IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 25, NO. 11, NOVEMBER 2016 5199
The Bitonic Filter: Linear Filtering in an
Edge-Preserving Morphological Framework
Graham Treece
Abstract— A new filter is presented which has better edge
and detail preserving properties than a median, noise reduction
capability similar to a Gaussian, and is applicable to many signal
and noise types. It is built on a definition of signal as bitonic,
i.e., containing only one local maxima or minima within the filter
range. This definition is based on data ranking rather than value;
hence, the bitonic filter comprises a combination of non-linear
morphological and linear operators. It has no data-level-sensitive
parameters and can locally adapt to the signal and noise levels
in an image, precisely preserving both smooth and discontinuous
signals of any level when there is no noise, but also reducing noise
in other areas without creating additional artifactual noise. Both
the basis and the performance of the filter are examined in detail,
and it is shown to be a significant improvement on the Gaussian
and median. It is also compared over various noisy images to
the image-guided filter, anisotropic diffusion, non-local means,
the grain filter, and self-dual forms of leveling and rank filters.
In terms of signal-to-noise, the bitonic filter outperforms all these
except non-local means, and sometimes anisotropic diffusion.
However, it gives good visual results in all circumstances, with
characteristics which make it appropriate particularly for signals
or images with varying noise, or features at varying levels. The
bitonic has very few parameters, does not require optimization
nor prior knowledge of noise levels, does not have any problems
with stability, and is reasonably fast to implement. Despite its
non-linearity, it hence represents a very practical operation with
general applicability.
Index Terms— Gaussian filter, median filter, morphology, noise
reduction, edge preservation.
I. INTRODUCTION
T
HE removal of unwanted noise corrupting a digital signal
is a very common operation. In many contexts the signal
is not known apriori, except perhaps for some general
expectations concerning its overall form, and can contain both
smooth regions and discontinuities, or ‘edges’. In this latter
case, noise removal usually also leads to blurring of these
edges, and many algorithms have been proposed which seek
either to preserve them during noise removal, or restore them
after blurring.
This paper addresses the preservation of signal discontinu-
ities, in a way which still allows noise removal at the edge,
rather than simply lessening (or disabling) the noise reduction
where an edge is detected. Ideally this would still be the case
Manuscript received December 14, 2015; revised June 10, 2016; accepted
August 18, 2016. Date of publication September 2, 2016; date of
current version September 16, 2016. The associate editor coordinating
the review of this manuscript and approving it for publication was
Prof. Oleg V. Michailovich.
The author is with the Department of Engineering, University of Cambridge,
Cambridge, CB2 1PZ, U.K. (e-mail: gmt11@eng.cam.ac.uk).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIP.2016.2605302
at fairly high noise levels, where the discontinuity is less than
the level variation due to noise: this is in contrast to techniques
which rely on thresholds to distinguish between an edge to be
preserved and noise to be reduced.
The difficulty of removing noise without corrupting the
signal relates to the overlap in their respective definitions.
Noise can usually be defined as a random component, whilst
an unknown signal can be regarded as something which is
either repetitive or smooth or simply ‘not random’. In the
absence of specific training, however, it is usually presumed
that the noise has a greater, or higher, frequency content and
hence ‘smoothing’, or the removal of these frequencies, is used
as a synonym for ‘noise removal’. If the actual signal is smooth
(contains low frequencies) or repetitive (is relatively sparse in
frequency) then some noise can be removed without damage
to the signal. However, signal edges are often not repetitive,
and contain very high frequency content, hence removing
the noise leads to blurring, or other forms of corruption,
of the edge. Edge-preservation during noise removal hence
implies the removal of high frequency noise but not of high-
frequency signal: however, at least in the frequency domain,
these components are not distinct.
Here an alternative is considered, in which the signal is
regarded as ‘anything locally bitonic within a given range’.
A bitonic sequence (defined in the context of sorting [1] as an
extension of monotonic) is one which increases monotonically
(or not at all) to a peak then decreases monotonically (or
not at all), i.e. it has at most one local maxima. A signal
which when cyclically shifted meets this definition is also
bitonic. A slightly simpler definition is used here such that
a signal is deemed locally bitonic if it has either only one
local maxima, or only one local minima, or no maxima nor
minima.
1
This concept of local bitonicity equally encompasses
smooth signals and those containing edges, since only the data
rank matters, not the level. Many real signals hence might be
expected to exhibit local bitonicity over a reasonable range.
