An IIR median hybrid filter

An IIR median hybrid filter - 1068 IEEE TRANSACTIONS ON...

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Unformatted text preview: 1068 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40, NO. 5. MAY 1992 An HR Median Hybrid Filter Peter H. Bauer, Member. IEEE, Michael A. Sartori, and Timothy M. Bryden Abstract—A new class of nonlinear filters, the so—called class of multidirectional infinite impulse response median hybrid fil- ters, is presented and analyzed. The input signal is processed twice using a linear shift-invariant infinite impulse response fil- tering module: once with normal causality and a second time with inverted causality. The final output of the MIMH filter is the median of the two-directional outputs and the original input signal. Thus, the MIMH filter is a concatenation of linear fil- tering and nonlinear filtering (a median filtering module). Be- cause of this unique scheme, the MIMH filter possesses many desirable properties which are both proven and analyzed (in— cluding impulse removal, step preservation, and noise suppres- sion). A comparison to other existing median type filters is also provided. I. INTRODUCTION N this paper, a class of filters, the so-called class of multidirectional infinite impulse response median hy- brid (MIMH) filters, is presented. The MIMH filter is a novel concatenation of a linear filtering module and a nonlinear filtering module. In this paper, the linear filter— ing module consists of an infinite impulse response (IIR) structure, and the nonlinear filtering module consists of a simple standard median filter. The input signal is assumed to be of finite extent. The linear shift-invariant (LSI) IIR filter first scans the signal in one direction. The causality of the filter is inverted, and the input signal is scanned in the other direction. A three-point standard median filter then operates on these two-directional outputs and the original input signal at each particular point of the input signal. Hence, the output of the IIR filtering module is used as the input to a three—point standard median filtering module. In [1], a class of filters, the linear median hybrid (LMH) filters, was presented. This approach to filtering is a com— bination of both LSI filtering and median filtering and is a “relative” to the MIMH filter proposed here. The LMH filter and the MIMH filter, however, are distinctly differ- ent in many of their properties. One of the subclasses of the LMH filters, which were extensively studied by Hei- nonen et al., are the finite impulse response median hy— brid (FMH) filters [2]. Further extensions of the FMH filter concept are recorded in [3]. Manuscript received August 24, 1990; revised February 7, 1991. This work was supported in pan by NASA under Grant NAG3-l 186. P. H. Bauer and M. A. Sanori are with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556. T. M. Bryden is with IBM, Poughkeepsie. NY. IEEE Log Number 9106565. Besides the class of MIMH filters introduced here and the class of LMH filters of [2], another means to remove impulse noise and to alleviate the problem of signal edge degradation caused by LSI filters is the nonlinear standard median filter, which was originally introduced .in [4]. Among other properties, it is capable of preserving signal steps and eliminating impulses. The standard median filter processes a signal by choosing as the output the me- dian of the amplitude values in a sliding fixed length sym- metrical window. Several properties and many applica— tions of the standard median filter have been reported in the literature [5]—[7]. The standard median filter has also been used as a basis for extensions in the development of new filters with improved performance [2], [8], [9]. In Section 11, properties of the MIMH filter are derived. In Section III, the MIMH filter is compared to other re— lated filtering concepts via comparisons of properties, computational costs, and simulations. In Section IV, the uses for the MIMH filter are examined, and the advan- tages and disadvantages of using the MIMH filter are dis- cussed. II. THE MIMH FILTER In this paper, discrete finite length signals are assumed. The input signal of length (2L + l) for the MIMH filter is denoted as x(n) for —L s n s L. The output corre- sponding to scanning the input signal with a LSI IIR filter in the direction of increasing n is denoted as y“ (n) for —L S n s L. The output corresponding to the opposite scanning direction of decreasing n is given as y,l (n) for —L s n s L. The relationships between yH (n) and x(n) and between y_1(n) and x(n) are given by the recursive difference equations: Y+1(") = amY+1(n * m) + ' " + 004101 — 1) + b0x(n) + b1x(n — l) + -" + bkx(n — k) ( 1) and y—1(n) = amy71(n + m) + + thy—101 +1) + b0x(n) + b]x(n + 1) + ' -' + bkx(n + k). (2) The opposite scanning direction of (2) is realized by in— verting the causality of the [IR filter in ( 1) and leaving the coefficients unchanged. The final output of the MIMH fil— 1053»587X/92$03.00 © 1992 IEEE BAUER e1 01.: AN IIR MEDIAN HYBRID FILTER ter y(n) is given by 3’01) = median {y+l(n)! x(n)! y*l(n)}a for —L S n S L. (3) From now on, all