METHOD OF RECOGNIZING IMAGE STRUCTURES

Abstract: In order to solve problems with the construction and processing speed of a conventional neuron network, an optimum construction of a neuron network employing a synapse structure based on the biophysiological knowledge or inferred from this knowledge is determined to materialize advanced information processing functions including a characteristic extracting function, a characteristic integrating function and a memorizing function. This enables the neural network to be applied to image recognition and movement control by making use of its robust recognition capacity, or to the problem of optimalization and large-scale numeric value solution by making use of its parallel processing capacity.

Description: TECHNICAL FIELD The present invention relates to a method for constructing a neural network capable of solving problems such as recognition, which are difficult to solve by means of a prior art computer as well as the application thereof to pattern recognition, initial visual sensation processing, movement control, numerical analysis, etc. BACKGROUND ART Heretofore, learning, memorization, identification, etc. are discussed in "Parallel Distributed Processing I and II" by Mcclelland and Rumelhart (MIT Press, 1986). However neither knowledge on the cerebral physiology of living body, which is developed in the highest degree, is reflected therein nor discussion is done on the structure of the network, the speed of calculation, etc., which are problems, in the case where a practical application thereof is premised. In addition, no method for constructing the network for an object depending on the time is described therein. On the other hand, a method for solving a neural network as an energy minimizing method is described in "Hop-field & Tank" (Science, Vol. 233 pp. 625-633 (1986)). However the neural network dealt with there is restricted to a monolayer and any solution cannot be obtained within a practical calculation time. In their paper "Modification System for a Robot", published in IEEE Transactions on Systems, Man and Cybernetics, No. 4, July 1973, pp. 425-428, J. Koplowitz et al. describe a model of an intelligent machine and its training procedure. A linear polynomial is suggested as a means for evaluating choices based on motivations, and procedures to adjust the polynomial are disclosed. E. Mesrobian et al. in a paper entitled "Discrimination of Natural Textures: A Neural Network Architecture" published in IEEE First International Conference on Neural Networks, San Diego, California, June 21-24, 1987, pp. IV-247-258, consider ways of recognising local differences in texture elements of a woven cloth. The proposed architecture includes feature extraction, local boundary detection and higher-order texture discrimination networks. DISCLOSURE OF THE INVENTION An object of the present invention is to provide a high order information processing method using a neural network capable of determining the optimum structure of the neural network, in which a synapse structure based on the physiological knowledge on the living body or inferred from the knowledge is taken-in, to realize a high degree information processing function such as feature extraction, feature unification, memorization, etc. and making possible applications to pattern recognition, movement control, etc. making efficient use of the robust recognizing power thereof as well as applications to optimization problem, large scale numerical analysis, etc. making efficient use of the parallel processing power thereof. The problem for achieving the above object is to construct concretely a neural network effecting 1. feature extraction, 2. feature unification, and 3. memorization. In the cerebrum the above processes are effected successively. As physiological knowledge the information processing system for the visual sensation on 1, and the plasticity of the synapse coupling on 3. are known only slightly. 2. is a presently active research field, but it has not yet been achieved to obtain any unified understanding. The high order information processing method by means of the neural network according to the present invention is an information processing method simulating a cerebrum. Concerning the process 1. a neural network is constructed on the basis of the physiological knowledge on the information processing system on the visual sensation and concerning the process 3. a memory circuit utilizing the plasticity of the synapse coupling is constructed. Although all the processes 1. to 3. are constructed by neural elements having a basically same function, they have different meanings representing the state of the neural elements. A concrete method for constructing them will be described in the item "Best Mode for Carrying Out the Invention". BRIEF DESCRITPION OF DRAWINGS Fig. 1 is a scheme showing the conception of an embodiment of the present invention; Fig. 2 is a scheme showing a method for constructing a neural network; Fig. 3 is a scheme showing a network for the feature extraction; Fig. 4 is a scheme showing a network for the feature unification; Fig. 5 is a scheme showing a network for the memorization; Fig. 6 is a scheme illustrating an example of application to the image recognition; Fig. 7 is a scheme illustrating an example of application to the movement control; Fig. 8 is a scheme illustrating an example of application to the optimum control; Fig. 9 is a scheme illustrating an example of application to a non-stationary partial differential equation; Fig. 10 is a scheme showing the conception of the whole algorism of a minimum and maximum searching device, which is another embodiment of the present invention; Fig. 11 is a scheme showing a calculation method of the simulated annealing; Figs. 12 and 13 are schemes illustrating examples of application of the present invention; Fig. 14 is a scheme of animage processing system, in the case where the present invention is used for an image processing. BEST MODE FOR CARRYING OUT THE INVENTION At first the principle of the neural network based on the present invention. A neural network for the feature extraction is a network hierarchically constructed, as indicated in Fig. 3(c). It is supposed that neural elements 331 are arranged two-dimensionally on each of layers 332 and that there exist couplings 333 between different neural elements only between two layers adjacent to each other. When a concrete circuit construction is determined, physiological knowledge is referred to. What is well known up to present is the MT field concerning the visual sensation field and the movement. Here a report entitled "Receptive fields, binocular interaction and functional architecture in the cat's visual cortex" (J. Physiol, London, Vol. 160, pp. 106-154, 1962) by Hubel and Wiesel, which relates to the extraction of the feature of the former. Fig. 3(a) shows an experimental result indicating the orientation selectivity in the visual sensation region (the NL field) of the cerebral cortex. When an electrode 32 is inserted obliquely in a cerebrum from the surface 31, it is known that a group of cells reacting at a special value (with a 10° interval in the experiment) of the inclination of a light slit traversing the recepting field of the retina form a layer. In the direction perpendicular to the surface cells linked with each of the left and the right eye are concentrated. Therefore it can be seen that the cells are arranged in a module structure (Fig. 3(b)). Except that information from the left eye and that from the right eye appear alternately, the primitive features are extracted hierarchically in the visual sensation field by this module structure. For example, to each of the sides of a figure one layer corresponding thereto reacts to extract it. Further, although the layers have no inhomogeneity and are constructed by a completely same kind of neural elements, as information is transmitted, they extract sequentially different information as the result of the self organization. Now, if this procedure is expressed mathematically, it can be written as follows; F(ℓ) = S[K ℓ (q,q ℓ )G(q)] where ℓ represents the number of the layer in the network; G(q) input information (figure) having a Fourier wave number q, which is rotationally symmetric with respect to the orientation of the optical slit or corresponding to that orientation; K ℓ (q,q ℓ ) is a kernel function extracting an orientation q ℓ in a layer ℓ; S a mathematical operation executed in the network; and F(ℓ) a function, which is O except for the extracted layer. q ℓ (ℓ = 1, 2, ---) gives angles with an interval of 10° in the example described above. On the basis of the physiological knowledge as described above, an artificial neural network, which can realize the function expressed by Eq. (1), will be constructed. An example of the coupling structure between neural elements is indicated in Fig. 3(d). The state of elements in a layer of higher rank is determined as a function representing the state of 4 neural elements adjacent to each other in a layer of lower rank. In the following, a concrete method for determining the state of elements. Now the state of elements at a two-dimensional position r = (x,y) on a layer ℓ is represented by f ℓ (r). Then the coupling between elements on layers adjacent to each other is represented by a following state equation; f ℓ (r) = F ℓ ({f ℓ-1 }) + ξ ℓ , (ℓ = 2, 3, ...) where {f ℓ-1 } represents the elements on the layer ℓ-1 all together; ξ ℓ added noise; and F ℓ a general function for expressing the coupling between elements. It is supposed that f 1 (r) is given for the input layer (ℓ = 1). In a prior art neural network a non-linear saturation function such as a sigmoid function is given as the function F ℓ . However, in the information processing in the cerebrum, as a result of concurrent and competitive operation of a number of elements, universal processing independent of microscopic element functions should be executed. Therefore, in