COMPUTER
VISION,
GRAPHICS,
AND IMAGE
PROCESSING
&236-251
(1984)
Image Enhancement Using the Median and the
Interquartile Distance
IRWIN SCOLLAR AND BERND WEIDNER
Image
Processing
Laboratory
Rheinisches
Landesmuwum,
Bonn,
West Germany
AND
THOMAS S. HUANG
Coordinated
Science
Laboratory,
University
of Illinois,
Urbana-Champaign,
Illinois
61801*
Received October 30,1982; revised December 3,1982
Enhancement and noise cleaning algorithms based on local medians and interquartile
distances are more effective than those using local means and standard deviations for the
removal of spikelike noise, preserve edge sharpness better, and introduce fewer artifacts around
high contrast edges present in the original data. They are usually not as fast as the mean-standard deviation equivalents, but all are suitable for large data sets to be treated in small
machines in production quantities.
1. INTRODUCTION
Many effective image enhancement algorithms make use of local statistics in a
window of moderate size [l}. Such techniques are useful for noise cleaning, crispening, and adaptive contrast and brightness correction. In most cases the local
statistics used are the mean and standard deviation of the pixels in a square or
rectangular window. It is the goal of the present paper to show that, for certain
classes of images, better results can be obtained if we replace the local mean and
standard deviation by the local median and interquartile distance, respectively.
Specifically, this replacement is desirable if the image contains high contrast edges or
noise with heavy tailed probability density functions (pdf), especially spike- or
pointlike noise, or its two dimensional analog which we will call thread noise
because the noise looks like fuzz or threads in the image. It will be seen that the use
of the median and the interquartile distance is more effective in removing such noise,
in preserving edge sharpness, and in avoiding the introduction of artifacts on high
contrast edges (light or dark banding).
The images for which our enhancement algorithms were developed have inherently high resolution and large numbers of pixels. They are oblique aerial photographs of rural areas which show the outlines of buried archaeological sites as
discolorations in growing crops or soils; metallographic X-ray pictures of corroded
objects from archaeological excavations prior to restoration, infrared and ultraviolet
images of paintings and other art objects; and thermal infrared scans made from the
air to map buried ancient field systems. The users of such imagery demand hard
copy of the highest quality without visible scan lines and with full photographic gray
scale. To achieve this, most pictures are scanned with between two and four
thousand lines at equivalent pixel resolution, resulting in data which are typically 4
*On leave at the Image Processing Laboratory, Rheinkches Landesmuseum, Bonn, under the auspices
of the A. von Humboldt Stiftung, BOM, West Germany.
236
0734189X/84 $3.00
c~pyri%t
0 1984 by Academic Press, Inc.
All rights Of reproduction in any form reserved.
ENHANCEMENT
237
USING THE MEDIAN
to 16 Mbytes per image. Production quantities must be processed at our installation.
In such images, high spatial frequency but low contrast information is of great
interest, but high contrast edges of objects of no interest are widely present. These
edges may be modem field boundaries, roads, or structures, while the objects of
interest have low contrast and are near the edges.
Most of our pictures were obtained under unique conditions in the past and they
cannot be repeated. Defects in such images are typically due to dirt and stratches on
negatives because of improper storage or lack of care during processing, under- or
overexposure in spatially varying amounts, very low or very high brightness with the
information of interest restricted to a small part of the range, positionally dependent. Most pictures are on black and white aerial or X-ray film, some are direct
recordings on magnetic tape from nonfilm sensors, some on 35mm film of all
possible types. We have not processed the color pictures directly because we lack
suitable hardware, but black and white color separation negatives from color slides
have been treated.
Apart from the above, noise in our images is also due to film grain or scanner
photomultiplier
noise, and these are usually Gaussian. Most types are usually
present in an image at once. Signals are usually fine linelike structures with complex
geometries or irregularly shaped blobs, all identifiable as man-made when seen by a
trained observer. The aim of enhancement is to help him make the appropriate
pattern recognition decisions by reinforcing the ability of the human eye to detect
faint geometric structures in noisy backgrounds. Because of the very high resolution
demanded and given the quantities of pictures to be processed, algorithms must be
fast enough to keep up with picture input-output.