signals are assumed to be of finite sup— port, although the length of the signal will not be explic— itly mentioned in the following derivations. Although this assumption is restrictive, there exist many applications which have finite support data, such as video signals, im— age signals, and recorded data. The use of difference equations in (l) and (2), instead of some other represen— tations of HR filters (e. g., transfer functions), aids in the derivations of the properties of the MIMH filter and does not afiect the proofs or the results of the properties. In computing the endpoints of the signal, both the input sig- nal and the recursive components of the IIR filter are pad— ded such that there is no transient response at these loca- tions. In other words, with y+t(n) = x(n) = X(—L) V n < -L (4) and y—x(n) = x(n) = x(L) V n > L (5) the IIR filter is initialized to avoid a transient response (see assumption 1) in Section II—A). For LSI filters, many methods for spectral domain de- sign exist. For nonlinear filters, similar methods for spec- tral domain design still need to be developed. Because of this, the design of the MIMH filter relies upon the time domain properties of the filter (i.e., step and impulse re— sponse, rise time, zero frequency gain, and filter order). Considering that the MIMH filter does use an IIR filter, those design techniques which are available for IIR filters can be used for the design of the HR filter module of the MIMH filter. Since the main purpose of the developed filter is the removal of impulse and Gaussian noise while preserving signal edges. the discussion is restricted to only low—pass IIR filters for the linear filtering module. A. Deterministic Properties of the MIMH Filter In order to simplify the proofs of the properties, the following definitions are introduced: Definition 1: A step is defined as () 0 forn<0 (6) un = 1 fornzO. Definition 2: A unit impulse is defined as forn = 0 1 6(n) = { (7) 0 otherwise. Definition 3: A unit pulse of width T 2 1 (containing T + 1 nonzero values) is defined as 1101) = W!) - u(n - T — 1). (8) l069 Definition 4: The time constant 7-, denotes the smallest integer time at which the output of the IIR filter first pro— duces y+1(7r) 2 r (9) for the unit step input signal, x(n) = u(n), where O S r s 1. Let h(n) denote the impulse time response and H (0:) denote the frequency response of the IIR filter. The fol- lowing assumptions are made for the IIR filter module: 1) The IIR filter has a O-dB zero frequency gain, i.e., H (0) = 1. 2) The IIR filter has a low—pass characteristic with an aperiodic impulse response, i.e., h(n) 2 O for n 2 0. 3) The IIR filter’s impulse response h(n) is equal to zero forn = 0, i.e., 11(0) = 0 and therefore b0 = 0 in (l) and (2). Property 1: If for the input x(n) the output of the MIMH filter is y(n), then for the input k1x(n) + k2 (10) the output of the MIMH filter is k|y(n) + kg. (11) Proof: It is well known that median {k1a, klb, klc} = k, median {11, b, c}. (12) Furthermore, median {a + k2, b + k2, c + k2} = median {(1, b, c} + k2. (13) Using the linearity of the IIR filter module and (12) and (13), the output of the MIMH filter is median {k.y+1(n) + k2, k1x(n) + k2, k]y_,(n) + k2} ll median {klI'HU’lL [CI/W”), k1)’~t(")} + k2 2 k1 mCdian “+1009 x(n), Y—1(n)} + k2 Remark: Property 1 does not in any way imply linear- ity of the MIMH filter. If for the input x] (n) the output of the MIMH filter is yl (n) and for the input x2 (n) the output of the MIMH filter is yz (n), then for the input k1x1(”) + k2x2(") (15) the output of the MIMH filter is not k1)’1(") + k2y2(")- (16) Property 2: The MIMH filter preserves a constant sig- nal. Proof: The proof follows directly from property 1 with x(n) = 0. I Property 3 : The MIMH filter preserves a step. Proof: Let the input signal be x(n) = u(n). Due to the causality of the [IR filter y,1(n) = x(n) = l for n 2 0. (17) 1070 The output of the MIMH filter for n 2 0 is unity, i.e., y(n) = 1 forn 2 0. (18) Next, y+1(n) = x(n) = 0 forn < 0 (19) and hence y(n) = O forn < 0. I (20) Property 4: The MIMH filter removes an impulse. Proof: Let the input signal be x(n) = 6(n). Due to the causality of the IIR filter x(n) = y-1(n) = 0 forn > 0 (21) and thus y(n) = 0. Next, x(n) = y+.(n) = 0 forn < 0 (22) and hence y(n) = 0. Finally, y—1(0) = Y+|(0) = 0 (23) since h(0) = 0, so y(0) = 0. Therefore, y(n) = 0 V n. I (24) Remark: If assumption 3) for the [IR filter in property 4 cannot be satisfied for some reason, i.e., h(0) =1: 0, then the impulse is attenuated and y(n) = 0 for n at 0 and y(0) = h(0) at: 0. With the aperiodicity condition and H (0) = 1, it follows that h(0) < 1, which proves the attenuation of the impulse. Property 5: The MIMH filter preserves a unit pulse of length T = 27, with a maximum relative error of e = 1 — r. The resulting pulse has an m-shape with the maxi— mum error occurring at the center of the pulse (Fig. 1). Moreover, the length and the position of the pulse are not altered. Proof: From the unit pulse definition «$00 = 1401) - u(n — T — l) for T 2 l. (25) The output of the MIMH filter is H") = M”) = y+.(n) = 0 forn < O (26) and y(n) =x(n) = y_.