Recursive learning of image parameters and restoration of images using EM based learning algorithm

Birçok klasik görüntü onarım tekniği bulanıklık işlevinin bilindiği varsayımı altında çalışır. Ancak, gerçek hayat problemlerinde sadece gözlem verisi elde edilebilmekte bozucu sistemler hakkında yeterli bilgi sağlanamamaktadır. Bu yüzden görüntü onarımının ilk adımı bozucunun öğrenilmesi (tanınması) işlemidir. Geçmişte, görüntü ve bulanıklık parametrelerinin öğrenilmesi Enbüyük Olabilirlik (EO) problemi olarak ele alınmış ve Beklenti Enbüyükleme (BE) yordamı ile çözülmüştür. Özellikle BE yordamının E adımında kapalı yapıda bir çözüm bulunması bu yordamı daha cazip bir hale getirmektedir. Görüntü ve bulanıklık parametrelerinin tüm görüntü verisi kullanılarak öğrenilmesi geçmişte çalışılmış olmakla birlikte, parametrelerin yinelemeli BE’ye dayalı öğrenilmesi daha önce çalışılmamıştır. Yinelemeli teknikler dinamik işlem yetenekleri sayesinde tüm veri üzerinde işlem yapan yöntemlere nispetle çok daha az bellek ihtiyacı duyarlar. Daha az bellek ihtiyacı ise özellikle görüntü işleme alanında çok önemlidir. Bu çalışmada yeni bir eşzamanlı yinelemeli parametre öğrenme ve görüntü onarım yöntemi sunulmuştur. Dinamik Bayesçi Ağ (DBA) yapısında yeni bir çözüm önerilmiştir. Sunulan yöntem EO parametre tanıma ve durum kestirimi için en iyi Kalman yumuşatma ifadelerini içerir. Kalman yumuşatma ifadelerinin yoğun hesaplama gerektirmesi sebebi ile Kalman süzgeç yaklaşıklığı kullanılmıştır. Aynı zamanda, onarılmış görüntü eş zamanlı olarak bu süzgeç çıkışından elde edilmektedir. Görüntü ve bulanıklık parametrelerinin BE öğrenme problemi kapalı yapıda çözümlenmesi başarılmıştır.  Yöntemin başarımı gerçek görüntüler üzerinde yapılan benzetim ve denemeler ile verilmiştir.  Anahtar Kelimeler: Beklenti enbüyükleme, bulanıklık ve görüntü tanıma, yinelemeli işleme, kalman yumuşatma ve süzgeçleme.The image restoration problem can be defined as the general problem of estimating the ideal image from its blurred and noisy version. Many classical image restoration techniques have been reported under the assumption that the blur operation is exactly known. In real life applications, the corruption mechanism of any system is not known because only observed data is available, so it is necessary to handle uncertain events and observations. The image restoration problem is in general ill-posed; a small perturbation on the given data produces large deviations in the solution. The direct inversion of the blur transfer function usually has a large magnitude at high frequencies, therefore excessive amplification of noise results at those frequencies. Clearly, this is not an acceptable solution for noisy images. To overcome the noise sensitivity problem of the inverse filter, some filters have been developed based on the least-squares structure. The Wiener filter is based on batch processing which is usually implemented in the frequency domain. The Kalman filter is based on recursive processing which is usually implemented in the spatial domain. Both solutions only work when blur, image and noise parameters are known. The first step for image restoration is the identification of degradation. Consequently, modeling uncertain relationships among many kinds of variables and learning (identification) such variables are important topics. The blur and image parameter identification problem was formerly formulated as a constrained Maximum Likelihood (ML) estimation procedure which was based on optimizing the probability density function (pdf) of the observed image with respect to the unknown parameters. But, the direct optimization of the likelihood function is not feasible, because of its highly nonlinear character. The Expectation Maximization (EM) algorithm is a very popular and widely used algorithm for the computation of ML estimates. There are two steps in EM algorithm, as E (Expectation) and M (Maximization). The EM algorithm finds the conditional expectation of the log-likelihood of complete data given the observed incomplete data. In the E-step, the conditional expectation of the "hidden variables" is calculated.  In the M-step, this expectation is maximized with respect to the parameters. The advantage of the EM method is such that it avoids operating directly on the nonlinear likelihood function. The EM algorithm becomes more attractive if its maximization step can be formulated analytically. Even though batch processing of the EM based blur identification and restoration problem needs large memory size, recursive techniques allow dynamic processing with modest storage requirements. Although the EM learning was applied to learning of unknown image and blur parameters based on batch image processing before, recursive EM learning of unknown image and blur parameters has not been studied as much as necessary. Many time series models, including the Hidden Markov Models (HMM) and Kalman Filter Models (KFM) used in filtering and control applications, can be viewed as examples of Dynamic Bayesian Network DBNs. Since, a Bayesian Network is a graphical way to represent a particular factorization of joint distribution; we propose that state space image model can be represented as a DBN. In this work, we introduce a new simultaneous recursive parameter learning and image restoration method based on the ML parameter identification and state estimation for images. We present a new formulation which is given in a Dynamic Bayesian Network (DBN) framework. We focus on the problem of learning the parameters of a Bayesian network. This technique incorporates optimal Kalman smoothing equations for ML parameter identification and state estimation. The use of Kalman filtering instead of Kalman smoothing is employed because of the computationally extensive processing of smoothing. In addition, a restored image is obtained simultaneously as the output of the Kalman filter. We manage to solve the EM learning problem for image and blur parameters in closed form. Although our proposed method processes huge data, because of the recursive structure it does not need large size storage. Performance evaluation of the method is given based on experiments carried out upon real images.  Keywords: Expectation-Maximization, Blur and Image Identification, Recursive Processing,  Kalman Smoothing and Filtering