Modeling the Periphery of the Perception System
Abstract Generated abstract
This paper develops a model for the peripheral stage of a perception system by specifying operators that transform an input signal into a measured quantity and establish a signal-dependent coordinate system. Assuming a linear sensor for the first transformation, the authors define the second operator through successive convolution-based averaging and subtraction, using Gaussian weighting functions to decompose a signal into frequency components while preserving the original signal over its bounded spectrum. The resulting filter structure is generalized to two-dimensional signals such as images and is compared with known models of retinal coupling and auditory basilar membrane behavior. The authors argue that this correspondence suggests a potentially general framework for modeling peripheral structures in human perception systems.
Full Text
MODELING THE PERIPHERY OF THE PERCEPTION SYSTEM
V. P. BONDARENKO, V. M. RAZIN
In [1], modeling of the periphery of the perception system is reduced to the search for a sequential series of operators: 1) an operator \(L_1\), transforming the input signal into the measured quantity, 2) an operator \(L_2\), forming a coordinate system and reducing the measured quantity to it, and 3) a measurement algorithm. In the present work the operators \(L_1\) and \(L_2\) will be considered.
Let us consider the problem for the case of a simultaneous signal \(f(t)\), where \(t\) is time. We characterize the signal by the parameters [1]:
\[ E_k=\frac{1}{T}\int_T f_k^2(t)\,dt \leqslant E<\infty, \]
where \(k\) is the number of the signal realization, \(E\) is the upper bound of the signal energy, and
\[ \omega_{\mathrm{B}k}\leqslant \omega_{\mathrm{B}}<\infty,\quad \omega_{\mathrm{H}k}\geqslant \omega_{\mathrm{H}}>0, \]
where \(\omega_{\mathrm{B}}\) and \(\omega_{\mathrm{H}}\) are the upper and lower frequency limits of the energy spectrum of the signal.
The operator \(L_1\), i.e., the transformation of the signal into the measured quantity, is realized with the aid of a sensor [2]. As a rule, the input and output of a sensor are related by a linear one-to-one dependence. Consequently, the operator \(L_1\) transforms the input signal into the measured quantity without distortions.
The origin of the coordinate system (reference points) [2] is the zero of the scale of the measured quantity; for example, the melting point of ice is taken as \(0^\circ\mathrm{C}\), etc. Usually the reference points are conventional, i.e., they are fixed under definite conditions of the surrounding medium. To specify reference points analogously in our case is practically impossible. This requires additional, often very large expenditures for maintaining the constancy of zero.
We shall take as the basis for constructing the coordinate system the hypothesis of the relativity of reference points [3], which is equivalent to the following: the coordinate system for the measured quantity must be its own for each specific signal. As a result of the action of the operator \(L_2\), the signal will take the form:
\[ \varphi_1(t)=f(t)-m_1(t), \tag{1} \]
where \(m_1(t)\) is the reference level, and \(\varphi_1(t)\) is the signal arriving at the input of the measuring unit.
We shall define the function \(m_1(t)\) as a certain average value of the input signal by means of a convolution operator
\[ m_1(t)=\int_0^t f(\tau)h_1(t-\tau)\,d\tau, \tag{2} \]
where \(h_1(t)\) is a weighting function satisfying the conditions [4]:
\[ \int_{-\infty}^{\infty} h_1(t)\,dt=1,\qquad h_1(t)\geq 0. \tag{3} \]
Apart from conditions (3), no additional restrictions are imposed on the form of the weighting function; therefore, for simplicity of analysis, we choose as the function \(h_1(t)\) a function of the form
\[ h_1(t)=\frac{1}{\sqrt{2\pi}\sigma_1}\exp\left[-\frac{(t-d_1)^2}{2\sigma_1^2}\right]. \tag{4} \]
The relation between the parameters \(d_1\) and \(\sigma_1\) is determined from the condition of physical realizability of the operator (2). A sufficiently good approximation takes place when
\[ d_1^2=(3\div 5)\sigma_1^2. \]
Let us consider function (4) as the impulse response of a filter. The transfer coefficient of the filter will be equal to
\[ H_1(j\omega)=\frac{1}{\sqrt{2\pi}}e^{-\frac{\omega^2\sigma_1^2}{2}}e^{-j\omega d_1}. \]
We shall find the cutoff frequency of the filter as the value \(\omega_{1\mathrm{B}}\) at the level \(0.5\) of the maximum value of the square of the amplitude-frequency characteristic \(H_1\),
\[ \omega_{1\mathrm{B}}=\frac{0.7}{\sigma_1}. \]
Let us denote
\[ \sigma_0=\frac{0.7}{\omega_{\mathrm{B}}},\qquad \sigma_1=\alpha\sigma_0, \]
where \(\alpha>1\).