(n) = 0 forn 2 T+ 1. (27) For 0 s n s T, at least one of the two outputs, y_, or y“, is nonzero which yields a nonzero output y(n) for 0 s n s T. Hence, the width and location of the pulse are not changed. For Teven and t = 21,, obviously the center of unit pulse is T/2 = 7-,. Because of symmetry (i.e., two identical IIR filters applied in the yr. and y_, directions) y+|(Tr) = y—](Tr) = r- With the aperiodicity condition of assumption 2), y+1(n) is monotonically increasing and y_l (n) is monotonically IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40, NO. 5, MAY l992 1.2 1 , _ 0.8 — 0.6 1 0.4— 4 0.2 - _ 00 3 10 15 20 25 30 Fig. 1. For an input of a unit pulse (indicated by ————), the MIMH filter output for a = 0.5 after 1 pass. decreasing for 0 s n s T. Therefore, W!) = 1 > y—1(n) = you 2 r 2 MM) T for 0 s n S E (29) and mu = 1 > y+1(n) = you 2 r 2 L100 for 5T5 n S T (30) which yields T y_l(n) forO S n s E T y(n) = y+,(n) for; s n s T (31) 0 otherwise. Clearly, the maximum error occurs at T/ 2 = 7, and its value is 1 — r, where y<g>=y+l<g>=y-l<§>=r- Since the impulse response is aperiodic from assumption 2), it follows that for a step input (32) y+1(n) S y+.(n + 1) 0 S n S T (33) and y_1(n) 2 y_,(n +1) 0 s n s T. (34) Thus, with (31), (33), and (34) T y(n) 2 y(n +1), forO s n s 5 — 1 (35) and T y(n) S y(n + l), fora S n S T. (36) The output of the MIMH filter has an m-shape for 0 S n s T. For Todd, the results are similar. I BAUER et aL: AN llR MEDIAN HYBRID FILTER Property 6: If the input is symmetric, then the output of the MIMH filter is symmetric. Proof: Without loss of generality, symmetry is as- sumed around n = 0 for x(n), i.e., x(n) = x(-n) V n. (37) For the two endpoints of the signal (i.e., n = L and n = —L), (1) and (2) are Y+1(*L) = amy+1(—L —— m) + ... + boX(—L) + b.x(—L — 1) + ... + 01}’+1(—L — 1) + bkx(~L — k) (38) and y_l(L) = army—AL + m) + "- + air—.(L +1) + b0x(L) + b.x(L + 1) + --- + bkx(L + k). (39) With (4), (5), and symmetric initial conditions for the ini— tial values y+1(—n) = y—1(n) for" > L (40) (39) becomes >LI(L) : amY+1(‘L — m) + "’ + a1)’+1(—L — 1) + b0x(~L) + b.x(—L — 'l' ' ' ‘ + bkx(—L — k). (41) Thus, (38) and (41) are equal: y+1(—L) = y—1(L)- (42) The same argument can be used for the next recursion (i.e., n = L — 1) and for all further recursions. Hence, y+.(n) = rid-n) V n (43) and the output of the MIMH filter is symmetric, i.e., y(n) = median {y+i(n), x(n), y-1(n)} = median {yv1(—n),x(*n)7 y+1(—n)} = Y(—n) I (44) Property 7: The output of the MIMH filter is bounded by the two-directional IIR filter outputs. Proof: Property 7 can be rewritten as Vn. }'+1(") 2 fl") 2 )7—1(") for Y+I(n) Z Y—i(n) (45) or MM) 5 rm) 5 31—101) for MW) 5 L100 (46) being true. Obviously, since y(n) = median {y+1(n), x(n), y_, (n)}, the output has to be one of the following three cases: i) y(n) = y+l (n), ii) y(n) = x(n), or iii) y(n) = y_. (n). Cases i) and iii) directly yield property 7. For case ii), y(n) = x(n) = median {y+l(n), x(n), y_,(n)} also yields y(n) to be bounded between y+1(n) and y—1(n)- I 1071 Remark: Property 7 is useful in the analysis of the MIMH filter output to noisy input signals since it de- scribes the use of the [IR filter outputs as bounds for the MIMH filter response. Property 8: For the sinusoidal input signal x(n) = A sin (am + 45), the magnitude of the MIMH filter output is bounded by A | H (w)| , where lH(w)| is the magnitude re— sponse of the IIR filter module. Proof: With the input x(n) = A sin (am + 45), the directional outputs for all n are IY+1(n)l s AIH(w)I (47) and ly—1(n)| S A|H(w)|. (48) Using property 7. it follows that |y(n)| S AIH(w)|, V n. I (49) Property 9: The MIMH filter preserves a monotonous slope represented by: N s(n) = A aku(n — k) (50) where all ak for 1 S k S N are either nonnegative or nonpositive. Proof: Due to the aperiodicity condition of the IIR filter module, the output of the IIR filter with noninverted causality, y+1 (n), for a step input satisfies y+1(n) S u(n) for all n. Due to linearity and shift invariance, for the input of s(n), the output y+l (n) is bounded by N yflm) s [210(2an — k) v n (51) forak 2 0,1 S k S N, and N y+i(n) 2 goakutn — k) v n (52) for ak S 0, l S k S N. In the same way, it follows from (5]) and (52) that for the IIR filter with inverted causality with an input of s (n), N y_1(r1) a go aku(n — k) v n (53) forak 2 0,1 S k S N, and N Z} aku(n — k) v n k=0 IA 37—100 (54) forak S 0,1 S k S N. Hence, whenak 2 0f0rl S k SN, M100 5 SO!) 5 L101) V n (55) andwhenak S Dforl S k S N, L10!) 5 WI) 5 y+1(n) V n. (56) IOT.’ Thus , fl") = median {)’+1(n), S(n), L100} = 501)- I (57) Remarks: 1) Property 3 is actually a special case of property 9 withN = Oand ac =1. 2) If the aperiodicity condition in assumption 2) is dropped, properties 2—4 and 6—8 still hold true. For prop- erty 5, it can be shown that location and length of the pulse are not changed and that the overshoot of the MIMH filter is always smaller than or equal to that of the IIR filter module. 