Method description
Our LKF detection algorithm consists of three parts: (1) a preprocessing step
that transforms the input deformation data into a binary map of pixels that
mark LKFs by using different filter steps, (2) a detection routine that
splits the network of LKF pixels in the binary map into the smallest possible
segments, and (3) a reconnection instance that estimates the probability of
different segments belonging to one feature and then connects all segments of
a LKF. The general structure of the algorithm follows ,
although individual parts have been modified substantially. The main
enhancements of the algorithm are a parallel detection of segments with a
stronger constraint on the curvature and the introduction of a probability-based reconnection. Further, the entire algorithm was rewritten in Python
(Python Software Foundation, http://www.python.org, last access: 15 February 2019) to avoid license
issues with the previous code that was based on the commercial software IDL.
Filter sequence from (a) input field total deformation to
(f) final output of the morphological thinned binary map of LKF
pixels. Intermediate steps are (b) logarithmic deformation,
(c) histogram equalization, (d) difference of Gaussian
filter, and (e) the thresholded output of the DoG filter. RGPS
deformation data for 1 January 2006 are used for this example.
![]()
Data preprocessing and filtering
The standard input data of the LKF detection algorithm is the total
deformation rate of sea ice
ϵ˙tot=ϵ˙I2+ϵ˙II2,
where ϵ˙I is the divergence and ϵ˙II the shear
that can be derived from both satellite data and model output
(Fig. a). LKFs are defined by regions of high
deformation rates because they are located along the boundaries of ice floes
where most deformation takes place. The actual magnitude of deformation along
an LKF, however, varies with the background deformation and the spatial
scale because of its multi-fractal properties . Thus a
simple thresholding of deformation rates is not sufficient to filter LKFs.
Instead, we are interested in detecting deformation that is notably higher
than the local environment. As LKFs are lines of high deformation, we need to
detect edges in the deformation field.
Prior to the edge detection, we take the natural logarithm of the input field
(Fig. b) and perform a histogram equalization
(Fig. c). Both highlight the local differences
across different scales and enhance the contrast in regions of low
deformation rates.
We use a difference of Gaussian (DoG) filter following for
the edge detection (Fig. d). The DoG filter
subtracts two filtered versions of the same input data: the first is smoothed
with a Gaussian kernel of radius r1=1 pixel corresponding to a half-width
σ1=0.5, and the second is smoothed with a Gaussian kernel of radius
r2=5 pixels (half-width σ2=2.5), where r1<r2. The smaller radius
r1 corresponds to the smallest scale of features that will be detected by
the DoG and the second radius r2 provides the upper limit of the scales
detected. Note that this scale limitation applies to the width of the LKFs as
well as to their lengths. For edges of scale r1<L<r2, the DoG-filtered
values are positive because the local deformation rate is higher than in the
environment of radius r2. Pixels are marked as LKFs when the DoG-filtered
pixels are larger than a positive threshold dLKF. The result is a
binary map in which pixels with a value of 1 belong to LKFs (see
Fig. e for a threshold of 15). The threshold
dLKF does not have a unit as it describes the difference of two
histogram-equalized images, for which the highest deformation rate corresponds to
a pixel value 255 and the lowest to a value of 0.
At this point, LKFs in the binary map are still represented in their original
width. To detect which pixels belong to which LKF we add a further level of
abstraction and reduce the width of all LKFs to 1 pixel
(Fig. f). To this end, a morphological thinning
algorithm reduces the binary map to its skeleton. We use the
skeletonize function of the open-source Python package
scikit-image based on .
Skeletonization was used before to detect leads in original or classified,
that is, preprocessed and charted, SAR images
.
Segment detection
We detect small segments of pixels that contain parts of one LKF in the
binary map. Based on the morphological thinned binary map, groups of pixels
that form a line are detected. In this first detection step, we want to
guarantee that all pixels of a detected segment belong to the same LKF.
Therefore, we detect the smallest segments possible that are in the simplest
case the points in between intersections of lines in the binary map
(Fig. f).
