# On Mathews Correlation Coefficient and Improved Distance Map Loss for
Automatic Glacier Calving Front Segmentation in SAR Imagery

Amirabbas Davari  Saahil Islam  Thorsten Seehaus  Matthias Braun  Andreas
Maier  Vincent Christlein * AmirAbbas Davari and Saahil Islam contributed
equally to this work.A. Davari, S. Islam, A. Maier and V. Christlein are with
the Computer Science department at Friedrich-Alexander University Erlangen-
Nürnberg, 91058 Erlangen, Germany (email: amir.davari@fau.de).T. Seehaus and
M. Braun are with the Geography & Geosciences department at Friedrich-
Alexander University Erlangen-Nürnberg, 91058 Erlangen, Germany.

###### Abstract

The vast majority of the outlet glaciers and ice streams of the polar ice
sheets end in the ocean. Ice mass loss via calving of the glaciers into the
ocean has increased over the last few decades. Information on the temporal
variability of the calving front position provides fundamental information on
the state of the glacier and ice stream, which can be exploited as calibration
and validation data to enhance ice dynamics modeling. To identify the calving
front position automatically, deep neural network-based semantic segmentation
pipelines can be used to delineate the acquired SAR imagery. However, the
extreme class imbalance is highly challenging for the accurate calving front
segmentation in these images. Therefore, we propose the use of the Mathews
correlation coefficient (MCC) as an early stopping criterion because of its
symmetrical properties and its invariance towards class imbalance. Moreover,
we propose an improvement to the distance map-based binary cross-entropy (BCE)
loss function. The distance map adds context to the loss function about the
important regions for segmentation and helps accounting for the imbalanced
data. Using Mathews correlation coefficient as early stopping demonstrates an
average $15\text{\,}\mathrm{\char 37\relax}$ dice coefficient improvement
compared to the commonly used BCE. The modified distance map loss further
improves the segmentation performance by another $2\text{\,}\mathrm{\char
37\relax}$. These results are encouraging as they support the effectiveness of
the proposed methods for segmentation problems suffering from extreme class
imbalances.

###### Index Terms:

semantic segmentation, class imbalance, Mathews correlation coefficient (MCC),
improved distance map loss

##  I Introduction

The mass loss of the Polar ice sheets and other calving glaciers has
accelerated in the last two decades. This is one of the main causes of global
sea level rise. Currently, the sea level contribution from thermal expansion
is exceeded by the contribution from ice sheets, glaciers, and ice caps [1].
The retreat of the glacier fronts can produce a re-enforcing effect of ice
loss. For example, the retreat from the pinning points can reduce the lateral
buttressing forces, which further destabilize the glaciers’ ice discharge into
the ocean [2]. A significant frontal retreat can further cause a retreat of
the grounding zone leading to further destabilization of the glacier flow [3].
This makes the calving front, the region where the glacial ice breaks off into
the ocean. Hence, the calving front position (CFP) is an important variable
for monitoring, but also ice dynamic modeling.

The detection of the calving front locations is commonly carried out using
optical imagery or Synthetic Aperture Radar (SAR) imagery. Cloud covers and
polar nights affect the optical imagery and the data coverage is strongly
curbed. On the other hand, SAR acquisitions are unaffected by such situations
and can be carried out all year round, making SAR imagery more advantageous
for monitoring purposes in polar regions. The common approach to detect CFPs
has been visual analysis and manual delineation on the obtained SAR images.
However, manual digitization is tedious, subjective, expensive, and time-
consuming. Moreover, sea ice and calved-off icebergs in the oceans make it
often difficult to reliably differentiate between ice mass and these chunks of
floating ice, making manual delineation subjective. This leads to a lack of
temporally updated data on the position of the calving front, which is
considered an essential climate variable (ECV). Automatic detection of CFPs
would close this monitoring gap. In the past, various algorithms of automatic
and semi-automatic detection of the calving front positions had been proposed.
Edge detection algorithms such as Roberts edge extractor has been applied to
ERS-1 SAR images by Sohn _et al. missing_ [4]. Seale _et al. missing_ [5] have
exploited Sobel filter and brightness profiling to detect the CFPs on daily
MODIS images. A detailed overview of various automatic and semi-automatic
approaches on this application is provided by Baumhoer _et al. missing_ [6].

