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Proceedings of the 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. V, pp. 57-60.

(Minneapolis, Minnesota, 27-30 April 1993.)

LINEAR COMBINATIONS OF MORPHOLOGICAL OPERATORS:

THE MIDRANGE, PSEUDOMEDIAN, AND LOCO FILTERS

Mark A. Schulze and John A. Pearce

Department of Electrical Engineering and Biomedical Engineering Program

Engineering Science Building 610

The University of Texas at Austin

Austin, Texas 78712

ABSTRACT

Morphological image processing filters preserve shapes

related to the structuring element shape of the operator.

The basic morphological operators are minimum (erosion)

and maximum (dilation) operations performed on the pixels

within a structuring element. Although these operators

(and the compound operators formed from them) are able

to smooth noise, they also introduce a statistical and

deterministic bias, which is unacceptable in some applica-

tions. However, since every morphological operator has a

complementary operator that is equally and oppositely

biased, we propose averaging the complementary

operators to alleviate the bias. Of the three filters formed

by averaging the standard morphological operators, two

are the previously-defined midrange filter and

pseudomedian filter, while one is a new filter, which we call

the LOCO filter. Under most conditions, the LOCO filter is

the best of these at reducing impulses and noise.

1. INTRODUCTION

The techniques of mathematical morphology provide

shape-based methods for image processing. The basic

morphological operators have been shown to be effective

at reducing various types of noise while preserving shapes

compatible with the structuring element of the operator.

However, the basic morphological operators introduce a

statistical and deterministic bias to signals that they

process [1-3]. For many applications, such as

segmentation, this bias is not a problem. However, in

applications where preservation of intensity levels is

important, such as quantitative infrared thermography,

biased morphological operators may not be used to

process images.

Since morphological operators are defined in

complementary pairs that are equally and oppositely

biased, one potential solution to the biasing problem is to

average the complementary operators. We show in this

paper that of the three filters formed by taking the

averages of complementary morphological operators, two

of them are equivalent to the previously-defined midrange

and pseudomedian filters [4-6], while one is an entirely new

filter. This new filter, which we call the LOCO filter, is the

best of the three at reducing most types of noise,

especially impulse noise.

2. MORPHOLOGICAL FILTERS

The two basic morphological operators are dilation and

erosion. Let f(x) denote an n-dimensional function (the

image) and N denote a compact k-dimensional set (the

structuring element) with k = n. Also, let Ñ denote the 180°

rotation of N, and let the translation of the set N by a point

z be denoted by a subscript: N

z

. The morphological

erosion of f by N is defined by the following equation:

( f Θ Ñ ) (x) = )}({

inf

y

x

y

f

N∈

(1)

Morphological dilation is defined by:

( f ⊕ Ñ ) (x) = )}({

sup

y

x

y

f

N∈

(2)

For discrete images and structuring elements, the

infimum (inf) and supremum (sup) are equivalent to the

minimum and maximum, respectively. Because these

operators output extreme order statistics, it is obvious that

they introduce statistical and deterministic bias to the

functions.

Morphological opening is a compound operator

consisting of erosion followed by dilation; similarly,

morphological closing is dilation followed by erosion.

Opening is defined by:

Open{f(x); N} = [ ( f Θ Ñ ) ⊕ N ] (x). (3)

Closing is defined by:

Close{f(x); N} = [ ( f ⊕ Ñ ) Θ N ] (x). (4)

The doubly compound morphological operators open-

close (OC) and close-open (CO) are defined as opening

followed by closing and as closing followed by opening,

respectively:

OC {f(x); N} = Close{Open{f(x); N}; N} (5)

CO {f(x); N} = Open{Close{f(x); N}; N} (6)

Maragos and Schafer [2, 3] have demonstrated the

deterministic bias introduced by the morphological

operators by proving the following inequalities:

(f Θ Ñ) = Open{f; N} = f = Close{f; N} = (f Η Ñ) (7)

OC (f; N) = med

∞

(f; W) = CO (f; N), (8)

where med

∞

(f; W) denotes the median root signal (that is,

a signal invariant to further median filtering) achieved by

repeatedly median filtering f(x) with window W = N Η Ñ.

Stevenson and Arce [1] illustrate these properties

statistically by deriving the distribution function of the

output of the CO operator, which is biased toward higher

values than the input distribution. The output distribution

of OC is analogous to CO, but biased toward smaller

values. The other morphological operators have more

severely biased distributions, as would be expected from

(7) and (8).

