SUMMARY EDGE ADAPTIVE KALMAN FILTERING Restoration and edge detection are fundamental pr obi eras ini mage anal ysi s. Restoration of images can be defined as the gene ral problem of estimating a two dimensional C2-D3 object from a degraded version. The cause of these degrada tions are genarlly the imperfections in the electronic or photographic medium. In this thesis, we deal with pixels of monochromotic images representing the gray levels, degraded by an additive Gaussian white noise with zero mean. We assume that there is no blurring in these degraded image and try to recover the original image from noisy observations. Therefore, the degrada tion can be modelled as, rCm,n)=sCm,n3+VCm,n) CIS. 13 where rCra, nD, sCm, rO, and VCrn,nD represent the observed image, the orijinal image of size MXM, and additive whi te Gaussian noise, respectively. The estimation procedures are carried out using some statistical knowledge on the orijinal image, and the observation noise. In this thesis, recursive image restoration techniques based on Kalman approach has been used. Kalman filtering for the restoration of degraded images has some advantages when compared to other stoc hastic filtering techniques. First, it is recursive in the spatial domain, it does not require too much storage. Therefore, it is suitable for implementation on micro computers. Second, less computations are required due to recursibility. In Kalman filtering, it is necessary to have app ropriate initial conditions so that the overall estimate will be optimal. This can be especially important for the first few data points. Likewise, in two dimensional C2-DD Kalman filtering, it is necessary to properly -vi -consider t, he possibly random boundary conditions. An image is generally characterized by its auto correlation function. It is assumed that autocorrela tion function of images can be generally factor i zed. Autoregressi ve CARD model parameters are found from autocorrelation function. First order AR scalar image model corresponding to seperable and exponential auto correlation function is as follows, sCm,n)=c. sCm~l,n3+c. sCm,n-13-c. c. sCm-l,n-13 1 2 12.+uCm,n3 CS. 23 where ci and cz are the correlation of adjacent points in the image in horizontal and vertical dimensions, res pectively. uCm,n3 is white Gaussian noise with zero mean and with, variance <r. u In this thesis, autocorrelation method is used for the estimation of AR parameters. Sample autocorrelation function for separable model and degraded model are as follows, M-l-kN-l-L R Ck, 13=1 /CM. bDr rsCm,n).sCm+k,n+l) CS. 33 BS M-±-kN~l-l /s R Ck,13=l/CM. M3r £ rCm, n3. rCm+k, n+13 CS. 43 rr rn-O rt=0 Then, AR paremeters and variance of driving noise are calculated from separable model as follows, c =R C1.03/R C0.03 CS. 53 1 s s s s c =R C0,13/R CO, 03 CS. 63 2 s s ss ar =Cl-c23Cl-cZ3. R C0.03 CS.73 u 1 2 ss VI 1 -Finally, the paremeiers are estimated from degra ded image as follows, c =R C1,CD/CR CO,03-0`23 CS. 83 i rr rr v c =R C0,13/CR CO.OJV) CS. 93 2 rr rr v <y2=Cl-c23Cl-cZ3. CR CO, 03 -or2 3 CS. 103 u i 2 rr v where RrrC. D is autocorrelation function of degraded images, <j is variance of observation noise. V These coefficients are used in the state space representation of the model. Then, the Kalman filter is designed which yields the best estimates in the minimum mean square sense. The equations of the Kalman filter ar e gi ven as fol 1 ows, sCm,n3=c. sCra-l,n)+c. sCm,n-l)-c. c.sC m-1, n-1 3 1 2 12 CS.113 rCm,n3=sCm,n3 + VCm,n3 CS. 123 Prediction Ctime update} equations, sCm,n3=c. sCm-1, n3~t-c. sCm,n-13-c.c. sCm-l,n-13 1 2 1 2 CS. 133 MC m, n3 =c. PC m-1, n3 +c. PC m, n-1 3 -c. c.PC m-1, n-1 3 1 2 12 CS. 143 Filtering C measurement update3 equations, sCm,n3=sCm,n3+KCm,n3. ErCm,n3-sCm,n3 3 CS. 1S3 -1 KCm,n3=MCm,n3. CMCm,n3+r3 CS. 163 -1 PCm,n3=MCm,n3. r EMCm,n3+r 3 CS. 173 -viii-where r is observation noise variance. When the Kalman filtering is applied for noisy- image, filter smooths out the edges and reduces the con- t r ast, whi eh i n var i abl y r esul t