140 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 1, JANUARY 1990

U 1 -A

Fig. 2. Nine possible stopbands that can be obtained using the configuration in Fig. l(b).

and

represent two 1-D bandpass filters, then

1 H ( e J u l ~ l , e l u 2 7 2 ) I = ( 0, 1, 0 1 2 G ~ 1 3 and ~ 2 2 G 0 2 B 0 2 3

otherwise

and, therefore, the configuration of Fig. l(b) realizes a 2-D filter whose amplitude response is complementary to that of the band- pass filter obtained by cascading the two 1-D band-pass filters. If each of the two cascade filters is allowed to be a low-pass, high-pass, or band-pass filter, the nine rectangular stopbands depicted in Fig. 2 can be obtained.

The configuration of Fig. l(b) will complement the amplitude response of the two 1-D filters connected in cascade only if the phase angles of the numerator polynomials N,(z,) and N2(z2) are linear, according to (3). Following the approach of Antoniou [4, pp. 218-2191, it can be shown that (3) will be satisfied under the following circumstances:

The coefficients of N,(z,) and N2(z2) have mirror image symmetry, that is, a,, = a,( ,,-,) for i = 0,1,2; . ., n, and k = 0 in Fig. l(b). The coefficients of Nl(zl) and N2(z2) have mirror image antisymmetry, that is, a,, = - a,(,-,, for i = 0,1,2; . . , n, and k = 0 in Fig. l(b). The coefficients of Nl(zl) [or N2(z2)] have mirror image symmetry and those of N2(z2) [or Nl(zl)] have mirror image antisymmetry, and k = - 1 or + 1 in Fig. l(b).

used to show that the nine possible stopbands can be realized by using standard 1-D continuous low-pass, band-pass, or high-pass transfer functions that can be derived from classical analog-filter approximations.

REFERENCES [ l ] J. M. Costa and A. N. Venetsanopoulos, “Design of circularly symmetric

two-dimensional recursive filters,” IEEE Truns. Acowt. , Speech, Signul Processing, vol. ASSP-22, pp. 432-443, Dec. 1974. K. Hirano and J. K. Aggarwal, “Design of two-dimensional recursive digital filters,” IEEE Truns. Circuits Syst., vol. CAS-25, pp. 1066-1076, Dec. 1978. A. Antoniou, M. Ahmadi, and C. Charalambous, “Design of factorable lowpass 2-dimensional digital filters satisfying prescribed specifications,” Proc. Inst. Elec. Eng., vol. 128, pt. G, pp. 53-60, Apr. 1981.

[4] A. Antoniou, Digital Filters: Anulysis und Design. New York: McGraw-Hill, 1979.

[2]

[3]

A Model-Based Approach for Filtering and Edge Detection in Noisy Images

A. RANGARAJAN, R. CHELLAPPA, AND Y. T. ZHOU

Abstract -We consider the problem of enhancement and edge detection on noisy, real-world images. The restoration and edge detection framework is based on an auto-regressive (AR) random field model. An edge is detected if the first and second direchonal derivatives and a local estimate of the variance at each point satisfy certain criteria. Due to the modeling assumptions, the directional derivatives are functions of the model parame- ters and of the neighboring pixels in a 3 x 3 window. When noise is present, a good estimate of the original from the noisy images improves the signal-to-noise-ratio and this results in better estimates of the direc- tional derivatives. To avoid excessive computation, the problem of estima- tion of the original image and the model parameters is presented as a combination of a reduced update Kalman filter (RUKF) and an adaptive least squares parameter estimation algorithm. The restoration process is completed with a min-max replacement scheme to enhance edge strength.’ Since the edge detector operates on the processed image, restoration and edge detection cannot be performed simultaneously and the edge detector lags behind the restoration filter.

An orientation sensitive detector resulting from the use of an AR model may not detect edges of significantly different orientations. This is par- tially overcome by running four edge detectors on the four interior pixels of a 4 X 4 window; this corresponds to rotating the window in successive multiples of 90”. These intermediate results are stored at each point and the final result is the union of the outputs of the four edge detectors. Finally we present the results of applying this edge detector on noisy, real images. Comparisons with Haralick’s facet model edge detector, Nevatia-Babu’s line finder and Canny’s edge detector are also given.

