Image Compression using Singular Value Decomposition

Math 320Kristen Cunanan

Michael Tzen

Reading a matrix into MatlabCommand:

data=imread(“title”,”format”)

datanmp

Singular Value Decomposition

data USV T

Snm

1 0 00 2 0 0 n0 0

SVD in Matlab

we must do svd p times on each “page” of the array.

Command:

= svd(data(:,:,:i)i=1…p

datanmp

Ui,Si,Vi

Selecting for the number of Singular Values

S*nm

1 0 00 * 0 0 0

data* US*V T

Using 1 Singular Value

Using 11 Singular Values

Using 31 Singular Values

Using 51 Singular Values

Using 91 Singular Values

269 Singular Values

Eigenfaces/Facial Recognition

Process

• Is person X in the “training” group of M=50?

• SVD on manipulated pictures

• Is Euclidean distance in the threshold?

Prep Work

Image I Data matrix

“Training Set” of images

},...,,{ 21 MS

0

Subtracting the Mean

• Compute the Mean of S = Training set

• Subtract Mean from ea. face/vector in S

},...,,{ 21 M

Covariance Matrix

• Get the Covariance Matrix of S

SVD on C =

• SVD method on C to get the eigenvalues/eigenvectors

• Gives us the “important” values/vectors corresponding to each difference vector

Eigenvector = Eigenface

• The eigenvectors obtained, are called Eigenfaces

• Any can be written as a linear combination of the eigenfaces

=.87 + .2

+ .10 + .1

Euclidean distance

||||min 0l

lre

Conclusion

• If Value (Specified)– Human Face ~ 1500 range

• McMillen = Bradd Pittrere

re

Otherwise McMillen = Will Smith?