As a starting point of the segment detection we use LKF pixels that have only
one neighboring cell also marked as LKF. Within each iteration, the
detection algorithm proceeds to the LKF neighbor and again checks the number
of neighboring cells for LKFs. If the new cell also only has one
neighboring LKF cell (neglecting the cell from the prior iteration) the
search is continued. If the new cell has more than one neighboring LKF cell,
that is, it is a junction, the detection cycle is stopped and these
neighboring points become the new starting points. In addition to the number of
neighboring LKF cells, a change in direction compared to the orientation of
the last 5 pixels can also terminate the detection cycle: if the angle
between the line connecting the centers of the potential new cell and the
current cell and the linear fit to the previous 5 pixels of the segment
exceeds 45∘, the detection cycle is interrupted and the new cell is
marked as a new starting point. If the segment is still shorter than 5 pixels, all available pixels are taken into account. We use 5 pixels in
contrast to 2 pixels used by to impose a stronger
constraint on the curvature. As in the 2 pixel case, a 90∘ shift of
direction is possible within two steps of the detection. Within each cycle of
the detection algorithm, pixels that have been assigned to a segment are
removed from the input binary map to prevent double assignments. This
procedure is repeated until no new starting cells are found.
(a) Detected segments and the results of (b) the
first reconnection step and (c) the second reconnection step for
RGPS deformation data from 1 January 2006. Each color denotes a different
segment or LKF. Due to a limited number of colors, different segments or LKFs
can have the same color. The output of the second reconnection step is the
final output of the LKF detection algorithm.
![]()
After removing all linear segments, the remaining binary map contains only
non-LKF pixels or LKFs forming closed contours with no starting points. The
closed contours are opened by arbitrarily marking pairs of two neighboring
LKF pixels (every i⋅100th and i⋅100+1th LKF pixel for i=1,2,3,…) as starting points. Then the
segment detection is repeated until no new starting points are found. The
initialization step to open closed contours is then repeated until all
LKF pixels in the binary map are assigned to a linear segment. All segments
that were detected are shown in Fig. a for
1 January 2006.
Reconnection
The reconnection instance is designed to connect multiple detected segments
that belong to the same LKF. Two segments belonging to the same LKF should
have a similar orientation and deformation magnitude and they should be in
close proximity to each other. Thus, we compute the probability for all
possible pairs of segments to be part of the same LKF based on their
distance, their orientation, and their deformation rates. The two segments of
the pair with the highest probability are connected and the probabilities of
pairs containing one of the two are updated. These steps are iterated until
no new matches are found. This part of the algorithm represents a new feature
compared to .
The central element of the reconnection step is the probability matrix
P∈IRN×N that
stores the probabilities for all pairs of segments with N being the number
of segments. The rows and columns correspond to single segments, for which
Pm,n gives the probability of segment m and n
being part of the same LKF. The probability is given by
Pm,n=(ΔDm,nD02+ΔOm,nO02+Δlog10ϵ˙m,nϵ˙02)‾,
with the elliptical distance ΔD between the two segments, the
difference in orientation ΔO, and the difference of the logarithm of
the total deformation rate Δlog10ϵ˙. Here we use the
common logarithm, i.e., log base 10, in contrast to the natural logarithm used
in Sect. to directly describe the difference in the
order of magnitude in the total deformation of two segments. The difference
in orientation ΔO is determined by the angle between the two
segments, which are represented in this computation by a line connecting the
start and the end point. The elliptical distance describes the distance
between both segments, but also takes into account the alignment of the
segments. In doing so, we decompose the vector connecting both ends of the
segments va→b into an orthonormal basis with one vector
parallel to the first segment a∥ and one vector perpendicular to
it a⟂ as shown in Fig. . In the same way,
vb→a is decomposed into a basis for the second segment
b∥, b⟂,
va→b=a∥,a⟂⋅α=!vb→a=b∥,b⟂⋅β,
where α and β are the coefficients of the vector decomposition.
In the computation of the length of the connecting vector v, the component
perpendicular to the segment is weighted with an elliptical factor e>1 and
the distance computed by both bases is averaged to obtain a symmetrical
elliptical distance, that is, v=va→b=vb→a:
‖v‖e=12‖va→b‖e+‖vb→a‖e=12αT⋅100e⋅α+βT⋅100e⋅β=ΔD.
We consider only pairs of segments in which the starting point of one segment
lies in the direction of the other segment, that is,
α0≥0 or β0≤0. Thus points with
the
same elliptical distance lie on a half ellipse centered at the endpoint of
the segment as denoted in Fig. . The computed
probability for a generic pair of segments
Pm,n=Pn,m is symmetric, because both the
elliptical distance, the orientation difference and the deformation rate
difference are symmetric. Thus, we only compute Pm,n with m<n, where P is simplified as an upper diagonal matrix.
