Fractal dimension and the navigational information provided by natural scenes

Recent work on virtual reality navigation in humans has suggested that navigational success is inversely correlated with the fractal dimension (FD) of artificial scenes. Here we investigate the generality of this claim by analysing the relationship between the fractal dimension of natural insect navigation environments and a quantitative measure of the navigational information content of natural scenes. We show that the fractal dimension of natural scenes is in general inversely proportional to the information they provide to navigating agents on heading direction as measured by the rotational image difference function (rotIDF). The rotIDF determines the precision and accuracy with which the orientation of a reference image can be recovered or maintained and the range over which a gradient descent in image differences will find the minimum of the rotIDF, that is the reference orientation. However, scenes with similar fractal dimension can differ significantly in the depth of the rotIDF, because FD does not discriminate between the orientations of edges, while the rotIDF is mainly affected by edge orientation parallel to the axis of rotation. We present a new equation for the rotIDF relating navigational information to quantifiable image properties such as contrast to show (1) that for any given scene the maximum value of the rotIDF (its depth) is proportional to pixel variance and (2) that FD is inversely proportional to pixel variance. This contrast dependence, together with scene differences in orientation statistics, explains why there is no strict relationship between FD and navigational information. Our experimental data and their numerical analysis corroborate these results.