
\chapter{Discussion}
% here please philosophical possibly far-fetched arguments
After some wanna-be philosophical inspirations and some thoughts on the computation of divergence, we discuss the results of the study. 

\section{Some general thoughts}
The differences we focused on were more on a bottom-up level than the differences that had just been remarked on. However, there are some other lines of thought that could be worth investigation, which the author wants to shortly explore. 

Many experiments in developmental psychology show that people focus their attention differently on unfamiliar and familiar objects. It is also theorized (e.g. \cite{Stentiford}) that attention is optimized by evolution to reduce ambiguity in the environment, warn of danger, and signal availability of food. If that is true it has to be asked how behaviorally relevant stimuli are to monkeys (cf. also event-coding theory). Rhesus macaques are, like humans, diurnal animals, but mostly herbivorous (feed e.g. on leaves, roots, pine needles and sometimes insects and smaller animals), both arboreal and terrestrial. It can be speculated how cognitive differences to humans influence attention. 

As it is expected that there is a strong correlation of interest and familiarity with the scene and order of fixations, some words on interest and familiarity of participants with the stimuli. As for the rhesus macaques in question, it can be safely assumed that the scenes of wild animals (leopards, tigers, zebras, deer, African elephants, birds) and humans engaging in sporting activities as presented with the pictures are largely unfamiliar, because their natural environment (where they grew up) is southeast Asia. We consider human faces and faces of other primate species also less familiar (than to humans). 

In our study we had mostly German and other European participants, but also several participants from India and one from Pakistan. Most European participants were unfamiliar with the stimuli, some noted they lacked interest in the birds shown. Evident was that European participants looked differently at wild animals, especially predators. Many pictures showed eyes very prominently in the center and attracted many fixations there. 

Several cultural differences with some animals (elephants), familiarity with animals could be observed. Most commented, birds were boring and reported �too many birds� (Tang's image set). 
The general feedback was that the pictures are \textit{interesting}. Europeans tended to look more at fur than at pawns and teeth of animals. \cite{Nisbett} discussed differences of background/foreground focus between Westerners (Americans and Europeans) and Easterners (Chinese and Japanese), concluding that Easterners look, as a rule of thumb, more than Westerners at the background of scenes. 

\cite{BoothRolls} have established that monkey's inferotemporal cortex (IT) neurons can become turned to objects in the environment. They concluded that monkeys should be capable of comprehending natural scenes as such in photographs as they could not find differences in recordings of object-sensitive neurons between presentation of real objects and images of them. 

It is open to speculation, whether the presumably cognitively advanced perceptual system in humans allows them for more high-level control of perception, respectively, whether the supposed lack of higher developed systems would lead to comparatively more influence of bottom-up features in monkeys. Differences in visual processing between humans and monkeys could, in the long run, shed light on the influence of top-down mechanisms in perception and on the evolution of cognition. 

Similarly, it could also be argued that different settings activate particular strategies or cognitive frames (e.g. \cite{Minsky}), and thereby trigger bottom-up processes particular to a setting. The author conjectures that by way of epigenetic and evolutionary tuning, bottom-up mechanisms have been geared to requirements of vital tasks we dependent on and this way salient features have come to reflect characteristics of items of our daily life. 

Einh�user et al. (2006) argued that the extend to which a scene is perceived as being natural could differ over the species, facilitating the use of higher-order processes by different degrees with obvious consequences to the importance of image properties. Along this line, the author wants to point out that the set of images used had more semantic content than \textit{pure} natural scenes, as it contained e.g. faces, humans engaging in complex behaviors, man-made objects as bicycles etc., humans, monkeys, and other animals. 

Peters et al. (2005) avoided images such as close-up faces or well-known locations, because they considered them highly influenced by specialized top-down processes. In fact, Parkhurst and Niebur (2003; see also Schumann 2006;Honey, 2006) showed that saliency of features differs across different types of natural settings. Einh�user et al. (2006) argued further that monkeys could lack knowledge on relations of feature properties in natural settings, which could indeed account for some of the differences, given that we couldn't make out any animal from the natural environment of Rhesus Macaque (macaca mulatta) in the experimental image set but - on the other side - many scenes familiar to humans. Eye traces indicated different task interpretations in monkeys and humans for some scenes (e.g. monkeys looked at trees in a forest scene, while monkeys looked at the bicycle driver), also affecting saliency attribution to image properties. 