For 1-D discrete transfer functions obtained by applying the I. INTRODUCTION bilinear transformation to classical 1-D continuous transfer func- tions with zeros on the jw-axis or at the origin of the s plane (e.g., low-pass, high-pass, and band-pass transfer functions ob- tained from the Bessel, Butterworth, Chebyshev, or elliptic ap- proximation) one of the above conditions is always satisfied, as can be easily demonstrated, and, consequently, any one of the

Edge detection is an important topic to most researchers in the area of image understanding. A number of different techniques exist in the literature. A summary of many approaches to edge detection can be found in [l]. In this paper, we are interested primarily in edge detection on noisy images. When noise is

nine stopbands in Fig. 2 can be readily achieved. Manuscript received May 4, 1988; revised February 22, 1989. This paper

was supported in part by the Joint Services Electronics Program under Con- tract F49620-85-C-0071. This paper was recommended by Associate Editor H.

The authors are with the Signal and Image Processing Institute. University of Southern California, Department of Electrical Engineering-Systems, Los Angeles, CA 90089.

IEEE L~~ Number 8931728.

IV. CONCLUSIONS A modified version of a configuration proposed by Hirano and

Aggarwal for the design of 2-D digital filters with rectangular stopbands has been suggested and sufficient conditions for its correct operation have been derived. These conditions were then

Gharavi,

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present in an image, it is necessary to perform some form of preliminary image smoothing using standard presmoothing tech- niques such as median filtering, etc. Presmoothing is then fol- lowed by a standard edge detection algorithm. Oftentimes, the choice of a specific edge detector has no relation whatsoever to the presmoothing algorithm.

We propose a model based framework for edge detection and smoothing. The noise free image is assumed to be adequately represented by a 2-D AR space-variant model. An edge is de- tected if the first and second directional derivatives and a local estimate of the variance at each point satisfy certain criteria. Due to the modeling assumptions, the directional derivatives are func- tions of the model parameters and of the neighboring pixels in a 3 x 3 window. In the absence of noise, the model parameters can be adaptively estimated at each point from the underlying image itself. When noise is present, restoration improves the SNR and consequently edge detection itself as the directional derivatives also depend on the neighboring pixel values. This problem of simultaneous estimation of model parameters and pixel values can be formulated as an extended Kalman filter (EKF). Since this is computationally too intensive we have used an RUKF to perform image restoration and an adaptive least-squares (LS) technique for parameter estimation. The RUKF implemented along the lines suggested by Woods 121 nevertheless differs from it in that the parameters of the AR model are estimated adap- tively.

We define a state vector called the global state vector which is composed of M previous lines for a nonsymmetric half-plane representation of the noise-free image using an M X M model. This enables us to separate the past from the future which then gives us the state vector that is involved in a general Kalman filter iteration. When a finite order model is used, a subset of the global state, called the local state, can be used. Using this subset for the updating process is the reason for the name reduced- update. For the sake of convenience, the local state is chosen to correspond with the support of the model itself. When this is the case, the parameter estimates obtained in the restoration process can be directly used for the edge detector model parameters. Edge detection could then be performed simultaneously with the RUKF. However, this did not provide adequate filtering and the support was consequently increased to four pixels. Min-max replacement is then performed recursively to enhance edge strength. This forces edge detection to follow restoration.

The AR model results in an orientation sensitive edge detector and may not adequately detect edges whose orientations are markedly different from the edge detector itself. This problem is partially overcome by running four quarter-plane (QP) edge detectors [3] on rotated versions of the restored image. Since detection follows restoration, the four QP detectors are run on rotated versions of a 4 x 4 window. The final output is taken as the union of the outputs of the four QP edge detectors. The synthesis of the four QP edge detectors is called the full-plane (FP) edge detector.

The computational demands of this model based approach arise mainly from the RUKF. However, the amount of computa- tions required is less compared to a recent optimization ap- proach.

11. DESIGN OF THE FULL-PLANE (FP) EDGE DETECTOR

In this paper, edge detection is performed recursively and follows the restoration process. This means that our image model for edge detection is a representation of the restored image and

not the original image. The FP edge detector then is applied to the restored image. As mentioned in Section I, the image model is a 2-D AR space variant model. Let S (̂x,, yo) be an estimate of the gray level at position (xo, yo). The details of this restoration process are given in Section 111. The model is expressed as follows:

5( XI3 3 yo) = c1q xg - 1, Y o ) + xo, yo -1)

+ c 3 S ^ ( x 0 - 1 , y 0 - 1 ) + n ( x 0 , y ~ ) .