Given the financial limitations, a
large fast machine may not be used. The algorithms discussed in this paper possess
the required properties and have been used successfully for a number of years.
2. PROPERTIES
OF THE MEDIAN
AND THE INTERQUARTILE
DISTANCE
Let us first establish our notation.
f(i, i) = brightness at point (i, j) in the image.
Local mean (window size (2k + 1) x (2k + 1)):
Local standard deviation:
l/2
a(i, j) =
1
.
(2k + 1)’
Let pij(f)
be the sample cumulative distribution
{f(i
function of
- u, j - u)lu = -k ,..., 0 ,..., k; u = -k ,..., 0 ,..., k}.
Local median:
M(i,
j) = pFc,
such that Pii
= l/2
238
SCOLLAR,
Local interquartile
WEIDNER,
AND
HUANG
distance:
Q(i, j) = Y - A
where Pii
= 3/4 and Pii
= l/4. Note that if p is not unique, we take M(i, j)
as the midpoint of the range of ~1;similarly for h and V. Note also that for the sake
of simplicity we have used a square window, but generally the window may be of
any shape: rectangular, cross, ring, etc.
It is well known that the local median has the following properties [2]:
(1) It tends to preserve sharp edges. In particular,
the edges in an image are
much better preservedin M(i, j) than in m(i, j) (seeSection8).
(2) It tends to eliminate spike noise. More generally, the situation is as follows:
let us consider the simple case where the signal part of the image is constant and
zero, and the noise is zero mean, independent from sample to sample, and identically
distributed with some probability density function. Then
E[+, j)] =f(i, j) = E[M(kAl
and the variances ui and I& can be used as measures of the noise cleaning power of
the running mean and median filters. The smaller the remaining variance is, the
more the noise filter has removed.
It is well known that for Gaussian noise
for a window of fixed size, of course, while for Laplacian noise (double-sided
exponential) it has been observed empirically that uz > &. Generally, it has been
observed empirically that when the tails of the probability density function of the
noise become heavier, the running median filter becomes more effective relative to
the running mean filter. A particular example is the case when the probability
density function of the noise is of the form
where A, a, and (Y are positive constants; then it can be readily shown that as (Y
decreases (i.e., the tails of the pdf become heavier), (uz - I&) increases.
A similar comparison exists between the variabilities of the local standard
deviation and the local quartile distances. As the tails of the noise pdf become
heavier, (u,’ - yu4) increases (where y is a constant inserted to make E[dyQ] equal
to the standard deviation of the noise pdf. As shown in Section 8 of this paper, the
interquartile distance is used as a good approximation to the absolute deviation from
the mean, a measure of dispersion which is the order statistics analog of the standard
deviation. The interquartile distance is much easier to compute than the absolute
deviation, and for the data in question they seldom differ by more than a few
percent, judging from empirical observation. A more detailed analysis of the
advantages of order-based measures with regard to spike noise is given in Section 8.
ENHANCEMENT
3. COMPUTATIONAL
USING
THE
MEDIAN
239
REQUIREMENTS
Assume that we are calculating m(i, j), a(& j), M(i, j), Q(i, j) for the entire
image. Let the window size be A x B. It is well known that the local mean m(i, j)
can be calculated using four additions per output point in a recursive manner
independently of the window size [3, 41. One takes advantage of the separability of
the two dimensional mean and does the calculation in one direction at a time; this is
done recursively which requires only two additions per output point.
The local standard deviation a(& j) can be calculated similarly. The recursive
operation also needs four additions per output point, but we also need to subtract
m(i, j) from each input point and square the result. After the recursion we need to
take the square root, so that the total number of operations required per output
point are five additions/subtractions,
one multiplication
for squaring, and one
square root, independently of window size.