The quantity \(\alpha\) will determine the frequency range of the signal \(\varphi_1(t)\) and the upper bound \(E_1\) of the energy of \(\varphi_1(t)\). The requirements on the complexity of the measuring unit [1] impose certain restrictions on the value \(E_1\) and on the width of the frequency range of the function \(\varphi_1(t)\). We shall construct the operator in such a way as to have the possibility of varying the indicated quantities, i.e., the value of \(\alpha\), over wide limits. The exact value can be found only when analyzing the measuring unit.
Transformation (1) gives an error in the representation of the signal equal to the value [1]
\[ \int_0^T m_1^2(t)\,dt=\delta_1. \]
To reduce the value \(\delta_1\), for a given \(\alpha\), we introduce the function \(m_1(t)\) into the measuring unit through a parallel channel, bringing it to its own reference level. By successively using such an input one can cover the entire frequency range of the signal; in this case, for the \(i\)-th transformation (1), function (4) will be characterized by the parameter
\[ \sigma_i=\alpha^i\sigma_0. \]
In this case the signal arriving at the input of the measuring block will be represented by the sum
\[ \varphi(t)=\sum_{i=1}^{l}\varphi_i(t), \tag{5} \]
where
\[ l=\frac{\ln \frac{\omega_{\mathrm{v}}}{\omega_{\mathrm{n}}}}{\ln \alpha}, \]
and will be equal to the input signal \(f(t)\). Indeed, by the construction of the sum (5) one can write
\[ \varphi_i(t)=m_{i-1}(t)-m_i(t). \]
Substituting \(\varphi_i(t)\) into (5), we obtain
\[ \varphi(t)=f(t)-m_1(t)+m_1(t)-\ldots-m_e(t)=f(t)-m_e(t), \]
where \(m_e(t)\) coincides with the constant component of the signal, which for \(\omega_{\mathrm{n}}>0\) is equal to zero.
The formation of the components \(\varphi_i(t)\) of the sum (5) can be regarded as filtering the input signal by a filter with impulse response
\[ y_i(t)=g_{i-1}(t)-g_i(t), \tag{6} \]
where \(g_i(t)\) is the successive convolution of the functions \(h_\kappa(t)\), \(\kappa=1,2,\ldots,i\).
Taking (3), (4) into account, on the basis of convolution of normal distribution laws [5], one can write:
\[ g_i(t)=\frac{1}{\sqrt{2\pi}\,b_i}\exp\left[-\frac{(t-a_i)^2}{2b_i^2}\right], \tag{7} \]
where
\[ b_i^2=\sigma_0^2(1+\alpha^2+\ldots+\alpha^{2i}) =\frac{\sigma_0^2[\alpha^{2(i+1)}-1]}{\alpha^2-1}, \]
\[ a_i=d_0(1+\alpha+\ldots+\alpha^i) =\frac{d_0[\alpha^{i+1}-1]}{\alpha-1}. \tag{8} \]
Expressions (7), (8) describe the sought operator \(L_2\). It is possible to construct the operator \(L_2\) by tying it to the frequency \(\omega\gg\omega_{\mathrm{v}}\). In this case, discarding all \(y_i(t)\) whose frequency pass band does not coincide with the particular range of the signal, the expressions for \(b_i\) and \(a_i\) can be approximately written in the form
\[ b_i^2=\frac{b_0^2\alpha^{2(i+1)}}{\alpha^2-1},\qquad a_i=\frac{a_0\alpha^{i+1}}{\alpha-1}, \]
where \(b_0\) is determined by the frequency \(\omega_{\mathrm{v}}\), \(a_i^2=(3\div5)b_0^2\).
For \(1<\alpha<2\), the operator \(L_2\) can be generalized to the case of any weighting function (4) satisfying conditions (8), since the convolution of such functions, for sufficiently large \(i\), will converge to a function of the form (7) [5].
The operator \(L_2\) is easily generalized to the case of a two-dimensional signal, such as images. In this case \(a_0=0\), and expression (6) takes the form
\[ y_i(x,y)=g_{i-1}(x,y)-g_i(x,y), \]
where
\[ g_i(x,y)=\frac{1}{2\pi b_i^2}\exp\left[-\frac{x^2+y^2}{2b_i^2}\right]. \tag{9} \]
Comparison of expression (9) with the expression for the coupling function of the retinal network \[6\] indicates their good agreement. Similarly, the characteristics of filters with impulse responses (6), for \(1<\alpha<2\), agree quite well \[7\] with characteristics describing the properties of the basilar membrane of the organ of hearing.
The correspondence between the literature data on the organization of the peripheral structures of the human visual and auditory perception systems and the obtained operator \(L_2\) gives grounds for hope that the approach presented here to modeling the periphery of the perception system is sufficiently general and makes it possible, from the positions set forth, to consider the structure of human perception systems.
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