In applications of (nonlinear) model predictive control a more and more common approach for the state estimation is to use moving horizon estimation, which employs (nonlinear) optimization directly on a model for a whole batch of data. This paper shows that horizon estimation may also be used for joint parameter estimation and state estimation, as long as a bias correction based on the Kalman filter is included. A procedure how to approximate the bias correction for nonlinear systems is outlined.

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The mechanism of endogenous circadian photosynthesis oscillations of plants performing crassulacean acid metabolism (CAM) is investigated in terms of a nonlinear theoretical model. Blasius et al. used throughout continuous time differential equations which mode adequately reflect the CAM dynamics. They showed that the membrane effectively acts as a hysteresis switch regulating the oscillations. The model shows regular endogenous limit cycle oscillations that are stable for a wide range of temperatures, in a manner that complies well with experimental data. The circadian period length is explained simply in terms of the filling time of the vacuole. In this paper, we discuss the nonlinear dynamical model of CAM from the control theoretical viewpoint. In particular, we present an adaptive observer to estimate the states and the nonlinear function in the dynamics of the tonoplast order with the slow manifolds approximation.

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In this paper, we are concerned with a problem of optimal selection of the gain matrix of a linear observation for the stationary Kalman filter. In the previous works of the author, we introduced an information theoretic criterion based on a generalized Water Filling Theorem to obtain a gain matrix which minimizes the stationary error variance. The merit of this approach is that analytical and numerical solutions are rather easily obtained compared with the case of quadratic cost functions on the estimation error and the gain matrix. In this solution process, however, the Riccati equation of the error covariance matrix reduces to a quasi linear equation, and the property of the solution is somewhat different from that of the usual Riccati equation. This paper is concerned with the case of a quadratic cost function, and we obtain an expression of the condition of optimality. Also, a method of numerical solution is proposed

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A stochastic realization of Markov model on the time-frequency plane with the sub-space method is discussed to apply for identification of instruments and keys in a music transcription system that listen to sounds and display notes. The system projects music signal onto the time-frequency plane applying Gabor wavelet transform and estimates actually played pitch names on the plane by stationary Kalman filter that refers instruments and keys as parameters.

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In this paper, we consider simultaneous estimation of the state variables and unknown parameters of Induction Motors (IMs) using nonlinear filters. Simultaneous estimation is the most general method for sensorless controlled IMs, and an Adaptive Observer is used as its estimator generally. At present, by the advance of computer processors, nonlinear filters have been applied to various occasions, so we describe the method for applying nonlinear filters to Induction Motors model, and consider its estimate performance by simulations. Simulation results showed that nonlinear filters have more accuracy estimate performance than the Adaptive observer, and the excellent noise immunity.

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In this paper, a consistent estimation method for “errors-in-variables” (EIV) models is studied. The extended generalized least-correlation (EGLC) method has been proposed for the EIV models identification in the case where input and output measurements are corrupted by white noise. To obtain more stable and accurate estimates, we introduce the prefilter and the extended vectors. It is expected that the proposed EGLC method can give more stable and more accurate estimates because the regression vector is more correlated with the filtered and the extended regression vector . The results of a simulated example indicate that the proposed algorithm provides good estimates.

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This paper addresses identification of Hammerstein nonlinear ARMAX model by using a numerical algorithm for subspace state space system identification method. When the static memoryless nonlinear block of Hammerstein models is considered as the sum of some known functions, the ARMAX model of the linear part can be estimated as a multi-input single-output (MISO) system with the subspace identification method. The ARMAX model can be derived from the estimated state space model. The simulation results show that the proposed algorithm is effective.

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In this paper, a coal type selection method for a coal fired boiler in a thermal power plant is proposed. Coal combustibility and fly ash fusibility are derived by fuzzy inference using the coal and fly ash properties. Then, combustion state for each coal type is evaluated by the coal combustibility and fly ash fusibility. Applicable coal type is selected by the evaluated combustion state. The proposed coal type selection method is evaluated by the data collected from test furnace and an actual thermal power plant.

