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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 shiftinvariant 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 twodirectional 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 socalled 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 shiftinvariant (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 threepoint standard median ﬁlter
then operates on these twodirectional 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 NAG3l 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 OdB 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
ky(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) = y1(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 mshape 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>=yl<§>=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 mshape 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) = ridn) V n (43) and the output of the MIMH ﬁlter is symmetric, i.e.,
y(n) = median {y+i(n), x(n), y1(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 twodirectional 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 AH(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 threepoint 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): MIMII 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 threepoint
standard median ﬁlter. In addition, the output pdf for K
= 1 is the output pdf of the threepoint 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 secondorder 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
socalled 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
OdB zero frequency gain, lowpass 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 threepoint 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 lowpass 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 eroot signal: Deﬁnition 5: An output signal y(n) is an eroot 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 eroot 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 eroot signal for e = 0.4332 is in
any sense close to the e—root for a signiﬁcantly smaller 6.
The eroot 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 IIIB. 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 threepoint 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
ﬁrstorder 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 eroot 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 FIRmedian 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 bandlimited 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 OdB 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 ﬁrstorder
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 threepoint 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 threepoint 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 threepoint standard
median ﬁlter is shown in Fig. 2, where the MIMII ﬁlter
curve for K = l is the same as the threepoint 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 IIA, the aperiodic impulse response of the
IIR ﬁltering module ensures that K < 1. If a bubble sort routine is used, the threepoint standard
median ﬁlter needs 3 compare/swaps, and the ﬁvepoint
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 ﬁrstorder 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. Fivepoint 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 ﬁvepoint standard median ﬁlter. The
performance of the ﬁrstorder MIMH ﬁlter is therefore
compared with a three—point and ﬁvepoint standard me
dian ﬁlter. For the noisy input signal given in Fig. 5, the
threepoint and ﬁvepoint 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
MIMIl ﬁlter for K < 1 as shown in Figs. 2 and 3, where
the pdf plot for the threepoint 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 threepoint 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 MIMII ﬁlter to twodimensional 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. ASSP35, no. 6. pp.
832—838, June 1987. [3] P. Heinonen and Y. Neuvo, “FIRmedian hybrid ﬁlters with predic
tive FIR substructures,“ IEEE Trans. Aroust., Speech, Signal Pro
cessing, vol. ASSP36, no. 6, pp. 892~899. June 1988. [4] J. W. Tukey, Exploratory Data Analysis. Menlo Park, CA: Addi
sonWesley, 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. ASSP29, 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. COM32, 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. ASSP31, 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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