By this definition, noise is anything which is not bitonic
over the given range (or equivalently anything which is only
bitonic over a shorter range). Whilst the range does effectively
impose a lower frequency limit on the noise, and an upper
frequency limit on repetitive signals, it crucially does not
impose an upper frequency limit on edges, so long as the
overall shape is locally bitonic. The definition encompasses at
least all noise types which are zero mean and uncorrelated
over neighbouring samples (i.e. white noise), save for the
1
This is a reasonable re-definition since a true cyclically bitonic signal will
always be bitonic by this alternative definition over at least half of its length:
hence the difference is purely one of the range of the bitonicity.
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5200 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 25, NO. 11, NOVEMBER 2016
lower frequency limit. Impulsive, or ‘Salt & Pepper’ noise is
more ambiguous: an isolated impulse would be regarded as a
signal; several impulses within the range as noise. The ability
to preserve impulses whilst removing other types of noise is
a considerable advantage in some scenarios; but in any case it
will be demonstrated that it is possible to relax the bitonicity
criteria slightly to allow for good rejection of impulsive noise
as well.
In the context of filtering, interpreted here in the sense of
replacing each value in the signal with some combination of
the surrounding values within a specific range, a bitonic filter
would hence be one which seeks to preserve any signal with
bitonicity over the range of the filter, but reject anything else.
Such a filter would naturally preserve both edges and smooth
features in a signal.
II. T
HE BITONIC FILTER
A. Definition
Since bitonicity is concerned with the ordering, rather than
the value, of the data, it is natural to turn to rank filters [2], also
known as order-statistic filters. The median is the most well
known example, where the data in a local window is ranked
andthemid(50
th
centile) value is output. The median filter is
commonly used to eliminate impulsive noise, whilst preserving
edges well, at least if there is no more than one edge within
the range of the filter. A rank filter is a generalisation of the
median where any centile can form the output. Such filters can
be considered as monotonic in that they preserve signals which
are monotonically increasing or decreasing, and indeed this
leads naturally to impulsive noise reduction, since impulses
are bitonic rather than monotonic. For two-dimensional (2D)
data, the shape of the window used to form the set of ranked
data, in morphology known as the ‘structuring element’, has
some impact on which features can be preserved. Here we use
a circular disk for 2D image data to ensure isotropic behaviour.
Using a rank filter of 100
th
centile (or maximum, known
as a dilation) and immediately following this with another
of 0
th
centile (minimum, known as an erosion) results in a
morphological closing operation, which preserves signals with
a local maximum, whilst rejecting any signal with a local
minimum. Reversing the order of these filters results in a
morphological opening which has the opposite action. Such
filters have many uses in processing the shape of data, par-
ticularly in granulometry [3]. Figure 1 shows some examples
of opening and closing operations on one-dimensional (1D)
signals. Here a robust opening operation is used, with a small
centile c rather than the minimum, and (100 − c) in place of
the maximum. This is not the same as either the slightly better
known rank-max opening [4] nor the soft-opening [5], but it
has been previously suggested by Kass and Solomon [6]. The
use of a small centile allows some control over impulsive noise
rejection, since any impulse which takes up less of the filter
range than c will be rejected. The robust opening O
w,c
(x ) and
closing C
w,c
(x ) of a signal x can be defined as:
r
w,c
(x ) = c
th
centile
i∈w
{
x
i
}
(1)
O
w,c
(x ) = r
w,100−c
(r
w,c
(x )) (2)
C
w,c
(x ) = r
w,c
(r
w,100−c
(x )) (3)
where r
w,c
(x ) is a rank filter, w is the filter window (or
structuring element in 2D),
|
w
|
is the window length (or
number of elements in 2D) and c the chosen centile, which
will generally be a fairly low percentage. The bitonicity can
be seen in Fig. 1 (centre-left). However, robust opening and
closing operations only do half of what is required, since they
are not self-dual (symmetric in data value): they only preserve
local minima or maxima respectively. In addition, it can be
seen from Fig. 1 that the opening and closing operations do
not preserve mean signal values in the case of a noisy signal,
which would clearly be a vital property of a practical filter.