3) Assumptions l)—3) are not needed for property 7 to be true. B. Properties of the MIMH Filter in the Presence of Noise In this section, properties of the MIMH filter are de- rived for the case when either impulse noise or Gaussian noise is present. Property 10: An input of two unit impulses, Le. , x(n) = 6(n) + 6(n — T) (58) causes a nonzero output signal of the MIMH filter for 0 S n s T. Specifically, the output of the MIMH filter at location n is given by h(T) for n=0 and n=T min {h(n), h(T — n)} '(n) = 59) y for 0 < n < T ( 0 otherwise. Furthermore with 1, = T/Z Osy(n)<1—r forOSnST (60) where y+1 (1,) is defined as in definition 4. Proof: For n > Tor n < 0, the MIMH filter output y(n) is obviously zero, as the input signal x(n) and one of the IIR filter outputs are zero. From assumptions 1)— 3), it is known that h(O) = 0 and 0 s h(n) < l for all n. Therefore, y(0) = y(T) = median {0, 1, h(T)} = h(T). ForO < n < T, y(n) = median {0, h(n), h(T — 11)} = min {h(n), h(T — n)} and (59) results. For every n in O s n s T, at least one of the two IIR filter module responses, yH (n) or y_l(n), produces an impulse response value h(n) 2 0 with n 2 r, + 1. With h (n) 2 0 for an aperiodic impulse response and 0—dB zero frequency gain from assumptions l)—3), (61) (62) Z h(i)=1—r. i=ry+l (63) IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 40. NO. 5. MAY 1992 Obviously, h(n) < l —— rforn 2 r, + 1. Since x(n) = 0for0 < n < T, (64) At the location of the two impulses, y+1(0) = 0 and y,l(T) = 0. Thus, 05y(n)<l—r for0<n<T. OSy(n)<1-—r forOSnsT. I(65) Property 11: For the input signal x(n) = 5(n) — 5(n — T) the output of the MIMH filter is zero for all n. Proof: For n < 0 and n > T, the MIMH filter out— put y(n) is obviously zero. For 0 < n < T, (66) Y+1(") > x01): 0 > )HW- (67) So, y(n) = 0, for O < n < T. At the occurrences of the two impulses, y(0) = median {0, l, —h(T)} = 0 (68) and y(T) = median {h(T), —1,0} = 0. (69) Thus, y(n) = O, V n. I Next, properties of the filter driven by Gaussian noise are examined. From [2], given independent and identi- cally distributed (i.i.d.) input noise, the probability den- sity function of a three-point standard median filter is given. Using this result, the inverted causality of the two- directional outputs of the MIMH filter, and linear filter theory, the following two properties for the probability density function of the MIMH filter are stated without . proof. Property 12: If f(x) denotes the probability density function of the input signal, F (x) denotes the probability distribution function of the input signal, f,(x) denotes the probability density function of the HR filter output, and F ,(x) denotes the probability distribution function of the IIR filter output, then the probability density function of the MIMH filter output is fy(X) = 2F1(X)f(x)(1 — F100) + 2F1(x)f1(x)(l — Fm) + 2F(X)fl(x)(‘1 - F100). (70) Property 13: For Gaussian noise with g(x) denoting the standard Gaussian probability density function for 02 2 l and G(x) denoting the probability distribution func- tion of g(x), the probability density function of the MIMH filter output is fy(x) =. 20 g(x)G + a a e> g <a> are) BAUER e! 0].: AN llR MEDIAN HYBRID FILTER 1.6 r " 1.4 1.2 l 0.8 0.6 0.4 0.2 0 V - - 0 0.5 1 1.5 2 2.5 3 3.5 4 Fig. 2. Probability density functions. (—): MIMH filter with K 2 0|. (————): 11R filter with K = 0.1. (- r - -): MIMH filter with K = 0.4. ('-----): IIR filter with K = 0.4. 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ~ , 0 0.5 1 1.5 2 2.5 3 3.5 4 Fig. 3. Probability density functions. (——): MIMH filter with K = 0.7. (————): [1R filter with K = 0.7. (NH): MIMI-I filter with K 2 1.0. (-—'—~—): IIR filter with K =1.0. where on 211’ K = 210 h2(i) = —1— S0 |H(w)|2 do) (72) Zr is the energy content of the impulse response. Remarks: 1) Both properties 12 and 13 do not require assump— tions 1)—3) as described in Section [LA to be true for the IIR filter. 2) Equation (72) is useful for the evaluation of the noise output power of the HR filter modules. Figs. 2 and 3 show the probability density functions (pdf’s) of the MIMH filter (and the IIR filter) for different values of K. The Gaussian pdf is equivalent to the pdf of the HR filter with K = 1 and is included in these plots for comparison. Both figures refer to the pdf’s of the output signal of the MIMH filter when it is driven by i.i.d. Gaussian noise with a standard deviation of one and a mean of zero. The pdf’s of the outputs depend only on K as defined in (72) and are therefore applicable to all MIMH filters with the same value of K and a three-point standard median filter. In addition, the output pdf for K = 1 is the output pdf of the three-point standard median filter. In comparing the MIMH filter curves and the HR filter curves for the same values of K, the output pdf of the MIMH filter is always “narrower” than the output pdf of the single LSI IIR filter. This indicates that the MIMH 1073 filter produces a lower second-order moment which im- plies better noise power suppression capabilities. III. COMPARISON WITH SIMILAR FILTERS In this section, the MIMH filter is compared with the following filters: two subclasses of the LMH filters, the so-called FMH filter and HR2 median hybrid filter as in- troduced in [1], and the standard median filter. The com- parisons are based on the operation, the computational cost, and simulation