At this point it is assumed that the parameters c1, c2, and c3 are known. The methods for the estimation of these parameters from noisy images are discussed later in the section. The direc- tional derivatives can be evaluated once the parameters have been estimated. The details of how the first and second direc- tional derivatives are approximated as above may be found in [3].

It is assumed that an edge is detected if a) the second direc- tional derivative in the direction of the estimated gradient is negative. This ensures that there is only one output per edge and in this case, the pixel corresponding to the top of the step edge is marked as an edge pixel b) the magnitude of the first directional derivative is above some threshold c) and the first derivatives along the x and y directions are non-negative. This criterion results in a QP edge detector. The final edge output is decided based on the union of four QP edge detector outputs. To avoid multiple responses to an edge, we must restrict the domain of each QP edge detector to its appropriate quadrant. To the above, a final condition is added, viz., d) the local estimate of the sample variance (4 X 4 was used) must be greater than a threshold. After restoration the noise level is considerably reduced but is still present. This results in spurious peaks in the estimated gradient. Thresholding is important in real, noisy pictures in order to reduce the effects of residual noise.

The parameters c1, c2, and c3 are estimated from the restored image. Since real images are non-stationary, it is preferable to estimate these parameters adaptively. The nature of the model allows us to use a recursive LS approach [5] and the details can be found in [3].

The QP edge detector is an orientation sensitive edge detector. In the noise-free case, the edges corresponding to the QP's are detected by running a QP edge detector on the original image and three rotated versions of the original, the rotations being in successive multiples of 90". This is called the FP edge detector and further details on this method of implementation are dis- cussed in [6]. This method is not suitable for our approach which combines restoration and edge detection recursively. Instead, we describe a technique where rotations need not be performed on the whole image. Let us assume for the moment that restoration has been completed up to and including the point (xo + 1, yo + 1). Since the recursions are performed in a raster scan manner, the point (xo + 2, yo + 1) and beyond have not yet been processed. Now consider a 4 x 4 window whose top left comer is given by (xo - 2, yo - 2) and the bottom right hand comer by (xo + 1, yo + 1). The QP edge detector is now performed on this (4 X 4) window and is centered on the point (xo, yo). This yields the first of the four outputs needed at each point before the union can be taken. Now, the (4x4) window is rotated. This results in the point (xo - 1, yo) moving to the point (xo, yo) and the process can be repeated, resulting in a QP output corresponding to the point (xo - 1, yo) and for a rotation of 90". The window is then rotated twice and QP edge detection is also performed twice. The outcome of the entire process on the window is four QP outputs at the four interior points in the window (( xo, yo), ( xo - 1, yo), (xo - 1, yo - 1) and (xo, yo - 1)) for four different rotations (OO,

142 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 31, NO. 1, JANUARY 1990

90°, 180°, and 270°), respectively. The variance estimate is common to all the QP detectors in this window and is the sample variance of the pixels contained in the window. Now, the raster scan moves forward by one pixel. The point ( x o , y o ) now as- sumes the role of the point ( xo - 1, yo) and a different QP output can be obtained for the same point. Similarly, when the recursion is one line below this one, the remaining two QP outputs are obtained for the same point and now their union can be taken.

The recursive LS approach for parameter estimation can be extended to the other three QP's as well. We simply apply the same LS technique successively on the rotated versions of the 4 x 4 window. All the pixels in this window have been operated on by the RUKF and so we have the restored values of the image intensity at these locations. There is a tradeoff between the convergence of the LS routine and the non-stationary nature of the image. If a large amount of data is taken for the initialization step, the parameter values do not change much over the image.

111. IMAGE RESTORATION USING A RUKF

Given our emphasis on noisy, real images, the restoration process is particularly significant as far as the edge detector is concerned. The particular technique used here is the RUKF and this method and other Kalman filter approximation methods are fully discussed by Woods in [2]. Woods assumed a non-symmet- ric-half-plane (NSHP) model driven by white noise to represent the original image.

The model is written as

s( rn, n) = C y ' , % ( rn -1, n) + cy.'%( rn +1, n - 1)

+ C:'~'~'~)S(rn)n-1)+c~~"'s(rn-l,n-l)+ w ( r n , n )

(2) where w(rn, n) is the white process noise field (w(rn , n) =

Woods [2] refers to (2) as a 1 X 1 model and it can be seen that the form of this model is similar to the space invariant version of (1). The intensity process s ( m , n) in (2) refers to the original noise-free image. The image is assumed to be corrupted by additive white Gaussian noise (U( rn, n)) with variance U,:. In the equations that follow r(rn, n) represents the noisy image and is the only data available to the RUKF.

r( rn, n) = s( rn, n) + u ( rn, n).

