11/15 Notes
===========

Face Recognition
----------------

 #. Appearance Based - using raw pixel intensities, completly agnostic to what the actual picture is
      * Subspace analysis
         * Linear subspace analysis
            * PCA - Principle component analysis
            * LDA - Linear Discriminate analysis
            * ICA - Independant component analysis 
         * Non Linear subspace analysis
            * KPCA - Kernel Principle component analysis
            * KLDA - Kernel Linear Discriminate analysis
            * KICA - Kernel Independant component analysis 

 #. Model Based Methods - model individual parts of the face to understand picture
      * EBGM - Elastic bunch graph matching - specific for face
      * AAM - Active appearence model - can be used for more

 #. Texture Based Methods - dont care if face or not, extract features from image
      * SIFT - Scale invariante feature transformation
      * LBP - Local Binary Pattern
      * LBQ - Local Phase Quantization
      * BSIF - Binarized statistical image feature

DeepFace - deep learning
------------------------

Turk and Pentland
 * Created PCA 


Most face Recognition algorithims take only eyes, nose, mouth region into account

In Face Recognition, image becomes vector

Let's say the length of this vector is d

d = the dimention it is in = N x N = pixel count

Dimentionality Reduction is used in PCA
 * Face Images can be represented by a point in NxN space = the vector
 * can drop info from high dimentions to lower dimentions
 
PCA
---

Input face image = sum of many different face images (always the same faces) (eigenfaces)

Weight associated with faces (eigenfaces) changes for each input face

Finding Eigenfaces
------------------

*Step 1* - Given a set of training images, comvert images into vector X1, X2, ... , XN where N is the total training images

Each vector becomes a column vector of size d by 1


*Step 2* - compute the mean face *mui* vector

*mui* = Xbar of vectors

*mui* is  a vector of d by 1 = column vector


*Step 3* - Compute the mean-subtracted faces vectors

Matrix X = [X1 - *mui*, X2 - *mui*, ..., XN - *mui*]

Remember that Xi - *mui* is still a colunm vector so matrix X is d by N

this shifts the origin to the center of face vectors


*Step 4* - Compute the covarience matrix of matrix X

This captures the varience of the the faces

Matrix Cov = X times XTranspose

Matrix Cov is d by d

This creates a matrix that each diagonal element is the varience of that dimention

THe off diagonal elements tell us how the points are distribted based on the 2 dimetions

so in row 2,column 10 in the matrix, it is comparing dimetion 2 with dimetion 10


*Step 5* - compute the eigenvectors of Matrix Cov

Use eigs in matlab - this solves Cov x E = lambda x E

Gives you 2 values

Output = [e_1, e_2, e_3, ... , e_d] which is d by 1

where e_i is an eigenvector of 1 by d

so output is d by d

for each e_i you also get an eigen value which is a single value

the larger the eigenvalue, the greater the spread of points along that eigenvector

So relationship tends to be v_1 > v_2 > ... > v_d


All of this is to get the weight vectors for each eigenface



TO get compact space, take eigenvectors and select a new matrix e_k such that
it only contains the top K eigenvectors. THe top K are determined by v_i. Higher is better

95% Rule
--------

If you take the eigenvalues and keep adding eigenvalues until you get .95 from the equation below

(v_1 + v_2 + ... v_k) / (v_1 + v_2 + ... v_d) = .95


v_k is the value you get to 0.95 on

All of this gives us the compact space

Testing Phase
-------------

Consider 2 face images

FA and FA (both are matricies)

 #. convert them to vectors: XA and XB (both are d by 1)
 #. YA = XA - *mui*    YB = XB - *mui* (both Y are d by 1)
 #. Compute the feature vector
      * WA and WB - these are the weights 
      * WA = E_kTranspose (this is the top K eigenvectors, is k by d) x YA  ---- so WA is k by 1
      * same process for WB, also gives k by 1 matrix
 #. Compute match score
      * ScoreAB = Euclidian distance between WA and WB