Deep learning-based methods such as Convolutional Neural Networks (CNNs) [7]
are a better alternative to the traditional automatic delineation of CFPs.
Over the past two decades, CNNs have been seen to outperform the above-
mentioned manually designed filter-based methods and many such traditional
computer vision techniques in the field of image processing [8]. The CNN
architecture (the type and number of layers and their order) is chosen
depending on the application and the training data. The parameters in a CNN
are fine-tuned based on optimizing a predefined objective function, also
referred to as cost function or loss function.

Deep convolutional networks can be used on SAR images to automatically
classify the image pixels, belonging to semantically different regions, into
distinct classes. Such a task is referred to as semantic segmentation.
Architectures such as SegNet [9] and U-Net [10] are commonly used for semantic
segmentation applications [7]. The use of the U-Net architecture for SAR image
segmentation was first exploited by Mohajerani _et al. missing_ [11] on
Landsat-5, 7, and 8 images to automatically detect calving fronts. Zhang _et
al. missing_ [12] have used U-Net on higher resolution TerraSAR-X images of
Jakobshavn Isbræ glaciers to separate the whole ice mélange regions from non-
ice mélange regions, which are subsequently post-processed to manually
delineate the calving fronts.

Despite being powerful learning tools, the high number of parameters in Deep
Neural Networks (DNNs) comes with the risk of over-fitting to the training
data [13], i. e., it “memorizes” the non-predictive features of the training
data instead of “learning” its underlying distribution In this paper, one
contribution focuses on proper exploitation of early stopping to prevent the
network from over-fitting [14].

Early stopping suggests stopping training based on the error on independent
validation data to maintain high performance on new unseen data from the same
distribution. Since the training gets terminated based on the error on the
validation data, it is crucial to have the right performance metric as a
stopping criterion. The traditional method of early stopping comprises
monitoring the loss [15]. However, the significance of the performance metric
increases in cases of class-imbalanced data [16]. An imbalanced dataset
comprises classes with an unequal number of class elements. The majority of
performance metrics are challenged in the presence of severely imbalanced
data. For example, accuracy is not an appropriate metric in applications with
class-imbalance as incorrect classifications of the minority class could be
overlooked. On the other hand, other metrics such as f1-score greatly
diminishes the effect of class imbalance. Mathews Correlation Coefficient
(MCC) is theoretically argued to have a higher tolerance over class imbalance
than f1-score and confusion entropy [17, 18] and was for example used to
compare between a vast range of classifiers for evaluation of models on micro-
array gene expression and genotyping data [19]. In this paper, we are
suggesting (a) the use of early stopping to prevent over-fitting and (b) using
MCC instead of the classical BCE loss as stopping criterion.

While early stopping handles over-fitting with the right performance metric,
it is also crucial to objectively handle the learning of the network on the
imbalanced dataset. The imbalanced dataset introduces a bias towards the
majority class during the training of the network while the minority class is
the point of interest in many real-world problems. Common solutions are for
example data re-sampling [20], cost-sensitive learning, one class learning, or
ensemble learning methods [21]. While these methods could be effective, they
do not guarantee to be so in all kinds of tasks. For example, over-sampling
undergoes a risk of early over-fitting because the network uses many samples
more than once while under-sampling reduces the size of the training data.
Artificially balancing the class distributions was not very successful in some
tasks [22] and many ensemble-based methods are resource-demanding and time-
consuming [21].

