Wiener süzgeci ile görüntü onarımında yeni bir yaklaşım

A New approach in image restoration by wiener filtering

Tez No: 46283
Yazar: MELİKE DİŞBUDAK
Danışmanlar: Y.DOÇ.DR. MEHMET ERTUĞRUL ÇELEBİ
Tez Türü: Yüksek Lisans
Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
Anahtar Kelimeler: Görüntü restorasyonu, Wiener filtreler, Image restoration, Wiener filters
Yıl: 1995
Dil: Türkçe
Üniversite: İstanbul Teknik Üniversitesi
Enstitü: Fen Bilimleri Enstitüsü
Ana Bilim Dalı: Belirtilmemiş.
Bilim Dalı: Belirtilmemiş.
Sayfa Sayısı: Belirtilmemiş.

Özet

ÖZET Bu çalışmada, Wiener süzgeci ile görüntü onarımı konusuna değişik bir açıdan yaklaşılmış ve toplamsal beyaz Gauss gürültüsü ile bozulmuş görüntülerin onarımı, nedensel Wiener süzgeci ile gerçekleştirilmiştir. Gözlem işaretinin iki boyutlu özbağlanımlı kayan ortalamak (ARMA) model ile temsil edildiği varsayımına dayanarak, özgün görüntünün, en küçük ortalama karesel hata kestirimi, uzamsal domende özyinelemeli olarak elde edilmiştir. Böylece onarım, fazla işleme ve belleğe gerek duyulmadan gerçek zamanda gerçekleştirilebilmektedir. Görüntü onarımı alanında günümüze kadar yapılan çalışmalarda, özgün görüntünün sıfir ortalamalı beyaz Gauss gürültüsü ile sürülen iki boyutlu özbağlanımlı (AR) süreç ile modellendiği varsayımı yapılmıştır. İki boyutta faktorizasyon mümkün olmadığı için, bu varsayım altında, görüntü onarımı nedensel Wiener süzgeci ile gerçekleştirilemez. Bu yüzden onarımda genellikle, nedensel olmayan Wiener süzgeci veya Kalman süzgeci kullanılmıştır. Özgün görüntü modeli ile ilgili herhangi bir varsayımın yapılmadığı bu çalışmada, ilk olarak, değiştirilmiş Yule-Walker denklemleri kullanılarak ARMA gözlem model parametreleri tanımlanmıştır. Daha sonra, özgün görüntü, gözlem gürültüsünün değişintisi ve bu kestirim değerleri kullanılarak onarılmıştır. Benzetim sonuçlan, çeşitli gürültü seviyelerinde incelenmiş ve çalışmaya eklenmiştir.

Özet (Çeviri)