The indexes rn and n refer to the position of the RUKF. The coefficients of the model are not space invariant as in [2], but are estimated adaptively. Ideally, we could estimate both the parame- ters and states using a EKF but the computational burden would be very intensive.

3.1. Prediction

N(O, 0,; 1).

The RUKF consists of two major processes, prediction and update. Before prediction can be performed, in our modification of the RUKF, we need the space-variant model at that point. This is done by a LS recursion. The pixel value of the point ( m , n) is now predicted based on the model whose parameters have been estimated. Since the model is causal in the raster-scan- ning sense, it makes use of the restored values of the previous points in the recursion. Along with the predicted value of the point (rn, n) denoted by ;,,( rn, n), the prediction error covariance m a t h 9 L m . 1 1 ) ( i , j ; k , I ) which denotes the prediction error be- tween the pairs of points (i, j ) and ( k , I ) is also estimated. This is carried out over the global state region (2';:,")). It is worth pointing out that the point (rn, n) alone is predicted. This means

- ~~

that the prediction error covariances not involving the point (rn, n) are the same as the filtering error-covariances of the previous recursion step and only the pointers corresponding to these points have to be adjusted depending on, the mode of storage used.

The predicted value of the pixel intensity at ( m , n) can be written as follows.

( m - l , n ) ;Jn'.n)( rn, .) = C i m . n ) ; ; m - l , n )

+ C l m , n ) S l ; m - l . " ) ( r n + l , n - l )

+ C~m.n);;m~l.n)(m,n_l)

+ c$".") iU( m - 1, n)( rn - 1, n - 1) (3) and the details regarding the prediction error-covariance matrices can be found in [2].

3.2. Reduced Update

The Kalman filter if implemented optimally is computationally too intensive at present. One of the schemes to overcome this problem is the RUKF. The major modification here is that the update is restricted to the local state. This is because updates are significant only in a local neighborhood of the point (rn, n). In this case the local state is used as the update region. The Kalman gain vector and the update of the prediction state vector (iJm-")) is restricted to the local state region (a&: ")).

(4)

and

RI"'-")( i, j ; k , I ) = Rim.")( i , j ; k , I )

- k("',")( i, j ) R P . ' ' ) ( rn, n ; k , I) (5) for

( i , j ) E 9"'') and ( k , I) E 2 ' & ! . ' l ) .

Initial conditions must be prescribed for the filtering error covariance matrix ( R u ) . The initial error covariance was chosen to correspond with that of the white noise process (U:) which must be estimated.

The model used in the RUKF differs from the one used in the design of the FP edge detector. In this case, the support is four pixels, and this means that four and not three parameters have to be estimated. The process is identical in every other aspect aside from the initialization scheme to the parameter estimation pro- cess [3]. The same form of the adaptive LS technique described in the design of the FP detector can be used for estimation of the model parameters. Since the initial parameters are to be esti- mated from the noisy image, a bias-compensated LS routine is used [7].

We obtain initial estimates of c by performing a bias-com- pensated LS fit over fifty lines of the original image. This is different from the typical choice of ten lines as in the case of parameter estimation for the QP edge detector.

The process noise variance (U:) can be estimated from the noisy image. The estimation procedure used is the same em- ployed by Woods [2].

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 1, JANUARY 1990 143

(a) (b) Fig. 1. (a) Original airport. (b) Noisy airport

(4 (b)

Fig. 2. (a) Restored image using RUKF and min-ma. (b) Model based edge detector on 5-dB noisy airport.

We added a min-max replacement scheme to enhance edge strength prior to edge detection. This is performed by moving a 3 x 3 window over the entire restored image. The pixel values in the window are rank-ordered and the value of the center pixel is modified to the maximum or minimum value based on proximity of rank. The min-max replacement also contributes towards suppression of multiple responses to a single edge.

The flow of operation from the RUKF to the FP edge detector is explained below. When the RUKF is centered at ( xo + 2 , yo + 2) , the min-max replacement technique has processed up to and including the point (xo + 1, yo + 1) and the FP edge detector performs four QP operations centered at ( x o , yo), ( x , , - 1, yo), (x , , - 1, yo - 1) and (x,,, yo - l), respectively.