Fortunately both these drawbacks can be overcome by the
same means. It is fairly clear, by comparison of the original
signal with each of the opened and closed signals, which is
the most appropriate output for each part of the signal. We can
hence use such a comparison to weight a combination of the
opening and closing operations. However, a weighting based
on a point-wise comparison would simply return the original
signal,
2
so instead the differences between the original signal
and each of the rank-filtered signals are smoothed with a linear
filter. A Gaussian filter (i.e. a linear moving-window filter with
Gaussian weights) is used for this purpose, since it is known
to have good noise reduction properties. The filter length is
determined experimentally to match the noise reduction from
the rank filters, so that the standard deviation σ = 0.33l where
l is the window length in 1D, or the diameter of the structuring
element in 2D. This smoothed error can be seen in the middle
column of Fig. 1.
Defining the Gaussian linear filter as G
σ
(x ),thisisused
to weight the results of the opening and closing operations as
follows:
O
(x ) =
G
σ
x − O
w,c
(x )
(4)
C
(x ) =
G
σ
C
w,c
(x ) − x
(5)
b
w,c
(x ) =
O
(x )C
w,c
(x ) +
C
(x )O
w,c
(x )
O
(x ) +
C
(x )
(6)
where
O
(x ) and
C
(x ) are smoothed opening and closing
errors, and b
w,c
(x ) is the output of the bitonic filter
3
:the
errors, weights and filter output are shown in the middle
and right-hand columns of Fig. 1. This seems to have the
required properties, i.e. the preservation of any bitonic signal,
and reduction of noise in all regions, including across edges:
the opening and closing operations effectively remove bitonic
signals, leaving the Gaussian to reduce any residual noise
signal everywhere to the same extent.
B. Analysis
In fact, the noise reduction capabilities of the bitonic are
very similar to the Gaussian. In high noise environments,
the opening and closing operations will result in relatively
constant signals (see Fig. 1(c)), such that O
w,c
(x ) ≈ k
O
and
2
This can be seen by replacing G
σ
(x) with x in eq. (4) and following.
3
In the particular case that
O
(x) and
C
(x) are both zero at x,theyare
replaced by the arbitrary value 0.5: this would normally imply that both
opening and closing returned the exact original signal and hence how they
are weighted is inconsequential.
TREECE: BITONIC FILTER: LINEAR FILTERING IN AN EDGE-PRESERVING MORPHOLOGICAL FRAMEWORK 5201
Fig. 1. Stages of the bitonic filter. In each case the original signal (left) is both opened and closed (centre-left). The difference between the original and
each of these signals is filtered using a Gaussian (centre) and the smoothed error applied as a weighting (centre-right) to the opened and closed signals
to generate the final signal (right). The mathematical symbols relate to the equations in the main text. The morphological filters act to detect bitonicity
in the data so that the Gaussian filter only eliminates the residual noise. This preserves edges exactly in no noise (a) and very well in low to medium
noise (b). Even with noise as high as the signal level, some edge preservation is possible (c), whilst for very high noise the output tends to that of a Gaussian
filter.
C
w,c
(x ) ≈ k
C
,wherek
O
will generally be lower than the
mean value, and k
C
higher, in which case:
O
(x ) = G
σ
(x ) − k
O
C
(x ) = k
C
− G
σ
(x )
b
w,c
(x ) =
(
G
σ
(x ) − k
O
)
k
C
+
(
k
C
− G
σ
(x )
)
k
O
k
C
− k
O
= G
σ
(x ) (7)
i.e. in very high noise the bitonic filter reduces to a simple
Gaussian smoothing of the signal. However, if there is any
bitonic structure in the signal, this will be picked out and
given appropriate weight according to how well it matches
the original signal.
In fact eq. (7) demonstrates that, with increasing noise, the
bitonic filter will tend towards the performance of whatever
linear filter is used in its definition, eqs. (4) and (5), and hence
this could be seen as a framework for combining any linear
filter with rank filters in order to preserve signal edges.
