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\textbf{\Large{International Journal of Artificial Intelligience and Cognitive Psychology (IJAICP)}} \\\;Volume 1, Issue 1,\\ ISSN: 3043-5803\\  \; Website: \href{https:///ijaicp.aijournals.org}{https:///ijaicp.aijournals.org}
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  \textbf{\Large Parallel Associativity in Scan-path and Fixations on Usability Short-Term Events} \\[5pt]%
  \textbf{\small Fatima Isiaka \orcidlink{https://orcid.org/0000-0002-6455-776X}}\\%
 \textit{\scriptsize Department of Computer Science, Nasarawa State University, Keffi, Nigeria}\\%
\textit{Email: isiakafatima@nsuk.edu.ng}\\%
\textbf{\small Zainab Adamu}\\%
 \textit{\scriptsize Department of Computer , Ahmadu Bello University, Zaria, Nigeria.}\\%
\textit{Email: zadmu31@gmail.com}\\%
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\scriptsize{\it{\color{brown}{A Published Article of International Journal of Artificial Intelligience and Cognitive Psychology}}}

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\textbf{Abstract}
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In usability studies, when observation is made on a particular image or object on visual stimulus online, the perception of the user depends upon preceding expectations, the cognitive abilities and user memory. Hypothesis is usually set that cognitive processes is in the form of bottom to up processing or a top to down processing, and this could be determined through analysis of the fixation points on scan-path made by the eye movement on screen. Normally, to access the variations on the scan-path, user data from eye movement predictions is analysed using a four by four grid pattern which is superimposed on the visual stimuli in order to determine the user’s eye fixation dispersal pattern. This is mostly based on assumption that the visual interface is viewed as a 2-dimensional visual field (Cartesian plane). As part of the contribution to usability studies, this paper approaches predictions of fixation and scan-path, by determining the Euclidean distance between fixation points, determine the shortest scan-path between fixation points using the Dijkstra’s Algorithm for momentous cognitive response to visual stimuli in from of dynamic contents on webpages and to predict fixation points of eye movement based on constriction and dilation of the pupil response. 

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\textbf{Keywords:} Visual field, Cognitive response, Fixation points, Scan-path, Shortest path, Euclidean distance, Webpage visual field
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\small{Accepted: 12th September,  2022} \\
\small{Revised: 16th October,  2022}\\
\small{Published:10th November, 2022}\\


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\small{\textbf{Corresponding Author: Fatima Isiaka}} \\
\small{Correspondent Email:}\\
\small{fatima.isiaka@outlook.com}\\
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\section{INTRODUCTION} 

In the early seventy’s, scan-path on visual field was introduced by Noton and Stark \textcolor{blue}{\textit{(\citet{1,2})}} in their papers published as vision research and science program which features as saccadic eye movement while viewing and recognising patterns and scan-path in eye movement in a pattern perception. 
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The researchers claimed that when a distinct visual pattern is viewed, certain sequence of eye movement is executed and these sequences are very important in accessing the visual memory for the patterns.
Though their theory is not authentic enough to form a reliable bases for their result on the postulate of natural images that the human gaze selects for informative details i.e. the fixation points tend to fall on the nearest significant details of visual stimulus like the presented image in the paper. 
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Also, this arise from the interest in scan-paths as a number of powerful techniques that has been developed which explores the statistical properties of eye movement patterns as related to important characteristics of some visual stimulus.
  Some research \textcolor{blue}{\textit{(\citet{3, 4})}} considers scan-path to be any eye-movement data collected by a gaze-tracking device, which can be archived and retrieved  as recorded information about the trajectories (paths) of eyes during scanning on the visual field by viewing and analysing the visual information. Some of this information is data that consists of gaze-direction, duration and fixation position for saccade duration. Other broader definition by some work views scan-path and fixation points as sequential data that can allow usage from different experiments and various applications in a single analytical framework that deals with the visual analysis of natural and system generated images and videos. This papers looks at some algorithms that can predict eye movement to determine the shortest and significant cognitive perception of users on dynamic webpage stimulus. To achieve this, the pupil response (dilation and constriction) is set as a gauge for the eye movement variations and tracking of the fixations made by the cognitive response of the pupil changes of the users. To understand the mechanics the proceeding section discusses on the pupil response of the eye movement.


\section{Literature Review}

Some research \textcolor{blue}{\textit{(\citet{3, 4})}} have found that the pupil responds to three different kinds of stimuli such as they constrict in response to brightness, constriction in response to near fixation of the pupil response and also dilates in response to increase in arousal, cognitive and mental workload which is triggered by an external stimulus in a spontaneous event. The three pupil responses can be described by how they are related to high-level cognition to the neutral pathways that controls them. The functional relevance of pupil responses is how the pupil responses lead to better view of visual stimulus and the world in general. The pupil responses serve many functions which are fully understood as one of the important functions to optimise vision for acuity and depth of field i.e. small pupils seeing sharply at a wider range of distances and also for sensitivity which is a large pupil that detects faint stimuli. The pupils change their sizes to further optimise vision for a certain situations. In general, the pupil responses are identical to eye movements that include the saccadic and smooth gaze pursuit. They have properties of reflexive and voluntary action that are part of active and visual exploration. 

\begin{figure*}{ht!}
\centering
\includegraphics[height = 10cm, width = 15cm]{1.png}
\caption{A two-dimentional Cartesian Space}
\label{1}
\end{figure*}