Sketch of two segments A and B illustrating the principle behind
the elliptical distance. In the distance computation the component pointing
in the direction perpendicular to the segment is weighted by a factor of e.
Within the shaded area the elliptical distance is below a threshold D0. If
the endpoints of both segments lie within the area where both shaded
half ellipses overlap, they are considered for reconnection. The dotted lines
indicate the orientation of lines connecting the start and end points of both
segments and the angle ΔO is the difference in orientation. v
is the vector connecting the endpoints of both segments as defined in
Eq. (). a∥ and a⟂ are basis vectors
aligned in the direction of segment A, respectively b∥ and
b⟂ for B.
![]()
List of parameters used in the LKF detection algorithm. For each
parameter the lower and upper bounds of the optimization are given along with
the final choice of the parameter value.
Parameter name
Symbol
Lower
Upper
Final
Unit
bound
bound
choice
DoG filtering threshold
dLKF
0
30
15c
–
Max. elliptical distance
D0
1
7
4b
pixels
Elliptical factor
e
1
5
2
–
Max. difference in orientation
O0
25
65
35
∘
Max. difference in deformation
ϵ˙0
0.5
1.25
1.25
log(day-1)
Min. length
lmin
3
7
3
pixels
Min. radius of DoG
r1
1ab
pixels
Max. radius of DoG
r2
5ab
pixels
a Parameters that have not been optimized but taken
from . b Parameters are related to length
scales and need to be scaled with the spatial resolution of the input data.
c Parameters are used to suppress noise in the input data and
need to be adapted individually to input data.
The parameters D0, O0, and ϵ˙0 not only normalize the
individual components of Eq. () but also serve as an
upper threshold for these components. If for one pair the elliptical distance
ΔD, the difference in orientation ΔO, or the deformation rate
Δϵ˙ exceeds the threshold D0, O0, and
ϵ˙0, it is not considered for the reconnection. The threshold
values will be determined in Sect. and are
given in Table .
After initializing the matrix P, the pair of segments with the
highest probability Pm,n is connected and the
connected segment
(m n) replaces the
old segment m. Thereby the number of segments is reduced by one and the
nth row and nth column are removed from the probability matrix
P. The elements of the probability matrix that correspond to the
segments m need to be updated and thus the mth row and mth column of
P are reevaluated based on the new connected segment
(m n). This
process is iterated, and in each iteration the pair of segments with the
highest probability is connected, until no pair is left that satisfies the
threshold values D0, O0, and ϵ0˙.
In the final step of the LKF detection algorithm, features that fall below a
minimum length lmin are removed because most small features are
artifacts of the thinning algorithm and do not represent LKFs. With
increasing minimum length, the number of detected LKFs decreases and the
field of the detected LKFs shows a higher degree of abstraction. The minimum
resolution of the detected LKFs, however, is determined by the minimum length
used for the DoG filtering. The presented reconnection procedure shows better
results for longer segments because the orientation and mean deformation is
more sensitive for smaller segments. In theory, the best input would be
segments containing all the points in the binary map that lie in between
“junctions” of the lines, assuming that all those points belong to the same
LKF. The segment detection instance, however, yields smaller segments due to
the parallel detection that has been implemented to increase computational
efficiency. Thus, we apply the reconnection algorithm twice: the first
instance is meant to compensate for the tendency of the segment detection
algorithm to divide segments into smaller pieces although they actually
belong to the same inter-junction segment. Thus, we use very a restrictive
set of threshold values (maximum distance D0=1.5 pixels, maximum difference
in orientation O0=50∘, maximum difference in deformation rate
ϵ˙0=0.75log (day-1), elliptical factor e=1, and
minimum length lmin=2 pixels) to ensure that only segments are reconnected
that are not separated by more than 1 pixel and no segments are removed
because they are short (Fig. b). The second
reconnection instance with a different set of parameters is then used to
reconnect segments across junctions and to generate the final LKFs shown in
Fig. c. The choice of the set of parameters
used in the second reconnection instance is discussed in
Sect. .
Parameter selection
There are a number of parameters in the detection algorithm. For some of them
the range of possible choices can be narrowed down with information from
field and satellite observations as well as theory of ice fracture, but none
of them are strictly constrained. Therefore, we attempted an optimization of
the set of parameters, mainly of the reconnection step, given in
Table .