To calculate the local median using the Huang-Yang-Tang
histogram update
algorithm [5] requires approximately 2A comparisons and 2A additions per output
point. If we calculate the interquartile distance Q(i, j) at the same time we need 4A
additional comparisons. If we calculate the separable median [6], we must compute it
in one dimension horizontally, then vertically on the result. The computational
requirement is then nearly the same as for the mean: four additions and four
comparisons per output point independent of window size. However, the separable
median is similar but not identical to the two dimensional median. For the same
window size, the separable median removes less noise but preserves sharp comers
better. Storage requirements are much greater, as we shall see in Section 4.
Algorithms based on means and standard deviations are the fastest known for
local enhancement and filtering, sometimes called box filtering [4]. Median methods
may soon equal these since several companies are constructing VLSI chips for
video-rate two dimensional median filtering. The initial work involves only small
3 x 3 or 5 x 5 windows, and interquartile distances have not been mentioned.
When completed, with larger windows and the interquartile distances, this effort will
bring down throughput times to video rates.
A multipass algorithm for local enhancement using the median in place of the
mean and the median absolute deviation (MAD) as a dispersion measure is suggested in [7]. To calculate the MAD it is necessary to produce a median filtered
image, subtract it from the original, and then median filter the absolute differences.
This requires three passes through the data plus two calculations of a median per
point. A reduction in calculation can be obtained by using only the absolute
deviation from the median as the dispersion measure instead of the MAD. Since the
algorithm of [5] lends itself to simultaneous calculation of the interquartile distance
as well as the median in one pass through the data, there seems to be no reason to
use a multipass method in a nonhardware implementation.
For very high resolution images (greater than 1500 lines, 1500 pixels per line)
where the eye cannot see the scan structure at normal viewing distances, algorithms
based on these techniques are the only practical ones for serial computers of modest
speed and storage capability. The results cannot usually be distinguished by laymen
from those obtained by more complex algorithms which can be implemented only on
large machines when throughput time is important. Mean-standard deviation methods can usually be used if more than 400 pixels are available for the moving window.
For 400 pixels or more defined to 256 levels, the median approaches the mean very
240
SCOLLAR,
WEIDNER,
AND HUANG
closely for nearly all typical pdf’s encountered except at very sharp broad edges.
Hence it is seldom of interest to use median based methods for windows larger than
19 X 19 unless the window is sampled. We have -obtained much faster computing
times with larger windows, rather than the reverse, once the window contains more
than 400 values, by sampling at suitable intervals.
4. STORAGE
REQUIREMENTS
In the case of the separable median, the computer must have enough directly
addressable memory to store the entire input image or a very large part of it. In the
normal two dimensional median technique using the Huang-Yang-Tang
algorithm,
a portion of the image in the vertical direction equal in size to the window height
plus one additional line must be stored in a circular buffer at least. In the separable
mean-standard deviation approach, only the upper, center, and lower picture lines
of the strip in which the window moves need be stored, together with two accumulator arrays, one for the column sums of the moving window and one for the sums of
squares. The latter usually needs twice the word length of the first. In all cases only
one output line need be stored. Use of asynchronous input/output
techniques which
in our installation materially increase the speed of throughput require additional
storage capabilities for the input and output data lines. Given the modest storage
and word lengths of many micros and lower range minis, there is usually no way to
implement the separable median for high resolution images directly. It can be done
in two passes from the disks but the extra I/O time usually wipes out any
advantage. Only the property of comer preservation may be a reason for its
occasional use. We have programmed all algorithms described in this paper to work
with at least 2000 line/pixel images and in some cases with 4000 line/pixel images
in only 64K bytes of storage, including buffers for asynchronous input-output.
5. ALGORITHMS
FOR NOISE CLEANING
Let f( i, j) be the input image and g( i, j) the output image. We first list a number
of simple algorithms for noise cleaning using local mean and standard deviation.
Examples are shown in Fig. 1.
Algorithm
Nl, simple averaging:
g(i, i) = m(i, i).
Algorithm
N2, averagingwith
g(i, i) = 44
fixed threshold [8, 91:
i)
= f (i, i)
if If(i,
j) - m(i, j)l > t,
if lf(i, j) - m(i, j)l 5 t
where t is a threshold supplied interactively.
