widths being generated by the SVD pseudoinverse caluculation were not correct. Replaced with pseudoinverse method based on Gaussian elimination of a matrix.
  • Let A be the matrix of the activation values from the input vectors, w be the neuron weight vector, and d be the desired output vector.
  • Ideally we would like Aw = d.
  • If the inverse of A, inv(A), exists (e.g. A is non-singular), we could simply solve for our desired weight vector: w = inv(A) d
  • However, since A is singular, we must approximate inv(A) by its pseudoinverse A*.
  • Letting ' represent the transpose operator, the pseudoinverse cen be defined as: A* = inv(A' A) A'
  • Now we can approximate our desired weight vector as w = A* d, or expanding the expression as w = (inv(A' A) A') d.
  • Gaussian elimination is used to invert (A' A), which is guarenteed to be non-singular.
• Undertook a drastically rewrite of the gabor2D Gabor wavelet preprocessor
  • Wrote GaborA3 class which performs Gabor A3 preprocessing just as specified in Howell 1995.
  • Significantly expanded the number of Gabor mask calculations and their corresponding output coefficients.
    • 100x100 pixel window centered around the nose is first subsampled to 25x25 pixels.
    • Data is sampled at 25x25, 13x13, 7x7, and 3x3 pixel scales, with a non-overlapping scheme used.
    • Each size mask is applied at 0, 120, and 240 degree orientations.
    • Since the Gabor function is complex, both real and imaginary (sine and cosine) components are calculated.
    • A 25x25 image yielded 510 coefficients.
  • Wrote make_masks.m MATLAB program which calculates all the Gabor mask matrices and writes them to files which are then used by the gabor2D program.
  • Generalized input format of program to take read in multiple S-expressions like the one below, preprocessing a single image for each.
    • (
(infile /home/bfeinste/cs152/faces/alexp_pos1)
(outfile /home/bfeinste/cs152/gaborfaces_A3/alexp_pos1)
(width 384)
(height 287)
(map 384)
(nose (169 189))
)
  • Wrote make_ifile.pl Perl script which generates a master input file for the preprocessor, containing S-exps for each input image. The script fetches the nose coordinates for each image from a file generated by the NosePicker Java application which I wrote for the project during fall 1999.
• Experimented with two methods for automatically calculating widths for the pro and anti neurons. The width determination methods were hinted at in Dr. Howell's published works. I initiated an email exchange with Dr. Howell, during which he briefly explained the ways he performs width determination. A variation was also used in which the widths were normalized in a fashion described in Howell 1995.
  • Global avgerage - Calculates a global average euclidean distance betwee all pairs of the unique neurons. The widths of the neurons are set to this global value.
  • Individual averages - To set each neuron's width, the euclidean distance to all the other neurons is averaged.
  • Default values - Just set the pro and anti neuron widths to some default value.
    

Results

• Results were still problematic, with wild variations in classification performance across different face_net's. The culprit is believed to be the excessive sensitivity of the network to its neuron widths. Future work could involve exploring algorithms for calculating how wide the neurons should be in relation to one another.
• Most of the goals I set out to accomplish were met, although the major goal of a high performance face recognition system was not. The system as it currently stands represents a good foundation for future work by myself or others.
• gabor2D Gabor wavelet preprocessor works well, and could be adapted to preprocess input for a variety of applications, in particular neural networks.
• I learned a great deal about neural networks, linear algebra, and programming during the course of my research.

References

• Chen, S. et al. (1991). "Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks." IEEE Trans. on Neural Networks 2(2), 302-309.
• Duahman, J. G. (1988). "Complete Discrete 2-D Gabor Transforms by Neural Networks for Image Analysis and Compression." IEEE Trans. on Acoustics, Speech, and Signal Processing 36(7), 1169-1179.
• Duvdevani-Bar, S., Edelman, S., Howell, A. J. & Buxton, H. (1998). "A similarity based method for the generalization of face recognition over pose and expression." (available online).
• Howell, A. J. & Buxton, H. (1995). "Invariance in Radial Basis Function Neural Networks in Human Face Classification." (available online).
• Howell, A. J. & Buxton, H. (1995a). "A Scaleable Approach to Face Identification." (available online).
• Howell, A. J. & Buxton, H. (1995). "Receptive Field Functions for Face Recognition." (available online).
• Howell, A. J. & Buxton, H. (1996). "Improving Generalization in Radial Basis Function Networks for Face Recognition." (available online).
• Howell, A. J. & Buxton, H. (1998). "Automatic face recognition using radial basis function networks." (available online).
• Howell, A. J. & Buxton, H. (1998). "Towards Visually Mediated Interaction using Appearance-Based Models." (available online).
• Moody, J. & Darken, C. J. (1989). "Fast Learning in Networks of Locally-Tuned Processing Units." Neural Computation 1, 281-294.
• Musavi, M. T. et al. (1989). "On the Training of Radial Basis Function Classifiers." Neural Networks 5, 595-603.

Acknowledgments

• Dr. A. J. Howell, School of Cognitive and Computing Sciences, Univ. of Sussex, UK.
  • Provided sample face images and published the original papers this work is based on.
• Prof. Robert Keller.
  • Provided the C code for Gaussian elimination on a matrix.
  • Provided Polya polymorphic list library in C++.
  • Provided 2d and Nd matrix allocation and deallocation functions in the form of templated C++.
  • Valuable guidance and advice regarding this project.

Code Directory

• All code Copyright © 2000 Benjamin S. Feinstein. Released under the GNU General Public License.
• My code directory can be found here.
• The following are the significant programs I've written in the process of completing this project:
  • FaceUnit class and face_net application in C++, implementing a scalable face unit network.
  • GaborA3 class and gabor2D program in C++, performing image sampling and Gabor wavelet preprocessing.
  • Wrote make_masks MATLAB program to generate Gabor mask matrix files for use by gabor2D.
  • Wrote make_infiles Perl script to generate input files for the face_net program.
  • Wrote test_infiles Perl script to batch run the input files through the face_net program.
  • Wrote make_ifile Perl script to generate input file for gabor2D preprocessor