Fast median search: an ANSI C implementation
Nicolas Devillard - ndevilla AT free DOT fr
July 1998
Median filtering is a commonly used technique in signal
processing. Typically used on signals that may contain outliers
skewing the usual statistical estimators, it is usually
considered too expensive to be implemented in real-time or
CPU-intensive applications. This paper tries to gather various
algorithms picked up from computer literature, offers one
possible implementation for each of them, and benchmarks them on
identical data sets to identify the best candidate for a fast
implementation.
Let us define the median of N numerical values by:
- The median of a list of N values is found by sorting
the input array in increasing order, and taking the middle
value.
- The median of a list of N values has the property that
in the list there are as many greater as smaller values than this
element.
Example:
input values: 19 10 84 11 23
median: 19
The first definition is easy to grasp but is given through an
algorithm (sort the array, take the central value), which is not
the best way to define such a concept. The latter definition is
more interesting since it leaves out the choice of the algorithm.
Intuitively, one can feel that sorting out the whole array to
find the median is doing too much work, since we just need the
middle value and not the whole array sorted. However, most of the
job must be done in order to assess that the median value is
effectively the one in the middle.
Applications of a median search are many. In digital signal
processing, it may be used to remove outliers in a smooth
distribution of values. A median filter is commonly referred to
as a non-linear shot noise filter which maintains high
frequencies. It can also be used to estimate the average of a
list of numerical values, independently from strong outliers.
In image processing, a median filter is computed though a
convolution with a (2N+1,2N+1) kernel. For each pixel in the
input frame, the pixel at the same position in the output frame
is replaced by the median of the pixel values in the kernel.
Image processing morphological filters are well described in any
reference about image processing, such as [Russ].
Every ANSI compliant C library comes with an implementation of a
quicksort routine. This standard routine is however usually very
slow compared to the most recent methods, and overkill in the
case of median search.
The main interest of this function is to give a reference of what
the slowest way to achieve median search can be. On the other
hand, it is extremely simple to write, portable, and easy to
read.
This method makes use of a faster routine for array sorting,
taken from literature and slightly optimized in ANSI C. It gives
much better results than the Solaris qsort() for example.
This routine is reasonably portable, should not be a problem for
any ANSI compliant C compiler, can be encapsulated like the
standard qsort() and used straightaway for many other
purposes.
This algorithm has been taken from [Aho et al]. In
pseudo-code, it can be described as:
S is a list of numerical values
k is the rank of the kth smallest element in S.
procedure select(k,S)
if |S|=1 then return single element in S
else
choose an element a randomly from S;
let S1, S2, S3 be the sequences of
elements in S respectively less than,
equal to, and greater than a;
if (|S1| >= k) then
return select(k,S1);
else
if (|S1|+|S2| >= k) then
return a;
else
return select(k-|S1|-|S2|, S3);
procedure find_median(S)
return select(|S|/2, S)
This algorithm makes use of the fact that only the kth smallest
element is searched for in S, thus does not need to sort
the complete array. It is much faster than a complete array
search, and potentially useful for other applications
interested in getting other ranked values than the median.
This algorithm is recursive, and it needs to allocate a copy of a
part of the input array at each iteration. For big input arrays,
this puts serious memory requirements and virtually no possible
limit on the quantity of potentiatlly used memory. In the worst
case, the amount of memory needed to work could reach N*(N-1)/2
which would most likely bring memory allocation failures or
program crashes. Even if the probability of hitting such cases is
low, it is not recommended to use this algorithm on big arrays or
in any program for which strong reliability is a keyword.
This is an interesting method for educational purposes, but
unrealistic to use in any other environment, because of the
recursivity constraints. Furthermore, the other methods described
here have both the advantage of being faster and
non-recursive.
NB: "Choose an element randomly in S" has been modified in the
proposed implementation to "take the central element in S" to
avoid a call to the random generator and a modulo division at
each iteration. This brings in the fact that some arrays will be
very bad cases for this method, needing a lot of iterations.
Random picking would have ensured to stay (almost) always within
reasonable bounds but constrains to 2 expensive calls at each
iteration.
This algorithm has been taken from [Wirth].
The pseudo-code for this algorithm is given in the book, and in
wirth.c (see code section below) a literal translation is
done from Pascal to ANSI C. It does not try to sort out the
complete array but browses through the input array just enough to
determine what is the kth smallest element in the input list. It
is not recursive and does not need to allocate any memory, nor
does it use any external function. As a result, it gains a factor
25 in speed compared to the qsort() based method.
Apparently, it is the same algorithm as the AHU median, but
implemented in situ. The advantage is obviously that it
gets rid of recursivity, the price to pay is an initial copy of
the input array because it is modified during the process.
The median search is defined as a macro on top of the function
which finds the kth smallest element. It defines the median for
an odd number of points as the one in the middle, and for an even
number the one just below the middle. See the discussion below
about finding the median of an even number of elements.
