Abstract — A multilayer neural network based on multi- valued neurons (MLMVN) is a neural network with a traditional feedforward architecture. At the same time this network has a number of specific properties and advantages. Its backpropagation learning algorithm does not require differentiability of the activation function. The functionality of MLMVN is higher than the ones of the traditional feedforward neural networks and a variety of kernel-based networks. Its higher flexibility and faster adaptation to the mapping implemented make possible an accomplishment of complex problems using a simpler network. The MLMVN can be used to solve those non-standard recognition and classification problems that cannot be solved using other techniques. In this paper we use the MLMVN as a tool for the blur identification problem. A prior knowledge about the distorting operator and its parameter is of crucial importance in blurred image restoration. I. INTRODUCTION multilayer neural network based on multi-valued neurons (MLMVN) has been introduced in [1] and then investigated and developed further in [2]. This network consists of multi-valued neurons (MVN). That is a neuron with complex-valued weights and an activation function, defined as a function of the argument of a weighted sum. This activation function was proposed in 1971 in the pioneer paper of N. Aizenberg et al. [3]. The multi-valued neuron was introduced in [4]. It is based on the principles of multiple-valued threshold logic over the field of the complex numbers formulated in [5] and then developed in [6]. A comprehensive observation of the discrete-valued MVN, its properties and learning is presented in [6]. A continuous-valued MVN and its learning are considered in [1],[2]. The most important properties of MVN are: the complex-valued weights, inputs and output coded by the K th roots of unity (a discrete-valued MVN) or lying on the unit circle (a continuous-valued MVN), and the activation function, which maps the complex plane into the unit circle. It is important that MVN learning is reduced to the movement along the unit circle. The MVN learning algorithm is based on a simple linear error correction rule This work was supported in part by the Collaborative Research Center for Computational Intelligence of the University of Dortmund (SFB 531, Dortmund, Germany) and by the Academy of Finland, project No. 213462 (Finnish Centre of Excellence program (2006 - 2011). Igor Aizenberg is with Texas A&M University-Texarkana, P.O. Box 5518, 2600 N. Robison Rd. Texarkana, Texas 75505 USA, e-mail: igor.aizenberg@tamut.edu Dmitriy Paliy and Jaakko T. Astola are with Tampere International Center for Signal Processing, Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland, email: firstname.lastname@tut.fi and it does not require differentiability of the activation function. Different applications of MVN have been considered during recent years, e.g.: MVN as a basic neuron in the cellular neural networks [6], as the basic neuron of the neural-based associative memories [6],[7]-[10], as the basic neuron in a variety of pattern recognition systems [10]-[12], and as a basic neuron of the MLMVN [1],[2]. MLMVN outperforms a classical multilayer feedforward network and different kernel-based networks in the terms of learning speed, network complexity, and classification/prediction rate tested for such popular benchmarks problems as the parity n, the two spirals, the sonar, and the Mackey-Glass time series prediction [1],[2]. These properties of MLMVN show that it is more flexible and adapts faster in comparison with other solutions. In this paper we apply MLMVN to identify blur and its parameters, which is a key problem in image restoration. Usually blur refers to the low-pass distortions introduced into an image. It can be caused, e.g., by the relative motion between the camera and the original scene, by the optical system which is out of focus, by atmospheric turbulence (optical satellite imaging), aberrations in the optical system, etc. [13]. Any type of blur, which is spatially invariant, can be expressed by the convolution kernel in the integral equation [14],[15]. Hence, restoration (deblurring) of a blurred image is an ill-posed inverse problem [16], and regularization is commonly used when solving this problem [16]. There is a variety of sophisticated and efficient deblurring techniques such as deconvolution based on the Wiener filter [13],[17], nonparametric image deblurring using local polynomial approximation with spatially-adaptive scale selection based on the intersection of confidence intervals rule [17], Fourier-wavelet regularized deconvolution [18], expectation-maximization algorithm for wavelet-based image deconvolution [19], etc. All these techniques assume a prior knowledge of the blurring kernel, characterized by the point spread function (PSF), and its parameter. When the blurring operator is unknown, the image restoration becomes a blind deconvolution problem [20]- [22]. Most of the methods to solve it are iterative, and, therefore, they are computationally costly. Due to the presence of noise they suffer from the stability and convergence problems [23]. The original solution of blur identification problem that is based on the use of MVN-based neural networks was proposed in [12] and [24]. Any blur specifically distorts the Multilayer Neural Network based on Multi-Valued Neurons and the Blur Identification Problem Igor Aizenberg, Member, IEEE, Dmitriy Paliy, and Jaakko T. Astola, Fellow, IEEE A 0-7803-9490-9/06/$20.00/©2006 IEEE 2006 International Joint Conference on Neural Networks Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada July 16-21, 2006 473