Principle Component Analysis (PCA)     

Principle component analysis has been widely used for analyzing the image data. It is applied to biometric systems as a classification design. PCA enables us to measure the difference between two images while allowing expression changes. In the vector space, PCA identifies the major directions and corresponding strengths, of variation in the data. PCA performs these by computing the eigenvectors and eigenvalues of the covariance matrix of the image data. Keeping only a few eigenvectors corresponding to the largest eigenvalues, PCA can be also used as to reduce the size of the data while hold the major variation of data. For example, we may reduce the database size by using only first twenty eigenspace while our whole eigenvectors are 250.

Principle Component Analysis (PCA), also known as Eigen-XY analysis is a standard statistical technique for finding directions of maximum variations in data. These directions, called the principle components, can be used to reconstruct all of the information within the set and can be tested to which level a test image couples with an image of the training set. Principle components with smaller associated magnitudes can often be omitted, as they contribute less to the overall reconstruction of each data element. This allows sufficient representation of the original data set with a reduced set of principle components.

Principle components are found by computing the eigenvalues and eigenvectors of the covariance matrix associated with the data. Eigenvectors with largest associated eigenvalues are the principle components that describe the most variation in the data set. Principle component coefficients of data elements are found by projecting each datum onto the eigenspace of the covariance matrix.

-Working Principle of PCA

         In order to use PCA for image processing, firstly it is required to convert RGB (Red – Green – Blue) picture to gray scale (black and white). It is also possible to use RGB picture but instead of using 3D matrices, 2D matrices are better to evaluate. Given a gray scale image, that can be represented as a matrix of intensity values. We can convert it into a vector by placing one column of the image on top of another. Eye image is a two-dimensional array of size x * y, where x and y are width and height of the image. Each image can be represented as a vector of dimension x * y.

-Training Phase of PCA

             Let ?_n be an image from the collection of M images in database. P1, P2, P3 to Pn  refers vector form of each picture.

 Average image ? is defined as:

 After getting average column vector matrices, we required to build difference matrix. Each image ?_i differ from the average image ? by the vector:

The covariance matrix of the data set is defined as:

Where ? is

                                                           ?= [?1 ?2 … ?M]            

After getting covariance matrix, we can build the eigenvectors.

If we consider eigenvectors vi of ?T? such that 

                                                               ?^T? v_i = µ_i v_i                             

When we multiply both sides of eq. (4.1.4) by  ? we obtain:

                                                             ? ?^T? v_i  = µ_i ? v_i                           

Hence, we can construct M x M matrix:

                                L=?^T?                    

and find M eigenvectors vl   of L. If we now define eigenvectors of C as ul.

Eigenvectors ul, are actually images, and they are called eigenfaces. Among those eigenfaces ones with the higher eigenvalues are more useful. Therefore those M ' << Meigenfaces that are most significant are used for constructing the “eye subspace” for image projections which are used in iris identifying, classifying or recognizing.

If we reshape the eigenfaces, they give us the ghost pictures. In Fig.5, we can observe first twenty ghost pictures for eigenfaces.

Figure – 5: First 20 eigenfaces with highest eigenvalues.

-Test Phase of PCA

             After finishing training phase and getting eigenfaces, let us move on the test phase to proof our PCA theory.

            Although we have M pictures and eigenfaces in our database, it is not required to use all of these eigenfaces. We may use M ' of them (M ' << M).

            In order to test any picture to whether it is a sample of the training part or not, we should to transform that picture to eigenface form by the following operation:

?k is the k-th coordinate of the ? in the new “eye space”. Eq. (4.3.1) describes point-by-point image multiplications and summations, resulting in the scalar value (dimension 1x1), previously defined as a weight that describes each face. Those weights form a vector  

gives us the projection vector.

Vector ? describes contribution of each eigenface in representing the input (test) image, where the eigenfaces represent basis set for eye images.

This vector is used in finding the class that input image belongs to if there are more than one images describing a person, otherwise it is used to find to which single image is most similar. This will be done by building Euclidian distance between test image and projection image classes. If we compare the distance between by using ? values for training and test phase, it will give us the distance of the test picture according to the training phase.

                                                             ?_k= ||?-?_k ||2                                

Here ?_k describes k-th eye class, which is average of the eigenface representation of eye images of each individual. An eye will be classified to some class if the minimum ?_k is below some specified threshold, otherwise it will be classified as “unknown” picture.