3. LINEAR COMBINATIONS OF

MORPHOLOGICAL FILTERS

One potential solution to the bias problems of the basic

morphological operators is to form new filters that take the

average the complementary operators. The three filters

formed by this method are the midrange, pseudomedian,

and LOCO filters. They are described in more detail below.

3.1. Midrange Filter

Morphological erosion simply returns the minimum

value within its structuring element, while morphological

dilation returns the maximum value. The average of the

erosion and dilation is therefore the midpoint of the range

of values in the structuring element. This is the midrange

filter [6, 7], a well-known estimator in the theory of order

statistics. It is the minimum variance unbiased estimator of

the median of a uniform noise distribution [7].

The response of the midrange filter is typically not desir-

able for image processing, because it destroys edges. Its

performance is good for constant signals in the presence

of uniformly distributed noise; however, for other noise

distributions and for images with edges, other filters

perform better.

3.2. Pseudomedian Filter

The pseudomedian filter was first developed by Pratt,

Cooper, and Kabir [4] in 1985 to mimic the response of the

median filter. In one dimension, their definition for the

pseudomedian filter of window width 5 (PMED

5

) was

PMED

5

{

a, b, c, d, e

}

=

1

2

max

{

min{a,b,c}, min{b,c,d}, min{c,d,e}

}

+

1

2

min

{

max{a,b,c}, max{b,c,d}, max{c,d,e}

}

. (9)

Pratt [5] later called the two halves of this definition the

"maximin" and "minimax" functions. Schulze and Pearce

[8] defined a two-dimensional pseudomedian filter

analogous to the 1-D definition (9).

Morphological opening consists of erosion followed by

dilation. Erosion finds the minimum over a particular struc-

turing element, and the dilation that follows this erosion

finds the maximum of the previously-computed minima in

the structuring element. Opening is thus exactly the same

as the "maximin" portion of the pseudomedian definition

(9). Similarly, morphological closing is dilation followed by

erosion, so it finds the minimum of the maxima and is the

"minimax" portion of (9). Therefore, a new definition of the

pseudomedian filter (PMED) that is completely equivalent

to the previous definitions is

P

M

E

D

{

f

(

x

)

;

N

}

=

1

2

O

p

e

n

{

f

(

x

)

;

N

}

+

1

2

C

l

o

s

e

{

f

(

x

)

;

N

}

,

(

1

0

)

where f(x) is the (n-dimensional) signal to be filtered and N

is the structuring element of the morphological operators.

The response of the pseudomedian filter is indeed

similar to that of the median filter, with two very important

exceptions. First, the pseudomedian filter does not

completely remove isolated impulses, either high-valued or

low-valued, but reduces their amplitude to one-half the

original values. This can be verified by noting that

opening preserves negative impulses but removes positive

impulses, whereas closing preserves positive impulses and

removes negative impulses. Second, while the output of

the median filter is restricted to values that appear in the

original signal, the pseudomedian filter output may take on

values that do not appear in the original signal because it

takes the average of two values. This is important when

working with signals that are quantized. The output of the

pseudomedian filter may have to be rounded or truncated

to restrict it to the same levels. Other differences between

the pseudomedian and median filters, notably the effect of

the structuring element shape, are described in [8]. The 1-

D finite-length root signal set of the pseudomedian filter is

shown to be identical to that of the median filter in [9].

Fig. 1. Noisy original image.

Fig. 2. Midrange-filtered image (N = 3x3 square).

Fig. 3. Pseudomedian-filtered image (N = 3x3 square).

3.3. LOCO Filter

Open-closing and close-opening have been shown to

have many desirable properties [1-3]; for example, either

operation reduces a 1-D signal to a median filter root signal

in one pass. Open-closing (OC) is simply opening

followed by closing, while close-opening (CO) is closing

followed by opening. Unlike opening and closing, OC and

CO are able to remove both positive and negative

impulses. After considering the midrange filter as the

Fig. 4. LOCO-filtered image (N = 3x3 square).

Fig. 5. Median-filtered image (5x5 square window)

Fig. 6. Image after Close-Opening (N = 3x3 square).

average of erosion and dilation and the pseudomedian

filter as the average of opening and closing, it is logical to

form a filter from the average of OC and CO. We call this

filter the LOCO filter, for Linear combination of OC and CO.

LOCO

{

f(x); N

}

=

1

2

OC

{

f(x); N

}

+

1

2

CO

{

f(x); N

}

(11)

Since both OC and CO remove positive and negative

impulses, the LOCO filter is much less susceptible to

impulse noise than the pseudomedian filter. In this

characteristic, the LOCO filter resembles the median filter

even more than the pseudomedian filter does. The LOCO

filter is also not restricted to values in the original signal.

