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## Components of the Covariance Matrix

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Instructions

Components of the Covariance Matrix

Question 1 – Face recognition

Perform a face identification task. For training, you are given 4 facial images of a person DG and 4 facial images of a person KS. For testing, you are given a facial image X that is known to be of

either DG or KS. The task is to determine the likelihood ratio (𝑋|𝐷𝐺) 𝑝(𝑋|𝐾𝑆)

and on that basis to

decide which of the 2 persons is depicted by X and to state the level of confidence you have in the decision.

To begin, use the Matlab function call load([folder,’images.mat’]) to load the 9 images into Matlab, where folder is the folder on your PC where the file images. Mat resides, for example folder=’C:\Users\Kate\Documents\Uni\Biometrics\’. The 9 images are stored in a Matlab cell array images, where bitmaps dg1…dg4 are accessed as

images{1}…images{4}, ks1…ks4 as images{5}…images{8}, and xx1 as images{9}.

Each bitmap is a 70×50 matrix of uint8 values representing grey-scale pixels. The faces have already been extracted from larger images and normalised such that the person’s right pupil is in a standard position. (You may also inspect the bmp images to confirm your result, but visual inspection and subjective judgment alone do not attract any marks!) In parts a) to c) of the question, you perform principal-component analysis on the training data.

- a) Load the 8 training files into Matlab and convert each 70×50 matrix to a 3500×1 vector of double values by stacking the 50 matrix columns. Convert all values to double. Each vector then has all the 3500 components of the corresponding matrix, but double instead of uint8. [Hint: Use the Matlab functions load( ), reshape( ) and double( )]
Print the first 5 components of each (transposed) vector in your homework. [1.5 marks]

- b) Determine the 3500×1 mean vector and the 3500×3500 covariance matrix of the 8 training vectors, then compute the first 7 principal axes of the feature space. [Hint: Combine the 8 training vectors into a 3500×8 matrix, then use the Matlab functions mean( ), cov( ) and eigs( )]
Print the first 5 components of the (transposed) mean vector and the top-left 5×5 components of the covariance matrix in your homework. [1.5 marks]

- c) Project the training vectors onto the principal axes, thereby reducing each vector from 3500 components to 7 components. [Hint: Multiply the matrix of mean-normalised training vectors from the left with the transposed matrix of eigenvectors. Use the Matlab function repmat( ).]
Print ALL components of each (transposed) projected vector in your homework. [1 mark]

At this point, principal-component analysis is complete. Next, you have to calculate the models for Person DG and Person KS.

- d) From the 4 projected training vectors of DG, compute the 7×1 DG mean vector and the 7×7 DG covariance matrix. Then diagonalise the covariance matrix. Do the same for KS. Together, the mean vector and diagonal covariance matrix of each person form that person’s face model. [Hint: Use the Matlab function diag( ).]
Print ALL components of the 2 (transposed) mean vectors and ALL components of the diagonals of the 2 covariance matrices in your homework. [1 mark]

At this point, training is complete. Finally, you need to calculate the PDFs of the test face given, respectively, the DG model and the KS model.

- e) Load the test file and convert it to a 3500×1 vector of type double as in Q1a).
Print the first 5 components of the (transposed) vector in your homework.

- f) Project the test vector onto the principal axes as in Q1c). [0.5 marks]
Print ALL components of the (transposed) projected vector in your homework. [0.5 marks]

- g) For each of the 2 models, determine the probability density (PDF) that the test vector is
produced by that model and calculate the likelihood ratio 𝑝(𝑋|𝐷𝐺) 𝑝(𝑋|𝐾𝑆)

.

[Hint: Use the Matlab function mvnpdf( ).]

Print the 2 resulting PDFs, the likelihood ratio and the log10-likelihood ratio in your homework. [1 mark]

- h) Finally and assuming equal priors and one-zero decision costs, state whether your statistical evidence supports the hypothesis 𝐻(𝐷𝐺|𝑋) that the test face belongs to Person DG or the hypothesis 𝐻(𝐾𝑆|𝑋) that the test face belongs to Person KS. How confident are you of the decision? Use the likelihood ratio you calculated to support your conclusion.
Print your statement in plain English in your homework. [1 mark]

RUBRICExcellent Quality95-100%

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Average Score50-85%

40-38 points More depth/detail for the background and significance is needed, or the research detail is not clear. No search history information is provided.

83-76 points Review of relevant theoretical literature is evident, but there is little integration of studies into concepts related to problem. Review is partially focused and organized. Supporting and opposing research are included. Summary of information presented is included. Conclusion may not contain a biblical integration.

52-49 points Content is somewhat organized, but no structure is apparent. The use of font, color, graphics, effects, etc. is occasionally detracting to the presentation content. Length requirements may not be met.

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75-1 points Review of relevant theoretical literature is evident, but there is no integration of studies into concepts related to problem. Review is partially focused and organized. Supporting and opposing research are not included in the summary of information presented. Conclusion does not contain a biblical integration.

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Components of the Covariance Matrix

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