IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 40. NO. 2. FEBRUARY 1992

482

TABLE I STATISTICS OF THE QUANTIZATION ERRORS

Storage error

f

If Some Coefficients

If All Coefficients are

Are Nonzero

Present

+

U,’

FIB uSlB

Multiplication roundoff error

7

U(a

U;

u:(cy

Coefficient quantization error

f,

0

U;

uf(auf

Input quantization f error U;

e(N I)’B uf(N 1)’D

+

~

p)B p)D

+

+ pu;)D

FA/B ufH

0 2oS(N 0 uf(N

+

+

[9] A. N. Oppenheim and R. W. Schafer, Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975. [IO] R . A. Robert and C . T . Mullis. Digital Signa[Processing. Reading. MA: Addison-Wesley, 1987. [ I I ] L. B . Jackson, Digital Filters and Signal Processing. Dordrecht. the Netherlands: Kluwer, 1986. [I21 H. 1. Buttenveck, J. Ritzerfeld, and M:, Wetter, “Finite word length effects in digital filters,” Arch. Elek. Ubertragung, vol. 43, pp. 7689, 1989.

1)’D

I)’(O:

+u~)D On the Gradient Inverse Weighted Filter

-

eA/B

u ~ H

Xin Wang

a and 6 are the numbers of the coefficients present in the numerator and denominator polynomials of H(:,, z ’ ) . l = I , + I?. I , is the number of nonzero pairs of a,, and b,, ( i > 0 ) . l2 is the number of nonzero pairs of a,), and h,,,, , . A , B , D . and H a r e defined in (23)-(26).

Abstmcr-In this correspondence, we recommend a new algorithm for the gradient inverse weighted filter. Examples show that it can preserve edges and smooth noise more efficiently.

+

I. INTRODUCTION The statistics of the four kinds of errors, viz., the storage error, the multiplication roundoff error, the coefficient, and the input quantization errors are given in Table I. V . CONCLUSION The quantization effects are analyzed for the systolic structure of a 2-D IIR digital filter proposed by Sid-Ahmed [4]. Expressions are derived for the total combined error at the output of each PE as well as at the output of the filter for a given time. In the systolic realization the roundoff accumulation error, the coefficient quantization error, and the input error expressions are the same as in the direct form realization, but an additional error called storage error is present at the output of the systolic realization. One can afford to have only a small additional error, totally avoiding the overflow due to addition. Finally, w e would like to mention that the error analysis of the systolic realizations of I-D IIR and 1-D FIR and 2-D FIR filters can be taken as special cases of the 2-D IIR filter error analysis.

Wang et al. [ I ] presented a gradient inverse weighted (GIW) filter. The proposed noise smoothing scheme is based on the observation that variations of gray levels inside a region are smaller than those between regions. Let x ( i , j ) be the pixel gray level at coordinate (i, j ) . Define

d(i,j;k, I ) = /x(i

+ k,j + I)

-

x(i,j)l

(1)

where k , 1 = - I , 0, I , but k and 1 are not equal to zero at the same time. W e denote this vicinity as V ( i ,j ) . The absolute inverse of the gradient at (i, j ) is defined as

Then the output of the GIW filter has the form

f(i,j) where

y(i, j )

=

=

O.Sx(i,j) + 0.5y(i,j)

,z, ,)

Wk. Ox(i

+ k, J + 1)

(3)

(4)

and

REFERENCES R. C. Gonzalez and P. Wintz, Digital Signal Processing. Reading, MA: Addison-Wesley, 1977. S . G. Tzafestas, Multidimensional Systems. Techniques and Applications. New York: Marcel Dekker, 1986, ch. 5, pp. 233-266. H . T. Kung, “Why systolic structures‘?.” Comput. Mag., vol. 15. pp. 37-46, Jan. 1982. M. A. Sid-Ahmed, “A systolic realization for 2-D digital filters.” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 560565, Apr. 1989. M. D. Ni and J . K. Aggarwal, “Two-dimensional digital filtering and its error analysis,” IEEE Trans. Comput., vol. C-23. pp. 942-954, Sept. 1974. P. Agathoklis, E. I. Jury, and M. Mansour, “Evaluation of quantization error in two-dimensional digital filters,” IEEE Truns. Acoust.. Speech, Signal Processing, vol. ASSP-28, pp. 273-279. June 1980. L. M. Roytman and M . N. S . Swamy, “Determination of quantization error in two-dimensional digital filters,” Proc. IEEE. vol. 69, pp. 832-834, 1981. B. G. Mertzios and A. N. Venetsanopoulos, “Combined error at the output of two-dimensional recursive digital filters,’‘ IEEE Trans. Circuits Syst., vol. CAS-31, pp. 888-891, Oct. 1984.