Although the OC and CO individually yield a root signal in

one pass, since these roots need not be identical (and

usually are not), the LOCO filter does not always yield a

root signal in one pass. The LOCO filter, like the

pseudomedian filter, preserves edges and is not

susceptible to fast-fluctuating periodic signals to which

the median filter is susceptible.

4. EXAMPLES

Examples of an image filtered by linear combinations of

morphological operators are given in the figures. Fig. 1 is

the noisy original image, with white Gaussian noise (SNR ˜

19 dB) on the left side, negative impulse noise (10% of

pixels) at the lower central and right parts, and positive

impulse noise (10%) in the upper right portion. Fig. 2

shows the results of 3x3 midrange filtering (N = 3x3

square). Fig. 3 is the result of 5x5 pseudomedian filtering

(N = 3x3 square), and Fig. 4 is the result of LOCO filtering

with N = 3x3 square. For comparison, the result of 5x5

square median filtering is shown in Fig. 5 and the result of

the CO operator (N = 3x3 square) is shown in Fig. 6.

These figures illustrate the differences among the

various linear combinations of morphological operators.

The midrange filter is neither good at removing impulse

noise nor at preserving edges. The pseudomedian filter

preserves edges but does not reduce impulse noise or

Gaussian noise very well. The LOCO filter is the best of

the three at reducing impulse and Gaussian noise,

although it is still somewhat susceptible to nearby

impulses. The median filter (Fig. 5), in contrast, does an

excellent job of removing impulse noise and has reduced

the Gaussian noise quite well. However, it also distorts the

object shapes in the image somewhat, particularly at sharp

corners. This effect is not observed in Figs. 3 and 4

because the morphological structuring element is square.

Finally, the result of CO in Fig. 6 shows the bias toward

high (bright) values of this operator for both the positive

impulse noise and Gaussian noise.

5. CONCLUSIONS

In this paper, we have shown how linear combinations

of morphological operators may be formed to alleviate the

bias introduced by the individual morphological operators.

Two of the filters formed by averaging complementary

operators, the midrange and pseudomedian filters, were

previously defined in non-morphological terms, but the

LOCO filter is a new filter. The examples given in the paper

along with the known properties of the constituent

operators (OC and CO) illustrate the potential superiority

of the LOCO filter over the midrange and pseudomedian

filters for many image processing applications.

Linear combinations of morphological operators allow

the shape control of morphological filters (exerted by the

selection of a structuring element) without introducing

bias. For example, the LOCO filter with a square

structuring element preserves 90° corners in an image

while reducing noise almost as well as the square-shaped

median filter, which rounds off such corners. These new

filter definitions provide a way to perform transformations

similar to those of mathematical morphology on images in

which preservation of intensity levels is important.

REFERENCES

[1] R. L. Stevenson and G. R. Arce, "Morphological filters:

Statistics and further syntactic properties," IEEE Trans.

Circuits Syst., vol. 34, pp. 1292-1305, Nov. 1987.

[2] P. Maragos and R. W. Schafer, "Morphological filters--Part

I: Their set-theoretic analysis and relations to linear shift-

invariant filters," IEEE Trans. Acoust., Speech, Signal

Processing, vol. 35, pp. 1153-1169, Aug. 1987.

[3] P. Maragos and R. W. Schafer, "Morphological filters--Part

II: Their relations to median, order-statistic , and stack

filters," IEEE Trans. Acoust., Speech, Signal Processing,

vol. 35, pp. 1170-1184, Aug. 1987.

[4] W. K. Pratt, T. J. Cooper, and I. Kabir, "Pseudomedian

filter," In Architectures and Algorithms for Digital Image

Processing II, Proc SPIE, vol. 534, F. J. Corbett, Ed., pp.

34-43, 1985.

[5] W. K. Pratt, Digital Image Processing, 2nd ed. New York:

Wiley, 1991.

[6] A. Restrepo and A. C. Bovik, "Adaptive trimmed mean

filters for image restoration," IEEE Trans. Acoust., Speech,

Signal Processing, vol. 36, pp. 1326-1337, Aug. 1988.

[7] H. A. David, Order Statistics, 2nd ed. New York: Wiley,

1981.

[8] M. A. Schulze and J. A. Pearce, "Some properties of the

two-dimensional pseudomedian filter," in Nonlinear Image

Processing III., Proc SPIE, vol. 1451, E. R. Dougherty, J.

Astola, and C. G. Boncelet, Jr., Eds., pp. 48-57, 1991.

[9] M. A. Schulze, Mathematical Properties of the

Pseudomedian Filter, M.S. thesis, University of Texas at

Austin, 1990.