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In our previous research, we have developed the methods to estimate the VTEC (Vertical Total Electron Content) and to provide the ionospheric delay maps for certain local region such as the sky over Japan. The estimated local models of the VTEC over Japan were also applied to the PPP (Precise Point Positioning), and the positioning accuracy was improved effectively. On the other hand, for the accuracy of PPP, satellite orbit errors and satellite clock errors are also factors should be properly corrected. In this paper, a method to estimate the satellite orbit and clock errors is proposed by extending the algorithms in our previous researches. In the method, the broadcast ephemerides are assumed as the nominal satellite orbits and clock errors. Then satellite position errors and clock errors with respect to the nominal values are modeled by first order Markov processes or Brownian motion processes, and they are estimated by the Kalman filter together with the other unknown quantities such as tropospheric delays, hardware biases and ambiguities. The generated correction information can be easily applied to not only PPP but also any kinds of GNSS positioning algorithms. The proposed method can work by using the GPS data collected by reference stations such as GEONET (Gps Earth Observation NETwork) provided by the Geographical Survey Institute (GSI) of Japan.

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In [1], the authors have developed a method of estimating the local models of an ionosphere VTEC (Vertical Total Electron Content) [2] based on GNSS (global navigation satellite systems) regression models (abbreviated by GR models) and we have shown the ionospheric delay (or advance) in the sky over Japan using the Gps Earth Observation NETwork (GEONET) data provided from the Geographical Survey Institute (GSI) of Japan [3]. In the GNSS positioning, the ionospheric delays are predominant error factors. Therefore, it is important to correct them for higher accurate positioning. In this paper, we present a method of estimating an ionospheric TEC (Total Electron Content) based on GNSS Regression Models. The method of estimating the local models with power series in Spherical Cap Harmonics Expansion (SCHs) [4] for TEC at the ionospheric pierce points in the ionospheric single layer model is discussed.

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In this study, a revised radial basis function (RBF) network is proposed and applied to the identification problems of the nonlinear system and the interactive art system. In the revised RBF network, the structural parameters such as means and variances of the radial basis functions in the neurons are determined automatically and so revised RBF network can be easily applied to the practical complex problems such as the interactive art system. The interactive art system outputs the art expressions such as the sound and graphics using the artificial sensibility surfaces that are identified using the revised RBF network.

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It is well known that the learning effects of agents are important on the diffusion process of rumors in social science. Though these processes can be simulated by agent-based simulations, we cannot disregard the learning effects of agents. In this paper, we deal with an agent-based problem including Q-learning, a kind of reinforcement learning. At first, we design action rules and Q values of agents. Agents decide their attitudes by attitudes which neighbor agents express, and then Q values are updated. Finally, by performing and analyzing simulation studies the effect of proposed algorithm is confirmed.

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In this study, we propose a revised radial basis function (RBF) network algorithm and apply this algorithm to the recognition of the lung region. The recognition results are compared with those obtained using the conventional RBF network and the conventional sigmoid function neural network trained using back propagation algorithm. It is shown that the revised RBF network is accurate and useful method because the parameters are automatically determined. Then, the revised RBF network is applied to the recognition of the pulmonary vessels and the bronchial trees of the lung. Furthermore, we applied the revised RBF network to the computer aided diagnosis (CAD) of the lung cancer.

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In this study, the multi-layered Group Method of Data Handling (GMDH)-type neural network self-selecting optimum neural network architecture is applied to the computer aided image diagnosis (CAD) of the cancer of the liver. The GMDH-type neural network algorithm has an ability of self-selecting optimum neural network architecture from three neural network architectures such as sigmoid function neural network, radial basis function (RBF) neural network and polynomial neural network. The GMDH-type neural network also have abilities of self-selecting the number of layers, the number of neurons in hidden layers and useful input variables. This algorithm is applied to CAD and it is shown that this algorithm is useful for CAD of the cancer of the liver and is very easy to apply practical complex problem because optimum neural network architecture is automatically organized.

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Tn this paper, we propose a new Gaussian sum filter [1] based on the stochastic equivalent linearization technique [2] for suboptimal state estimation of discrete time nonlinear systems. The derived nonlinear filter will be expected to be a low computational cost and high quality for recursive state estimation.

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Blind estimation of single-input multiple-output (SIMO) finite-impulse-response (FIR) channels has been extensively studied over the recent years. However, due to the difficult nature of the blind problem, there are certain classes of channels that cannot be successfully estimated by existing blind algorithms. Such examples include FIR channels with trailing (or preceding) close-to-zero coefficients, driven by correlated input signal. In this case, classical methods based on the second-order statistics fail to provide accurate blind channel estimates. Moreover, if the channel is not minimum phase, the estimation task is more complicated. In this contribution, we explore an alternative approach utilizing hierarchical Bayesian model together with variational approximation. With this approach, blind estimation error for the difficult channels can be significantly improved, which was verified via numerical simulation.