s in poor vi sual qual i t y. So, edge adaptive filters provide a better match to lo cal image statistics, thus helping to preserve edges with greater noise reduction in nonedge regions. The second problem of fundamental importance in image analysis is edge detection. An edge in an image is boundary or countour at which a significant change occurs in some physical aspect of an image, such as the surface reflectance, illumination, or the distances of the visible surface from the viewer. Edge points can be thought of as pixel locations of abrupt gray level change. As there is a direct relationship between the edges and properties of a scene, much of the scene in formation can be recovered from an edge image. This edge detection converts gray scale image into a binary edge image which may have direction information. The trans formation preserves a great deal of the useful informa tion in the orijinal image. A difficulty with edge detection is that.the de tected edges often have gaps in them at places where the transitions between regions are not abrupt enough. Moreover n they may be detected at points that are not part of region boundaries, if a given picture is noisy. The detection of the sharp changes in image intensity requires the evaluation of several different derivati ves of the noisy data, however, this is an unstable pro cess since it amplifies the noise. Thus, edge detection algorithms first employ a noise suppression proces prior to the differentiation process. ac*çr Our aim suggestion is that both filtering and edge detection should take place at the same time. The way of the doing this is by statistical theory of hypothesis testing. According to this aim, we used two different edge adaptive filtering techniques. The first method is binary model edge adaptive Kalman filtering which has two hypothesis, Ho and Hi. Ho is defined as a nonedge model which has a high corre lation with the neighboring pixels. Hi also is defined edge model which has low correlation. The decision test which is based on the minimum probability of error, and determines the optimum model. According to this deci sion, either Kalman filtering or independent estimation equations are used. ~ix-The second method is multiple model edge adaptive Kalman filtering which consists of five cases. Four ed ge models corresponding to major correlation directions of O, 45, 90, and 13S degrees, plus a nonedge models are used to represent the image at each location. When the design five cases, one for each models, prior to filte ring. In the filtering a decision is made at each pixel, based on minimum probability of error decision logic, to estimate the edge orientation. The pixel locations at which the model transitions take place are a priori unknown, and have to be detected from the noisy observations during the filtering procedure. In this case, five a priori image models were identified over the orijinal image within the respective regions. The best model for each pixel is obtained during the decision test, and then suitable Kalman filtering equation is applied according to the obtained model. As a result, we can say that multiple model edge adaptive Kalman filtering methods is very efficient than the o t her Ka 1 man f i 1 1 er i ng a 1 gor i t hms. -x-

ÖZET Bu çalışmada görüntü işleme konusunda temel öneme sahip görüntü onarımı ve kenar tanıma işlemlerine yer verilmiştir. Toplamsal beyaz Gauss gurultulu görüntülerin onarımı ele alınmıştır. Burada özbağlanımlı (AR) model kullanılmıştır. önce toplamsal beyaz gurultu ile bozul muş görüntünün onarımında Kalman filtreleme metotu kullanılmıştır. Filtrelerine sonucu görüntüdeki detayların yok olması nedeniylede ikinci olarak hem kenarları sezen hem de filtrelemeyi yapan bir onarım metotu tasarlanmış tır. Bu metot için ikili ve çoklu model kenar uyumlu Kalman filtreleme olmak Üzere iki algoritma geliştirilmiştir. Son olarakta, incelenen bu görüntü onarımı metotlarının gurultulu ortamlardaki performansları karşılaştırılarak, görüntü için en optimum metot saptanmıştır. -v-

60