this step, no special functional form is presumed for F ℓ . Eq. (2) can be rearranged as follows; f ℓ (r) - f ℓ-1 (r) = 1 T ℓ-1 δH({f ℓ-1 }) δff ℓ-1 + ξ ℓ where H({f ℓ-1 }) is a function determined, when F ℓ is given; and T ℓ-1 is a positive constant. Further δ/δf ℓ-1 represents a functional differentiation with respect to f ℓ-1 . Although, as it can be seen later, the function H represents the energy of a system on the analogy of a physical system, H doesn't exist always, when F ℓ is given. In the following, the formulation is executed, supposing that Eq. (3) is given Eq. (3) is a difference equation with respect to the number of layer ℓ, from which a so-called Fokker-Plank type probability differential equation can be deduced, in the ease where ℓ is great. If P ℓ (f ℓ (r)) represents the probability that the state of elements at a position r on a layer ℓ is f ℓ (r), the following equation is valid; P ℓ (f ℓ (r)) - P ℓ-1 (f ℓ-1 (r)) = 1 T ℓ-1 δ δf ℓ (r) {∫∫ δH({f ℓ-1 }) δf ℓ-1 (r') P ℓ (f ℓ-1 (r'))dr' + D δ 2 δf ℓ-1 (r)δ ℓ-1 (r) P ℓ (f ℓ-1 (r)) where D represents a dispersion of the added Gauss type noise ξ, but hereinbelow, for the sake of simplicity, it is put at 1. The stational solution P ℓ s of Eq. (4), in the case where ℓ is sufficiently great, can be given by; P ℓ s ∼ exp (-H({f ℓ-1 })/T ℓ-1 ) That is, when inputted information (signal) has passed through a sufficiently great number of layers, the distribution thereof approaches that described by Eq. (5). The distribution expressed by the above equation is called Bolzmann or Gibbs distribution, in which H corresponds to the energy of the system and T to the temperature. Now, using the probability distribution expressed by Eq. (5), the relationship in the state of elements between different layers expressed by Eq. (2) is defined as follows; exp(-H({f ℓ })/T ℓ ) = F exp(-H({f ℓ-1 })/T ℓ-1 where F is an operator defined below and executed by the neural network. The basic operation burdended by the operator F is coarse-grained roughening in a feature extracting circuit 21. That is, the average value of the state of elements located in lower layers is propagated to upper layers. The summation in the left member is executed over the elements existing around the position r. Owing to thus visualization roughening, local fluctuations in f ℓ-1 become smaller. However, F should be defined so that the feature buried in H({f ℓ-1 }) is not lost. The intuitive interpretation of the transformation formula (6) of the probability distribution is to demand that the probability distribution consisting of feature components doesn't vary, even if noise components are eliminated. Since the layer space is reduced once by the visualization roughening (Fig. 3(e)), the action of returning the size to the initial one should be also included in F. The procedure described above is a process called rememorization group transformation. It is more convenient to express the relationship by using the frequency region g than Eq. (6). exp(-H({f ℓ })/T ℓ = ∫...∫exp(-H({f ℓ-1 })/T ℓ-1 )Πdf| f ℓ-1 (q) → λπf ℓ (2q) where f ℓ is a value obtained by Fourier-Transforming f ℓ (r) and 2q represents a transformation of enlargement by a factor of 2 in the frequency space, because the summation in Eq. (7) is executed only in the closest neighborhood. Further λ ℓ-1 is a constant and by the operation f ℓ-1 (q) → λ ℓ-1 f (2q) no noise components are mixed. If a concrete energy H is given, the operator F satisfying the demand described above, i.e. the relationship among the constant λ ℓ-1 , T ℓ and T ℓ-1 , can be defined. The energy H({f ℓ }) represents the coupling relation among elements in the layer ℓ. In general, H can be written as follows; H = Ω∫∫(▿f ℓ-1 ) 2 dr + Ω 2 ∫∫(▿ 2 f ℓ-1 ) 2 dr + ··· + r∫∫f ℓ 2 -1 dr + u∫∫f ℓ-1 4 dr + ··· where Ω 1 , Ω 2 , r and u are constants. Now, if the state of elements is expressed by +1 (ignition) and -1 (pause), since it can be thought that H is invariant with respect to the inversion of all the states of elements {f ℓ } → -{f ℓ }, H contains only terms of (even number)-th order of (f ℓ }. That is, this is because the definition by +1 is made only for the sake of convenience. The first term of Eq. (9) represents the coupling relation among the dosest elements. It represents the most important local features of the coupling among elements in term of the energy. Consequently, it is desired that this term is unchanged for all the layers. According to the demand described above, substituting the Fourier transformation of Eq. (9) for H in Eq. (8), executing the transformation of the right member, λ ℓ-1 = 4/ T ℓ /T ℓ-1 can be obtained. What is understood from Eq. (10) is that the temperature schedule expressed by T ℓ = 4T ℓ-1 is at a certain critical value. This is because, since at this time λ ℓ-1 = 1, which corresponds to a simple averaging operation, as the signal propagates in the layers, the spatial distribution thereof becomes more and more uniform and finally only a signal having a uniform distribution can be obtained. This is an extreme smoothing processing, which means that all the information is lost. Therefore, introducing an extremely small quantity ε ≡ 4 - (T ℓ /T ℓ-1 ), the non-linear terms in Eq. (a) are left. In this way, calculating Eq. (8), two equations expressing the relationship among the coefficients; r ℓ = h 1 (r ℓ-1 , u ℓ-1 , ··· , T ℓ-1 ) u ℓ = h 2 (r ℓ-1 , u ℓ-1 , ···, T ℓ-1 ) are obtained, where h 1 , h 2 , --- are non-linear functions. The behavior of the solution of Eq. (11), when ℓ is great, r ℓ = O(ε) < O, u ℓ = O(ε) > O, where O(ε) means a value of order of magnitude of ε. All the terms such as Ω 2 , which don't appear in Eq. (11), are extremely small quantities of order of magnitude of O(ε 2 ), which can be neglected. In the result, when the temperature schedule of T ℓ ≒ 4T ℓ-1 is supposed, for the layers, whose ℓ is great, H approaches a universal energy given by; H = Ω 1 ∫∫(▿f ℓ-1 ) 2 dr + r ℓ-1 ∫∫f ℓ-1 2 dr + u ℓ-1 ∫∫f ℓ-1 4 dr A concrete coefficient, in the case where ℓ is great, is given by; r ℓ = - Λ 2 9 ε, u ℓ = Λ 4 144·C 1 T ℓ · ε where Λ represents the maximum frequency (= 2π/▿, ▿ being the spatial resolution) and C is a constant. Since the temperature T ℓ increases as T ℓ ∼ 4 ℓ , when ℓ is great, u ℓ is extremely small. Taking this into account, the energy given by Eq. (12) can be expressed, as follows, by using Fourier components F ℓ-1 (q); H = ∫∫[q 2 F ℓ-1 (q) 2 + r ℓ-1 F ℓ-1 (q) 2 ]dq Here, for the sake of simplicity, the procedure is normallized with Ω 1 . The essential feature extracting function is never changed. From Eq. (5) representing the probability distribution and H given by Eq. (13) it can be understood that the component F ℓ-1 ( - r ℓ-1 ) of the Fourier frequency, which is q = - r ℓ-1 , gives the maximum of the probability. That is, in the layers ℓ-1 only the component F ℓ-1 ( - r ℓ-1 ) is extracted. Now, when the initial value r 1 of r 2 is determined, based on the maximum frequency Λ, r approaches the value given by Eq. (13) in the order of; r 1 < r 2 < r 3 ··· < r ℓ That is, it is possible to extract high frequency components in the lower layers (in the case where ℓ is small) and low frequency components in the upper layers (in the case where ℓ is great) (Fig. 3(f)). It is confirmed that the physiological experimental facts described previously can be simulated with the feature extracting network constructed as described above. The light slit is a viewed sensation object, which is symmetric with respect to a certain point. A copy 372 of the light slit 371, which copy has a given direction with respect to a certain direction (e.g. vertical direction), is prepared as indicated in Fig. 3(q). Then a group of slits including the slit can be defined unequivocally as a periodical function in the peripheral direction. The slit prepared in this way is inputted in the network stated above. If it is thought that the frequency q in Eq. (13) is the frequency in the peripheral direction, it is possible to take out successively specified frequencies. 2. Feature unifying network Primitive information extracted by the feature extracting network 414, e.g. the contour of a figure, etc., is inputted in a feature unifying network (Fig. 4(a)). Fig. 4(a) shows the process of unifying information by 3 layers as an example of the feature unifying network. Each neural element 417 located in a first layer 413 bears respective primitive information. Since each group of information is exclusive, all the neural elements in the first layer are coupled with negative values with each other. That is, if an element corresponding to a certain group of primitive information is in the ignition state, the other elements should be in the pause state. Of course, in the case where a number of groups of primitive information are inputted simultaneously, since the corresponding elements are turned to the ignition state, no negative coupling is necessary among these elements. In general, the coupling is not always necessary among the elements in the first layer. The neural elements 416 located in a second layer 412 are made correspond to information constructed by the primitive information from the first layer, e.g. figures. Consequently they are joined with the elements in the first element corresponding to the sides constructing each of the figures with a positive value 418 and with the other elements with a negative value 419. Since each of the figures is exclusive, the elements in the second layer are coupled negatively. The neural elements 415 located in a third layer 411 are made correspond to information of higher order constructed by the information from the second layer, e.g. composite figures. Consequently they are joined positively with the elements in the second layer corresponding to a figure constructing each of the composite figures and negatively with the other elements. The feature unifying process described above is not confirmed physiologically, but many alternative propositions