Algorithm N3, averaging with variable threshold depending on contrast:
g(i, j) = m(i, j)
=f(i, j)
where B is a constant.
if If(i, j) - m(i, j)l > Mi, j>,
if If (k j) - m(i, Al 5 Ba(i, i)
ENHANCEMENT
USING
THE
MEDIAN
FIG. 1. Noise cleaning using local averaging (algorithms Nl-3);
threshold lo%, N3: B = 1.5.
241
2000 line images, 7 x 7 window. N2:
Algorithm Nl is used to smooth random Gaussian noise. It blurs edges. Algorithm
N2 can be used with a window much larger than the noise features to smooth spike
or thread noise. If t is well chosen, it may not blur edges as much as Nl. Algorithm
N3 is a sign&ant improvement on N2 since the usual outlier criterion of 95%
confidence limits in elementary statistics is satisfied when /I = 2. In N2, if t is chosen
too large, then noise cleaning will not be good; if too small, edges will be blurred.
N3 adapts to local contrast in the data, and sharp edges will hardly be blurred at all,
providing that the window is large enough. More complicated algorithms have been
proposed which can preserve edge sharpness, but they do not have the computational advantage of Nl, N2, or N3 [lo].
We propose here an alternative approach to high contrast edge preservation with
small windows in the presence of spike or thread noise. In algorithms Nl-3 replace
the local mean m(i, j) by the local median M(i, j) and the local standard deviation
242
SCOLLAR,
FIG.
2.
WEIDNER,
AND HUANG
Noise cleaning using local median (algorithms N’l-3).
Parameters as in Fig. 1.
u by the local interquartile distance Q(i, j) as an approximation to the local
absolute difference. We call the three new algorithms Nl’, N2’, and N3’. Results
using this substitution are shown in Fig. 2 for the data of Fig. 1. Algorithm N3’ is
superior to algorithm N3 in removing spike noise and preserving edge sharpness
since smaller windows can be used, the median and quartile differences being less
alkted than the mean and standard deviation. Empirically we have observed that
two- or threefold size reductions are possible. Algorithm Nl’ is superior to Nl in
preserving edge sharpness, but the relative abilities of Nl and Nl’ depend on the
noise pdf. Both Nl’ and N3’ will remove spike noise, but edge sharpness is better
preserved by N3’, while Nl’ will remove more Gaussian noise. If large windows are
used, comers will be better preserved with N3 than with N3’ using the two
dimensional median. Performance is roughly the same using the separable median.
For large windows N3 is the preferred algorithm; for small windows N3’. Artifacts
with large windowed N3 runs may sometimes be objectionable on very large edges.
ENHANCEMENT
USING
6. ALGORITHMS
THE
FOR
MEDIAN
243
CRISPENING
By crispening we mean an apparent sharpening of the image without change of
local brightness or contrast [ll]. A very effective algorithm which we have used most
extensively because of its high speed is a variant on box filtering with roll-off control
and brightness normalization.
Algorithm
C
g(i, j) = w{ f(i,
j) - am(i, j))
where (Y is a constant, 0 < a < 1, and w is a brightness normalization constant
which depends on the window size and the value of (Y. The frequency response of
this linear spatially invariant filter depends on the constant (Yand the window size
used in computing the local mean m(i, j). The larger the value of (Ythe greater the
crispening effect. The larger the window size, the more the effect is obtained over
broader edges. However, for very high contrast edges already present in the picture,
there is some overshoot which results in light or dark banding which may be
objectionable when large window sizes are used. This is shown in Fig. 3. A large
window size should be used to obtain stable means in noisy areas, but then the effect
is most noticeable.
To overcome this we recommend replacing the local mean by the local median to
get our algorithm C’. For the same value of a and the same window size, algorithm
C’ will crispen the image slightly less but this is hardly observable (Fig. 3). It is
possible to combine crispening and noise cleaning in one pass. Algorithm C’ can be
combined with N3’, C with N3, or mixtures of the two types used. For example, see
the following.
