Ймовірність виконання візуальної задачі як сигмоїдна функція

Abstract

The execution of the visual task of data acquisition by a multi-channel target sightseeing system of a sample of armored vehicle in  the interests of target awareness situation, target tracking and destruction of enemy armament and military facilities on maximal distances (within the limits of action of basic regular armament of samples of the armored vehicles) is considered. The criterion for the effectiveness of execution of the task of detection / recognition / identification of the target is the probability of execution of the corresponding visual task. The analysis of modern approaches to the assessment of the effectiveness of the task of data acquisition on the target, based on the Johnson and TTP (Targeting Task Performance) models is performed.
The empirical parameter introduced by Johnson by its physical content is the size of the image of the target, which provides a fifty-percent-probability of execution of the corresponding visual task. Obviously, by itself a certain size of the image cannot guarantee a  desired probability of solving the corresponding visual problem at low contrast values (visibility of the target). Conversely, even the very high contrast of the target is not sufficient to successfully execute the visual problem: the size of the target must be sufficient to be capable to detect / recognize / identify a target. Thus, when using Johnson’s approach, one inevitably faces a contradiction.
Namely: on the one hand, Johnson’s approach assumes that the size of the image of the target, described by the number N50task, corresponds to the fifty-percent-probability of performing the corresponding visual task, while on the other hand, it is clear that at low visibility even a fairly large size of the image does not provide even a fifty percent probability of performing the corresponding visual task. At sufficiently low values of the visibility of the target, the probability of performing the corresponding visual task falls to zero. In Johnson’s approach, this contradiction is overcome implicitly, assuming that the number N50 task that shows the number of pairs of lines, corresponding to the fifty-percent-probability of performing the corresponding visual task is determined from the experiment and depends on the conditions of visibility of the target. For different visibility conditions, different values of the number N50 task are obtained for the same target.
The experimental data available in the literature suggest a linear relationship between the integral number of line pairs in the TTP model and the number of line pairs in the Johnson model, but there was no theoretical explanation for this fact.
This paper theoretically proves that the corresponding variables in both models are indeed linearly related. Estimation of the coefficient of linearity leads to the need to solve the equation with exponential functions, the solution for which is not known. To proceed with the solution of this mathematical problem, we used the property according to which in both models (Johnson and TTP) he probability of performing the visual task is a sigmoid function, which in turn provides the possibility of approximating the probability function by a linear dependence in the range between two corresponding thresholds in the Johnson model and the TTP model. To determine the corresponding lower and upper threshold values, we use the double-tangent method. The threshold nature of visual perception leads to the conclusion that the probability of performing a visual task should also be threshold. The threshold nature of human visual perception shows up in the fact that the targets (for visual observation) or their images (when viewed on the monitor of the target sightseeing system) can be detected by the operator only if the size and contrast for a sufficient duration of the target image are higher than the corresponding threshold values associated with the threshold nature of human visual perception.
Taking into account that because of the threshold nature of human visual perception, the probability of execution of the visual task should also be of threshold character, we conclude that the double-threshold linear approximation of the sigmoid function is more appropriate to describe the probabilities of execution of the visual task than the sigmoid functions given by equations, used in the Johnson and TTP models. Notice, that the functions describing the probability of the performance of execution of the visual task are empirical dependencies that satisfactorily describe the experiment, while their smooth character contradicts with the threshold nature of human visual perception. In turn, human visual perception is based on the response of the corresponding neurons. The response of each individual neuron is, firstly, threshold, and secondly not smooth, being in the form of a step from 0 to 1. These features are different from the properties of the sigmoid function. Therefore, for modeling the probability of execution of the visual task, any sigmoid function will be convenient, but worse approximation in comparison with the step function, for example, a function of the type of ReLU activation function. The double-threshold approximation proposed in this paper for the sigmoid function is more adequate model representation than the sigmoid functions themselves.