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A method to reduce Peak to Average Power Ratio (PAPR) in OFDM systems with null subcarriers is considered. For a given optimal or suboptimal pilot set designed for channel estimation and/or synchronization, deliberate introduction of pilot phases can lead to minimum PAPR of the time domain OFDM signals. In this paper, we presented a novel algorithm based on cross entropy (CE) optimization techniques to design phase to each pilot in order to minimize the peak pilot tones. Compared to the other exhaustive search methods, the proposed algorithm converges fast to the near optimal phase. Due to its high convergence rate, proposed scheme has the potential to make practical design of phase for different pilot subcarrier sets. The effectiveness of the proposed pilot phase design is demonstrated with several examples including the one based on the IEEE 802.11a.

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Rayleigh scattering in an optical fiber has the property that its spectrum shifts in proportion to strain and/or temperature changes, which makes distributed sensing of strain/temperature possible. To estimate the spectral shift, cross-correlation between spectra before and after the shift can be used. However, when the shift is large, the overlap portion of the spectra becomes small and false detection of a correlation peak readily arises. In this paper, we analytically obtain the probability of false alarm for cross-correlation coefficients and determine the threshold that gives a specified constant false alarm rate (CFAR). It is shown that stable estimates of spectral shifts are obtained by using the CFAR threshold.

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It is known that Electroencephalograph (EEG) signal shows specific responses according to the event (e.g. visual stimulus, cognition and motor imagery). Especially, by classifying short time EEG signal, features are used to control an electronic device (such system is called brain computer interface: BCI). Generally, in feature extraction from EEG signal, these features are extracted by using linear method such as FIR filter and wavelet transform etc. Though, linear method is not suitable because impulse noise distorts the important features. To avoid this, the morphological analysis method with non-linear characteristics has been used in this fields. In this paper, we propose a design method of structuring function that determine the filter characteristic of morphology. The morphological method is compared to discrete wavelet transform (DWT) from the view point of filter characteristics. We apply our method to real data observed from visual stimulation.

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Human motion has been analyzed in the field of welfare, care worker, sports and so on. Motion capture system is one of useful measurement equipments of human motion. ^[1] It measures reflection of light from some markers which are attached on a body and calculates their positions based on a coordinates which is set beforehand. Numerous multivariable time series data is obtained by this system when human motion is measured. In order to describe the motion mathematically, processing multivariable time series data is needed. In this paper, we deal with modeling a human motion to steer a unicycle.

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This paper discusses the brain activity evaluated from mapping technique ^[1] based on hierarchical decomposition analysis (HDA) ^[2]. Hemodynamic data corresponding to brain activity is measured by using a multi-channel near-infrared spectroscopy (NIRS). The experiment concerned with the sense of memory is carried out for healthy subjects. The difference between two similar tasks becomes clear by proposed technique.

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In this paper we present the localization using received signal powers for sensor networks. The purpose of this study is to develop the indoor positioning systems utilized IEEE std 802.15.4 based wireless sensor networks.The distance between nodes can be presumed by using the RSSI (received signal strength indicator) to use the wireless for the data communication in the sensor network. Therefore, it becomes possible for the position of the sensor node to presume by using the RSSI from a certain sending source if a number of base stations already measured the position.The variation of the RSSI is large because of the influence by the measurement environment. Therefore, it is necessary to acquire a lot of the RSSI data to improve the position estimation because a lot of error margins are included in the presumption distance. In this paper, we discuss a distance transformation model applying Friis equation or experimental results model. Namely, the signal powers distribution are modeled as the Rayleigh distribution. We propose an optimal distance model from the probability density functions of the signal powers.First, we describe its implementation on a ubiquitous device. Next, we examine the estimation of distance I with measurement values, and a method for estimating locations. Based on the above method we show the experimental results. Finally we mention a brief summary and future work.

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An infectious disease is still a menace to our health even in high-technogical and advanced information society. A prevention of prevalence of the infectious disease is one of important problems in epidemiology. In such a problem, the mathematical model which describes the spread of the infectious disease has a very important role in the prediction and the analysis of prevalence of the infectious disease. In this paper, we study the modeling of the infectious disease in populations consisting of four populations: susceptible, infective, recovered and vaccinated ones. Noting that the real infectious disease contains some kinds of random fluctuations caused by changes in the environment and the weather, we propose the stochastic infectious model. By using the proposed stochastic model, we analyze the influence of the random fluctuation and vaccination on the spread of the infectious disease by numerical simulations.