are conceivable. For example, although a neural network consisting of 3 layers is used in this example, it may consist of 2 or 4 layers, depending on the object. Further the state represented by each of the neural elements may correspond to 1 group of information and also 1 group of information may be dispersed to a number of elements. Fig. 4(b) is a conceptual scheme for calculating concretely the state of the neural elements in each of the layers. The state of a marked element i in a layer 421 is represented by x i 422. The variable x i (i = 1, 2, ----, N) is either +1 (ignition state) or -1 (pause state). The input to the marked element is a sum of a signal from an element j 423 in the same layer and a signal from an element k 424 in another layer. Since the former has generally a negative effect, it has a coupling -W ij 1 (<0) 425 and the latter has a coupling W jk 2 426, which can be either positive or negative. That is, the total input can be written as follows; If the total input is greater than a certain threshold value, the element is ignited and otherwise it is in a pause state. Although it is possible to determine the state of each of the elements in each of the layers by this procedure, a little more elegant method will be described below. Forming a product of Eq. (15) and x i , since this product is maximum in both the states, ignition and pause, it is sufficient to obtain the state, in which is minimum, where W ij = -W ij 1 + W ij 2 ,  being a threshold value. The method, by which the state of elements is given in this way as the state, in which the energy function equation (16) is minimum, is disclosed in "Computing with neural circuits" by Hopfield & Tank (Science Vol. 233, pp. 625-633, 1986). However it doesn't deal with neural elements existing in a number of layers as in the present invention, but it takes only elements in a single layer into account. By this method, calculations are executed not successively from the lowest layer, as indicated previously, but the state of elements in all the layers can be calculated parallelly all together. Consequently the formulation according to Eq. (16) is an algorism suitable for parallel calculations. It is in fact very difficult to obtain the minimum value of the energy represented by Eq. (16), because, since the state x i is two-valued, i.e. ±1, a number of minimum values appear and the true smallest value cannot be well obtained. On the basis of such a background Kirkpatrick, Gelatt and Vecchi have invented a simulated annealing method published in "Optimization by simulated annealing" (Science Vol. 220, pp. 671-680, 1983), which is a smallest value retrieving method by repeation utilizing the probability. The essential point of the present invention consists in that it is possible to escape from a minimum value owing to fluctuations given to the state by introducing a parameter, which is the temperature. Hopfield and tank have found further to be able to obtain a lower energy, if a problem of the discrete quantity of x 1 = ±1 is transformed into a problem of a continuous quantity y 1 (- ∞ < y i < ∞) through a transformation expressed by x i = tan h (y i /constant). A disadvantage of this method is that it takes a very long time. A minimum and maximum retrieving method improved from this point of view will be described later. The minimization of the energy equation (16) is not restricted to the method as described above, but as an alternative method, e.g. the following method is known. By this method, since it is not necessary to introduce the tanh function for making the variable continuous, differing from the method described above, an excellent calculating property can be obtained. By the simulated annealing method, the maximization of the probability exp(-E/T) is taken into account instead of the minimization of the energy, where T is a positive parameter. This probability can be rewritten as follows, introducing a continuous variable z i (-∞ < z i < ∞); This can be easily proved by using the following equation; Further, (W) ½ ij means a (j, i) component of the square root of a matrix W. For the sake of simplicity it is supposed here that the threshold value  is 0 and that the coupling constants W ij are symmetric with respect to the suffixes (W ij = W ji ). The essential feature of the algorism is not lost by this supposition. When the kernel function of the integral of Eq. (17) is considered as a function of x i , the smallest value thereof can be obtained clearly to Further, when the kernel function is a function of z i , since it is a second order function convex downward, gives the greatest value of the kernel function. Here  is a stepwise function, which is 1, if the argument is positive, and -1, if the argument is negative. Consequently is valid and this expresses the basic function of the neural elements. That is, it is sufficient to execute the maximization of the kernel function given by Eq. (17) with respect to the continuous variable z i . Fig. 4(c) indicates the relation between the initial neural network, in which the neural element state x i i 431 is coupled with an adjacent element x i 432 through W ij 433 and the network equivalent thereto, in which the continuous variable z i 434 represents the state of elements according to Eq. (17). All the coupling constants in the equivalent network are 1. The state of elements x i 439 is determined from the variable z i 434 calculated by using the equivalent network through a convolution operation 437 thereof with the square root (W) ½ of the coupling constant and a comparing operation 438. The feature of the equivalent circuit thus constructed is that the calculation time (CPU time) is short, because it is not necessary to introduce newly the tanh function for making the function continuous, as Hopfield and Tank have done. Further, since the kernel function expressed by Eq. (17) is a second order function with respect to z i , it is possible to preestimate an approximate value of the state z i giving the smallest value thereof, and in addition, since there exists no minimum values, it is possible to estimate the convergence from the initial state to the state, where the smallest value is given. In the minimization problem, since x i is two-valued, it was extremely difficult to determine the initial value, because numberless minimum values appeared, and in many case no suitable initial value was determined and no state of the smallest value could be obtained. A concrete algorism for realizing the method described above is indicated in Fig. 4(d0. [Algorism) 1 ○ Start of the calculation. 2 ○ The square root of the given coupling constant W is obtained. As an example, x ij = (W ½ ) ij is determined (block 441) by obtaining the solution of 3 ○ The initial value of the continuous variable z i (i = 1, 2, ----, N) is set (block 442). 4 ○ Based on z i , the neural element state x i is determined from (block 443). Here  is a stepwise function, which is 1, if the argument is positive, and otherwise it is -1. 5 ○ Based on x i determined in 4 ○, z i , which makes the kernel function expressed by Eq. (17) the greatest, is calculated e.g. by Monte Carlo method (block 444). 6 ○ The convergence is judged. If it is not, 4 ○ and 5 ○ are executed repeatedly. If it is, the process proceeds to the succeeding step (block 445). 7 ○ Termination of the calculation. 3. Memory network The high order information such as the figure unified by the feature unifying network is stored in a neural network, as indicated in Fig. 5(a). An input pattern 514 of the high order information is inputted in the lowest input layer and propagated to upper layers so that an output pattern 515 is outputted by the highest output layer. Neural elements 512 located in each of the layers 511 are located one- or two-dimensionally, corresponding to the input pattern. Further it is supposed that the state of each of the elements takes only 2 values, i.e. 1, if it is in the ignition state, and -1, if it is in the pause state. In the case where it is multi-valued, it can be dealt with by increasing the number of elements. The principal function of the memory network is to store the relation between the in- and output patterns by learning 518. For example, into which class the input pattern is classified, as in the classification (output patterns constituting classes) is stored, or the relation between hand-written letters and correct letters corresponding thereto, as in the recognition of hand-written letters, is stored. Or it is also possible to control appropriately a control object, whose behavior is unknown, by learning. Such a method for making a multi-layered neural network learn has been already developed in "Parallel Distributed Processing I and II" (MIT press, 1986). However, because of the following disadvantages, the use thereof in practice is limited to a small field. (1) Synapse coupling structure By the prior art method as disclosed in the publication stated above, from the point of view of dispersing the memory to all the synapse couplings, synapse couplings are spread over all the elements. Therefore the amount of information supported by each of the synapses is small so that even if incomplete information is given, complete information can be remembered as associated memory. However, since the time necessary for modifying the synapse couplings, depending on the learning, is proportional to the total number thereof, enormarous calculation time is necessary and therefore it is an undesirable structure in practice. (2) Learning algorism The prior art learning algorism disclosed in the publication stated above is a back-propagation. By this method, at first, an appropriate initial value is set for the synapse coupling. Based on this initial synapse coupling, the state of elements is calculated from the lower layer to the upper layer one after another. In general, since the outputted value is different from a teacher pattern 516, the difference 517 therebetween is obtained. Then the synapse coupling is modified 519 so as to reduce the difference. The procedure described above is repeated, unit1 the difference becomes 0. Such a method, by which the feedback function is the essence thereof, is intuitively understandable and the programming thereof is easy. But, on the other hand, the efficiency is not high from the point of view of the calculation time. (3) Physiological knowledge on the memory The memory described above, referring to the plasticity of the synapse coupling, corresponds physiologically to a long term memory. According to psychological experiments, it has been clarified that there exists a short term memory, for which no plasticity of the synapse is presumed, apart from the long term memory. A probability to obtain a memory method having a performance higher than prior art one is hidden in simultaneous technological application of the two memory mechanisms. On the basis of the background described above, the present invention gives a new method for the memory, which is a variation thereof. In the following the long term memory and the short term memory will be discussed separately. 