Iv. RESULTS AND DISCUSSIONS The RUKF filter/FP edge detector was run on a 5-dB noisy

airport image. This image and the original image are shown in Fig. 1. The restored image (RUKF) and the edge output from the RUKF/FP edge detector are exhibited in Fig. 2. The FP edge detector requires two thresholds to be set and we chose the best possible thresholds from a visual inspection of the outputs. The edge outputs are compared with three other techniques. These three are the Canny operator [8], the Nevatia-Babu line finder [9], and the Haralick facet model approach [lo] and the results obtained from these operators are displayed separately in Figs. 3-5, respectively.

The Canny edge detector has achieved considerable popularity in the last few years and consequently we have compared the performance of our edge detector with that of Canny. In order to facilitate comparisons between the two edge detectors, we chose to obtain an intermediate output from the Canny edge detector. This edge output is obtained after the derivative estimation step

(4 (b) Fig. 3. (a) Intermediate output of Canny edge detector on Fig. l(b).

(b) Final output of Canny edge detector on Fig. l(b).

Fig. 4. Haralick edge detector on 5-dB noisy airport

(a) (b)

Fig. 5. (a) Unthinned Nevatia-Babu on Fig. l(b). (b) Nevatia-Babu on Fig l(b).

and is shown in Fig. 3(a). The final output is shown in Fig. 3(b). The process of obtaining the best outputs is as follows. Gaussian filters with several different standard deviations (U) were tried. The pair of thresholds corresponding to Canny’s thresholding with hysteresis (possible edges previously passed over by a high threshold are now marked as edges if they are part of an edge contour and if their gradients are above a lower threshold) which yielded the best visual output were chosen for each U. The final output was chosen based on visual comparisons across the differ- ent U ’ S . The intermediate output was then obtained by threshold- ing the gradients for t h s choice of U. Once again the best threshold was chosen. Comparisons between this intermediate output and our edge output are meaningful as they highlight the role played by the filtering and derivative estimation processes. It can be seen from Fig. 3(a) that this intermediate output is very thick. Many details that are “rescued” by the hysteresis thresh- olding step are missing. The final edge output (which incorpo-

144 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 1, JANUARY 1990

rates thinning, non-maximum suppression and thresholding with hysteresis) is impressive. This shows the importance of thinning and thresholding with hysteresis. Our model-based edge detector does not employ a thinning step but the min-max replacement and the single threshold (on the gradient) contribute towards suppression of multiple responses and noise respectively. We are currently involved in incorporating some form of thresholding with hysteresis and edge extension into our model based ap- proach.

The Haralick edge detector fits a facet model over a (7 x 7 was used) neighborhood. The edge output obtained from this edge detector is shown in Fig. 4. When this technique was applied on the noisy airport image, certain features help in improving our understanding of this approach. Compared to our scheme, the Haralick edge detector output (after suitable thresholding in order for the two outputs to be comparable) is consistently thicker. This is to be expected because the facet model obtains its parameters by averaging over a larger neighborhood than the FP detector.

The Nevatia-Babu edge detector output is shown in Fig. 5. The results obtained through the Nevatia-Babu edge detector prior to thinning are also presented to clarify the role played by the thinning process. The results obtained before thinning are shown in Fig. 5(a). The final edge output is shown in Fig. 5(b). The final output seems to have more continuity in some of its edges (by this we mean that even without linking, some linear features are recognizable) but the price paid is in the thickness of the edges. Viewed in this context, our edge detector is capable of yielding thin edges while performing comparable smoothing (through the RUKF).

The FP edge detector works well on noise-free data [3] showing that a model-based approach results in adequate estimates of derivatives using only a 3 X3 window. Based on the work of Marr, Hildreth, Canny and others and drawing upon physical intuition, it is easy to see that this operator will not perform well on noisy images. The advantage of the technique described here is that the RUKF/min-max operation provides adequate restoration enabling the 3 X 3 derivative operator to be effective (it operates on the restored image). This shows that edge detec- tion in noisy images does not necessarily require derivative opera- tors with a large support.

Our work can be extended to the case of blurring in addition to pure additive noise. The RUKF has to be modified to ensure that adequate restoration is performed.

REFERENCES [ l ] Y . T. Zhou, A. Rangarajan, and R. Chellappa, “A unified approach for

filtering and edge detection in noisy images,” Tech. Rep. 109, Univ. of Southern Calif., 1987.