Algorithm
C’N3’
g(i, i) = M(i, j)
= w{f(i, j) - aM(i,i)}
if V(i, j) - M(i, j)l > PQ(i,j),
iflf(i, j) - M(i,j)l < PQ(i,i).
It has been observed that crispening followed by noise cleaning in a two-pass
algorithm is superior to the above or to the reverse order, because spike noise is first
exaggerated by C’ and moved to the outer tail of the pdf where it can be removed
more easily by N3’. It is useful to use C instead of C’ in the first pass to save time.
Because of the variant data structures required, it is not useful to program this
combination for a single pass.
In general, it is desirable to use different window sizes for the crispening and
cleaning parts of the treatment. Usually a smaller window for the crispening is
desirable. This increases the storage requirements for CN3 since extra image lines
have to be buffered and multiple accumtdators are needed. This is not the case for
C’N3’ where a large circular buffer for the big window is available anyway. Despite
the slower computational speed, the I/O saving is considerable. Multiple window
versions of these algorithms can be combined to give bandpass and Gaussian
filtering and are computationally effective, especially with the possible variants of
244
SCOLLAR,
WEIDNER,
AND HUANG
FIG. 3. Crispening using local averaging and local median; 2000 line images, 7 X 7 window.
algorithm C. There is some advantage, as pointed out in [12], in the sequential
combination CC’ with values of a! equal to 1 and a larger window size for C than for
C’ as a noise insensitive edge extractor.
We also wish to note that a simple modification of C or C’ can be used for
reduction of linear motion blur if it is uniform over the image and more or less at
right angles or parallel to the scan direction. It is merely necessary to displace the
central pixel in the window by a small amount depending on the direction of blur.
Usually it is advantageous to use a rectangular window with the long axis parallel to
.the blur direction. Blur which is not -parallel to the scan direction ought to be
reducible with displacement at some angle from the true center. Large amounts of
blur cannot be dealt with by this technique. Blur which is not uniform over the
image might be dealt with by varying the amount and direction of offset over the
image, but I we have not tried this. ..Here there will be a complex problem in
determining the needed displacement and varying window sizes.
ENHANCEMENT
7. ALGORITHMS
FOR LOCAL
USING THE MEDIAN
CONTRAST
245
AND BRIGHTNESS
ENHANCEMENT
These algorithms are needed for exposure and development correction, especially
in spatially variant cases which cannot be treated by simple histogram transformation of the entire image. One approach to enhance local contrast is to normalize the
local standard deviation to some constant, a kind of automatic gain control [13].
Another possibility is raising the local standard deviation by multiplying the
mean-pixel difference by a constant and replacing part of the output pixel [14]. The
most satisfactory scheme which we have used judging by user acceptance has been
that due to Wallis [15], as follows.
Algorithm
W
g(i, j) = Au(if;
+
ud { f(i,
j)
-
m(it
d}
+
amd
+tl
-
a>m(i?
j)
where md = a desired mean, ed = a desired standard deviation, A is a gain or
contrast expansion constant (usually 4 < A < 20), and a is a brightness forcing
constant, usually 0 < a < 0.3. This algorithm can be improved by limiting either the
maximum observed standard deviation a(i, j) or the value of the pixels which
contribute to it as suggested in [l]. The first is computationally preferable. Performance on high contrast edges is helped by this modification, reducing the edge
artifacts somewhat; see Fig. 4.
Experimentally, it has been found that good results which do not add noise to the
image require windows of the order of 5 to 10% of the side length. Noise effects
which have been noted [l] are due to the use of windows which are too small, giving
consequent instability in the standard deviation in the presence of heavy-tailed pdf’s.
However, if the image contains high contrast edges, then a large window will
introduce dark or light bands along the edges; see Fig. 4. This artifact can be
considerably reduced (Fig. 4) if we replace m and u by M and Q giving algorithm
W ‘. The local contrast and brightness correction of W ’ are indistinguishable from W
for the same window sire, but since smaller windows can be used, banding effects are
less. Noise effects are the same since such large windows must be used for W. Very
small windows produce a bit more noise in W than in W’, but on very high
resolution images it is sometimes difficult to see the difference except in highly
uniform gray areas.