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In this paper we study the fixed-interval optimal smoothing problems for a class of linear continuous-time systems with impulsive effects. We adopt a maximum likelihood (ML) approach and derive a maximum likelihood noncausal estimator on the fixed time interval. We derive an extension of the Fraser's algorithm for the linear discrete-time smoother to the impulsive systems utilizing the filtering theory of the impulsive systems.

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In this paper we study the H_∞ smoothing problem for a class of linear continuous-time systems with impulsive effects and Gaussian white noises. We adopt the forward-backward filtering theory as a solution of this problem. Two types of Riccati differential equations depend on the H_∞ bound, i.e., γ value. The theory presented in this paper can be restricted to the one for discrete-time systems. Finally we give numerical examples.

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This paper tries to evaluate the flow of passengers towards the emergency exit in system simulation based on an autonomous agent and multi-agent system. For the representation of behavior of a panicking passenger, an internal model that reflects the changing mind of a passenger who is experiencing unstable emotions during an aircraft accident is introduced into the system simulation. By this internal model, an agent of a panicking passenger exhibits non-adaptive behavior, which means not going along with social rules toward queuing in line at the emergency exit in the cabin. From the simulation studies, it was found that the selfish and abnormal behavior of a passenger, attributed to panicking emotions, caused a slowdown in the evacuation flow in the cabin that resulted in a delay in the evacuation egress time from an aircraft after an accident.

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In this paper, we analyze a new discrete-time GeomX/G/1 queue model with multiple vacations. We obtain the Probability Generating Function (P.G.F.) of the queue length by using the method of an embedded Markov chain, and the mean of the queue length by using L'Hospital rule. We also derive the P.G.F. of the busy period and the probabilities for the system being in a busy state or in a vacation state. Moreover, we derive the P.G.F. of the waiting time based on the independence between the arrival process and the waiting time. Finally, we show some numerical results to compare the means of the queue length and the waiting time in special cases.

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In mobile networks, how to control energy consumption is one of the most important issues for the battery-powered mobile stations. The standard energy-saving mechanism for conserving power of the mobile stations is “sleep mode”. Based on the operating mechanism of the sleep mode in the conventional power saving class (PSC) type III, and taking into account the self-similar nature of a massive-scale wireless multimedia service, in this paper, we build a batch arrival queueing model as a Geom^X/G/1 queuing model to suppose the batch size as a random variable following a Pareto(c,α) distribution. By using a discrete-time imbedded Markov chain, we analyze the probability generating functions of the queueing length and the waiting time of the system. Then we give the formula for performance measures in terms of the response time of data frames and the energy saving ratio. Moreover, we consider a sleep-delay into the system to consider how the sleep-delay impacts on the network performance. In numerical results, we evaluate the system performance when considered with different degrees of self-similarity and different sleep window sizes. We also compare the average performance measures for systems both with sleep-delay and without sleep-delay by using numerical results.

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In this paper we study the H_∞ tracking problems with preview for a class of linear continuous-time Markovian jump systems. The systems are described by the switching systems with Markovian mode transition. The necessary and sufficient conditions for the solvability of the H_∞ tracking problem are given by coupled Riccati differential equations with terminal conditions. Correspondingly feedforward compensators introducing future information are given by coupled differential equations with terminal conditions. In this paper we focus on the derivation method of preview compensator dynamics from the point of view of dynamics constraint. We derive the pair of coupled preview compensator dynamics and coupled Riccati differential equations by calculating the first variation of the performance index under the dynamics constraint.

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A method of generating artificial wind load is proposed incorporating with the recommendations for statistics of the random wind offered in the architecture community. The method consists of two steps; the first one is to generate the wind velocity from the computer simulated white noise, and the second is to produce the wind pressure from the wind velocity using the aerodynamic admittance. An inverse problem is further investigated for estimating wind load acting on the building structure from the measurement data which is made on the response of the structure.

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Under multi-source and multi-microphone configurations, it is clarified that the decomposed spectrum is uniquely expressed in terms of the product of a source and its transfer function to the corresponding microphone. We propose a new method for solving the permutation problem inherent in FDICA (frequency domain independent component analysis) by utilizing prior information about the relative position of sound sources. The proposed method can also solve the channel selection problem for the target sound estimation, which is critical to a practical application of FDICA. The proposed methods have been verified by several experiments in a real environment.

