This assignment is for coursework of AV493 - Machine Learning for Signal Processing(MLSP) done under the guidance of Dr. Deepak Mishra, by Samvram Sahu, SC15B132.
In this assignment we will implement the Eigenface method for recognizing human faces. You will use face images from The Yale Face Database B, where there are 64 images under different lighting conditions per each of 10 distinct subjects, 640 face images in total. With our implementation, we will explore the power of the Singular Value Decomposition (SVD) in representing face images.
The data set can be found here. The data set is a file as face.zip, which contains all the training and test images; train.txt and test.txt specifies the training set and test (validation) set split respectively, each line gives an image path and the correspond- ing label.
Load the training set into a matrix X: there are 540 training images in total, each has 50 × 50 pixels that need to be concatenated into a 2500-dimensional vector. So the size of X should be 540 × 2500, where each row is a flattened face image.
Here is an instance of loaded data, we get when we load from X:
A sum of all the rows, divided by the total number of rows gives us an Average Face which, serves as origin for component analysis.
Here is the Average Face evaluated in our code:
We subtract average face
μ
from every column in
X
. That is,
x
i
:=
x
i
−
μ
,
where
x
i
is the
i
-th column of
X
.
The mean subtracted face of tenth person is:
Perform Singular Value Decomposition (SVD) on training set X ( X= U Σ V T ) to get matrix V T , where each row of V T has the same dimension as the face image. We refer to v i , the i -th row of V T , as i -th eigenface . Here are the first 10 eigenfaces as 10 images in grayscale.
Since Σ is a diagonal matrix with non-negative real numbers on the diagonal in non-ascending order, we can use the first r elements in Σ together with first r columns in U and first r rows in V T to approximate X . That is, we can approximate X by ˆ X r= U [:, : r ] Σ [: r , : r ] V T [: r , :]. The matrix ˆ X r is called rank- r approximation of X . Here is a plot of the rank- r approximation error ‖ X − ˆ X r ‖ F 2 as a function of r when r= 1, 2, . . . , 200.
The top r eigenfaces V T [: r , :]= { v 1 , v 2 , . . . , v r } T span an r -dimensional linear subspace of the original image space called face space , whose origin is the average face μ , and whose axes are the eigenfaces { v 1 , v 2 , . . . , v r }. Therefore, using the top r eigenfaces { v 1 , v 2 , . . . , v r }, we can represent a 2500-dimensional face image z as an r -dimensional feature vector f : f= V T [: r , :] z= [ v 1 , v 2 , . . . , v r ] T z
In order to get F , multiply X to the transpose of first r rows of V T , F should have same number of rows as X and r columns; similarly for X test
We extract training and test features for r=10. We train a Logistic Regression model using F and test on F test . Now we see the plot of Accuracy vs r:
We note that the maximum accurcy saturates at 96%. Also there is NO dip at 200, it is a faulty representation
Also, the text output of code is:
r = 10
Testing Accuracy : 79.0 %
Thus at r=10, the accuracy on test data is 79%.