We have obtained a considerable speedup of W by using a regularly sampled
window for those larger than 20 x 20 pixels. Presumably, this approach could also
be applied to W’ but it does not appear to be worthwhile, since there is little
difference in performance for the two algorithms in the noise sense when windows
are so large. Only the difference in edge banding justifies the extra computational
cost of W’.
8. ERROR NORMS AND ENHANCEMENT
OPERATORS
The analogies between means, standard deviations, medians, and quartile differences used throughout this paper are difficult to understand with standard filtering
theory. We propose that they can best be understood in terms of error norms and
their associated dispersion measures. For this we must introduce a few theoretical
concepts.
246
SCOLLAR,
WEIDNER,
AND HUANG
FIG. 4. Local contrast and brightness enhancement (algorithms W and W’) W window 50 x 50,
av. = 127 (on 255 scale), (I = 50, a = 0.1, A = 20; W’ window 11 X 11. Same parameters for median and
quartile difference as in W.
A set is called a metric space if, for every pair of elements, a real distance is
defined with satisfies the following conditions:
4x9 Y) > 09
while d(x, y) = 0
ifandonlyifx
=y;
d(x, y) = &Y, x);
d(x, z) < d(x9 Y) + d(Y, z)
An important
triangle inequality.
class of distance functions is generated by norms. A norm 1111on a
ENMCEmT
247
USING THE MEDIAN
linear space is a real function which satisfies the following axioms:
PII h 03
II4 = lal
while llxll = 0
ifandonlyifx
. llxll for every number a
= 0
lb + YII 5 Ilxll + IIYII
4x9 Y) = lb - YIl*
In particular the L, or Minkowsky
L,(a,
norms are defined as
b) = ( jblx(t)l’dr)?
a
Now, for our statistical measures, let p be a pdf defined on the interval I:=
[-&in, -&ax ] and P be the corresponding cumulative distribution function or probability function. The expected value of a function f is then detined as
E(f):= if
dP.
Let the dispersion of p around y E I with regard to the L, norm be
d,(y):= {ax
lsr<ao.
- Yl’)I”’
The value
d,:= $nd,(
y)
is called the dispersion of p with regard to the L, norm or simply the r dispersion of
p. If there is a unique c, E I which minimizes d,(y) such that
it will be called the r center of p.
Let us now consider a few special cases:
r = 2 (the Euclidean Norm); E(x - y)’ = min,
* 9 = E(x),
the mean value of p and
d, = (E((x
- E(x))~))~‘~,
the standard deviation of p. This norm underlies all classical statistics and linear
transform theory.
r + co (Max Norm); in this case d,(y) converges to
d,(y)
= ~e~lx - YI
248
SCOLLAR,
WEIDNER,
AND
HUANG
independentlyof p suchthat
the half range,and
r = 1 (Manhattanor City Block Norm); E( lx - ~1) = min *
/x~h~(y-x)dP+~x=-~(x-y)dP=o*
[,dP=[-dP=i
IlUll
since@‘=~.
The valueci with
is just the medianand d, = E( Ix - ci I) is the absolutedeviationfrom the median.
Becausethesemeasuresare dehnedin terms of the L, norm, attemptsto study their
propertiesusingclassicaltransformtheory which dependson the L, norm will fail.
For d, anotherexpressioncan be derived:
d,= j”(q&in
x)dP+[;-(x-q)dP=
jX-xdPf,xdP.
Cl
OM
Now let
pl=
=
2P9
x 2 Cl
0,
x
>
md
Cl
pr = 0,
= 2P,
x 5
Cl,
x > Cl
be the left and right half distributions of p, respectively;then dl = l/2{ Epr(x) i.e., the differenceof the mean valuesof the two half distributions,which
can be written, usingthe notation c,(p) for the r centerof p,
E,,(x)},
4 = :(cz(~r)
- c,(~l))-
It now seemsnatural to define,in an analogousmanner,
Q:= t(c,<
pr) - c~( Al))
with ci( pr) and ci( pl) as the mediansof the half-distributionshavingthe obvious
properties
s(p”dp I c$dJ’ = cp
5 jxm” dp = 4
Jxmim
cl(Pr)
1
1
and are calledthe quartilesof p.