Reducing noise without smearing edges is the advantage of this nonlinear filter. As an improvement of the GIW filter, a new algorithm is introduced in this correspondence.

11. A NEW ALGORITHM OF

THE

GIW FILTER

The new algorithm of the GIW filter is based on the general form of ( 3 ) i ( i ,j )

=

K(i, j ) x ( i , j ) + [ l

-

K(i, j ) ] y ( i ,j ) .

(6)

Manuscript received January 14, 1991; revised July 19. 1991. The author is with the Department of Electronics, Shandong Polytechnic University, Jinan 250014, People‘s Republic of China. IEEE Log Number 9104861.

1053-587X/92$03.00 S 1992 IEEE

4x3

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 40. NO. 2, FEBRUARY 1992

W e suppose that x ( m , n ) ( m ,n = 0, I , 2, . . . ) are independent and identically distributed Gaussian random variables with variance U:, and consider y(i, j ) as an output of the filter defined by (4). For smoothing noise efficiently, the optimal coefficients in (6) should be [2]

K(i, j )

= ut/[uf

+ ut(;, j ) ]

28 1 24 20 16 12

(7)

where uf(i, j ) is the variance of y(i, j ) . If u t ( i , j ) = uf. from (7) w e have K ( i , j ) = 0.5. But in general cases, u t ( i , j ) is much less than 0:. Thus, it is not a good choice letting K(i, j ) = 0.5 to get (3). However, an exact expression of of(;.;) is not available. T o obtain an approximate value, we consider every W ( i ,j ) as a constant at point (i. j ) . Then we have

8

4 0

0

2

4

8

10 12

14 16 18 20

\

12

Substituting (8) in (7), we obtain

where

6

Fig. I . A noisy edge

- ,

,

,

2 4 6 8 10 12 14 16 18 20 Fig. 2. The smoothed edge by the GIW filter (old) with a three-point window. 0

As in the GIW filter, this new algorithm requires no preassigned parameters and involves only simple arithmetic calculations. For the sake of comparison, supposing that there is pulse noise at point (i, j ) , we have x ( i , j ) = h and x ( i k , j + I ) = 0, ( k , I E V ( i ,j ) ) . Using ( I ) , (2), ( 5 ) , and (IO), we obtain W(i,j ; k , I ) = D ( i , j ) = 1 /8. Hence from (3) and (6), the filtered outputs evaluated by using the two algorithms are h / 2 and h / 9 , respectively. In this case, the new algorithm has the same output as the mean filter.

+

24 i

111. EXAMPLES

For investigating the difference between the two algorithms in signal processing, two examples are given below. Example I : Fig. 1 shows a one-dimensional noisy edge. Figs. 2, 3 demonstrate the applications of the two algorithms to the curve with a three-point window. It is shown that the proposed new algorithm can smooth noise more efficiently. Example 2: A 6 4 x 64 pixel test image was generated to evaluate the performance of the algorithms. Fig. 4 is the simulated image showing a small square (gray level = 80) and a large dark ring (gray level = 30) [ l ] . In order to examine the noise smoothing and edge preserving ability of a filter, we divide the composite image into two parts. One is the flat region, the other is the edge region which only contains the boundary lines. The signal-to-noise ratio improvement of the two parts is denoted by (SNI),and (SNI),, respectively. Adding the white noise generated from N ( 0 , 2 5 ) , N ( 0 , loo), N ( 0 . 400), and N ( 0 , 900) to Fig. I , respectively, we obtain four noisy images. The results evaluated by applying the two algorithms of the GIW filters and the median filter to each of the four noisy images are shown in Table I. Table I demonstrates that the GIW filter gives a stable value of SNI. It means that the GIW filter can be used more than one time to obtain better results [l]. Fig. 5 shows the original image corrupted by Gaussian white noise with mean zero and variance 400. Figs. 6 to 8 are the images filtered with the two algorithms of the GIW filter and the median

Fig. 3 . The smoothed edge by the GIW filter (new) with a three-point window.