[2] J. W. Woods, “Two-Dimensional Kalman Filtering,” Two Dimensionul Digital Signul Processing- Topics in Applied Physics, (T. S. Huang, Ed.), vol. 42, pp. 155-205, Springer-Verlag, 1981.

[3] Y. T. Zhou, R. Chellappa. and V. Venkateswar, “Edge detection using zero crossing of directional derivative of a random field model,” in Intl . Con/. Acoust.. Speech. Signul Processing, Apr. 1986.

[4] A. Blake and A. Zisserman. Visuul Reconstruction. Cambridge, MA: MIT Press, 1987.

[5] B. D. 0. Anderson and J. B. Moore, Optimul Filtering. Englewood Cliffs, NJ: Prentice-Hall. 1979.

[6] Y . T. Zhou. R. Chellappa. and V. Venkateswar, “Edge detection using the directional derivatives of a space varying correlated random field model,” in Proc. C V P R Con/. 1986. pp. 115-121, June 1986.

[7] H. Kaufman. J. W. Woods, S. Dravida, and A. M. Tekalp, “Estimation and identification of two-dimensional images,” IEEE Truns. Automut. Contr., vol. AC-28, pp. 745-756, July 1983.

[8] J. Canny, “A computational approach to edge deletion,” I E E E Truns. Putt. Anul. Much. Intell., vol. PAMI-8, pp. 679-698, Nov. 1986.

[9] R. Nevatia and K. R. Babu, “Linear feature extraction and description,” Comp. Gruphics Image Process., vol. 13, pp. 257-269, July 1980.

[lo] R. M. Haralick, “Digital step edges from zero crossings,of second direc- tional derivatives,” IEEE Truns. Putt. Anul. Mech. Intell., vol. PAMI-6, pp. 58-68, Jan. 1984.

Cascade-Parallel Realization of 2-D Separable-Denominator Transfer Functions Using a

Variant Partial-Fraction Expansion

TONG-YI GUO AND CHYI HWANG

Absfracf -A direct procedure based on using a variant partial-fraction expansion (VPFE) is presented for finding canonical minimal realizations of cascade-parallel structure for 2-D transfer functions having separable denominator. A similarity transformating matrix which transforms a 2-D separable-denominator state-space model from a phase-variable canonical form to the VPFE canonical form is also presented. A numerical example is provided to illustrate the procedure.

I. INTRODUCTION

In recent years there has been a great deal of interest in two-dimensional (2-D) linear systems. This is particularly true of recursive 2-D digital filters and their applications to digital signal processing. Two-dimensional digital filters are often designed in a recursive form for the purpose of saving computation and mem- ory costs [l]. In the frequency domain, recursive 2-D digital filters are represented by 2-D proper rational transfer functions. In the spatial domain, however, several different state-space models, such as Attasi’s model [2], [3], Forasini-Marchesini’s model [4], [ 5 ] , and Roesser’s model [6], for 2-D systems can be used to model recursive filters. It has been shown by Kung [7] that Roesser’s model is the most general recursive 2-D state-space model for the class of quarter-plane-causal filters and other models can be embedded in this model. The one-dimensiopal (1-D) notions of controllability, observability and minimality have been generalized and extended to Roesser’s model [6], [7]. However, the fact that local controllability and observability of a general Roesser’s model does not guarantee its minimality causes the failure of the theorem of Kalman’s canonical structure in two-dimensional systems [8].

Recently, a class of recursive 2-D digital filters that are repre- sented by two-dimensional transfer functions having separble denominator has gained growing interest and has been under investigation in connection with state-space realization [9]-[13], observer design [14], [15], [19], and the model reduction [16]-[18]. The reasons for the attraction of this class of filters are that the implementation and stability test is simple and the different state-space models realizing this class of 2-D filters, such as Attasi’s model [3] and simplified Roesser’s model [6], satisfy Kalman’s canonical structure theorem, and share many proper- ties of 1-D systems.

Manuscript received June 17, 1988; revised February 15. 1989. This work was supported by the NSC of the Republic of China under Grant NSC-77- 0402-EOO6-01. This paper was recommended by Associate Editor D. M. Goodman.

T.-Y. Guo is with the Department of Electrical Engineering, National Kaohsiung Institute of Technology, Kaohsiung, Taiwan 80761. Republic of China.

C. Hwang is with the Department of Chemical Engineering, National Cheng Kung University. Tainan. Taiwan 70101. Republic of China.

IEEE Log Number 8931730.

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