This algorithm was published in [Numerical Recipes].
Speedwise, it is a close tie with Wirth's method. On the average,
this one is faster, however. It works in situ and modifies the
input array, so the same caveats apply: the input data set must
be copied prior to applying the median search.
This method was pointed it out to me by Torben Mogensen.
It is certainly not the fastest way of finding a median, but it
has the very interesting property that it does not modify the
input array when looking for the median. It becomes extremely
powerful when the number of elements to consider starts to be
large, and copying the input array may cause enormous overheads.
For read-only input sets of several hundred megabytes in size, it
is the solution of choice, also because it accesses elements
sequentially and not randomly. Beware that it needs to read the
array several times though: a first pass is only looking for min
and max values, further passes go through the array and come out
with the median in little more time that the pixel_qsort()
method. The number of iterations is probably O(log(n)), although
I have no demonstration of that fact.
The methods described above are very useful to search for the
median value of many elements, but for small number of values
there are even faster methods which can be hardwired to produce
the median in the fastest possible time. In image processing, a
morphological median filter on a 3x3 kernel needs to find the
median of 9 values for each set of 9 neighbor pixels in the input
image. Code is provided here to get the median out of 3, 5, 7, 9
and 25 values in the fastest possible time (without going to
hardware specifics). Other sorting networks can be found for
different numbers of values, they are not provided here.
An article about fast median search over 3x3 elements can be
found at [Smith].
Image median filters using large kernels may use the redundancy
in the fact that the method is looking for the median of NxN
pixels, then going to the next kernel position (usually: next
pixel on the right) means taking out N pixels and adding N new
ones. For a 3x3 kernel, it is unefficient to use this redundancy
since only 3 pixels stay unchanged, but for large kernels e.g.
40x40, there are only 40 pixels less and 40 pixels more at each
iteration, but 1560 values which stay unchanged. In that case, it
may be more efficient to go to histogram or
tree-based methods. No implementation for these methods
is provided here, only the general ideas.
Building up a histogram, one can notice that the median
information is indeed present in the fact that pixels are sorted
out into buckets of increasing pixel values. Removing pixels from
buckets and adding more is a trivial operation, which explains
why it is probably easier to keep a running histogram and update
it than to go from scratch for every move of the running kernel.
Interested readers are referred to [Huang et al].
The same idea can be used to build up a tree containing pixel
values and number of occurrences, or intervals and number of
pixels. One can see the immediate benefit of retaining this
information at each step.
The following plots have been produced for all generic methods
(QuickSelect, Wirth, Aho/Hopcroft/Ullman, Torben, pixel
quicksort) on a Pentium II 400 MHz running Linux 2.0 with
glibc-2.0.7. It is interesting to note that all methods are
roughly proportional on this machine, with the following
estimated ratios to the fastest method (QuickSelect).
The basic method using the libc qsort() function has not
been represented here, because it is so slow compared to the
others that it would make the plot unreadable. Furthermore,
it depends on the local implementation of your C library.
The following ratios have been obtained for sets with increasing
number of values, from 1e4 to 1e6. The speed ratios have been
computed to the fastest method on average (QuickSelect), then
averaged over all measure points.
QuickSelect : 1.00
WIRTH median : 1.33
AHU median : 3.71
Torben : 8.95
fast pixel sort : 6.50
On the x-axis, the number of elements from which a median is
extracted, in thousands (goes from one thousand to one million
elements). On the y-axis, the time used in seconds.
| Elements |
QSelect |
Wirth |
AHU |
Torben |
pixel_qsort |
| 100,000 |
0.010 |
0.020 |
0.050 |
0.140 |
0.100 |
| 200,000 |
0.040 |
0.040 |
0.180 |
0.310 |
0.220 |
| 500,000 |
0.110 |
0.080 |
0.510 |
0.800 |
0.580 |
| 800,000 |
0.120 |
0.160 |
0.590 |
1.270 |
0.940 |
| 1,000,000 |
0.210 |
0.290 |
0.600 |
1.580 |
1.240 |
To reject outliers. When you have a signal distribution that is
smooth enough but contains crazy outliers, you might get into
trouble with fitting routines or statistical tools. That is the
case for astronomical detectors hit by a cosmic ray for example,
but there are many cases where you want to get an estimate of the
average signal value without bothering about outlier rejection. A
median can often be a fast and useful answer.
If you have little numbers of elements, e.g. for image processing
filter kernels, use the provided macros to find out a median out
of 3, 5, 7, or 9 elements. There is no faster way. If you are
applying a kernel with a large number of elements, see the
section above about large kernels.