results of the filters. First, simula— tions and the computational costs of the MIMH filter are presented. Next, the three related filters are examined in comparison to the MIMH filter. A. The MIMH Filter The simplest representative of the class of filters dis- cussed in Section II satisfying assumptions 1)—3) (i.e., a O-dB zero frequency gain, low-pass filter with a delayed aperiodic response) is described by the first—order IIR fil- ter. M") = 0M" — l) + (1 — (1)2601 — 1) forO s a < 1 (73) where the upper bound on a is imposed to guarantee sta- bility of the filter. In the case of a = 0, (73) becomes a degenerate FIR filter consisting of a single delay and the resulting MIMH filter is equivalent to a three-point stan— dard median filter. Obviously, the properties of the [IR filtering module can in this case be controlled by a single parameter a. The time constant 7, (see Section 11) there- fore depends only on a: Tr = log, (1 — r) (74) where 7-, is usually interpreted as being an integer. Clearly, for a fixed r, an increasing value of or causes 7, to increase as the bandwidth of the low-pass filter is lowered. In Fig. 4, an uncorrupted input signal, which is a line from a digital image of length 256 with values between 0 and 255, is shown. In Fig. 5, the input signal is shown corrupted with Gaussian noise with zero mean and a vari- ance of 10 and with i.i.d. impulse noise of 0.02 occur- rence probability with an amplitude of 50. Using Fig. 5 as the input signal, Fig. 6 shows the MIMH filter output for 01 = 0.3 after 1 pass, and Fig. 7 shows the MIMH filter output for 01 = 0.75 after 1 pass. The noise output power of the MIMH filter with a = 0.75 in Fig. 7 is clearly lower than that of the filter with at = 0.3 in Fig. 6sinceK =1/7 fora = 0.75 andK = 0.7/1.3 fora : 0.3. In both Figs. 6 and 7, the impulses are eliminated, although the filter with the larger passband (Le, a = 0.3) tends to produce more noise in the immediate neighbor— hood of an impulse and also preserves more of the original signal details of Fig. 4. An interesting phenomenon is that the filter with a = 0.75 removes noise less efiiciently at or near signal edges than the filter with a = 0.3. This is due to the increasing difference between y“ (n) and y_1(n) 1 074 250 r 200 150 100 50 .a. .4. L . 0 50 100 150 200 250 300 Fig. 4. Uncorrupted input signal, 200 150 100 50 0 .i . _ t . 0 50 100 ISO 200 250 300 Fig. 5. Input signal corrupted with Gaussian noise with zero mean and a variance of 10 and with i.i.d. impulse noise of0.02 occurrence probability with an amplitude of 50. 250 200 150 mo 50 t. A . .1 .— 0 50 100 150 200 250 300 Fig. 6. MIMH filter output for a = 0.3 after 1 pass. 250 w a 50 0 l i p .t. 50 l 00 l 50 200 250 300 Fig. 7. MIMH filter output for a = 0.75 after 1 pass. IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 40, N0. 5, MAY 1992 around signal edges for slowly responding linear filtering modules (i.e., filters with large 0:). In contrast to the standard median filter, the MIMH fil- ter may show a rather slow convergence rate to a root signal. This is due to the infinite impulse response length of the linear filtering module. It is therefore convenient to define an e-root signal: Definition 5: An output signal y(n) is an e-root signal to an input signal x(n) if max |y(n) — x(n)| < e (75) where e is a positive real number. In Fig. 8, the e-root signal is shown after 20 passes for e = 0.4332. In other words, the largest difference be- tween two signals belonging to two consecutive filtering passes at any point is smaller than 0.4332, where the total dynamic range is from 0 to 255. Although the output of two consecutive filtering passes is relatively close, it does not guarantee that the e-root signal for e = 0.4332 is in any sense close to the e—root for a significantly smaller 6. The e-root signal however shows signal features which are preserved totally or with a very small error. Fig. 8 demonstrates that sharp edges and step signals as well as monotonically increasing or decreasing signal slopes are preserved (see properties 3 and 9). For the filter described by (73), the value K in (72) is given by (76) For the permitted interval of a, 0 s 01 < 1, the constant K is similarly bounded as 0 < K s 1. Hence, even for a first—order IIR filter, the constant K can be made arbitrar- ily small as discussed at the end of Section III-B. The pdf of the IIR filtering module, and the overall MIMH filter output, can thus be made arbitrarily narrow to conform to any desired specifications. This implies that total control over the output variance of the MIMH filter is indepen- dent of the order of the IIR filter module. As discussed in the next section, this is in direct contrast to the FMH fil— ter. Next, the computational cost of the MIMH filter is de- rived. From [10], a bubble sort routine to sort it items needs n(n — l) / 2 comparisons and swap operations. So, the three-point standard median filter needs 3 compare/ swaps. Using an IIR filter described by M") = arty/('1 — m) + ' ' ' + aim" * 1) + b]x(n — l) + ' " + bkx(n — k) (77) the MIMH filter thus needs 3 compare/swaps + 2(m + k) multiplications + 2(m + k — l) additions. With the first-order IIR filter MOI) = 0M" — 1) + (1 r (1)1601 — 1) oz(y(n — 1) — x(n — 1)) + x(n — l) (78) BAUER et al.: AN IIR MEDIAN HYBRID FILTER 250 200» 150 0 50 flj 100 130 260 256 300 Fig. 8. MIMH filter e-root signal for a = 0.3 after 20 passes. 