Insect-Inspired Self-Motion Estimation with Dense Flow Fields--An Adaptive Matched Filter Approach

Abstract

The control of self-motion is a basic, but complex task for both technical and biological systems. Various algorithms have been proposed that allow the estimation of self-motion from the optic flow on the eyes. We show that two apparently very different approaches to solve this task, one technically and one biologically inspired, can be transformed into each other under certain conditions. One estimator of self-motion is based on a matched filter approach; it has been developed to describe the function of motion sensitive cells in the fly brain. The other estimator, the Koenderink and van Doorn (KvD) algorithm, was derived analytically with a technical background. If the distances to the objects in the environment can be assumed to be known, the two estimators are linear and equivalent, but are expressed in different mathematical forms. However, for most situations it is unrealistic to assume that the distances are known. Therefore, the depth structure of the environment needs to be determined in parallel to the self-motion parameters and leads to a non-linear problem. It is shown that the standard least mean square approach that is used by the KvD algorithm leads to a biased estimator. We derive a modification of this algorithm in order to remove the bias and demonstrate its improved performance by means of numerical simulations. For self-motion estimation it is beneficial to have a spherical visual field, similar to many flying insects. We show that in this case the representation of the depth structure of the environment derived from the optic flow can be simplified. Based on this result, we develop an adaptive matched filter approach for systems with a nearly spherical visual field. Then only eight parameters about the environment have to be memorized and updated during self-motion.

Fig 1. The averaged angle error, arccos(θ→Nest⋅θ→∣θ→Nest∣⋅∣θ→∣), between the estimation θ→Nest, which depends on the number of flow vectors N, and the true values θ→, is shown for translation (solid lines) and rotation (dashed lines).

Red curves show the errors for original KvD algorithm, green curves for the modified KvD algorithm. The errors are averaged over 40 trials (see methods 4.3). For each trial the true self-motion parameters are chosen randomly, equally distributed over the sphere, in such a way, that the resulting translational flow equals the resulting rotational flow in magnitude. The distances, also determined randomly, lie with equal probability between one and three in arbitrary units. The results for three different variances ${(△\overrightarrow{p})}^2$ are shown (from bottom to top): 1, 3, and 9 times the flow vector length, where the factor is interpreted differently in the two graphs. A) Results for non-equally distributed flow vectors with equal variance of the flow vector errors $△{\overrightarrow{p}}_i$ (the variance is matched to the mean flow vector). B) Results for equally distributed flow vectors, where the variance of the errors $△{\overrightarrow{p}}_i$ depends linearly on the length of the ${\overrightarrow{p}}_i$ (the variance is matched to the local flow vector).

Fig 2. An agent flies inside the sphere on a circular trajectory (see Fig 3A).

The center of the circle does not coincide with the center of the sphere to avoid symmetries in the depth model. The correct depth model is constant in this configuration, only the initialization of the depth model is tested. The y-axis shows the angle error $\text{arccos}\left(\frac{{\overrightarrow{\theta }}^{\mathrm{\text{est}}}\cdot \overrightarrow{\theta }}{\mid {\overrightarrow{\theta }}^{\mathrm{\text{est}}}\mid \cdot \mid \overrightarrow{\theta }\mid }\right)$, between the estimated self-motion axis of ${\overrightarrow{\theta }}^{\mathrm{\text{est}}}$ and the axis of the true self-motion values $\overrightarrow{\theta }$. The depth model is initialized with constant distances. With every update step of the depth model the error decreases exponentially for the translation (blue) as well as for the rotation (red).

Fig 3

A) shows a circular trajectory to analyze the initialization phase of the adaptive MFA. The height of the trajectory lies above the middle point of the sphere to avoid trivial depth models. Due to symmetry the depth model for this configuration is the same at every trajectory point. B) shows a sinusoidal trajectory. It is used to analyze the self-motion estimation error during adaptation. Again the height of the trajectory is lifted up against the middle point of the sphere to make the depth model more complex in relation to an agent, which flies along the trajectory.

Fig 4. Spherical harmonic functions from the expansion of the inverse distances μ _i.

A) The sum of the zeroth order function and a first-order dipole-function. B) The sum of the zeroth order function and a second-order quadrupole-function.

Fig 5. As in Fig 2 an agent flies inside a sphere (see methods 4.4).

The trajectory is a sinus curve with two full periods. The amplitude of the sinus curve is 0.5 of the radius of the sphere. The sinus curve is lifted in the sphere by 0.3 units in z-direction to avoid symmetries in the depth model. The agent performs 600 steps which result in rotation angles of up to 8 degrees per frame. The initialization phase is not shown. The left figures shows the error of the translation estimates and the right figures shows the error of the rotation estimates. The y-axes show the angle error $\text{arccos}\left(\frac{{\overrightarrow{\theta }}^{\mathrm{\text{est}}}\cdot \overrightarrow{\theta }}{\mid {\overrightarrow{\theta }}^{\mathrm{\text{est}}}\mid \cdot \mid \overrightarrow{\theta }\mid }\right)$, between the estimated self-motion axis of ${\overrightarrow{\theta }}^{\mathrm{\text{est}}}$ and the axis of the true self-motion values $\overrightarrow{\theta }$. The estimated angle errors have a pole, when the rotation gets zero at the inflection points of the sinusoidal curve (see methods 4.4). The small region around the poles are cut out in the figures. A, B) The two figures show the error of the adaptive MFA (red curve) and the original MFA (blue curve) with a constant inverse distant assumption for the original MFA. The adaptive MFA is updated every time step. The right y-axes of the figures show the averaged ratio between the rotational and translational optic flow. C, D) shows the adaptive MFA and the original MFA as in figures A and B, but with an error of 10% added to the optical flow. E, F) show different update frequencies of the depth model. All models rotate with the agent. The update frequencies are: black = 1 frame, green = 5 frames, blue = 10 frames and violet = 20 frames.