ENHANCEMENT
USING
THE
249
MEDIAN
2Q is then called the interquartile distance.
Let q1 = cr( pl), q2 = ct( p), q3 = cl( pr). To see what effect spikes in the data
have, we may interpret p as a frequency distribution of samples and consider what
happens if a small fraction h of samples is replaced by new samples with a fixed
value z. Let p’ be the distribution of samples replaced with /dP’ = h, the fraction
replaced. We also assume that the samples replaced are taken randomly, such that,
approximately,
and that z B cl.
p’ = hp
We then have
c; = (1 - h)c, + hz.
For the median c1
1
-=
2
1'; d(P - p') = /" d(P - P') + /“d(P
Xmin
= 9
XtN”
+(I
- h)l’;dP
- P’)
Cl
= 9
+(l
- h)P(c,
5 x 5 c;)
(‘1
=y+(l-h)(P(x+;)
-P(X~c;)=~~.
1
1
Analogous relations hold for the quartiles, so that for i = 1,2,3,
P(X 5 q() = $A
= $1
+ h)
for small h and
Q’
= $( P-‘(a(1
+ h)) - P-‘(a(1
+ h)).
All these relations are independent of z and give corrections only on the order of h,
whereas the change for c2 is of the order hz. For spike, thread, and comparable
“singularity” noise in pictures h, the fraction of spikes seen in a window is small
( < 0.1) whereas lz - ci 1 is usually quite big. In that case, as has been shown, the
median and interquartile distance give much more noise tolerant estimates of local
brightness and contrast than do the mean and standard deviation.
IDefIning an edge as two adjacent areas with Gaussian distributions having
different means and perhaps different standard deviations, let us examine what
happens in a window when it passes over this edge, starting well out in the first area.
When the window just touches the edge, the pixels belonging to the second
distribution appear as singularities in the sense defined above (spikes or threads).
250
SCOLLAR,
WEIDNER,
AND HUANG
Therefore the value of the median does not change immediately. It does change
when the number of new pixels from the second distribution becomes a significant
fraction of the total number in the window. This is why the normal type of median
filtering blurs the edge less than does filtering with the mean.
Given the advantages of the L, norm based measures, one would like to think that
a theory of filtering could be built up using this norm by analogy with linear Wiener
filtering which is based on an L, norm. L, norm filtering is based on the fact that the
L, norm is derivable from a scalar product such that the notion of orthogonality is
well defined. Unfortunately this is not the case with the L, norm. Classical filtering
(Fourier, Hadamard, etc.) is efficient because of the analytical tractability of the L,
norm approximations or estimates of a function. L, norm approximations to a
function can only be obtained with linear programming techniques. This is not
efficient, although filtering approaches using them are possible. As far as we know,
using the median, quartile difference approach is the only efficient way to take
advantage of the properties of the L, norm without using linear programming or
something still worse.
9. CONCLUSIONS
We have shown that, for certain types of images, the use of the local median and
the interquartile distance is more effective than the use of the local mean and
standard deviation at the expense of computation time unless the complete image
can be stored. In this case separable medians and quartile differences restore the
time advantage or even gain something since no squares or square roots are needed.
Specifically, algorithms using M and Q have the following general advantages:
(1) They are more effective in removing spike noise or noise with a heavy tailed
pdf.
(2) They preserve edge sharpness better.
(3) They introduce fewer artifacts around high contrast edges already present in
the original data.
On the negative side they are generally slower to compute in a small general purpose
machine. When VLSI chips for the median are available, video-rate operations will
be possible. Combined algorithms with one pass through the data are presently a
worthwhile improvement in serial machines, but the order of operations is important
and different effects can be achieved with multipass algorithms depending on what is
done first. The suite of algorithms is especially suitable for production quantities of
high resolution images without excessive amounts of noise in an application environment, without the use of high-cost machines.
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