50 the MIMH filter needs 3 compare/swaps + 2 multiplica- tions + 4 additions. B. The FMH Filter The FMH filter for M = 3 with identical tap coefficients is given by y(n) = median (x(n — k) + --- + x(n — 1)), x(n), % (x(n + 1) + --- + x(n + k))} (79) where M is the number of subfilters as defined in [2] and where k is the order of the finite impulse response (FIR) filter. Consider the MIMH filter described by (l)—(3) and set a,- = O for 1 s i s m. The two IIR filtering modules now become two FIR linear filtering modules, and so the resulting multidirectional FIR-median hybrid (MFMH) filter is given by y+1(n) = b0x(n) + blx(n — 1) + ~'~ + bkx(n — k) (80) y,](n) = b0x(rt) + b1x(n + l) + --' + bkx(n + k) (81) and y(n) = median {y+1(n), x(n), y_l(n)} for l s n S L. (82) If [)0 = 0 (see assumption 3)) and bi = l/k for 1 s i s k, the FMH filter in (79) is obtained as a special case of the MFMH filter. Some of the properties are now briefly investigated for the MFMH filter if the same assumptions are made as in Section [LA for the MIMH filter. Obviously, with similar arguments, properties 1—13 are still satisfied. In addition, the following properties can be formulated for the MFMH case. Property 14: For the MFMH filter, a pulse of length T 2 2k is preserved with a relative error of e = 0. Proof: From Definition 4, Tm = k for the FIR filter. Using this and following the proof of property 5, the pulse of length T 2 2k is not altered. I 1075 Property 15: If T > 2k, the response y(n) of the MFMH filter to the input signal of (58) is N") = 0 V n. (83) Proof: If the two FIR directional outputs do not overlap, then there is no partial output response. Using this and following the proof of property 10, both impulses are removed with no artifact developing between them for T > 2k. l Properties 14 and 15 can be interpreted in the following way. Due to the finitely extended impulse response of the FIR module, the MFMH filter is capable of producing a zero aliasing error for the response to two impulses if the impulses have a certain minimum distance to each other, which is a function of the number of taps of the FIR filter. The MIMH filter, however, will always produce time do- main aliasing of the directional responses of the two im- pulses. Clearly, the aliasing will decrease with an in- crease in the distance between the two impulses. This result has a well—known analogy in the sampling theorem of band-limited and infinitely extended signal spectra. For a FIR filter of order k, Y;(n) = a1x(n — 1) + a2x(n — 2) + --- + akx(n — k) (84) it is desired to optimally choose the coefficients a,- for 1 s i s k such that a O-dB zero frequency gain is achieved and the noise power at the filter output is minimized. From [2], this choice is shown to be 1 '=ak=; (85) which yields " 1 2 1 K... [:21 (k) k. (86) For an IIR filter, the minimization of K under the condi— tion of O—dB zero frequency gain does not depend on the filter order since the passband can be made arbitrarily small for any order filter. This then yields a K which is arbitrarily close to zero. Hence, for an eflicient suppres— sion of Gaussian input noise, the FMH filter has to use FIR modules of sufiicient order, whereas for the MIMH filter, a first—order selection of the IIR filter module is al- ways sufficient. With the FIR filter described by (84) and (85), the three— point FMH filter needs 3 compare/swaps + 2 multipli- cations + 2(k — l) additions. For the MFMH filter de- scribed by (80)—(82) with b0 = 0, the MFMH filter needs 3 compare/swaps + 2k multiplications + 2(k — 1) addi— tions. For comparison to a first order MIMH filter with a = 0.75 and K = 1/7, choose a seventh order FIR filter with K = 1 / 7 described by yF(n) = ;x(n — l) + - - - + %x(n — 7) =;(x(n—1)++x(n—7)) (87) l076 250 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 40, N0. 5. MAY 1992 200 150 100 500 ST 1‘66 ~__J_.. 150 260 250 300 Fig. 9. Three»point FMH filter output for k = 7 after I pass. to form a FMH filter which needs 3 compare/swaps + 2 multiplications + 14 additions, more operations than the MIMH filter with a first order IIR. In Fig. 9, using the noisy signal of Fig. 5 as input, the three—point FMH of order 7 after 1 pass is shown. This FMH filter has K = 1/7, which is the same value for K as the first-order MIMH filter with a: = 0.75. Comparing the outputs of the MIMH filter with a = 0.75 in Fig. 7 and the FMH filter in Fig. 9, both filters have roughly the same noise suppression capabilities, which is due to the identical K values. However, the MIMH filter tends to preserve the original signal shape slightly better. As an IIR filter response can be approximated by an FIR filter, exactly zero aliasing for the MFMH filter can be obtained, as discussed in properties 14 and 15, if the resulting truncation error created by the FIR approxima- tion can be accepted. Since the resulting FIR filter re- quires an infinite number of taps to recover the frequency