SUMMARY A NEW APPROACH IN IMAGE RESTORATION BY WIENER FILTERING One of the major areas in digital image processing is image restoration. The field of image restoration is concerned with the estimation of the original image from the degraded vesion. In many applications recorded images represent a degraded version of the original scene. The degradations may be due to blurring and observation noise. Two major causes of blurring are (1) relative motion between the imaging system and the original scene and (2) out-of-focus imaging systems. On the other hand, noise may be caused by the transmission medium, the recording medium, inaccurate measurement or quantization of the data for digital storage. An image is generally defined as a real or complex valued function of two space variables belonging to some support region. Although this support may be continuous, it is commonly sampled on a rectangular grid. This defines a set of pixels and the image can be represented by an array of pixel intensity levels. In this thesis, we deal with pixels of monochromatic images representing the gray levels, degraded by an additive Gaussian white noise with zero mean. The model of the image formation system for blurred and noisy image can be defined as, y(m,n)= E d(k,l)x(m-k,n-l) + v(m,n) (1) (k,l)eSd where x(m,n) and y(m,n) denote the original image of size MxM and observed image, respectively. v(m,n) represents the zero mean Gaussian white observation noise with variance ov. d(m,n) is the impulse response of a two-dimensional (2-D) finite impulse response (FIR) linear space-invariant system and is called point spread function (PSF) or blurring function. In this thesis, we assume that there is no bluring or another type of noise - like multiplicative noise- in the observed image and try to recover the original image from its observed version degraded by the additive Gaussian white noise. (vüi)Restoration of images requires some statistical kno wedge of both the original image x(m,n) and the observation noise v(m,n). Therefore, restoration can be redefined as estimating x(m,n) given the observed image y(m,n) and some statistiscal knowedge of x(m,n) and v(m,n). Image identification concerns with estimating this knowedge prior to the restoration. However, it should be pointed out that image identification and image restoration are separate problems and the use of image identification is not limited to only image restoration. The mentioned statistical knowedge of image is also useful in several other fields of image processing, e.g., image data compression, coding, filtering and image analysis. Several approaches has been used in image restoration. The first studies aimed to eliminate the effects of deterministic blurring, and the effects of noise weren't considered. But in later studies, it was discovered that it should be given importance to the effects of noise. To eliminate noise, 2-D least mean squares methods like Wiener filtering has been used. In recent studies, noncausal Wiener filter and Kalman filter have been most commonly used, which are based on minimizing the mean square error between the estimate and original image. For realization of Wiener filtering, it is necessary to take inverse of the covariance matrix of the observed image in order to estimate the original image. Therefore, Wiener filter uses generally two-dimensional discrete Fourier transform (DFT) and inverse discrete Fourier transform (TDFT) to avoid the operational difficulties in taking the inverse matrix. In this method, all image is processed at once (batch processing), and it is not suitable to process in the real time because of noncausality. There are two basic restrictions in constructing the Kalman filter which is recursive in the spatial domain. The first one is the difficulty in the construction of an appropriate recursive model and the second one is the need for a high dimensional state vector. In most practical situations, the first step in restoring an image is to identify degradation the image has suffered. In more recent work, the original image has been modeled by a two-dimensional autoregressive (AR) process, and the blurring has been modeled by a two-dimensional linear space-invariant system (LSI) with finite impulse response (PSF). In this thesis, it is assumed that the observed image degraded by the additive Gaussian white noise is represented by a two-dimensional autoregressive moving average (ARMA) model, and no model assumption is made about the original image. This is a new approach in using the Wiener filter for image restoration, and provides (ix)the causal Wiener filter which requires less computations and memory for implementation in the time domain. Therefore a real-time restoration is also possible. In order to construct a causal Wiener filter, one needs to factorize the power spectral density of the observed image and find the transfer function. Because of the 2-D factorization problem, and based on the assumption made in the previous studies, it is not possible to express the observed signal in the form of lower ordered 2-D polynoms. Assuming no blurring effect, we can express the observed image and its power spectral density as follows. y(m,n)=x(m,n)+v(m,n) (2) Sy(z1,z2) = Sx(zi,z2) + a^ = U(z1,Z2)U(z71,Z21) (3) Since it has been assumed that the observed image is represented by a 2-D ARMA model, its power spectral density function is factorized. The transfer function is given by U(zi,z2) = A( x = -j- t (4) A(Zl,z2) j_ ?