3.1 Long term memory In the neural network data or patterns are not stored, as they are, but they are dispersed to be stored in the network in the form of values of the synapse coupling. That is, they are coaded dispersedly. Now N data sets I i (i = 1,2, ----, N) are given as an input pattern. In general, the input pattern may be either one- or two-dimensional and either two-valued or multi-valued. In the case where they are multi-valued, since the input data I i can be transformed into two-valued data by increasing the number of data sets, hereinbelow it is supposed that I i is two-valued. The process, by which the input pattern I i 514 propagates towards an upper layer, can be formulated, as follows. For the non-linear function F written in the form of f = F(T x ), a sigmoid function having a threshold value, whose output f is ±1 in the saturated state, is representative. Denoting the output of the elements within the layer ℓ by f i (ℓ), a relational equation; is obtained, where W ij (ℓ) indicates the value of the synapse coupling between the element i in the layer (ℓ-1) and the element j in the layer ℓ. If W ij has some value for all the js, it represents that there exist couplings with all the elements in the layer (ℓ-1). The sum thereof over all the elements in the layer (ℓ-1) may be different from the number of the data sets N. When the relation given by Eq. (19) is applied successively from the input layer (ℓ-1) to the output layer (ℓ= L), can be obtained. Consequently memorization by learning is to determine the synapse coupling W ij (ℓ) (ℓ = 2, 3, ----, L) so that the output F i (L) of Eq. (20) is equal to the teacher pattern d i 516. However Eq. (20) represents a system consiting of N equations (here it is supposed that there are also N outputs) and if all the elements are coupled with each other, there are N 2 (L-1) unknown coefficients W ij (ℓ). That is, the unknown variables are excessively redundant. It can be thought that the amount of information per synapse has a magnitude of N/N 2 (L-1) = 1/N(L-1). If N or L is great, since each of the synapses supports only little information, flexible processing such as e.g. associated memory is possible. However, in a real cerebrum, since N is greater than 10 billions, the ratio stated above is substantially equal to 0. Further, in the cerebrum, all the neural elements are not coupled with each other by the synapse coupling. This suggests that there exists a kind of structure in the synapse coupling. Still further, when a uniform neural network is considered in the cerebrum, it is not conceivable that any object dependent type structure depending on the kind of given input is formed therein. According to the present invention, the optimum structure of the synapse coupling is determined on the basis of the cerebral physiological knowledge. In the prior art neural network aiming technological applications, each of the synapse couplings is changed by learning, but information supported by each synapse is completely uniform in the average. That is, the dependence of W ij (ℓ) on the suffixes i and j is not so great. For this reason, all the synapses should be modified, which makes it difficult to use it in practice from the point of view of the calculation time. At the present stage cerebral physiological experiments don't clarify so far the detailed structure of the synapse coupling. In the present situation rather only a kind of statistical, i.e. macroscopic structure is understood. However, since the macroscopic structure is determined by the structure of the synapse structure, it is possible to conjecture the microscopic synapse structure by the method described below. Fig. 5(b) indicates an example of the macroscopic structure obtained experimentally according to "A quantitative Study of Snaptic Reorganization in Red Nucleus Neurons after Lesion of the Nucleus Interpositus of the Cat" by Murakami et al (Brain Research, vol. 242, pp. 41-53, 1982). In the upper figure variations of the number of synapses T 522, to which degenerative terminals of cerebrum-red nucleus synapses are attached, are indicated in the function of the diameter R 524 of the tree-like protrusion and in the lower figure variations of the diameter R 524 of the tree-like protrusion are indicated in the function of the distance x 523 from the center of the cell having the tree-like protrusion. In the upper figure a relation represented by T ∼ R -α (α = 2 / 3) and in the lower figure a relation represented by R ∼ x -β (β = 1) are obtained, where the mark ∼ indicates a proportional relationship. The results described above relate to statistical quantities obtained by using a number of samples. It seems that the two figures are independent from each other at a glance. However, there exists an intimate relation therebetween as a conclusion from the structure of the synapse coupling. It is possible to infer the optimum structure of the synapse coupling by showing this fact. As indicated in Fig. 5(c), the branching of the tree-like protrusion is a bifurcation. By bifurcating n times, in total 2 n protrusions 531 are obtained. This branching is a branching method, which can be seen fairly generally. The signal propagating in the tree-like protrusion is a pulse train, which is a medium transmitting information. Consequently the information transmission efficiency should be different for every branch. For example, when the cross-section of the tree-like protrusion is approximated by a circle, the transmission efficiency depends on the diameter R. Now the diameter of a branch generated by n bifurcations is represented by R n 532 and the diameter of a branch generated by the succeeding bifurcation by R n+1 533. Information sets flowing through these branches are represented by I n 534 and I n+1 535, respectively. Here the flow of information is replaced by a flow of electric current in an equivalent circuit. The energy consumed in this branch is considered here. The energy corresponding to electric power is clearly proportional to I n 2 /R n 2 , where the fact that the resistance is inversely proportional to the area of the cross-section of the protrusion 4πR n 2 is used. The volume occupied in the space by this branch is equal to 4πR n 2 x (length of the branch) and it is conceivable that the living body regulates this volume so as to occupy a space as small as possible. Forming a summation of the energy stated previously and this volume over all the branches; where the coefficient is a positive constant for fitting the dimension of the terms. The minimization of this quantity expresses mathematically that transmission of a same quantity of information is achieved by utilizing energy as little as possible and a space as small as possible. Now, supposing that the length and the coefficient are constant, it is understood that the formula (21) is minimized, in the case where a relation; R n 2 ∼ I n is fulfilled, by putting the differential of the formula (21) with respect to R n at 0. Concerning the electric current flowing through the bifurcated branches, I n = 2 I n+1 is valid, and from the formula (22) together therewith R n+1 R n = 1 2 is obtained. That is, the diameter of the tree-like protrusion is reduced to 1/ 2 for every bifurcation. Denoting the diameter of the initial tree-like protrusion going out from the cell by R 0 , since R 0 / 2 n = R n , after n bifurcations 2 n = (R 0 /R n ) log 2 log(1 2 ) ∼ x 2β branches appear, where x represents the length of all the branches and a relation R n ∼ x -β is used. The formula (24) is a formula represneting the total number of branches at a distance x measured from the center of the cell. Although the formula (21) represents a hypothesis concerning the microscopic structure of the synapse coupling, it will be shown below that a series of resulting formulas (22) to (24) deduced on the basis of this hypothesis can explain Fig. 5(b) indicating physiological experimental facts. A sphere having a diameter L, as indicated in Fig. 5(c), is considered. The total number Q of the tree-like protrusions, which find their way from a cell within this sphere to the surface thereof, is calculated. Supposing that the cells are distributed uniformly in this sphere, the total number of the tree-like protrusions finding their way to the surface of the sphere can be calculated, as follows, using the formula (24); Q ∼ ∫ L x 2β xdx sin ddψ∼L 2β+2 where xdx sin ddψ represents an incremental volume element within the sphere and  and ψ independent angles in the polar coordinate system. The formula (25) indicates that the dependence of Q on the diameter L of the sphere is L 2β+2 . On the other hand, another formula is deduced for Q from another point of view. The total number of Q of the tree-like protrusions is related clearly to the diameter L of the sphere and the number T of the synapse couplings at