If you have a reasonable number of elements and are allowed to
modify your input element list, use QuickSelect or Wirth. The
choice between both should be done depending on your typical
input data sets. Try out both and pick one. If you are not
allowed to modify your inputs but the data set is sufficiently
small to hold in memory, I would go for QuickSelect or Wirth
again. Copy your input data set to memory, apply the
median-finder, and destroy the temporary data set.
If you are not allowed to modify your input data set and it is
large enough that copying it causes serious overheads, try out
Torben's method. It does not modify the input set thus avoiding
the overheads of a copy, but beware that it needs to run several
times through the set. Fortunately, browsing the set is done
sequentially, that gives more chances of being able to read the
set through bufferized inputs.
Hopefully yes! There is no library call of any kind, it should
work rightaway on any place that compiles C. The given code finds
out the median out of a set of elements, up to you to
specify the kind of elements you want to search. You could also
apply templates in C++ to use the same code for any numeric type.
There are several possible answers to this question. According to
the NIST web site (National Institute of Standards and
Technology):
Median definition: The value which has an equal number of values
greater and less than it. For an even number of values, it is the
mean of the two middle values.
The mean is, in this case, understood as the arithmetic mean of
both values (i.e. half of their sum).
In Introduction to algorithms [7], the authors
define two medians: the lower median and the upper median, and
state that for an odd number of elements both medians are
identical. In the rest of the paragraph dedicated to medians,
only the lower median is considered.
Example:
input values: 19 10 94 11 23 17
median: 17 if element n/2
19 if element n/2 +1
If you do not require the median value to be one of the elements
of the initial list, you can decide that the median is the
average of the two central elements. In the above example, the
median would be the average of 17 and 19, i.e. 18. You can of
course use whatever average you prefer, arithmetic or geometric,
since at this point it is up to you to check what suits your
application.
Taking the median out of a list of elements also does not mean
that the elements have to be numbers, they can be anything you
want provided they obey a sorting rule (i.e. it is always
possible to tell if one element is greater than, smaller than or
equal to another element). For some elements, taking the average
does not make sense.
The default behaviour in the routines provided on this page is to
take the element just below the middle (lower median). Taking an
average of the two central elements requires two calls to the
routine, doubling the processing time.
Notice that if you are taking the median out of a great number of
elements, and these elements tend to behave all more or less the
same except some outsiders, the median is likely to be exactly
the same whatever definition you use.
Many thanks to Torben Mogensen for his non-destructive median
method allowing to work on very large data sets, and for a number
of fruitful discussions. My gratitude goes to Martin Leese for
pointing out the histogram-based method and of course introducing
me to the Quickselect method, for which he provided a C++
implementation (the C version given in here is merely an
optimized translation of his code). Thanks to all contributors
from the Usenet, too many to quote here.
Bare-bone algorithms have been isolated into independent source codes:
The following snippets are placed in the public domain.
- quickselect.c implement the
QuickSelect method, the fastest one tested to date.
- wirth.c implements N. Wirth's method.
It is actually a tool which finds the kth smallest element from a
list of values, the special case of a median is implemented through
a macro.
- torben.c implements Torben's method
to find a median in an input set without modifying it.
- optmed.c implements the fast networks
for 3, 5, 7, 9 or 25 elements, particularly useful for morphological
filters in image processing.
You can benchmark the different methods by yourself:
- benchmed.c compares all
methods described above.
- optmed_bench.c compares the
hardwired median search over 3, 5, 7 or 9 elements against a
standard approach using qsort().
To compile the benchmark sources, just do:
% cc -o benchmed benchmed.c -O
% cc -o optmed_bench optmed_bench.c -O
If your compiler complains about undefined symbols or syntax
errors, it is likely to be non-ANSI compliant. On Sun4, use acc
or gcc. On HPUX, add the option -Ae to compile.
-
- 1
- The Image Processing Handbook, John C. Russ, CRC
Press (second edition 1995).
- 2
- Aho, Hopcroft, Ullman, The design and analysis of
computer algorithms (p 102)
- 3
- Niklaus Wirth, Algorithms + Data structures =
Programs (p 84).
- 4
- Numerical recipes in C, second edition, Cambridge
University Press, 1992, section 8.5, ISBN
0-521-43108-5
- 5
- John Smith, Implementing median filters in XC4000E
FPGAs, [url]http://www.xilinx.com/xcell/xl23/xl23_16.pdf[/url]
- 6
- T.S.Huang, G.J.Yang, G.Y.Tang, A fast two-dimensional
median filtering algorithm, IEEE transactions on
acoustics, speech and signal processing, Vol ASSP 27
No 1, Feb 1979.
- 7
- Thomas H. Cormen, Charles E. Leiserson, Ronald L.
Rivest, Clifford Stein, Introduction to algorithms,
MIT press.
(轉載自:[url]http://ndevilla.free.fr/median/[/url])
文章來源:
http://wintys.blog.51cto.com/425414/102975
[附件]:
中位數.pdf
posted on 2009-03-18 12:02
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