
\chapter{Discussion}
% here please philosophical possibly far-fetched arguments

\section{General thoughts}
It is assumed here, that (visual) attention and eye movements are tightly coupled. Most information comes during a fixation, not during a saccade (due to saccadic suppression). The central one or two degrees of the visual field (the fovea) provide the bulk of information, while the input from the larger periphery is less informative (investigated as covert attention, cf. Christian M�hl) and provide mainly information about movement. Hence, thelocations of fixations along the scanpath indicate what information from the visual scene was processed. 

Many experiments in developmental psychology show that people focus their attention differently on unfamiliar and familiar objects (e.g. toddlers tend to focus their attention on unfamiliar objects rather than familiar objects[citation needed]). Other people [Goldbach? see Stentiford] theorize 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 differences to humans influence attention. These factors make comparison of monkey fixations and human fixations difficult, because differences in top-down perception cannot be excluded. 

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 subjects 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). 

Most German and European subjects were unfamiliar with the stimuli, some noted they lacked interest in the birds shown. Evident was that European subjects 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. Some people thought birds were boring, reported �too many birds�. 
The general feedback was that the pictures are interesting. Europeans tended to look more at fur than at pawns and teeth of animals. Evidence of character differences in scanpaths? � 
Interesting focus: differences of background/foreground fixations 
(hypothesis: women look more at background than men, and Easterners more than Westerners
\cite{Nisbett}. Men seem also to repeat their fixations more frequently (backtracking strategy) but I didn�t have many women. What we could subjectively find were huge differences in speed of saccades, duration of fixations, and oscillation rates. 

These last points make one think that, given such differences between human observers, a comparison of just three monkeys has to be considered hopelessly skewed. However, the differences we focused on were more on a bottom-up level than the differences that had just been remarked on. 

Subjectively, Yarbus' \citeyear{Yarbus} finding that inter-subject consistency of fixations decreased in time when watching a stimulus was seen confirmed. 
First fixations are often very similar. 
This holds especially for fixations on pictures of humans, 
where fixations often showed an oscillation that started form eyes and breasts 
(surprisingly this behavior also extends to pictures of men) and came to incorporate with longer exposure more locations of the body, such as eyes, lips, breasts, legs. 

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 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. 

Comparison of monkeys with humans could facilitate the understanding of perception (or more general, cognition) of our own species and the evolution thereof. 

\cite{BoothRolls} concluded that monkeys should be capable of comprehending natural scenes as such in photographs. Einh�user et al. (2006) speculated however 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. Similarly, it could also be argued that different settings activate particular strategies or cognitive frames (Minsky, 1974), and thereby trigger bottom-up processes particular to a setting. We could conjecture 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. 
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, at least in part c. 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. Traces indicated different task interpretations in monkeys and humans for many scenes, also affecting saliency attribution to image properties. 


Correlations are difficult to compare for data sizes so different in size as for monkeys and humans 


\section{Differences found}
Differences in eye movement properties could be due to the definitions that differ between Osnabr\"uck and Montana. However, we think that fixations and saccades are accurately defined. We checked several definitions, looked at the match on the images, and confirmed that the definition in \cite{Lierhaus} best described eye movement events in monkeys. Further \cite{Tichacek} argued convincingly that EyeLink's definitions of fixations and saccades give a good match. Therefore, we think that our definitions are justified. 

We stated that monkeys make slightly faster, longer, longer lasting saccades than humans and fixate shorter than humans. 

Difference in eccentricity in pdfs between monkeys and humans due to missing/existant fixation crosses before trials. Same accounts for initial differences between humans and monkeys.  

Surprising 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. \cite{Acik} showed there are big differences between image category (cf. fig. \ref{imagecats}). 

\begin{figure}[htbp]
\label{imagecats}
\includegraphics[angle=90,width=\textwidth]{lumGlumCtrG81.eps}
\caption{Influences of features for different image categories}
This figure was provided by Christian Honey \citeyear{Honey}. Please note that, different from our plots, this figure gives controls along the ordinate and actuals along the abscissa. The bars denoting effect sizes in the last graph match the scatter plots. 
\end{figure}

For humans in LC and TC, characteristic distribution on the scatterplot. However monkeys -- don't they recognize a scene? To what extend are the experimental settings ecologically valid for monkeys?

 
More refined definition of controls were needed for nextfixation. 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. 

 
\section{Divergence}


Euclidean distance for eye movement divergence and feature effect divergence emphasizes great differences between features. 

inter-subject Euclidean distance less dependent on number of fixations on image than spread measures. 

For 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>1). \cite{Tatler2005} dedicated a very readable discussion to explanative frameworks as to why divergence increases over time between participants' fixation points. While the question to 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-subject 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. 

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. 
 
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 Mahalanobis distance, the KL-divergence, and the Jensen-Shannon divergence. 

% divergence: spread out euclidean based distance: 
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. So the measure could result in high values even if the fixations are subjectively highly similar only concentrated on two or more different regions. 

%divergence:
From this perspective an optimal measure would give information about how many regions of high interest there are in the image and how close are the points in that region to each other or to a common center in that region, not dissimilar to a clustering algorithm (e.g. k-means or self-organizing maps). Following this intuition, if we draw circles of radius r around each point, filled with ones, and look at an overlap, we would get perfect convergence if all circles are the same (in other words, if the average or minimum of all points of the circles>0 is equal to the number of fixation points). Complete divergence is, if there is no overlap (i.e. if the number of points>1 is equal to zero). If we imagine a histogram based on the values of points>0, we would expect an overwhelming number of ones, maybe some counts for two, and less for other values until the number of fixation points. If we want to derive a measure of convergence from these histograms we need to introduce some weight for overlaps of fixation points. Summing up all overlapping points would correspond to a weight of k for all bin counts of an overlap of k fixation points. Given a reasonable radius, this measure is more refined than inter-participant Euclidean distances in weighting inter-cluster distances and within-cluster coherence. 

\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}


\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. % Is this better for discussion than introduction?

data set of images more semantic content than "pure" natural scenes (e.g. faces, actions, man-made objects as bicycles etc., humans, monkeys, other animals)
 
 %features and eye movements
 While we expected that eye movements correlate positively with features contrasts, what we found is that the correlation is only marginal and sometimes negative. 