the surface of the sphere. This relation is expressed generally by; Q = f(L, T) The T dependence is conceived for taking the experimental facts indicates in Fig. 5(b) into account. Now a sphere of smaller scale having a diameter L' = b -1 L (b > 1) is considered instead of the sphere having the diameter L. Together with this transformation T and Q are transformed into values given by T' = b k1 T and Q' = b k2 Q, respectively. Since both T and Q are values defined at the surface of the sphere, clearly k 1 = 2 and k 2 = 2. That is, when all the quantities are measured with the unit of the small scale L', the quantities T and Q expressed two-dimensionally increase proportionally to the increase in the surface area with respect to the initial scale L. Even if the measurement is effected by using the quantity obtained by such a transformation, the relational equation expressed by Eq. (26) remains unchanged. Consequently, the following equations are obtained; Q' = f(L', T') f(L, T) = b -k2 f(b -1 L, b k1 L) Since the diameter L of the sphere is an arbitrary quantity, a function f(L, T) satisfying the equation stated above for all L should be obtained. The result is written as follows; f(L, T) = L -k2 f (L k1 T) where f and is a function, which cannot be defined only by Eq. (27). Now, from experiments, the number T of synapses depends on the distance x from the center of the cell as T ∼ R -2 ∼ x 2β . Using the formula (24), when the relational equation is rearranged by using the formula Q ∼ x 2β , another relational formula Q ∼ T 2 α is obtained. Although this has been obtained for one tree-like protrusion, it is supposed that it is similarly valid for an assembly of a number of tree-like protrusions. At this time the unknown function f and in Eq. (28) shows a dependence expressed by f (K k1 T) ∼ L k 2 α T 2 α . Consequently, Eq. (28) is transformed into; Q ∼ L -k 2 + k 1 2 α T 2 α This is another formula expressing the total number Q of the tree-like protrusions. From the dependence of the formulas (25) and (29) on L, 2β + 2 = -k 2 + 2k 1 α can be obtained. Substituting k 1 = k 2 = 2 in this relational equation. 2 α - β = 2 is obtained. This relational equation is valid just for the experimental formulas α = 2 / 3 and β = 1, which shows that the microscopic structure of the synapse coupling previously presumed is correct. That is, the optimum structure of the synapse coupling is determined so as to minimize the function given by the formula (21). Since information is made correspond to electric current, it is proportional to the area of the cross-section 4πR n 2 of the tree-like protrusion of the transmission medium (Formula (22)) and the amount of information, which can be transmitted for one distribution, is 1/2. Fig. 5(d) shows variations in the ratio of the amount of information 542, which can be transmitted, depending on the number of branchings 541. From this figure, after 6 branchings, it is reduced to an order of 1% of the initial amount of information, which means that the information transmission is substantially impossible. Further, after about 3 branchings, it is reduced to 10%. That is, in practice, it is sufficient to take 3 to 4 branchings into account. For example, when a case where a neural element 545 is coupled with elements in the succeeding upper layer 543, is considered, it is sufficient to take only the coupling with a group of elements 544 of 2 4 = 16 around an element, which is located just above the element 545, into account. Here it is supposed that a tree-like protrusion is bifurcated from the central element towards elements close thereto, in the order of increasing distance, one after another. Since the amount of information transmitted by the tree-like protrusion is expressed by the magnitude W ij (Eq. (16)) of the synapse coupling in an artificial neural network, the magnitude of W ij should be varied according to the Table indicated in Fig. 5(d). For example, if starting points are couplings of elements located at same positions in an upper layer and a lower layer adjacent to each other, W nn(i),i /W i,i = 0.5, W sn(i),i /W i,i /W i,i = 0.25, etc. are valid, where nn(i) represents the element closest to an element; and sn(i) the second closest element. It is a matter of course that, if the memory is taken into account, the synapse coupling should be modified by learning. However it is conceivable that the degree of modification effected thereby is small. Consequently it can be thought that the ratios described above don't vary so remarkably. Hereinbelow a concrete learning rule is considered. The back propagation method, which is a prior art learning method, is discussed in detail in "Parallel Distributed Processing I and II" (MIT press, 1986). The basic conception is that synapse couplings are determined from an upper layer to a lower layer successively so that the following square error e is minimized, using the output f i (L) given by Eq. (20); Concretely speaking de/dW ij (ℓ) = 0 (ℓ = L, L-1, ----, 2) are determined one after another by the method of steepest descent. By the back propagation method the synapse couplings formed among all the elements are corrected one after another, using Eq. (32). For this reason, since the necessary calculation time is proportional to the number of synapses N 2 (L-1), it is no practically efficient learning method. However, for example, if the structure of the synapse coupling is as indicated in Fig. 5(d), the total number of couplings is 16N(L-1), which is 16/N times as small as the number described previously. If 1000 elements are considered, the former is only 1.6% of the latter. Even if the number of branchings is 5, it is only about 3.2%. [Algorism] Hereinbelow the processing procedure will be explained, referring to Fig. 5(e). 1 ○ Start of calculation 2 ○ Set initial values for states of elements f i (ℓ) (ℓ = 1, 2, ----, L) and synapse couplings W ij (ℓ) (ℓ = 2, 3, ----, L). (Block 551) 3 ○ Either the states of elements f i (ℓ) are calculated, starting from the given input, from the lower layer to the upper layer one after another according to Eq. (19) or f i (ℓ) is determined by executing the minimization, using the formula (17). (Block 552) 4 ○ The structure of the synapse coupling according to the number of branchings is determined as indicated in Fig. 5(d) and for these synapses their coupling constants W ij (ℓ) are corrected successively from the upper layer to the lower layer so as to minimize the formula (32). (Block 553) 5 ○ The convergence is judged. If it is not convergent, 3 ○ and 4 ○ are repeated and if it is convergent, the process proceeds to the following step 6 ○, END. (Block 554) 6 ○ End of calculation Apart from the learning method considered in the above, there are alternative proposals as indicated below. Alternative proposal 1 The prior art back propagation method and the method indicated by the algorism in Fig. 5(e) are based on the suposition that all the synapse couplings W ij are independent from each other. Thus, by the learning method considered above, on the basis of the physiological knowledge, the structure of the coupling necessary and sufficient from the point of view of the transmitted information was determined. As the result, it was made possible to shorten the time necessary for the learning. However, in an artificial neural network it is possible to reduce the number of synapse couplings from another point of view. Now, it is supposed that all the neural elements are coupled between different layers. The synapse couplings W ij (ℓ) are generated by another variable ξ 1 (ℓ). That is, they are generated by a variable having a lower dimension, instead of the initial N 2 coupling variables. Denoting the dimension thereof by M, it is supposed that; W ij (ℓ) = W[ξ k (ℓ)], M < N 2 where ω is a generating function and ξ k (k = 1, 2, ..., M) is a variable of dimension M. E q . (33) represents a generalization of the method indicated in Fig. 5(a). A state, where an element i in a certain layer is coupled with each of the elements in the upper layer adjacent thereto, is considered. Supposing that the synapse coupling W ij represents an electric current and R ij a diameter of the cross-section thereof, R ij may be determined so as to minimize; where if R ij = 0, it is thought that there is no coupling between (i, j). Alternative proposal 2 By the methods described up to now, when a teacher pattern d i 516 is given, the sunapse couplings are corrected successively from the upper layer to the lower layer so as to minimize the square error the formula (32). Apart from such a repetition method, there is another method, by which the synapse coupling can be determined rapidly. That is, it is determined analytically. The function F expressed by Eq. (20) is a non-linear saturation function. For example a sigmoid function is used therefor. Roughly speaking, the sigmoid function can be divided into saturated parts and a part put therebetween, which can be transformed into a linear form. This linear transformation is approximated by F = A + Bx. Now, supposing a case where all the elements behave in this part, the synapse coupling is determined. In the present invention, a 3-layered neural network is considered as an example. When Eq. (20) is rewritten by using this approximation. A + B{AΣW ij (3) + BΣW ji (3) ΣW jk (2) I k } = d i is obtained. It is sufficient to determine W ij (3) and W ij (2) satisfying this equation. For example, supposing a separable type synapse coupling such as W ij (3) = n i (3)ξ i (2), they can be determined as follows; where ζ i is a random variable, whose average value is 0 and whose valiance is σ 2 . When the synapse coupling thus defined is used, it is necessary no more to repeat the procedure required by the prior art techniques. In practice, since all the elements don't work in their linear region, Eq. (36) is not valid therefor and they should be dealt with separately. Eq. (36) has another manner of utilization. In general, by the back