 
\section{Divergence}

In studying convergence, we want to compare n fixations (fixations number f) of n participants on the same image and say how different on how similar they are based on their location (always given $n \geq 2$). \cite{Tatler2005} dedicated a very readable discussion to explanative frameworks as to why divergence increases over time between participants' fixation points. 

While the question addressing the why is interesting, it is beyond the scope of the section here to give it due respect. It may be worth however to look at differences between several measurements of divergence. While Tatler used the Kullback-Leibler (KL) divergence, \cite{Einhaeuser2006} calculated inter-participant consistency between participant p1 and participant p2 on image i as the Euclidean distance between n-ths fixations on i of p1 and p2.\footnote{More in-depth discussions of similarity measures can be found in \cite{Comaniciu,Puzicha}. }{} 

In two-dimensional Euclidean space $\mathbb{R}^n$, the shortest distance between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ can be calculated by the Pythagorean theorem: \begin{displaymath}d=\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}\end{displaymath} 

The KL-divergence, for distributions $P$ and $Q$ of a continuous random variable, is defined as \begin{displaymath}D_{KL}(P\|Q) = \int_{-\infty}^{\infty} p(x) \log \frac{p(x)}{q(x)} dx\end{displaymath}, where $p$ and $q$ denote the densities of $P$ and $Q$.\cite{WikipediaKL}

Metrics are nonnegative functions $g(x,y)$ that describe the distance between points. A metric satisfies several conditions 
\begin{itemize}
  \item [a)] $g(x,y) + g(y,z)\geq g(x,z)$ (triangle inequality)
  \item [b)] $g(x,y) = g(y,x)$ (symmetry)
  \item [c)] $g(x,x) = 0$ 
  \item [d)] $g(x,y) = 0 \Rightarrow x=y$
\end{itemize}
Without d), $g(x,y)$ is called a pseudometric. \cite{WeissteinMetric}

Examples of common metrics are the city block metric \begin{displaymath}s=\sum_{i=1}^{n} |x_{i} - y_{i}|\end{displaymath}, the p-norm distance \begin{displaymath}\left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}\end{displaymath}, or the infinity norm distance \begin{displaymath}\lim_{p \to \infty} \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p} = \max \left(|x_1 - y_1|,  |x_2 - y_2|,  \ldots, |x_n - y_n| \right).\end{displaymath}

While Euclidean distances are true distance measures, in the sense of metrics, this cannot be said of the KL-divergence, which though always non-negative and equals zero only if $p_{k} = q_{k}$, is not symmetric and does not fulfill the triangle inequality. The Jensen-Shannon divergence is a symmetrized and smoothed version of the KL-divergence. It is defined by \begin{displaymath}\frac{1}{2}\Big(D_{KL}(P\| \frac{P+Q}{2}) + D_{KL}(Q\| \frac{P+Q}{2})\Big).\cite{Fuglede} \end{displaymath}

When looking at the formula of the KL-divergence the question of why not adding up the matrices and deriving a measure thereof should be addressed. It is a simple fact that divergence measures that are based on a sum of the probability functions of fixated points in an image are skewed, because --- given we take Gaussians --- the Gaussian is cut-off at the edge. This results in artifacts of eccentricity. 
 
We devised one Euclidean based measure that represents the medians of Euclidean distances to the center of gravity of all fixation points. This measures to what extends are spread out from a common center. It has the trade-off that if there are several conspicuous regions in the image, the distance depends on how distant they are. We also tried the inter-participant Euclidean distance measure, the KL-divergence, and the Jensen-Shannon divergence. We also thought of one heuristics that accounts for the number of points that have no overlap in Gaussian convolved fixation maps. 

