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Intro to 2D-LDA Experiments Conclusion 2D-LDA Face Recognition on ORL Dataset Garrett Bingham University of North Carolina Wilmington July 7, 2017 Garrett Bingham 2D-LDA 1
Intro to 2D-LDAExperimentsConclusion
2D-LDAFace Recognition on ORL Dataset
Garrett Bingham
University of North Carolina Wilmington
July 7, 2017
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Intro to 2D-LDAExperimentsConclusion
Table of contents
1. Intro to 2D-LDAMathematical Framework
2. ExperimentsMethodologyResults
3. Conclusion
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Intro to 2D-LDAExperimentsConclusion
Mathematical Framework
Intro to 2D-LDA
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Intro to 2D-LDAExperimentsConclusion
Mathematical Framework
Intuition
Similar to 2D-PCA, in 2D-LDA we project each (m × n) image Ai ontoX , obtaining the projected image
Y i = AiX . (1)
Here X is (n × d) and is made up of d vectors. We choose the vectorsthat minimize within class variance and maximize between class variance.This equates to maximizing the 2D Fisher criterion
J(X ) =XTG bXXTGwX
. (2)
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Mathematical Framework
Scatter Matrices
The between class scatter matrix G b is denoted as
G b =1
N
C∑i=1
ni (Ai − A)T (Ai − A), (3)
and the within class scatter matrix Gw as
Gw =1
N
C∑i=1
∑j∈C
(Aj − Ai )T (Aj − Ai ). (4)
Here N represents the total number of images, C the number of people,ni the number of images for the ith person, Ai the average image of theith person, and A the overall average image.
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Mathematical Framework
Finding Optimal X
Maximizing
J(X ) =XTG bXXTGwX
(5)
equates to finding the eigenvectors of G−1w G b. The columns of X are
then set to the d eigenvectors corresponding to the d largest eigenvalues.
In 2D-PCA we have discussed the bilateral version, where an image ismultiplied on the left and the right. The same applies to 2D-LDA, andwe can instead consider
Y i = ZTAiX . (6)
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MethodologyResults
Experiments
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MethodologyResults
Introduction to ORL
The ORL Database is a collection of 400 images taken between 1992 and1994. It features 40 distinct individuals, with 10 pictures per person.Lighting conditions, facial expression, and glasses/no glasses vary amongimages. All images are grayscale and 92 × 112 in size.
ORL is available at http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html
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MethodologyResults
Experiment Design
In each test, I compare the performance of 2D-LDA vs 2D-PCA. I usedthe bilateral version of both algorithms. A nearest neighbor (1NN)classifier was used for simplicity, and distances were calculated using theFrobenius norm for matrices (equivalent to Euclidean distance).
Some people’s images were intentionally set aside as unknown. For agiven test image, the algorithms had to then either assign the image to aknown person, or classify it as an unknown person if its nearest neighborwas not within a certain threshold. I varied this threshold over a widerange of values in order to visualize the tradeoff between sensitivity andspecificity.
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MethodologyResults
Parameters
There are many parameters that can be modified in this experiment.
• The number of images per person for training vs. testing.
• The number of people that are known vs. unknown.
• d1 and d2, the number of principal components kept, also thedimensions to which each image is reduced.
Selecting just four values for each parameter results in 256 differentscenarios. Uncovering the trend in the data will take some work, but fornow we can consider a few examples.
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MethodologyResults
An Example Scenario
Many graphs had a trendsimilar to this one. Itis possible that 2D-LDAclassifies individuals bet-ter than 2D-PCA becauseit is supervised. However,2D-PCA retains more ofthe variation in the data,so it is better able to cor-rectly identify unknownpersons.
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MethodologyResults
The Other 255 Scenarios...
...are all available for your analysis and viewing pleasure.
See my folder in Google Drive.
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Conclusion
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Conclusion
There is still a lot of work left to be done. In addition to understandingwhich of 2D-LDA or 2D-PCA is preferable in a given circumstance, thisexperiment needs to be generalized to larger and more complicateddatasets. Different methods for classification, as well as various distancemetrics could also be used to yield potentially important results.
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Reference
S. Noushath, G. Hemantha Kumar, and P. Shivakumara. (2d)2lda:an efficient approach for face recognition. Pattern Recognition,39(7):1396–1400, 2006. issn: 0031-3203.
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