propagation method, etc. according to the prior art techniques, a better result will be obtained, when the initial value of the synapse coupling is generated by using random numbers having small values. This is because, if the network is in the state most unstable in the initial state, it can be thought that the value converges rapidly into a stable state. That is, Eq. (36) can be used as the initial value for the synapse coupling. Alternative proposal 3 By the prior art method, although there were differences in the calculation algorism, it was divided into a part for calculating the state of elements and a part for modifying the synapse coupling by learning. However, apart from the easiness of the intuitive thinking, it is not necessary to calculate them separately. Here an algorism executing both of them simultaneously is shown. The learning as well as the part for calculating the state of elements are based, similarly to that based on Eq. (32), on the minimization of the energy according to the thought of Hopfield described above. The energy, for which both the state of elements and the learning are taken into account, is represented by; where k is a positive constant. When the procedure is formulated in this way, it is possible to determine simultaneously both the state of elements f i (ℓ) 561 and the synapse coupling w ij (ℓ) 562 (Fig. 5(f)). That is, since the repetition of the determination of state of elements → synapse coupling is no more necessary, but they can be executed simultaneously, this method is suitable also for implementing parallel computers. A simpler formula can be obtained, if the separable type synapse coupling introduced previously W ij (ℓ) = ξ i (ℓ)ξ j (ℓ-1) is introduced, where ξ i (ℓ) represents a new variable 563. If this formula is substituted for W ij (ℓ) in Eq. (37) is obtained and thus the number of variables, which are to be determined, is reduced to N(L-1), where F i (ℓ) = ξ i (ℓ)f i (ℓ). 3.2 Short term memory In a word, the long term memory is a mapping of information on synapse couplings or coding. The short term memory stores information by a method completely different from the mechanism of the long term memory. Fig. 5(g) shows results of a psychological experiment on the memory disclosed in "Human Information Processing: An Introduction to Psychology", 1977 by Lindsay and Norman. 19 subjects listened 30 words having no relation therebetween, which were presented them with a rate of 1 word per second. A time of 1.5 minites was given to each of the subjects for every termination of presentation of lists. They were asked to write all the words, which they could remember, in any order which they liked. 5 to 10 seconds after they had finished to write them a new list was presented. The procedure identical to that described above was repeated 80 times. A serial position curve 573 is a curve obtained by representing the reproduction ratio 571 at this time in the function of the serial position 572 indicating the order, in which the words have been presented. The feature of this curve is that the reproduction ratio decreases up to an about 7th word counted from the word most newly presented and that it remains almost equal for the words preceding it. The former part is made correspond to the short term memory 575 and the latter part to the long term memory 574. The short term memory is a mechanism, by which data given to a person are temporarily stored. Necessary data are selected among these data according to a certain criterion, for which data the procedure proceeds to the long term memory. The memory intending prior art technological applications was the long term memory and no attention was paid to the short term memory. consequently it is necessary to select previously necessary data among given data before storing them. That is, the judgment of the selection should be newly considered as a preliminary processing outside of the neural network. On the contrary, the neural network having the mechanism of the short term memory can incorporate such a selection mechanism in the network. Further the short term memory has an important aspect concerning the passage to the long term memory. As described previously, in the short term memory, the synapse coupling presents no plasticity, but it is constant. In order to be able to store information with unchangeable synapse coupling, the synapse coupling should be determined according to some rule. Therefore, when the procedure passages to the long term memory, if the coupling structure differs remarkably from that rule, the succeeding short term memory becomes impossible. That is, the approximate value of the synapse coupling is determined at the point of time of the short term memory. Therefore, in the following, a model, which can explain the psychological realization of the short term memory seen in the serial position curve, will be constructed and at the same time the rule imposed on the synapse coupling will be elucidated. An equation system representing the behavior of the state of the neural elements can be described by Eq. (16) or Eq. (19). Since both the equations are deduced from equivalent mechanisms, here Eq. (16) is considered. At first, it is considered how the memory in the short term memory can be expressed in Eq. (16). Since the synapse coupling W ij is invariant, it cannot be coded in W ij as for the long term memory. Therefore the short term memory is made correspond to the minimum of Eq. (16). According to the serial position curve 573, since it is shown that about 7 states can be memorized, corresponding thereto, a number of minimum values around that number are necessary. That is, the condition imposed on the synapse coupling therefor will be obtained. Now, supposing that W ij = const. (independent of i and j), in the case where the number of element is sufficiently great, depending on the sign of the threshold value , x i = -1 or x i = 1 gives the smallest value of Eq. (16). When the dynamic process of Eq. (16) is considered, there exist no minimum values other than these values, but the state where all the neural elements have a same value is only one memory. Consequently the supposition that W ij = const. (independent of i and j) cannot explain the serial position curve. The dynamic process to the minimum value of the energy is memorized by using the following probability equation. ∂ ∂t P({x}, t) = -Σ[ω(x j )P(..., x j ...,t)] where P represents the probability distribution, in which the state is {x} at a time t and ω the transition probability. The necessary condition in order that the probability equation stated above gives a stationary distribution exp(-E({x})) is expressed by Instead of integrating directly Eq. (39), the average value of the state x i is considered. Here it is supposed that the synapse coupling W ij takes the following values (Fig. 5(h)); Now, as the experiment for obtaining the serial position curve, the situation where words are presented to subjects one after another is considered. At this time the groups of neural elements 591 (Fig. 5(i)) corresponding to these words are turned to the ignition state one after another. Of course the groups of elements can be overlapped on each other. In any case, the fact that the words are presented one after another corresponds to that the number of elements coupled in each of the elements (group) increases. That is, (p + r) increases. Therefore the behavior of the average value <x i > of the state of elements in function of p+r will be examined. According to Thouless, Anderson and Palmer (Philosophical magazine, Vol. 35, p. 583, 1977), executing Eq. (40), the average value <x i > varies approximately according to the following equation; In general, since different <x i >s correspond to different stationary states, the number of stationary solutions of Eq. (41) is equal to the number of short term memories. As disclosed also in "Digital dynamics and the simulation of magnetic systems" (Physical Review B, Vol. 28, pp. 2547 - 2554, 1983) by Choi and Huberman, the result is indicated in Fig. 5(j). When the number of elements (p+r) 5101 is small, there exists only one stationary state 5102. The bifurcation takes place, as (p+r) increases, and thus the number of stationary states increases in the form of a tree 5103 as 2, 4, 8 and so forth. However, when the value of (p+r) increases further, exceeding that giving 8, there exist no stationary states, which gives rise a chaotic state 5104. That is, there don't exist more than 8 stationary states (minimum values of energy) in this neural network. From the result described above it was recognized that the short term memory can be realized by using a neural network having positive and negative random synapse couplings. Further, from the consideration described above, it can be understood that almost random synapse couplings may be presumed also for the long term memory. Hereinbelow some embodiments of the present invention will be explained, classifying them into several items for different objects of application. 