Another metric, the \textit{Mahalanobis distance}, is based on correlations between variables and is useful for determining similarity of an unknown sample set to a known one. The Mahalanobis distance from a group of values with mean $\mu = ( \mu_1, \mu_2, \mu_3, \dots , \mu_p )$ and covariance matrix $\Sigma$ is defined as: \begin{displaymath} D_M(x) = \sqrt{(x - \mu)^T \Sigma^{-1} (x-\mu)}.\end{displaymath} \cite{WikipediaMahalanobis} 

The memory and time requirements, however, are much more expensive than for, say, the Euclidean distance and we did not get satisfactory indicators of divergence. 

\begin{figure}[!htb]
\label{divergencemeasures}
\centering
\fbox{ \begin{minipage}{\textwidth}
\begin{tabular}{ccc}
\input{euclidspread.tex} & \input{jsdiv.tex} & \input{kldiv.tex} \\
\input{intersubeucliddist.tex} & \input{nonoverlap.tex} & { } \\
\end{tabular}
\caption{Comparison of different divergence/convergence measures}
\end{minipage} }
\end{figure}

We found that Inter-participant Euclidean distances reflected best the picture of convergence/divergence and therefore took this measure for the result part. 

\section{Differences found}
We found that humans tend to look rather up, while monkeys' fixations go rather downwards. Monkeys make slightly faster, longer, longer lasting saccades than humans and fixate shorter than humans. In humans, there is a trend that with shorter timed, more narrow, and slower saccades fixations become longer. With monkeys the opposite is true: longer timed, wider, faster saccades are followed by longer saccades. 

In humans, fixations become shorter and saccades speed up with increasing eccentricity. In monkeys, saccade duration and saccade length increase with eccentricity. In humans, saccade length increases only on the ordinate with eccentricity. Monkeys fixate more with increasing ordinate eccentricity and less so with increasing abscissa eccentricity. In humans, saccade speed increases over time, saccade durations decrease, and saccade lengths first increase until around the 8th fixation and then decrease. 

 While we expected that eye movements correlate positively with features contrasts, what we found is that the correlation is only marginal and sometimes negative. This could lead to questions concerning our new control definitions. More refined definition of controls were needed for analysis of saccade targets. We argued that our definition of controls makes more sense than the definition of controls in \cite{Tatler2006}, since eccentricity is by far the strongest determinant of feature contrast and taking fixations with similar saccade length is automatically skewed because of the low correlation of saccade length and eccentricity. Taking controls as average feature values over all trials on fixations location is unconventional. The effect sizes differ considerably and not in the same proportion for monkeys and humans from the usual definition. We think that the effect sizes with the new definition show therefore not be compared to differently calculated effect sizes. 

We were again surprised at the higher influence of feature contrasts in humans. From \cite{Lierhaus}, we expected higher influences of features in monkeys. However, we found arguably stronger influence of luminance and texture contrasts in humans for experiments 1 and 2. For experiments 3 and 4, the picture is inconclusive. Experiments 3 and 4 used the same stimuli, so differences in influences of features could stem from memory effects. Given that monkeys understand the pictures in both Montana and Bochum, some of the differences in feature effects could be explained by the different sets of images in experiments 1,2 and 3,4. Image category was studies by \cite{Acik}, who showed large differences in feature effects. 

Conditional probabilities and joint feature probabilities, as per \cite{Schumann}, evidenced an increased preference in monkeys for low extrema and a decreased one for high extrema in normalized feature space in the case of luminance and texture as compared to humans. This observation does not hold for barness, high values of which are slightly more salient to monkeys than to humans, while monkeys looked relatively less at the middle spectrum. 

Some differences in eccentricity in pdfs between monkeys and humans are due to missing or existent fixation crosses before trials. The same could account for initial differences between humans and monkeys in other dimensions. Mainly distinct image sets for the three monkeys render a comparison even more difficult. 

In a bootstrap comparison between monkeys and humans fixation maps we showed that monkey and humans look at different locations, though the degree of this difference was pronounced more for one monkey and less for the other two. 

At last we investigated inter-participant differences over time (divergence/convergence). While monkeys stay on the same level of differences, we found changes over time in humans. First we observed that while human fixations converge over time according to their locations. Then, by measure of their eye movements, we saw that initial high human differences decrease over time. For effect sizes, we found that humans diverged over time. As for eye movements monkeys were more consistent, while our other 
analyses found the opposite. 
 %features and eye movements