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Electroencephalogram (EEG) and electrocardiogram (EKG) signals are usually contaminated by artifacts. Characteristic analysis based on contaminated signals would lead to spurious results. In order to extract reliable characteristics from EEG and EKG signals, it is necessary to select available signals. In this paper, a real-time data classification system for EEG and EKG analysis was developed. Automatic artifacts detection is helpful for checking long-time recording, and real time classification is useful to obtain satisfactory recording and better understanding of artifacts.

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We study static hedges of knock-in/out options written on the price ratios of several risky assets. These options are knocked-in/out when one of the prices hit against another one. The proof is given only for the simple cases.

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This paper is a survey on the studies around the random walk hypothesis and related validity of principal component analysis done by the authors and their coworkers in recent years.

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In [4], we study mathematical formulation of an optimal execution problem in consideration of market impact and some properties of the corresponding value functions. But there are few studies, including [4], which treat the noise of market impact. In this study we construct a model with random market impact as a generalization of [4]. We consider the case where the noise of market impact in a discrete-time model is given as i.i.d. random variables, and then we derive a continous-time model as a limit in which the noise is described as a jump of a Lévy process.

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We consider a slight perturbation of the Schwartz-Smith model for the electricity futures prices and the resulting modified spot model. Using the martingale property of the modified price under the risk neutral measure, we derive an arbitrage free model for the spot and futures prices. As the futures price formula is based on the arithmetic average of the spot prices, it is highly non-linear. Hence, we use the particle filtering methodology as our identification method for estimation. The main advantage of the new model is that it avoids the inclusion of artificial noise to the observation equation for the implementation of the particle filter. The extra noise is build within the model in an arbitrage free setting.

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In an earlier paper Liang [1], the author applies the framework of Merton [2] to a market model that consists of a no-recovery defaultable bond as well as a riskless asset. In this paper, we extend the result of Liang [1] for a non-zero recovery bond.

We study an optimization problem of consumption and investment strategy for an investor who can invest in a riskless asset and a defaultable asset with a positive recovery. Merton [2] is the first to formulate and solve such an optimization problem, and many researchers have extended and generalized the model of consumption and investment. However, optimization problems with defaultable assets have been rarely studied, though it is so important in today's financial market. This paper applies the framework of original Merton's model to a new market model that consists of a recovery bond as well as a riskless asset. Under the assumption that the defaultable asset's price is modeled as a geometric Brownian motion with an unpredictable jump to a positive value, the optimal problem is reformulated and analytically solved.

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In this paper, we give a specific stochastic model of the integrated assessment for climate changes where very explicit solution is available. In doing this, we maintain the structure of DICE model as well as the explicit property of Ramsey model by considering a certain stochastic extension causing “parallel shifts”.

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We study an optimal execution problem in consideration of market impact as some regular stochastic control problem. We focus on mathematical formulation of such an optimization problem and study some properties of the corresponding value functions. We formulate our optimal execution problem as a discrete-time model and describe the value function with respect to a trader's optimization problem. By shortening the intervals of execution times, we derive a value function of a continuous-time model and study some properties of them. We show that the properties of the continuous-time value function vary by the strength of market impact. Moreover we introduce some examples of this model, which tell us that the forms of the optimal execution strategies entirely change according to the amount of the security holdings. As one of consequences, we observe some phenomenon by using typical examples of our model, that “the form of market impact function is concave or S-shape.”

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The present paper is a survey on our studies on pricing of an exotic warrant. The problem becomes a double stopping problem and is solved numerically by constructing “binary forest”.

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We analyze a money demand function in long equilibrium relation that is defined by a cointegration property among (money, gdp, interest rate). A wide sense of money ”M₂＋CD” consists of narrow money ”M₁” and wide one ”quasi currency＋CD”(denoted by q-money). Regarding money as an asset, a rigorous correspondence of money to interest rate requires that M₁ and q-money should be coupled with short-term interest rate and spread interest rate (long-term interest rate minus short-term one) respectively. Hence, The cointegration between M₂＋CD and gdp should be described by two kinds of interest rates stated above.Due to a deflationary economy in Japan, cointegration property is said to break down, because for future anxiety people save money and the balance between gdp and total amount of money is disturbed. Such saved money is called ”precautionary demand”. Assuming that precautionary demand is proportional to the magnitude of recession, we define adjusted moneys M_1adj ＝ M₁ － k₁ * |recession| and q-money_adj ＝ q-money － k₂ * |recession| respectively and can show that the cointegration property holds among (M₂＋CD, gdp, interest rates) by using M_1adj and q-money_adj with parameters k₁ and k₂ estimated.

