From Permutations to Horizontal Visibility Patterns of Periodic Series

Periodic series of period T can be mapped into the set of permutations of [T−1]={1,2,3,…,T−1}. These permutations of period T can be classified according to the relative ordering of their elements by the horizontal visibility map. We prove that the number of horizontal visibility classes for each period T coincides with the number of triangulations of the polygon of T+1 vertices that, as is well known, is the Catalan number CT−1. We also study the robustness against Gaussian noise of the permutation patterns for each period and show that there are periodic permutations that better resist the increase of the variance of the noise.

Human Visual Search Follows Suboptimal Bayesian Strategy Revealed by a Spatiotemporal Computational Model

Humans perform sequences of eye movements to search for a target in complex environment, but the efficiency of human search strategy is still controversial. Previous studies showed that humans can optimally integrate information across fixations and determine the next fixation location. However, their models ignored the temporal control of eye movement, ignored the limited human memory capacity, and the model prediction did not agree with details of human eye movement metrics well. Here, we measured the temporal course of human visibility map and recorded the eye movements of human subjects performing a visual search task. We further built a continuous-time eye movement model which considered saccadic inaccuracy, saccadic bias, and memory constraints in the visual system. This model agreed with many spatial and temporal properties of human eye movements, and showed several similar statistical dependencies between successive eye movements. In addition, our model also predicted that the human saccade decision is shaped by a memory capacity of around 8 recent fixations. These results suggest that human visual search strategy is not strictly optimal in the sense of fully utilizing the visibility map, but instead tries to balance between search performance and the costs to perform the task.Author SummaryDuring visual search, how do humans determine when and where to make eye movement is an important unsolved issue. Previous studies suggested that human can optimally use the visibility map to determine fixation locations, but we found that such model didn’t agree with details of human eye movement metrics because it ignored several realistic biological limitations of human brain functions, and couldn’t explain the temporal control of eye movements. Instead, we showed that considering the temporal course of visual processing and several constrains of the visual system could greatly improve the prediction on the spatiotemporal properties of human eye movement while only slightly affected the search performance in terms of median fixation numbers. Therefore, humans may not use the visibility map in a strictly optimal sense, but tried to balance between search performance and the costs to perform the task.

Computing the Global Visibility Map Using Slice Geometry for Setup Planning

This paper introduces a new method that uses slice geometry to compute the global visibility map (GVM). Global visibility mapping is a fundamentally important process that extracts geometric information about an object, which can be used to solve hard problems, for example, setup and process planning in computer numerical control (CNC) machining. In this work, we present a method for creating the GVM from slice data of polyhedron models, and then show how it can help determine around which axis of rotation a part can be machined. There have been various methods of calculating the GVM to date, tracing back to the well-known seminal methods that use Gaussian mapping. Compared to the considerable amount of work in this field, the proposed method has an advantage of starting from feature-free models like stereolithography (STL) files and has adjustable resolution. Moreover, since it is built upon slicing the model, the method is embarrassingly parallelizable in nature, thus suitable for high-performance computing. Using the GVM obtained by this method, we generate an axis of rotation map to facilitate the setup planning for four-axis CNC milling machines as one implementation example.