Uncertainty of Object Points Monoplotted from Terrestrial Images

TL;DR

This paper introduces methods to estimate the uncertainty of 3D points reconstructed from single images, which is important for analyzing historical glacier changes.

Contribution

The paper proposes three methods to estimate monoplotting uncertainty and evaluates their accuracy and speed for different use cases.

Findings

The unscented transform provides more accurate uncertainty estimates than variance propagation for manually selected points.

Monte Carlo simulation is recommended for accuracy despite slower performance.

Variance propagation is suitable for fast uncertainty estimation across entire images.

Abstract

With monoplotting, object points can be reconstructed from a single oriented image if a reference surface of the captured scene is available. While used extensively in environmental sciences, prior approaches fall short of describing the uncertainty of the reconstructed points.

In this paper, we estimate this monoplotting uncertainty using three different methods: i) Monte Carlo simulation, ii) unscented transform and iii) classical variance propagation with tangential approximation of the terrain. Our investigations are guided by two different use cases: i) For manually selected image points, the estimated uncertainty determines whether these monoplotted points are accurate enough for a subsequent research question (e.g. deriving glacier changes from historical terrestrial images). ii) Estimating the monoplotting uncertainty for each pixel of the whole image to get an overview of the expectable uncertainty, which will already be beneficial during the image orientation step. While for the first use case, the precision of the estimated uncertainty is crucial, the second use case requires a fast method. Furthermore, in both use cases, silhouettes must be considered because the estimates in their vicinity will not be valid. Therefore, we further investigate the derivation of silhouette masks, optimally exploiting the available information from the three different methods.

For evaluation, we use a selected historical terrestrial image showing a glacier in the Alps around 1900, where, for the first use case, we manually digitised individual vertices of a glacier outline. Using the Monte Carlo estimates based on 1000 samples as reference, the results from the unscented transform are closer to those (14.1% RMS) than the ones from variance propagation (24.7% RMS). Despite this good result from the unscented transform, our recommendation for this use case is nevertheless the Monte Carlo simulation, thanks to the speed of existing ray-casting routines.

However, for the second use case, where the monoplotting uncertainty is predicted for each pixel of the entire image to get a quick overview, the enormous amount of millions of ray-castings prohibits both Monte Carlo simulation and unscented transform. Here, we propose to use variance propagation because of its speed and still reasonable precision, yielding uncertainty estimates with an RMS of 7.8% in areas away from silhouettes.

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