response of the IIR filter correctly, which is not feasible, the resulting magnitude response is altered due to win- dowing. Using this approach, the resulting FIR filter is unfortunately of much higher order than the [IR filter and therefore computationally more expensive. C. The 1le Median Hybrid Filter A special t pe of LMH filter was introduced in [l], the so—called IIR median hybrid filter. The output of the IIR2 median hybrid filter is defined in [1] as )4") = median {¢L(x(")), ¢c(x(n)), ¢R(x(n))} (88) where <I>,,(x(n)) is defined as the output of the IIR low— pass filter L operating on the signal x(n), <I>R (x (n)) is de- fined as the output of the delayed IIR low—pass filter R operating on the input signal x(n), and <I>C (x (n)) is usually either x(n) or a delayed version of x(n). If the output expression (88) of the IIR2 median hybrid filter is com— pared with (3) of the MIMH filter, one major difference becomes apparent: the MIMH filter employs one IIR module applied in two opposite directions using the con— cept of inverted causality, whereas the IIR2 median hybrid filter is a causal system and different IIR modules can be used. Due to this difference, properties 3—6 and 9—13 do not hold for the [IR2 median hybrid filter, while properties 2 and 8 may hold under special conditions. D. The Standard Median Filter The three-point standard median filter is given by y(n) = median {x(n — 1), x(n), x(n + 1)}. (89) By using a single delay for the IIR filtering module, the three—point standard median filter becomes a special case of the MIMH filter. The standard median filter is not re— stricted to a three-point window, but this size is used for comparison since it is the one used in the nonlinear filter- ing module of the MIMH filter. Standard median filters of varying window sizes have been treated in previous works. When using the standard median filter, the only design parameter available is the window size, whereas when using the MIMH filter, all of the theory involving the design of LSI IIR filters is available. In previous publications, properties of the standard me- dian filter are derived which have parallels to the ones proved here. Namely, for a standard median filter with any window size, properties 1—4, 6, 9, and 11 hold. The pdf for Gaussian input noise of the three-point standard median filter is shown in Fig. 2, where the MIMI-I filter curve for K = l is the same as the three-point standard median filter curve. From comparing the graphs, it can be seen that the MIMH filter has superior Gaussian noise suppression capability for K < 1. With assumption 2) from Section II-A, the aperiodic impulse response of the IIR filtering module ensures that K < 1. If a bubble sort routine is used, the three-point standard median filter needs 3 compare/swaps, and the five-point standard median filter needs 10 compare/swaps. Assum— ing that a compare/swap operation takes more time than a floating point addition operation or a floating point mul— tiplication operation, the computational load of the MIMH filter using a first-order IIR filter is somewhat higher than BAUER et ul.; AN 11R MEDIAN HYBRID FILTER 250 a .. 200 — . 150 M _ 100 » J 500 Si) 160 13'0 20‘?) 257) 300 Fig. 10. Threeepoint standard median filter output after 1 pass. 250 200 — _ ISOI— 1 100 r aw] ] 50 fit . 1 r . 0 so 100 150 200 250 300 Fig. 11. Five-point standard median filter output after 1 pass. that of a three—point standard median filter (due to the ad- ditional arithmetic operations) and usually significantly lower than that of a five-point standard median filter. The performance of the first-order MIMH filter is therefore compared with a three—point and five-point standard me- dian filter. For the noisy input signal given in Fig. 5, the three-point and five-point standard median filter outputs after 1 pass are shown in Figs. 10 and 11, respectively. Comparing these outputs with signals of the MIMH filter shown in Figs. 6 and 7, the noise suppression'capability for the standard median filter is far inferior to that of the MIMH filter. This conclusion is reinforced by comparing the pdf plots of the three—point standard median filter and MIMI-l filter for K < 1 as shown in Figs. 2 and 3, where the pdf plot for the three-point standard median filter is equivalent to the one for the MIMH filter with K = 1. IV. CONCLUSION Finally, some of the advantages and disadvantages of using the MIMH filter are elaborated. First, the MIMH filter is a novel modularized method to combine both lin— ear filtering methods and nonlinear filtering methods such that desirable properties of the overall filter develop; a LSI IIR filtering module which uses inverted causality is concatenated with a three-point standard median filtering module to form the MIMH filter with both signal preser- vation and noise suppression properties. Because the con- 1077 catenation scheme is chosen, a simplification of the anal— ysis of the nonlinear MIMH filter is achieved. Since the linear filtering module is not specified, a different LSI IIR filter can be used for a particular signal or class of signals. This has the advantage of being able to use the plethora of available LSI IIR filter