1. Recognizing problem 1.1 Recognition of moving images 1.2 Initial visual sensation 2. Control problem 2.1 Control of movement 2.2 Optimum control 3. Mathematical problem 3.1 Method for solving non-stationary partial differential equations 1. Recognizing problem 1.1 Recognition of moving images In Fig. 2 a neural network for high order information processing was constructed, starting from an orientation selective extracting circuit in the visual sensation in the cerebral cortex. Here a neural network for recognizing moving images will be constructed by applying this network. According to psychological knowledge, when a person recognizes an object, concurrent or competitive actions of physical signals (images) of a body, which is an object to be recognized, and conceptions (images) are necessary. There are many cases where the recognition is not possible only by one of them. That is, features are extracted from a number of physical signals entering the system, which are unified, and a significant recognition is effected by matching them with the memory. Consequently, in addition to the general structure indicated in Fig. 2, a feedback mechanism from the memory to the feature unification is necessary (Fig. 6(a)). Primitive features are extracted from the input by means of a feature extracting network 611 and these primitive features are unified by means of a feature unifying network 612, which are matched with images stored in a memory network 613. Images 621 and 622 corresponding to 2 frames measured in a unit time are compared as input information to obtain variations in the gradation. This comparison is effected for every mesh in the images to confirm the presence of movement. However, since it is not known the direction of the movement, in the input image 623, the pixels, where variations have taken place, are marked. Further, since the direction cannot be judged by this alone, information on the approximate direction exists separately. The linear direction of the movement is extracted from the image thus prepared by means of the feature extracting network 611 indicated in Fig. 6. Further the feature unifying network 612 determines the direction of the movement by using the extracted linear direction as a group of continuous straight lines 631. However, in general, there is no unnatural movement as indicated by 631 in Fig. 6(c). That is, conceptions on the movement, concretely speaking, that the movement follows a smooth curve 632, etc. are incorporated in a memory network 613. In this way a smooth curve 632 is constructed by matching them. 1.2 Initial visual sensation Various processings such as the recognition of the movement direction, the recognition of the depth, etc. in the initial visual sensation can be formulated as an inverse problem of the problem of identifying the solution from the input data. That is, a system of equations deduced spontaneously from the problem, which is the object, and limiting conditions based on some apriori information are necessary, because the solution cannot be determined only by the system of equations. Now, denoting the variable to be obtained by x and input data by I, the problem can be formulated as a minimization problem as follows; E({x}) = E 1 ({x}, {x}) + λE 2 ({x}) where E 1 represents the energy corresponding to the system of equations, E 2 that corresponding to the limiting conditions, and λ a parameter indicating the ratio of them. Consequently the method described for Eq. (17) can be utilized efficiently. In the following, in order to consider a concrete formulation, the movement direction sensation will be examined. Since the input image (two-valued) I is constant with respect to the movement direction and remains unchanged, ∂I/∂t + (▿I)·V = 0 is valid, where ▿ = (∂/∂x, ∂/∂y), V = (V x , V y ), t represents the time, ▿ the differential vector, and V the velocity vector of the movement direction. The energy E 1 is obtained by integrating the square of the above formula. Next, as the apriori constraint conditions, in the meaning of eliminating the noise, it is supposed that E 2 = ∫∫{▿·V x ) 2 + (▿·V y ) 2 + d x d y Adding them together, E = ∫∫{I x 2 V x 2 + λ(▿·V x ) 2 + I y 2 V y 2 + λ (▿·V y ) 2 + 2∂I/∂t(I x V x + I y V y ) + 2I x I y V x V y }d x d y is obtained. Since the image is divided in the form of a mesh, when suffixes i and j, which are integers, are attached in the x and y directions, respectively, the above equation can be transformed into; where W ij x = (I xi 2 + 2)δ ij - 2λδ i.i-1 W ij y = (I yi 2 + 2)δ ij - 2λδ i.i-1 h i x = 2∂I/∂tI xi h i y = 2∂I/∂tI yi Further, by using V i = (V xi , V yi ), it can be summarized as follows: where W ij and h i can be deduced easily from the definition formula stated above. Utilizing Eq. (17) here, the problem can be lead to the maximization problem of the probability expressed by the following formula: The maximization of this formula can be executed by using the simulated annealing method, etc. 2.1 Movement control It is to determine the input u(t) to the system so as to follow the trajectory d d (t) of the target depending on the time, just as for the control of a robot manipulator, to effect the movement control (Fig. 7(a)). Explanation will be made, by taking a robot manipulator as an example. Now, denoting an n-dimensional articulation angular vector by , the movement equation can be given by; where the first term in the left member is an inertia term; the second term is a Coriolis' centrifugal force term; the third term is a gravitation term; and the right member represents the n-dimensional articulation torque. On the other hand, denoting the direction vector indicating the position of the handle of the manipulator by x, from the principle of the kinematics, x = f(), x = J()  are valid, where J() represents a Jacobian. From the movement equation with respect to x the vector P of the force acting on the handle, i.e. moment, can be expressed by; u = J T ()P where T represents the transposition of the matrix. It is to determine the time dependence of the articulation angle u(t) so that the position and the direction x(t) follows x d (t) to effect the movement control (Fig. 7(a)). However, for the control of the robot manipulator, the determination of a model with a perfect precision is almost impossible because of the nonlinearity of the dynamic characteristics and the undeterminacy of the parameter as well as the nonlinearity between the operation space and the articulation space, etc. Consequently it is necessary at first to identify the dynamic characteristics of the system by means of the neural network. Therefore the dynamic process of the neural network having the time dependence is defined as follows; y i (ℓ) = (x i (ℓ)) where τ is a time constant and  represents the threshold function. Further u is inputted in the input layer and for the output layer it is a component of x i (L) = x. The synapse coupling W ij (ℓ) has the time dependence or it is constant. As soon as the neural circuit network has learned and the learning is terminated, Fig. 7(b) indicates a simple feedforward control. That is, as long as the output x of the robot manipulator differs from the target trajectory x d (t), u T is calculated by using control rule of the articulation torque, depending on the difference x-x d (t). Further the articulation torque u N is calculated from the input x d (t) to the neural network, which has not yet learned and u = u T Lu N is inputted in the manipulator as an external force. When the learning is completely terminated, since u = u N (u T = 0), the torque u N from the neural network is inputted directly in the manipulator and the process proceeds from the feedback control to the feedforward control. The learning in the neural network is effected as follows. Just as the back propagation method, by one of the method the synapse coupling is determined so as to minimize the error expressed by; E = 1 2 Σ(x i (L) - X di (t)) 2 that is, the synapse coupling is varied in the course of time by using; W ij (ℓ) = -c ∂E ∂W ij (ℓ) As another method, the ε method proposed by Balakrishnan in the optimum control theory can be utilized. That is, it is to determine u, which minimizes However, for the input layer (ℓ=2), u is added thereto. A neural network for solving this problem will be described in detail in the following paragraph as a problem of the general optimum control theory. 2.2 Optimum control problem A useful application of the method described above to the optimum control problem is considered, making the most of the parallel processing power of the neural network. Since the optimum control problem can be formulated in general as the minimization (maximization) of a certain evaluation function, this method can be applied to fairly numerous problems. Here it will be shown as an example that this method can be applied to the ε method proposed by Balakrishnun. Now it is supposed that the dynamic process of the object system follows a differential equation; x (t) = f(x(t), u(t), t), where x(t 0 ) = x 0 and x(t) represents a state variable at a point of time t, f a given function, u(t) an operation variable at a point of time t, and = x 0 the value of x at the initial point of time t 0 . Here the minimization of; is considered, where y is a given function and t 1 indicates the final point of time. The problem is to determine the operation variable u(t) and the state variable x(t), which minimize the evaluation function for a system following the dynamic process described above. By the method the problem stated above is formulated as a problem of minimizing: where ∥----∥ represents the norm in a suitable space. Now, supposing that the inner product (x, f) ≦ c(1+∥x∥ 2 ), c being a constant, it is proved that the solution, which minimizes E by ε → 0, converges to the solution of the initial problem. In order to express the variables such as x, u, etc. with two-valued variables X i , U i , etc., a transformation expressed by; where X i (O) = ±1, U i (O) = ±1, is effected. Then, E is given by; If the time is divided in the form of a time mesh and each of the elements expresses X i or V i in order to execute the mapping on the neural network, a scheme indicated in Fig. 8 is obtained. Each of the layers represents X i and V i at a same point of time and the coupling between different layers depends on the given problem and the structure of f and g. INDUSTRIAL APPLICABILITY According to the present invention the following effect can be obtained. The calculation speed for problems of recognizing images, sound, etc., movement control, time dependent large scale numerical calculations, which were difficult to solve by means of a prior art computer, can be increased by taking-in internal structure based on living body physiological knowledge or presumed from that knowledge and by means of a neural network, whose basic principle is the parallel concurrent and competitive action of groups of neural elements.