Another way of cost-sensitive learning is to weigh the minority class higher
to punish miss-classification. In imbalanced segmentation tasks, the loss
function can also be trained to penalize more the contour of the segmentation
masks than the regions. Kervadec _et al. missing_ [23] used such a boundary-
based loss function for medical image segmentation. They argued that region
integrals of the segmentation masks highly vary over the classes and instead
trained on the space of contours. Another distance-based loss approach was
stated to segment knee bones in 3d MRI [24]. However, they employed the idea
of a distance map-based loss because most errors in segmentation are found to
be at the edges of the segmented masks. A distance map-based loss was also
used by Adhikari _et al. missing_ [25] to detect forest trails in optical
images of forests, which cover only small regions in the images. Therefore,
the trails were weighted higher in the loss function while the other regions
were weighted less based on the distance to the forest trails.

In this paper, we show that the use of MCC as an early stopping criterion is
superior to a binary cross-entropy (BCE) monitored network. The performance of
the network is further enhanced by an improved distance map-based BCE loss and
is compared with (a) the traditional BCE loss and (b) a weighted BCE loss. The
rest of this paper is organized as follows: Section II reviews the tools that
have been used in our contributions, i. e., Mathews correlation coefficient,
and the distance map loss. Section III describes our proposed pipeline and
explains the main contributions of this paper in detail. Next, the dataset
description, experimental setup, and the quantitative and qualitative results
are presented and discussed in Section IV. Finally, Section V concludes the
paper.

##  II Theoretical Background

###  II-A Mathews Correlation Coefficient

Binary cross-entropy is a popular loss function in binary classification
scenarios. A good loss is usually supposed not to produce very small or
extremely large gradients, which is one of the desirable properties of the BCE
loss [26]. We could choose to use the validation BCE loss also as an early
stopping criterion. However, BCE loss is very fragile in the presence of
class-imbalance [27]. Hence, it becomes an unfavorable early stopping
criterion for class-imbalanced segmentation.

On the other hand, MCC represents true positives (TP), false negatives (FN),
false positives (FP) and true negatives (TN) in a balanced manner [28]. It is
balanced in the sense that the rate of positive class predictions and negative
class predictions are given equal importance. MCC is defined as:

$\text{MCC}=\frac{\text{TP}\times\text{TN}-\text{FP}\times\text{FN}}{\sqrt{(\text{TP}+\text{FP})(\text{TP}+\text{FN})(\text{TN}+\text{FP})(\text{TN}+\text{FN})}}\enspace.$
(1)

It lies in the range $[-1,1]$ with $1$ representing the perfect prediction,
$-1$ denoting wrong, and $0$ suggesting random prediction. MCC takes positives
and negatives equally into account and provides very little “discrimination”
to random acts [17].

###  II-B Distance Map Loss

While BCE is the most commonly used loss function in most deep learning-based
methods, it is not favorable for class-imbalance problems. A common approach
to tackle the class-imbalance with BCE loss is to weigh each class with its
inverse class frequency. However, it does not always guarantee to produce
better performance [22]. Moreover, minority classes with higher weights can
lead to noisy gradients [25]. On the other hand, a distance map-based approach
adds context to the loss functions based on the regions of importance in the
segmentation task. Distance maps, also known as distance fields or distance
transforms, are derived representations of an image based on a distance
metric. In computer graphics, distance maps are often used for volume
rendering to generate offset surfaces or a blending between surface models
[29].

