Page 198 - The-5th-MCAIT2021-eProceeding

Where × ( ) is a function that maps ∈ ℝ × and ( ̅ ̅ ) denotes the -th principal eigenvector of

̅ ̅ .
Production of PCA filters initially can only accommodate images with single input (i.e., grayscale image)
thus, multi-channel PCA filters is produced to cater for colored input images (Chan et al., 2015). This is essential
if the filters generated are required to capture complex features from complex image database. Similar to the
existing step of constructing image matrix , an additional individual matrix of each RGB channel is created
̂
and is denoted by , , ∈ ℝ × ℎ ̂ , respectively. Production of PCA filters for RGB channels are then

similar to equation (4) with a slight variation which is defined as follows,
(5)

where ̃ = [ , , ] and × ×3 ( ) is a function that maps ∈ ℝ × ×3 into a matrix ∈ ℝ × ×3 .

3.2. Gaussian PCA (GPCA) filters
Generation of GPCA filters are similar to the generation of PCA filters with a slight variation to the sliding
window used which is the Gaussian window, , , . The Gaussian window weights can be defined as equation
(6) below,
(6)
In comparison to the rectangular window, Gaussian window are weighted with a two-dimensional Gaussian
function and is not weighted in unity. Thus, the gaussian parameter, represents the width of the Gaussian
function. The Gaussian window weights, , , will be used to obtain the overlapping sub-image by
,
multiplying the pixel value of test image B. The remaining process in generating GPCA filters are the same as
the process of producing PCA filters from equation (3) until equation (5).
3.3. Generalized GPCA (G-GPCA) filters

Generation of PCA and GPCA filters are required to be recalculated for each image which is a strenuous
process. However, this can be handled by approximating and defining a generalized equation for GPCA. This
enhancement provides an advantage of creating filters that are suitable for different images by only setting the
correlation parameters and gaussian parameter. The generalized GPCA (G-GPCA) filters can be defined as
follows,

(7)

Based on equation (7), GPCA filters is a product of two one dimensional eigenvectors ( ) and ( ).

Analytical formulation of the eigenvectors can be formed as follows,
(8)

Where, the parameter and ′ represents constant parameter while, parameters and ′ can be calculated
as follows,

(9)
Moreover, calculating the eigenvalues can be conducted as follows,
(10)

E- Proceedings of The 5th International Multi-Conference on Artificial Intelligence Technology (MCAIT 2021) [185]
Artificial Intelligence in the 4th Industrial Revolution