Different forms of algorithm [] have been used to computer the scan-path and shortest parth for eye movement and fixation predicts, one of the most basic one is the Euclidean distances between fixation points in a Cartesian plane (Figure \ref{1}), this is the length of the line segment between the points. It is computed from the fixation points using the Pythagoras theorem (Equation \ref{1}).

\begin{equation} D = \sqrt{X^2 + Y^2}    \label{1} \end{equation}

with $X$ and $Y$ as the two nearest points. Sometimes the Euclid distance is not represented as numbers but in degrees in the case of multi-dimensional plane. These distances between the gaze objects that are not termed as points is usually known to be the smallest or shortest distance among the two fixation points from the two point objects. Computational models are used to determine this short distances on the different types of dynamic contents that attracts the pupil responses. These techniques are applied in statistical and optimisation fields for visual optimisation and predictions. For a given one dimensional plane, the distance between the two fixation points are the real visual line is the absolute value of the numerical differences and relation of the numerical difference of the coordinates of the points on the visual field; i.e. if p and q are the two fixation points on the real line, then the computed distance between these points is given as: 

\begin{equation} d(p, q) = \sqrt{(p - q)^2}    \label{2} \end{equation}

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\includegraphics[height = 10cm, width = 15cm]{2.png}
\caption{A Multi-dimentional Cartesian Space, Courtesy: Google image}
\label{2}
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For a two-dimensional Euclidean plane (Figure \ref{1}), point $ p(p_1, p_2)$ and point $q(q_1, q_2)$. The distance is given as: 
\begin{equation} d(p, q) = \sqrt{(q_1 - p_1)^2 + (q_2 - p_2)^2}    \label{3} \end{equation}   


For multi-dimensional space (Figure \ref{2}) like a webpage visual field the different dynamic contents renders the field multi-dimensional due to multiple effects like the video and image contents on the visual field. The points given by the Cartesian coordinate points in an n-dimensional Euclidean space are given as the distance:
\begin{equation} d(p, q) = \sqrt{(q_1 - p_1)^2 + (q_2 - p_2)^2} + \dots +   \sqrt{(q_i - p_i)^2 + \dots + (q_n - p_n)^2} \label{4} \end{equation}   


\section{Method}

In usability testing for visual content optimisation, different types of techniques are used for the main purpose of facilitating the task and cognitive load and also to improve the visual aesthetics. The method applied here is to use the Dijkstra’s algorithm to determine the shortest distance between fixations points and be able to predict the significant cognitive response from eye movement behaviour made and webpage dynamic field on a visual field. The webpage contents different contents and is assumed to be a multi-dimensional Cartesian plane that contain multiple fixation points and scan-path. The cognitive response user can also be predicted from the user’s pupil changes based on dilation and constriction of the pupil sizes. The main objective is to determine the shortest possible and significant cognitive perception of the users on the webpage of the visual field. The visual field is the spatial display of visual sensation which is available to the observers in psychological experiments as the entire area which is seen by the eye inn a fixed straight line at a point. Experimental setup on visual field depends on the size and content of display.

\subsection{Task}

Sixty participants were recruited for the first stage of the experiment, they were given consent form to fill and a brief introduction the given task was briefly highlighted. The main goal is to generate data with eye movement behaviour of the participants recorded in an eye tracker (Tobii eye tracker) where they have to interact with the webpage content (Two business webpages), by simply surfing and browsing for object of interest that either involves a high cognitive load or a simple search. The process is the Dijkstra’s algorithm to predict the significant cognitive perception of the participants from the search and compute the shortest distance between the points. 

\subsection{Shortest Scan-path on Visual field}

To predict the shortest scan-path on visual field the problem is solved based on f(n, log n) time interval by using the recursive divide and conquer approach using the following steps.

\begin{enumerate}
\item Sort ponts on the $x$-Cartesian plane
\item Split the set of points to two different equally sized subsets using a vertical line $x = mid(x)$.
\item Compute the recursive problem in both side of subsets, with given minimum distance
	\item Determine the minimum distance $ min(DL)$ among pair of fixation points
	\item Determine the minimum distance $min(DL), min(DR) \text{and} min(DLR)$ of the left, right and among pair of points.
\end{enumerate}

To computer the shortest path between two fixations on a webpage visual field the data generated is set as a structured control for rudimentary optimisation. The eye movement on visual field is considered a plane with series of nodes connected by edges that can also be weighted edges with computed values and directional objects. The Dijkstra’s algorithm is used to update the shortest path between the current node and its neighbouring nodes (Fixation points). The direction is set to the node with the lowest distance after updating the distance of all its neighbours and the process is repeated with all the subsequent neighbours until the visual field is covered. The simple process adopted Equation \ref{3}

 \begin{align}
          \sin A \cos B &= \frac{1}{2}\left[ \sin(A-B)+\sin(A+B) \right] \\
          \sin A \sin B &= \frac{1}{2}\left[ \sin(A-B)-\cos(A+B) \right] \\
          \cos A \cos B &= \frac{1}{2}\left[ \cos(A-B)+\cos(A+B) \right] \\
          \end{align}

\begin{equation} \sqrt2x \text{ is not the same as } 2^{ax}\sqrt{2x}. \label{1} \end{equation}


\section{RESULT}

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\section{CONCLUSION}


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\section*{ACKNOWLEDGMENTS}
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