Thesizeofwindow
|
w
|
controls the amount of smoothing
of the filter in exactly the same way as would be expected
for any moving-window linear filter. In contrast, the centile c
determines the sensitivity to impulsive noise. c = 0% treats
any isolated impulses as signal, c ≈ 20% will reject impulses
similarly to a median filter (but with significantly less non-
linear distortion than the median), and c = 50% is the
upper limit, since in this case C
w,c
≡ O
w,c
. In nearly all
scenarios, c = 10% provides a very good balance, and
is used in all subsequent results except for the no-noise
environment in Fig. 2. No setting of w nor c can cause
instability, and they create useful output features across the
entire range. Both Gaussian and rank filters are constrained
5202 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 25, NO. 11, NOVEMBER 2016
Fig. 2. Filtering results on ‘seattle’ image with no added noise. All outputs were generated with the same filter length l, and hence similar smoothing in
areas regarded as noise at this scale, for instance the reflected trees at the base, and the top of the Seattle needle. The Gaussian (b) has uniform smoothing
but no sharp edges. Techniques sensitive to data value, (f) image-guided filtering, (g) anisotropic diffusion and (h) non-local means, retain edges above a
certain threshold but either remove or blur them elsewhere, for instance the panels at the top of the building. The median (d) has sharper edges but removes
or distorts high level details, as does the OCCO filter (e), whilst Gaussian levelling (i) preserves all edges. The bitonic filter (c) smooths similarly to the
Gaussian but with edges of all magnitudes preserved and much better handling of isolated detail than the median, though not as well as levelling.
to the convex hull of the input data, and hence the bitonic
filter will never generate values outside of the input data
range.
It should be noted that, whilst in practise (and as demon-
strated by the results, particularly see Fig. 7) the bitonic filter
certainly acts to preserve bitonic signals and remove anything
else, it is not shown here that bitonicity over the filter length l
is actually guaranteed by this filter. This could only be true
if the centile c is set to zero, in which case the opening
and closing operations are likely to be independently bitonic,
though the weighted combination is more complex and may
not be. In any case, better visual results are obtained by setting
c = 10% and in this case whilst the filter may still generate
a bitonic signal, it will certainly remove some components
(i.e. isolated impulses) which are themselves bitonic. However,
local bitonicity can still be advanced as the best summary of
what the filter seeks to preserve in a signal.
C. Implementation
The bitonic filter can also be implemented reasonably
efficiently. Since the first rank filters of each of eqs. (2) and (3)
both act on the original signal, the ranking only needs to be
performed once, following which both centiles can be selected.
In addition, the Gaussian filters are generally faster than a rank
filter and have little impact on the processing time. Hence the
bitonic filter takes roughly 3 to 4 times as long as a single rank
filter of the same window size. The efficient implementation
of rank filters has been well studied, with histogram-based
O(l) algorithms, as used here for image data, in common use.
A constant-time implementation has also been proposed for
the median filter [7] which is equally applicable to rank filters
using other centiles.
III. E
DGE-PRESERVING FILTERS
Whilst, to the author’s knowledge, there are no other filters
designed for the specific purpose of preserving bitonic sig-
nals, there are various morphological filters which have some
similarity.
Most of the variants of median filters, for instance adaptive
median filters [8] focus on better preservation of signals in
the context of reducing impulsive noise, often by some form
of signal or impulse detection. The filter can then either be
modified in extent, or some data weighted by repeating values
in the ranked list: the rank filter equivalent of multiplying by a
larger weight in linear filtering. Examples include the constant-
time weighted median [9], rank-conditioned rank selection
filtering [10], or permutation weighting. A comprehensive
discussion of such filters has been presented in [11]. These
filters vary in rank selection, but will return one of the
input values: hence they are limited in their power to reduce
non-impulsive noise, though more complex combinations of
morphological operations such as discussed by [12] have been
used to de-noise specific signals [13]. In particular, the OCCO
filter [26] takes the average of an opening-then-closing and a
closing-then-opening, which has the advantage of being self-
dual (symmetric in data value).
TREECE: BITONIC FILTER: LINEAR FILTERING IN AN EDGE-PRESERVING MORPHOLOGICAL FRAMEWORK 5203
Connected operators, like rank filters, are also sensitive
to data ordering rather than value, however they divide an
image into non-overlapping sets, and then operate on these
sets: a very helpful recent review is in [14]. In this context
area opening and closing remove features based on their
area rather than their shape [18]. The sets can be defined
inclusively, where the area of a component is considered to
be the total of all included sets, and there exist very efficient
algorithms for extracting such definitions, for instance the
tree of shapes [17]. Self-dual filtering can be achieved by
removing any sets which have areas below a threshold, either
by ‘pruning’ the connected tree of sets and reconstructing the
image, or by directly manipulating the original image as in
the grain filter [15], similar to level sets [16].