Claims: A method of recognition processing effected by introducing neural network structures based on living body physiological knowledge or presumed from such knowledge enabling parallel and competitive actions of groups of neural elements as a basic learning principle, consisting of (a) a feature extracting step in which primitive features (such as lines, edges or even figures) are extracted from composite information units (such as images), (b) a feature unifying step in which the extracted features are unified in a learning step to obtain information units of a higher order which facilitate the recognition process by matching recently arrived composite information units with composite information units already learnt, and (c) a storing step in which the higher-order information units are stored, wherein the feature extracting step (a) includes (a1) defining the state f ℓ (r) of neural elements (331) at a two-dimensional position r = (x, y) on a layer ℓ (332) of a neural network in dependence on a user-definable non-linear function F which expresses the strength of coupling (333) between elements of said neural network having linear and non-linear outputs and added noise ξ ℓ , as (a2) propagating the average value of the state of a group of neural elements located in a lower layer to an upper layer according to (a3) deducing from the equation given in step (al) a so-called Fokker-Planck type probability differential equation for where P ℓ (f ℓ (r)) represents the probability that the state of neural elements at a position r on a layer ℓ is f ℓ (r), the stationary solution of this equation in case ℓ is sufficiently great can be given by where T ℓ-1 is a positive constant and energy represents the coupling relation among neural elements in the layer ℓ-1 and can be expressed by using Fourier components F ℓ-1 (q) by the equation with q= - r ℓ-1 being the Fourier frequency giving the maximum of the probability, (a4) extracting, as to the frequency components of the Fourier-transformed energy H and said stationary solution, the features of the composite information unit hierarchically one after the other simultaneously with step (a2), and (a5) eliminating noise unnecessary for the feature propagation, thereby keeping the probability of the coupling strength between adjacent neural elements invariant, wherein the feature unifying step (b) includes (b1) arranging a neural network with a plurality of layers (411, 412, 413) each consisting of neural elements (415, 416, 417), (b2) according to a predetermined given intention coupling the neural elements within a layer and also neural elements of lower layers to neural elements of the next higher layer and depending on the degree of relation within each layer or between different layers, assigning to the respective element coupling a positive coupling value (418) if the relation of adjacent features positively contributes to the synthesis of composite features of a higher conceptional order, and a negative value (419) in the opposite case, and (b3) guiding the elementary features extracted in step (a) to neural elements of the input layer (413) whereby feature unifying processing is carried out by concurrent and competitive actions in the individual neural elements, and wherein the storing step (c) includes (c1) long-term storing high-order information units by coding them as a value of neural element coupling, (c2) memorising high-order information units which have been learnt during step (b), (c3) coupling the neural elements in the input layer according to the high-order information units, (c4) making the neural elements in the intermediate layers transmit information through the neural elements whose coupling strength already been learnt, and (c5) modifying the coupling strength of the neural network dependent on the difference between the determined output values from the output layer and its desired output values. The method of Claim 1 wherein said feature unifying step (b) consists of a processing, by which a problem of minimising the energy consisting of a product of the synapse coupling (W ij ) and a second-order term of the state (x i x j ) of elements and a product of a threshold value () and the state (x i ) of elements is solved, the elements having a function that, if all information inputted in a remarked neural element (i) is greater than the threshold value (), the element is ignited and, if it is smaller, it is in a pause state. The method of Claim 2 wherein said processing of solving said problem of minimising the energy includes a processing constituted by a hypothetical neural network (436), in which the synapse coupling is constant and the threshold value depends on the square root (W ½ ) of the original synapse coupling (433), by introducing a new variable (434) taking a continuous value (z j ) and deducing a new energy consisting of a second-order term of this new variable and a product of this variable, the square root of the synapse coupling and the original state of elements as The method of Claim 1 wherein said storing step (c) is executed by using means for realising memories having different mechanisms corresponding to a long-term memory (574) and a short-term memory (575), which are realised psychologically. The method of Claim 4 wherein said long-term memory processing is executed by using means based on a synapse coupling structure presumed from statistical experimental physiological facts including the dependence of the diameter (R 524) of the tree-shaped protrusion on the distance (X 523) from the centre of the cell body, which is a cerebral physiological knowledge, and the dependence of the number (522) of synapses attached to the tree-shaped protrusion on the diameter (R 524) of the tree-shaped protrusion at the position of the attachment. The method of Claim 5 wherein said structure of synapse coupling is so determined that the amount of information (542), which can be transmitted by the coupling between a predetermined neural element and the closest adjacent elements decreases successively in such a manner that the amount of the coupling between the predetermined neural element and a second adjacent element is 1/2 of the amount of the preceding information, the amount of the coupling between said predetermined neural element and a third adjacent element is 1/4 of the amount of the preceding information, and so forth, on the basis of the optimisation principle, by which the sum of the energy consumption necessary for transmitting information through axons serving as information transmitting medium and the space occupied by the axons is minimised. The method of Claim 4 wherein, in order to determine suitable values (W ij (ℓ)) for the synapses, said long-term memory processing is executed by expressing the synapse coupling between the input layer (ℓ = 1) of the neural network and the next-upper layer by a product of the input information (514) and a random variable; the synapse coupling between the output layer (ℓ = L) and the next lower layer (ℓ = L-1) by a product of the output information and a random variable; and the synapse coupling between the other layers by a product of two kinds of independent random variables as or generating the synapse couplings from two kinds of random variables and determining those random variables for the synapse coupling of the lower layer one after another, starting from the synapse coupled with the output layer, so that the difference between the output information and the target information is decreased. The method of Claim 4 wherein said short-term memory processing is executed by means that make the memory correspond to the minimum values of the energy for the neural network or stationary states of dynamic equations deduced from said energy. The method of Claim 8 wherein said short-term memory processing includes a processing for elucidating psychological facts, which makes stationary states of said dynamic equation appear by setting random values for all the synapse couplings and makes it possible to increase the number of the stationary states (5103) to a certain finite number by increasing the number of the synapse couplings coupled with neural elements, the state (5104) being chaotic, where no stable states can be realised, even if the number of couplings is increased, exceeding said finite number. The method of Claim 2 wherein said processing for solving the problem of minimising the energy consists of an initial visual sensation processing, such as the movement direction recognition, the depth recognition, etc., and a processing for minimising the sum of the energy based on a system of equations deduced spontaneously from the fact that the brightness of the image is kept unchanged with respect to the movement etc. and the energy based on a priori limiting conditions for unequivocally determining the solution. The method of Claim 1 wherein, in a movement control such as the control of a robot manipulator by means of said neural network, in order to control an object following a dynamic equation structurally determined so that the trajectory thereof follows a target trajectory, at first a neural network for identifying the behaviour of the object is newly constructed and the object is controlled by the neural network at the point of time where the output of the new neural network is equal to the output of the object. The method of Claim 2 wherein said processing for solving the problem of minimising the energy consists of a processing, by which the negative solution method is used for the time finite differences, for which the stability of the solution of non-stationary partial differential equations is secured; for the space the finite differentiation is effected by using the finite difference-finite element method; finite-differentiated variables are assigned to neural elements constituting the neural network; and the minimisation of the energy deduced from said equations is executed by a network multiply layered corresponding to the time finite difference.

Inventors: MATSUBA IKUO, MINAMI KEIKO, MATSUBA, IKUO, MINAMI, KEIKO
Applicants: HITACHI LTD, HITACHI, LTD.
Application year: 1989
Main classification: G06F15
IPC codes: G06T1, G06N3