Efficient Algorithms for Rotation Averaging Problems

The rotation averaging problem is a fundamental task in computer vision applications. It is generally very difficult to solve due to the nonconvex rotation constraints. While a sufficient optimality condition is available in the literature, there is a lack of a fast convergent algorithm to achieve stationary points. In this paper, by exploring the problem structure, we first propose a block coordinate descent (BCD)-based rotation averaging algorithm with guaranteed convergence to stationary points. Afterwards, we further propose an alternative rotation averaging algorithm by applying successive upper-bound minimization (SUM) method. The SUM-based rotation averaging algorithm can be implemented in parallel and thus is more suitable for addressing large-scale rotation averaging problems. Numerical examples verify that the proposed rotation averaging algorithms have superior convergence performance as compared to the state-of-the-art algorithm. Moreover, by checking the sufficient optimality condition, we find from extensive numerical experiments that the proposed two algorithms can achieve globally optimal solutions.

Examining Ethnic Exposure through the Perspective of the Neighborhood Effect Averaging Problem: A Case Study of Xining, China

An increasing number of studies have observed that ignoring individual exposures to non-residential environments in people’s daily life may result in misleading findings in research on environmental exposure. This issue was recognized as the neighborhood effect averaging problem (NEAP). This study examines ethnic segregation and exposure through the perspective of NEAP. Focusing on Xining, China, it compares the Hui ethnic minorities and the Han majorities. Using 2010 census data and activity diary data collected in 2013, the study found that NEAP exists when examining ethnic exposure. Respondents who live in highly mixed neighborhoods (with high exposures to the other ethnic group) experience lower activity-space exposures because they tend to conduct their daily activities in ethnically less mixed areas outside their home neighborhoods (which are more segregated). By contrast, respondents who live in highly segregated neighborhoods (with low exposures to the other ethnic group) tend to have higher exposures in their activity locations outside their home neighborhoods (which are less segregated). Therefore, taking into account individuals’ daily activities in non-residential contexts in the assessment of environmental exposure will likely lead to an overall tendency towards the mean exposure. Using Tobit models, we further found that specific types of activity places, especially workplaces and parks, contribute to NEAP. Ignoring individual exposures in people’s activity places will most likely result in misleading findings in the measurement of environmental exposure, including ethnic exposure.

Who Could Not Avoid Exposure to High Levels of Residence-Based Pollution by Daily Mobility? Evidence of Air Pollution Exposure from the Perspective of the Neighborhood Effect Averaging Problem (NEAP)

It has been widely acknowledged that air pollution has a considerable adverse impact on people’s health. Disadvantaged groups such as low-income people are often found to experience greater negative effects of environmental pollution. Thus, improving the accuracy of air pollution exposure assessment might be essential to policy-making. Recently, the neighborhood effect averaging problem (NEAP) has been identified as a specific form of possible bias when assessing individual exposure to air pollution and its health impacts. In this paper, we assessed the real-time air pollution exposure and residential-based exposure of 106 participants in a high-pollution community in Beijing, China. The study found that: (1) there are significant differences between the two assessments; (2) most participants experienced the NEAP and could lower their exposure by their daily mobility; (3) three vulnerable groups with low daily mobility and could not avoid the high pollution in their residential neighborhoods were identified as exceptions to this: low-income people who have low levels of daily mobility and limited travel outside their residential neighborhoods, blue-collar workers who spend long hours at low-end workplaces, and elderly people who face many household constraints. Public policies thus need to focus on the hidden environmental injustice revealed by the NEAP in order to improve the well-being of these environmentally vulnerable groups.

Dividing Splittable Goods Evenly and With Limited Fragmentation

A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares from at most F pieces. We call F the fragmentation and mainly restrict attention to the cases $$F=1$$F=1 and $$F=2$$F=2. For $$F=1$$F=1, the max–min and min–max problems are solvable in linear time. The case $$F=2$$F=2 has neat formulations and structural characterizations in terms of weighted graphs. First we focus on perfectly balanced solutions. While the problem is strongly NP-hard in general, it can be solved in linear time if $$m\ge n-1$$m≥n-1, and a solution always exists in this case, in contrast to $$F=1$$F=1. Moreover, the problem is fixed-parameter tractable in the parameter $$2m-n$$2m-n. (Note that this parameter measures the number of agents above the trivial threshold $$m=n/2$$m=n/2.) The structural results suggest another related problem where unsplittable items shall be assigned to subsets so as to balance the average sizes (rather than the total sizes) in these subsets. We give an approximation-preserving reduction from our original splitting problem with fragmentation $$F=2$$F=2 to this averaging problem, and some approximation results in cases when m is close to either n or n / 2.

Beyond Commuting: Ignoring Individuals’ Activity-Travel Patterns May Lead to Inaccurate Assessments of Their Exposure to Traffic Congestion

This research examines whether individual exposures to traffic congestion are significantly different between assessments obtained with and without considering individuals’ activity-travel patterns in addition to commuting trips. We used crowdsourced real-time traffic congestion data and the activity-travel data of 250 individuals in Los Angeles to compare these two assessments of individual exposures to traffic congestion. The results revealed that individual exposures to traffic congestion are significantly underestimated when their activity-travel patterns are ignored, which has been postulated as a manifestation of the uncertain geographic context problem (UGCoP). The results also highlighted that the probability distribution function of exposures is heavily skewed but tends to converge to its average when individuals’ activity-travel patterns are considered when compared to one obtained when those patterns are not considered, which indicates the existence of the neighborhood effect averaging problem (NEAP). Lastly, space-time visualizations of individual exposures illustrated that people’s exposures to traffic congestion vary significantly even if they live at the same residential location due to their idiosyncratic activity-travel patterns. The results corroborate the claims in previous studies that using data aggregated over areas (e.g., census tracts) or focusing only on commuting trips (and thus ignoring individuals’ activity-travel patterns) may lead to erroneous assessments of individual exposures to traffic congestion or other environmental influences.

The Neighborhood Effect Averaging Problem (NEAP): An Elusive Confounder of the Neighborhood Effect

Ignoring people’s daily mobility and exposures to nonresidential contexts may lead to erroneous results in epidemiological studies of people’s exposures to and the health impact of environmental factors. This paper identifies and describes a phenomenon called neighborhood effect averaging, which may significantly confound the neighborhood effect as a result of such neglect when examining the health impact of mobility-dependent exposures (e.g., air pollution). Several recent studies that provide strong evidence for the neighborhood effect averaging problem (NEAP) are discussed. The paper concludes that, due to the observed attenuation of the neighborhood effect associated with people’s daily mobility, increasing the mobility of those who live in disadvantaged neighborhoods may be helpful for improving their health outcomes.