ON THE MODELING OF HUMAN PERCEPTION SYSTEMS

V. P. Bondarenko, V. M. Razin

The creation of a general model of the perception of signals by humans—in particular, of the auditory and visual systems—encounters a number of difficulties. These consist in the fact that, on the one hand, perception systems have a high structural complexity and at present have been little studied, and, on the other hand, there is no general theory of the functioning of perceptual systems. In particular, the principal function of the periphery of the human auditory system (the basilar membrane) is considered to be the task of coarse spectral analysis of signals (frequency-coordinate transformation) [1, 2, 3], while that of the visual system (the retina of the eye) is the sharpening of contours in perceived signals [4, 5]. However, in the general scheme of perception these elements are located at approximately the same level and possess almost identical properties as linear filters, which is readily detected by comparing their impulse characteristics [3, 6, 7].

Apparently, models of perception of the visual and auditory systems should proceed from common premises that make it possible to explain their functional organization and the particular features of signal transformations. A certain tendency toward the development of a general approach is outlined in a number of works [8, 9, 10], where an analogy is drawn between recognizing systems and systems of human perception. But in these works questions are considered that are connected mainly with signal processing at higher levels, and the periphery of perceptual systems remains practically outside the field of view of such an approach.

The basic proposition in developing a general model of perception, evidently, must be the hypothesis of the functional identity of any human perception systems. This proposition is expressed in the uniformity of the tasks solved by the systems under identical conditions of optimality of the structure of the systems. The main task solved by any perception system is the task of extracting information from the input signal [10], and the optimality conditions are then expressed as follows: for a given amount of extracted information, the perceiving system must have minimal structural complexity and required energy consumption. Solving such a problem requires an exact definition of the concepts of signal, complexity, and quantity of extracted information.

It is necessary to define a signal in such a way that any signal arriving at the input of a perception system falls under this definition. The definition must be sufficiently general and at the same time sufficiently concrete so that it is possible to operate with it in solving the general problem. The basis for defining a signal may be the specification of the region of its existence, by analogy with the definition of the audibility region of sound signals [11]. As shown in [7], for this it is sufficient to specify the upper boundary of the ener-

signal energy $E$ and the limits of the frequency range $\omega_{\mathrm{B}}$ and $\omega_{\mathrm{H}}$. A specific signal may have narrower limits in energy and frequency, but not extend beyond the indicated bounds.

The amount of information extracted can be estimated by a functional associated with the inversely proportional dependence

\[ \delta = \int_S [f(X) - \varphi(X)]^2\,dX, \tag{1} \]

where $S$ is the region in which the signal is specified, $f(x)$ is the input signal,

\[ \varphi(X) = L^{-}[a(X)], \]

where $a(x)$ is the representation of the signal in the perception system, $L$ is the representation operator, and $L^{-}$ is the inverse operator.

In solving the general problem, the functional (1) must be evaluated over the upper bounds $E$, $\omega_{\mathrm{B}}$, and $\omega_{\mathrm{H}}$.

The complexity of the perception system can be estimated by analogy with the estimate carried out in [12]:

\[ S = K \sum_{i=1}^{n} m_i, \tag{2} \]

where $m_i$ is the number of equivalent distinguishable states (information capacity) of the $i$-th element, $n$ is the total number of elements of the perception system, and $K$ is a proportionality coefficient.

The energy consumed by the perception system will be determined by the number of elements and by the energy consumed by a single element, which is proportional to the number of its distinguishable states. Consequently, the total energy, up to a constant factor, will be determined by expression (2), and the problem of minimizing the consumed energy reduces to the problem of minimizing the energy $P$ required to transfer an element from one state to another.

At first glance, for any structure of the perception system there is no minimum of complexity, since this quantity is completely determined by the amount of information being processed. However, in a complex system the probability of failure of a single element cannot be reduced to zero, and, when considering the operation of the system, the influence of its own thermal noise cannot be excluded. These two factors require the introduction of structural redundancy and of a certain level by which the signals in the system exceed the level of thermal noise.

Thus, the problem of constructing a model of perception reduces to finding such signal-transformation operators $L$ that, for a given value of the functional (1), ensure a minimum of the sum (2) and of the energy $P$. The solution of this problem can be reduced to a number of particular problems [8, 10], namely: 1) forming the initial description of the signal, 2) coding the signal according to informative features, 3) classifying situations. The sequence of particular problems is largely determined by the requirement of minimizing the sum (2). The possibility of replacing the general problem by a number of particular ones indicates the hierarchical nature of the structure of the perception system, which is confirmed by data on the structure of human perception systems [1, 4, 8], as well as by works [8, 13], in which the advantages of hierarchical systems over single-level systems are proved.

The block responsible for forming the initial description will evidently contribute the largest share, compared with all the other blocks, to the value of the functional (1), and the smallest share to the value of the sum (2). Therefore it is more convenient to solve the first problem in the inverse formulation: for a given complexity of the block, minimize the error,

determined by functional (1). The solution of the second and third problems can no longer be carried out without taking into account all the activity of the brain, which in our case can be reduced to the process [10] of “transforming the aggregate of input signals into purposeful action.” This leads to a somewhat different functional that evaluates the quality of operation of the perception system.

A criterion of quality for the second and third problems may be a functional of the type of a loss function, risk, etc. [1, 9].

At present, the problem of classifying situations (images) [1, 9, 10] is being developed most intensively; less so, the problem of searching for informative features; and the problem of forming the initial description of a signal—or, in our interpretation, the first problem—is almost not considered.

Specifying the quality criterion in the form (1) is equivalent to the following: at the first stage the perception system removes some share of the uncertainty in the input signal, which is possible only in the process of measuring signal values [12, 14, 15]. Consequently, the creation of the first level of the perception system reduces to the creation of a measuring device satisfying the requirements set forth above. Consideration of this problem [7] shows that it reduces to the search for a number of signal-transformation operators: an operator transforming the input signal into a measurable quantity, an operator forming a coordinate system and bringing the measurable quantity into it, and a measurement operator.

Thus, modeling a perception system is carried out in parallel along three directions and reduces to solving three particular problems. To determine or specify the quality criterion of the system, for the second and third problems it is convenient to use loss functions, risk, etc., and for the first, functional (1), where the optimality condition reduces to minimizing the sum (2) and the energy \(P\). Signal transformations at the first level of the perception system are determined by the problem of measuring its values.

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