The main challenge of the optimization is the strong nonlinearity of the
detection algorithm. The main source of nonlinearity is the fact that
whether a feature is detected or not can depend sensitively on small changes
of a parameter. Another source of nonlinearity is the small number of
reference data. The strong nonlinearity of the problem constrains both the
definition of the cost function and the optimization method to make sure that
the optimized solution is the global minimum of the cost function.
We constructed different cost functions, ranging from simple counts of
falsely undetected features by the algorithm to a cumulative modified
Hausdorff distance (MHD) between all detected and handpicked features as a
cost function, in an effort to smooth the strong nonlinearity. In addition,
we used different nonlinear optimization routines including basin-hopping
and a nested brute-force implementation. No combination of
cost function and optimization method leads to a satisfying result because in
all cases the cost function was very sensitive to smaller variations in the
parameters. We concluded that finding a global minimum of the cost function
is impossible. Therefore, we use a set of parameters estimated with a simple
brute-force algorithm that minimizes the number of not-detected features for
the range of parameters given in Table ,
in which for each parameter five equally spaced values within its range are
used. We do not regard this set of parameters as the global optimum but
rather as a useful working basis given the strong nonlinearity of the
problem and the limited number of reference data. The performance of the
detection algorithm with this set of parameters is evaluated in detail in
Sect. .
Evaluation
In the evaluation we use all data for January 2006 (11 snapshots) from the
handpicked LKF data set, the LKFs detected by the algorithm presented in
this study, and LKFs detected by the original version of this algorithm
. The reference data of the original algorithm are generated
with the parameters used in the evaluation section of . In
this way, we evaluate the overall ability of the method to properly detect
LKFs but also check whether the modifications and new additions improve the
performance of the algorithm. We determine to what degree the algorithms can
detect the same features that were recognized by visual inspection and
furthermore provide detailed information about the similarity and differences
between automatically detected and handpicked features in an element-wise
comparison.
The principal idea behind this evaluation is that we compare the features
pairwise: one LKF from the handpicked data with the best-matching automated
detected LKF. We find the best matching automatically detected feature for
each handpicked feature by minimizing the MHD
between the handpicked features and all automatically
detected features. Next, we categorize all pairs by the degree of overlap of
the handpicked feature with its closest matching detected feature. The
overlap between two features is illustrated in Fig. .
The number of pixels for which the distance to the closest pixel in the
matching feature is smaller than dO=3 pixels is defined as overlap,
labeled as OA and OB in Fig. . To distinguish
between the overlap of feature pairs that have a similar shape but are
displaced and pairs that overlap only due to intersection of both features,
we compute the angle between overlapping parts of the matching pairs
θO. If this angle is smaller than
θO<25∘, the overlap of a matching pair is defined by
the minimum of overlapping pixels of both matching partners normalized by the
maximum length of both matching partners:
O=minlen(OA),len(OB)/maxlen(A),len(B),forθO<25∘.
Given this definition of overlap, we distinguish among three different
classes of pairs: (1) fully matching pairs that have an overlap
larger than 60 %, (2) partly matching pairs that have an overlap
smaller than 60 % but larger than 0 %, and (3) not matching pairs
with overlap equal to 0 %.
Illustration of overlap between two LKFs A and B.
![]()
The overall performance of the algorithm with respect to the overlap of the
detected features with the reference data set is given in
Fig. along with the number of pairs within each class.
We find that the features detected with our new algorithm overlap
significantly more with the handpicked reference data than the features
detected with the original version of the algorithm
(Fig. ). The original version of the algorithm is run with
the same parameters used in the evaluation section of . Our
modifications to the algorithm increase the number of fully matching LKFs by
66 % (from 314 to 522), along with a similar number of partly matching LKFs
(from 635 to 657) and a clear decrease of 66 % for the not matching LKFs
(from 347 to 117). This indicates a significant improvement of the original
algorithm.
Evaluation of the detection algorithm presented in this study and
the original algorithm against handpicked LKFs. The
cumulative frequency of occurrence of the overlap is given in the center
plot. The number of features is given in the bar plots for each class (fully
matching, partly matching, and not matching).
![]()
Statistics of all full matching pairs computed for the algorithm
presented here and the original version : (a) the
mean endpoint distance, (b) the line length error, and
(c) the modified Hausdorff distance (MHD). The background MHD refers
to MHD calculated for the handpicked features and the morphological thinned
binary field (Fig. f).