Fig. 4. The original image filter, respectively. From the figures we can see that if there are two neighboring pixels having the same value, both of them can be preserved by the GIW filter but removed by the standard median filter. It may be one of the reasons why the median filter is better than the GIW filter in smoothing noise. In other words, the GIW filter can preserve the details, for example, the thin lines of images. But those thin lines will be removed by the standard median filter.

IEEE T R A N S K T I O N S 0 1 1 S I G K A L PROCESSING. VOL 40. NO 2 . FEBRUARY 1992

4x4

TABLE I SNI (dB) Filter Name

0;

25

100

400

900

GIW filter (old)

(SNI), (SNI),

2.58 2.76

2.63 2.66

2.67 2.67

2.71 2.62

GIW filter (new)

(SNI), (SNI),

4.12 3.49

4.22 3.44

4.31 3.32

4.33 3.32

(SNI),

3.62 -2.54

5.68 1.02

6.39 2.93

6.55 4 06

Median filter ( 3

3)

X

(SNI),

Fig 8 The smoothed image by a 3 x 3 median filter window [4]. With this structurc. the thin lines may still be preserved. 2) The eight pixels in the vicinity of x ( i . j ) are replaced by the four medians of the pixels in the four sides of the square window. With this structurc. the thin lines will be removed but the edges are preserved. As more "details" are removed, with these substructures the GIW filter can perfomi better in smoothing Gaussian white noise.

I V . CONCLUSION Fig. 5 . The iniage corrupted by Gaus\ian white noise with variance 400.

In this correspondence. a new algorithm for the gradient inverse weighted filter is described. Of course, more calculations are needed in order to achieve the improvetncnt in smoothing noise. Since the median filter may create artifacts, the GIW filter would be more attractive in a number of applications. REFtRENCtS

[ I ] D. C C . Wang. A. H . Vagnucci. and C . C . Li. "Gradient inverse

Fig 6 The smoothed image by the GIW hltei (old)

Ii

Localization of Narrow-Band Autoregressive Sources by Passive Sensor Arrays

I

i

1!

j

Ilan Ziskind and Yeheskel Bar-Ness

i

i

I!

Abstract-Unique localization of multiple snurces by antenna arrays requires that the number of sensors be greater than the number of sources. When additional information is given this is no longer true. In this correspondence we assume that the signals' structure is of autoregressive (4R)tlpe. We present a localization procedure which con-

i

:

..".U

heighted scheme and the evaluation of it\ performance." Conipuf. Crciphics. Irncigo froc~~s.cir7g. \ o l . I S . pp. 167-181. 1981. [ 2 ] A. P. Sage and J. L . Melsa. Esrir?iation Theory. New York: McGraw-Hill. 1971 [3] P. Heinonen and Y. Neuvo. "FIR-median hybrid tilter." /E€€ Trans. Acousr.. S p t w h . SigmiI frocrs.\irig, \,oI. ASSP-35. pp. 832-838. June 1087. [41 G . R . Arce and M . P. Mcloughlin. "Theoretical analysis of the m a x i median filter." IEEE Trcinc. Acortsr.. Sprrch. Sigrial Processing. vol. ASSP-35. pp. 60-69. Jan. 1987.

.....

if

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 7 The smoothed image by the GIW filter (new)

To avoid the G I W filtcr taking as much noise a s the "details" to preserve, wc can choose some new substructures

1.71.

1 ) The eight pixels in the vicinity of.r(i. J ) are replaced by the four medians of the pixels in the four orientations in the squarc

Manuscript rcccived October 30. 1089: revised April 16, 1991. 1. Ziskind is w i t h the Department of Electrical and Computer Engineering. New Jersey Institute of Technology. Newark. NJ 07 102. on leave froin RAFAEL. Haifa. Isracl Y . Bar-Ness was with the Department of Electrical and Computer Engineering. New Jersey Institute of Technology. Newark. NJ 07 102. IEEE Log Number 9104865.