£ a^z/z,1 (k,l)eSa-(0,0) S a and Sj, are first quadrant plane causal image support regions, ay and by denotes the AR and MA model parameters. Before restoring, the observation model parameters should be estimated from the degraded image. Considering the first order observation model and first quadrant plane, we have this rational function from (4) as follows. TT, v b00 + b10z7 + b01z2 + bllz7z2,n U(zi,z2) = -j -j _1 _x (5) 1_ a10zi _ a01z2 ~ allzl z2 There are several methods to estimate ARMA model parameters. The method used here is based on estimation of the autoregressive (AR) and moving average (MA) parameters in two separate steps. First, the AR parameters are obtained as the solution of the so-called modified Yule- Walker (MYW) equations by using autocorrelation sequence. Then, the MA parameters are subsequently estimated by using the autocorrelation sequence and the estimated AR parameters are found in the previous step. In 2-D, the modified Yule-Walker equations are given by Ry(m,n)= S ay Ry(m - k, n - 1),(m,n)gSa (6a) (k,l) eSa-(0,0) (x)Z awRy(m-k,n-l)= Z by u(-m + k,-n + l),(m,n) e Sa (6b) <k,l)eSa (k,l)eSb where Ry(m, n) is the sample autocorreletion function of the degraded image and is estimated from all noisy image data, and is defined as follows., M-l-k M-l-1 Ry^n)^-^ E E y(m,n)y(m + k,n + l) (7) M2 k=0 1=0 u(m,n) the impulse response of the ARMA model and given by the inverse z transform of the transfer function in (5) as follows. E akıu(m-k,n-l)= E by 5(m - k, n - 1),(m,n)sSa (8) (k,l) eSa (k,l) eSb It is to be noted that {u(m, n) : (m, n) e Sa} depends on the parameters of the ARMA model in rather nonlinear fashion. It can be defined clearly as u(0,0) = b00 (8) u(0, 1) = aoıboo + boi (9) u(l,0) = ai0b0o + b10 (10) u(l, l) = 2a10aoib0o + anb0o + a10b0ı + a0ıbı0 + bn (H) Because of highly nonlinear character of the MYW equations, they will not be used here directly as a basis for estimating the ARMA parameters. However, the linear equations (6a) will indeed be used. So, the process is realized in two steps. In the first step, the AR parameters is estimated by using autocorrelation estimates from (6a). In the second step, these estimated parameters together with autocorrelation estimates is used to estimate the MA parameters from solving nonlinear equations in (6b). For separable model, the AR parameters are calculated by Ry(2,Q) Ry(Q,2) a10 _ RyOO) ' a°l = RyW) Ve an = ~ai°ao1 <12) where aio and aoi are the correlation of adjacent points in the image in horizantal and vertical dimensions, respectively. (xi)Substituting (8) -(1 1) in (6b), we obtain a nonlinear equation as follows. Ry(m, n) - aio%(m - 1, n) - aoiRy(m, n - 1) - auRy(m - 1, n - 1) = boou(-m, -n) +b iou(-m + 1, -n) + boiu(-m, -n + 1) +b nu(-m + 1, -n + 1) (13) The required MA parameters are then obtained by solving above nonlinear equation for (m,n) e Sa In this thesis, Gauss-Newton algorithm has been used to solve equations. After identifying the ARMA model parameters, the causal Wiener filtering can be realized which estimates the original image from degraded version. If we assume that x(m,n) and v(m,n) are samples of zero- mean stationary random processes that are linearly independent of each other and their power spectrum density functions are known, the optimal linear minimum mean square error estimation of original image is obtained by filtering degraded image with a Wiener filter whose transfer function is given by *«?>*=%$ (M) The filter H(zi,z2) is called the noncausal Wiener filter. The causal Wiener filter is also given by Hc(Zbz2) = 77777 U(zj,z2) Sx (z1;z2) UCzT^z-1) (15) [A]+ denotes the causal part of A. Substituting (5) in the above equation, we obtain Hc(zi,*2) = l -^ °? U(z1,z2)b00 = 1 *- a10zl1 - Hi7-!1' allzllz21 ^v b00 + blO2!1 + boi2^ + bll2!1^1 b°° (16) It can be shown that the causal Wiener filter can be realized by the difference equations in the spatial domain recursively. So, it does not require too (xii)much storage, and memory problem is alleviated. Also, there are no many calculations due to recursibility, and it is suitable to process in real time. The causal Wiener filter in (16) is recursively expressed in the spatial domain by these difference equations. x(m, n) = -r^x(m - 1, n) - r^x(m, n - 1) - r^x(m - 1, n - 1) + (1 - -^)y(m, n)+ Doo boo öoo öq0 b10 CTva10 _N b01 CTva01 bu gyanw_ oo b00 oo b00 oo b00 (17) x(m, n) denotes the estimate of the original image. It is not necessary to know all image data to estimate the original image due to the recursibility of the causal Wiener filter. In order to obtain the estimation of the observation in (m,n), it is sufficient to know a previous values of the observation and estimation. After the white Gaussian noise with zero mean is added the original Lena image with size 256x256, the noisy image is transformed to the image with zero mean, unit variance. Then, all those equations are applied step-by-step. For making comparasion, input signal-to-noise ratio (SNRI), output signal-to-noise ratio (SNRO) and improvement between input and output are used. (xiii)

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