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This paper considers the problem for designing of optimal vector smoothing splines with equality and/or inequality constraints. The vector splines are constituted employing normalized uniform B-splines as the basis functions. Then various types of constraints are formulated as linear function of the so-called control points, and the problem is reduced to quadratic programming problem. The performance is examined by some numerical examples.

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A primal-dual setting is presented for probabilistic feasibility of a robust Linear Matrix Inequality (LMI), where the coefficients of the LMI depend on uncertain parameters. The primal problem is to find decision variables satisfying the robust LMI for almost all uncertain parameter values, where a probability measure is introduced onto the uncertainty set. Then, a probabilistic dual formulation is introduced as a system of an integral inequality and integral equations. It is proved that the probabilistic primal problem is infeasible if and only if the probabilistic dual problem is feasible. As an application of this result, a standard LMI feasibility problem is also tackled.

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In this paper, we propose a novel differential evolution (DE) with growing population mechanism. We can apply the proposed DE algorithm to the optimization problems without prior adjusting of population size. The proposed DE algorithm starts an optimization task with the minimal population size, adjust population size to the searching process. Population size is gradually grown during optimization task. The proposed DE algorithm adds individual to population and deletes individual from population using search history in each individuals. The performance of the proposed DE algorithm is evaluated with two multimodal function optimization problems. The experimental results indicate that the proposed DE algorithm has an ability of global optimization without prior adjusting of population size.

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Nonlinear systems described by simple mathematical models often exhibit extremely complicated behavior called chaos. In this paper, chaotic behavior, exhibited by the one-dimensional difference equation, is investigated. A method for constructing nonlinear functions with the invariant density is newly demonstrated. Furthermore, the method for finding the invariant density of the nonlinear function is also presented. Nonlinear functions, treated in this study, are piecewise linear ones and the invariant density is piecewise uniform.

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To implement anticipative road following process, a stochastic perception-control coupling is introduced on satellite-roadway-vehicle network. Random shift of the boundary probability is detected via the saccadic perception process and interpolated by using stochastic growth dynamics to generate continuous trajectories subjected to human's unexpectable decision making. The validity of stochastic path generation scheme is verified via simulation studies.

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We propose a face-tracking system for AIBO by using skin colors. If AIBO finds a face, AIBO can acquire a situation of user and act more actively. In this paper, we focus on the human-face, which has many kinds of characteristic parts in a human (i.e. age, gender, expression). We detect a face using the radial basis function (RBF) network and the particle filter. First, we use the RBF network for the purpose of skin color recognition. Second, we use the particle filter in order to detect a face in skin color area, based on the motion pattern of a face. And, in order to show the effectiveness of the proposed method, we perform experiments. In various light conditions, we have relatively good results of the skin color recognition. Furthermore, we show the output value distribution in a color space. We can see a possibility of little false recognitions in unknown color, using the RBF network. Moreover, by applying moving images to the particle filters, we detect and track a face in various noisy environments by using the particle filter. Finally, we achieve the face tracking system for AIBO in a real system. It will be shown that AIBO can detect and track a face in many kinds of light conditions.

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In this paper, we present an algorithm to deblur the images degraded by the circular motion. We consider that the motion is driven by a constant acceleration and the point spread function is assumed to be unknown. Therefore, we perform motion estimation algorithms before deblurring the images. To deblur the images, we use a bidiagonalization algorithm employed together with Tikhonov regularization. Motion deblurring is perfomed at each circular motion path. The circular motion paths are obtained by using a digital circle generating algorithm. To obtain the complete deblurred images, deblurring results are combined. As the deblurred images are distorted by the empty pixels, the median filter is used to keep the images intact. Experimental results show the feasibility of the presented motion deblurring algorithm.

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In the deconvolution of motion blurred images, a lot of research has been done under the assumption of uniform motion. In case of uniform motion, necessary parameter for restoration is motion length of blurring. However, not only the blur magnitude, but also the coefficients of the blur kernel must be estimated. We will not restore by only motion distance. In this paper, we consider a method to estimate motion blur functions which deal the non-uniform motion case. The point spread function (PSF) is a cause for this degradation of images, which will be used for the deconvolution of blurred images. Therefore, the estimation of point spread functions from the blurred images is important.

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