![]()
We analyze the similarity of features of all pairs within each class to test
whether these improvements are made at the expense of the quality of the
detected features. In doing so, we define three metrics to determine the
similarity of the features: (1) the mean endpoint distance, (2) the line
length error, and (3) the MHD as metrics, at which the first two were introduced
by . The MHD is a measure of the general agreement of two
shapes. It takes into account changes in both orientation and length but
also a complete change in shape. In the sea ice context, the MHD is applied,
for example, to evaluate the ice edge position in sea ice forecasts
or to assess the predictability of LKFs
. Here we only focus on the class of full matching pairs.
For the mean endpoint distance, we determine the distance between both
endpoints of the detected and the handpicked features for each pair and
average them. For all full matching pairs with features detected by our new
algorithm the endpoint distance tends to be smaller compared to features
detected by the original version of the algorithm, which is indicated by the
shift in the distribution towards smaller errors
(Fig. a). The improved match because of the
modifications to the original algorithm is also reflected in the smaller mean
error (1.47 pixels as opposed to 1.96 pixels of the original algorithm). For 75 %
of the features detected with our algorithm, the mean endpoint distance is
smaller than 2 pixels whereas this is only the case for 60 % of features
detected by the original version.
The line length error is determined by the difference in length of the two
features in a pair normalized by length of the smaller feature of the pair.
For both algorithms, the distributions are similar with similar mean errors
of
∼18 % (Fig. b). For half of the pairs the
line length error is also lower than 15 % in both cases.
We find that our modifications to the original algorithm also reduce the
average MHD from 1.58 to 1.17 pixels (Fig. c).
A total of 73 % of these pairs lie within the fifth to 95th percentiles of the background
MHD defined as the MHD of the reference data and the morphological thinned
binary LKF field (Fig. f). Since all LKFs consist of
sets of pixels from this binary field, the background MHD is an upper limit
of how accurate a reconnection algorithm can become using this binary field as
an input value.
In conclusion our new version of the algorithm improves the original
algorithm in that it detects more features and also increases their agreement
with the handpicked reference data.
Discussion
Our adapted detection algorithm greatly improves the original version. The
total number of handpicked features that is reproduced by the algorithm
increased by 66 %. In addition, the quality of the detected features with
respect to their mean endpoint distance, the error in line length, and the MHD
is improved. We attribute these improvements to two changes that stand out
in addition to smaller adjustments in the code: (1) the introduction of a
probability-based reconnection and (2) the optimization of the DoG threshold
dLKF, which showed the highest sensitivity in the optimization. We use
this threshold to filter LKFs as regions that have high deformation rates
compared to the local environment. The deformation rates in RGPS are known to
be prone to grid-scale noise and uncertainties caused by tracking and
geolocation errors , which can lead to a
false classification of a pixel as LKF. Increasing the threshold slightly
suppresses this noise, albeit at the expense of losing features with smaller
deformation rate differences. Thus the threshold needs to be optimized to
balance both effects; we found dLKF=15 to be the best parameter choice
for the RGPS data set.
In our algorithm we reconnected segments to LKFs based on a probability
computed from the characteristics of the segments. Thereby those segments that fit “the best” are
reconnected, and in contrast to the
reconnection does not depend on the order in which the reconnection algorithm
runs over the list of segments. In doing so, we improve the quality of the
detected features and obtain a unique and consistent solution. Both
uniqueness and consistency are necessary ingredients for the ensuing
application of a tracking algorithm.
In this context, we found the elliptical distance ΔD and the
orientation ΔO to be the important contributors to the probability
function. The optimized threshold Δϵ˙0=1.25 for
differences in deformation rates is very high (as it is applied to the
difference of the common logarithm of the deformation rates, a difference
between deformation rates of a factor 101.25=17.78 is possible), so that
differences in deformation rates are normalized by a large value and do not
contribute much to the probability function. Omitting the deformation-related
part in Eq. () (equivalent to setting
Δϵ˙0=∞) does not change the results of the
evaluation of the optimized solution very much (not shown here). The small
influence of the threshold for deformation rates Δϵ˙0 on
the performance of the algorithm may also be caused by the noise in the RGPS
data along LKFs . Smaller segments, especially, are
affected by the noisy RGPS data so that segments that belong to the same LKF
may have different deformation rates.