design tools, which allows for a great design flexibility. The MIMH filter is also shown to reduce noise in signals with low density impulse noise and Gaussian noise. Due to the concatenation scheme used, some statistical properties of the MIMH filter could be derived. In addition, the MIMH filter has no direc- tional dependence, which is not the case for the LSI IIR filter or the IIR2 median hybrid filter. These advantages of the MIMH filter indicate a potential use for processing signals where constant regions and edges are to be pre- served and where impulses and other noise are to be sup- pressed. These requirements are often desired in image processing, where IIR filters perform poorly. A future di- rection of the work described here is the application of the MIMI-I filter to two-dimensional signals (i.e., im- ages). Next, some of the limitations of the MIMH filter are explained. If the exact properties of the noise in the signal are kn0'wn, there exist other methods besides the MIMH filter which can remove the noise very effectively. For example, if the signal is known to contain extremely high density impulse noise, the standard median filter is a bet- ter choice than the MIMH filter. Or, if the type of additive noise is known, optimal filtering techniques can be used. This obviously implies that there is either a priori infor— mation available or at least enough time to analyze the signal to obtain the required noise information. A poten- tial drawback of the MIMH filter is that it requires a finite length sequence over which to operate. This disadvantage might be overcome if the incoming signal of infinite length is separated into finite support sequences. REFERENCES [l] J. Astola. P. Heinonen, and Y. Neuvo, “Linear median hybrid fil- ters,” Trans. Circuits Sysr., v01. 36, no. 11, pp. 1430—1438, Nov. 1989. [2] P. Heinonen and Y. Neuvo, “FIR medianehybrid filters," IEEE Trans. Acoust, Speech, Signal Processing. vol. ASSP-35, no. 6. pp. 832—838, June 1987. [3] P. Heinonen and Y. Neuvo, “FIR-median hybrid filters with predic- tive FIR substructures,“ IEEE Trans. Aroust., Speech, Signal Pro- cessing, vol. ASSP-36, no. 6, pp. 892~899. June 1988. [4] J. W. Tukey, Exploratory Data Analysis. Menlo Park, CA: Addi- son-Wesley, 1971, 1977. [5] N. C. Gallagher. Jr., and G. L. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Arman. Speech, Signal Processing. vol. ASSP-29, no. 6, pp. 113641142. Dec. 1981. [6] T. A. Nodes and N. C. Gallagher, Jr., “The output distribution of median—type filters," IEEE Trans. Commun., vol. COM-32, pp. 5237 541. May 1984. [7] A. C. Bovik, T. S. Huang, and D. C. Munson, Jr.. “A generalization of median filtering using linear combinations of order statistics,“ IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, no. 6. pp. 134271350, Dec. 1983. 18] T. A. Nodes and N. C. Gallagher. “Median filters: Some modificav 1078 IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 40. N0. 5. MAY I992 Michael A. Sartori received the 8.8.. M.S.. and Ph.D. degrees in electrical engineering from the University of Notre Dame in 1987. 1989, and 1991. respectively. He worked for the McDonnell Douglas Elec- tronics Company during the summers of 1986 and tions and their properties." IEEE Tram. Ararat, Speech, Signal Processing. vol. ASSP—30. n0. 5, pp. 739—746, Oct. 1982. [9] Y. H. Lee, 5. K0. and A. T. Fam, "Efficient impulsive noise suppression via nonlinear recursive filtering.” IEEE Tram. Aroush, Speech, Signal Processing, vol. 37. no. 2, pp. 303—306. Feb. 1989. [10] R. Sedgewick, Algorithms. Reading. MA: AddisoneWesley, I988. Peter H. Bauer (S‘86~M‘87) was born in Bani» berg. Germany, in 1959. He received the Diplom degree from the Technical University Munich, Germany, in 1984 and the Ph.D. degree from the University of Miami in 1987, both in electrical engineering. During the years 1982 and 1984 he worked for MBB and Siemens AG, respectively. In May 1988 he joined the Department of Electrical Engineer ing at the University of Notre Dame as an Assis- tant Professor. He is currently affiliated with the Laboratory of Image and Signal Analysis (LISA) at the University of Notre Dame. His interests include multidimensional systems theory, nonlinear and robust system stability, digital filters, and signal processing. Dr. Bauer served as an Associate Editor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS from 1989 to 1991. autonomous systems. 1987 and for the McDonnell Douglas Missile Sys tems Company during the summer of 1989. He is currently with the U.S. Navy’s David Taylor Re- search Center. His research interests include neural networks, digital image processing, and Timothy M. Bryden received the 8.5. and MS. degrees in electrical engineering from the Univer- sity of Notre Dame in 1987 and 1990. respece tively. While at Notre Dame, his major focus was research in the area of distributed operating sys- tems as part of an ongoing. IBMefunded research project. He is currently employed by IBM in Pough» keepsie. NY. where his areas of interest are frameworks for CAD software, user interfaces for electronic CAD software. and computer graphics. ...
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