Combinations of linear and morphological filters are some-
what less common. Possibly the earliest is the alpha-trimmed
mean [21], which uses a rank filter to remove outliers before
taking the mean (effectively a simple linear filter) of the
remainder. Others simply represent linear combinations of
the output of a rank filter and a linear filter, for instance the
mean-median filter [22], or more general (and very similar)
morphological/rank/linear (MRL) filter [23] and hybrid order
statistic filter [24]. Such filters may require training to define
the many possible rank and linear weights [25]. They are also
limited by the use of a single rank filter which is fundamentally
monotonic: hence edge-preservation is only possible for a
single edge within the range of the rank filter, and also by
limiting any subsequent smoothing from the associated linear
filter.
An alternative combination of linear and rank filters is
given by [6]. In order to improve morphological operations
which can be implemented using histograms, they propose
local smoothing (by a linear filter) of the histogram of the data
values. Whilst this is rather different from the bitonic filter,
setting c = 20% or higher can offer a similar improvement to
the median as does histogram-smoothing on low-noise data.
It is also possible to combine self-dual levelling (effectively
creating larger, more single-valued regions in the image)
with linear filters [14], [19]. This can be implemented by
reconstruction, i.e. iteratively dilating and eroding according to
an independent mask: in this context the levelling re-introduces
edge information from the image into the mask. Hence if the
mask is a Gaussian filtered image as in [27] then the whole
process can be seen as edge-enhanced de-noising.
There are numerous options for noise suppression without
blurring edges which make no use of morphological filters,
a theoretical comparison of many of which can be found
in [28]. Three popular examples serve to provide a comparison
with the bitonic: the image-guided filter [29], anisotropic
diffusion [30], [31] and non-local means [28]. All of these can
preserve edges, but do so by different mechanisms. The image-
guided filter weights data according to difference in value as
well as location: this is similar to the better known bilateral
filter [32], but with improved performance at image edges.
Anisotropic diffusion uses the local curvature to discourage
averaging across steeper edges. Non-local means only averages
data with that from similar surrounding distributions. Each
of these has various derivatives but the fundamental form is
used here, since many of the alternatives, for instance local-
adaptation or iteration, could equally be applied to the bitonic
filter.
IV. R
ESULTS
Details of the tested filters are as follows, where the bitonic
filter length l (window length in 1D or diameter of the
structuring element in 2D) is chosen in each case to represent
the parameter which most controls the extent of the filter:
Gaussian Linear filter with Gaussian weights, with the
standard deviation σ set to 0.33l, and sufficient
filter length to cover up to ±2σ .
Median Median filter, in 1D with window length l,in
2D using an isotropic circular structuring element
with diameter l.
Bitonic Bitonic filter as previously described, with
length l,andc fixed at 10%, except for the no-
noise case in Fig. 2 where it is set to 2%.
Guided Image-guided filter, implemented using the
MATLAB
4
function imguidedfilter
5
[29], with
the local neighbourhood size set to l,andthe
degree of smoothing set to four times the added
noise variance in the image.
Diffusion Anisotropic diffusion, implemented for
MATLAB
6
[30], with number of iterations
set to l, the integration constant set to the
standard deviation of the added noise, the
gradient threshold set to twice the standard
deviation of the added noise, and the wide-
region conduction coefficient.
NLM Non-local means filter, implemented using a fast
algorithm for MATLAB
7
[28], with the window
and search length both set to l, and the filter
parameter h set to the standard deviation of the
added noise.
OCCO The average of a rank-based opening-closing,
with a closing-opening [26], except that the
centile-based operations C
w,c
, O
w,c
were used
with c fixed at 10% to improve performance.
l defined the diameter of the structuring element
as for the Bitonic filter.
Grain A self-dual grain filter based on area openings
and closings, implemented similarly to [15] and
[16], with an inclusive definition of areas [33].
In 2D, the minimum area was given by l × l.
Levelling A self-dual levelling based on reconstruction
using a Gaussian mask [19], [27]. l controlled
the standard deviation of the Gaussian, exactly
as with the Gaussian filter above.
Where relevant, the data was symmetrically extended at
the image edges, though very similar results are achieved by
simply extending the values at each edge. Guided, Diffusion
4
MATLAB R2014a, The MathWorks Inc., Natick, MA, 2000.
5
http://uk.mathworks.com/help/images/ref/imguidedfilter.html
6
MATLAB file exchange: Anisotropic Diffusion (Perona & Malik) by
Daniel Lopes,14 May 2007.
7
MATLAB file exchange: Fast Non-Local Means 1D, 2D Color and 3D by
Dirk-Jan Kroon, 28 Apr 2010.