A distance map-based loss trains the network by penalizing the predictions
with respect to its distance from the object of interest in the image. A
distance map-based loss is also beneficial for class imbalance. By forcing the
network to give more attention to the regions of interest (minority class), it
counters the bias of the network’s training towards the majority class. The
distance map loss works on the idea of weighting far away pixels (from the
semantic line) higher than the ones in its proximity [25]. The loss is
formulated as:

$\begin{split}J_{\text{DW}}&=\frac{1}{M\cdot
N}\sum\limits_{j=1}^{N}\sum\limits_{i=1}^{M}((1-g_{i,j})\cdot d_{i,j}\cdot
p_{i,j}\\\ &-g_{i,j}\cdot
d_{\text{max}}\cdot\log(p_{i,j}))\enspace,\end{split}$ (2)

with $g$ being the ground truth map and $p$ the prediction at pixel location
$i,j$. The distance map is denoted as $d$, with $d_{\text{max}}$ being the
maximum distance, and $M,N$ represent the spatial dimensions of the maps.

##  III Methodology

###  III-A Processing Pipeline

Figure 1: Proposed glacier front segmentation pipeline.

Figure 1 outlines our proposed segmentation workflow. We perform the
segmentation of calving fronts images using a U-Net architecture. The SAR
images are used as input to the network, which learns either to predict the
zones of ice mass regions or the calving front lines, depending on the ground
truth data presented to the model at training time. In the rest of this paper,
we refer to the ice/non-ice regions as _zones_ and calving front locations as
_front lines_ for convenience. In the variant that the network is trained to
predict the ice zones, the predictions are post-processed to extract the
boundaries and in this way obtain the front lines.

The images are first pre-processed to remove noise and then patches of a
predefined size were extracted from both the images and their corresponding
ground truth before they are fed to the network. Before extracting the
patches, the images are padded with zeros to the next larger image size that
is a multiple of the patch size. The predicted masks on the patches are later
stitched together to reconstruct the original size image, and the padding is
discarded.

###  III-B MCC as the Early Stopping Metric

The data is split into three subsets, namely training, validation, and test
set. We use early stopping to stop the training before it overfits. However,
the validation error is usually very noisy. Hence, to ensure that the training
terminates where the validation performance is optimum, we pre-define a
patience number. This is the number of epochs the early stopper waits to check
for improvement on the validation error before terminating the training when
no improvement is noticed. The common practice is to monitor the BCE loss on
the validation set. However, due to the severe class-imbalance, we propose the
use of the Mathews correlation coefficient (MCC) as an early stopping metric
as it is theoretically more robust towards class-imbalance.

###  III-C Modified Distance Map and Distance Map BCE Loss

We formulate the distance map in a generalized form to be used in different
loss functions. The computed distance map weighs the line locations higher
while the regions around the proximity of the front lines are weighed less
based on the distance to the lines. As we move progressively away from the
front line locations, the weights keep reducing until they become $0$. In
other words, true line locations are given high importance and the proximity
adds relaxation to the network. The points beyond which these weights become
zero are considered to be the background (majority class). In contrast to the
distance map-based approach by Adhikari _et al. missing_ [25], cf. Eq. 2, we
weigh the background differently depending on the class imbalance. In
particular, we calculate the distance map from the ground truth masks as
follows:

We first thicken the ground truth lines by morphological dilation
($\delta_{w}$) [30]. Then, we use the Euclidean distance transform (EDT),
divided by a constant factor of $R$. The strength of relaxation is determined
by $R$, i. e., how smoothly the weights fall off from the front lines to the
background. The morphological dilation of the ground truth images thickens the
front lines based on the structuring element used. We use a rectangular
structuring element with a size of $(w,w)$, where $w$ determines the amount of
thickening of the lines. It is further passed through a sigmoid function
$\sigma$ to obtain values in the range of $[0,1]$:

$y_{\text{dl}}=\delta_{w}(y)\enspace,$ (3)
$y_{\text{edt}}=\sigma\left(\frac{\text{EDT}(y_{\text{dl}})}{R}\right)\enspace,$
(4)

where $y$ denotes the ground truth. Finally, the distance map $d$ is obtained
by:

$d=y_{\text{edt}}+k(1-y_{\text{dl}})\enspace,$ (5)

where $k$ is a constant in the range of $(0,1]$ determining the weight given
to the background, with $k=1$ signifying utmost importance. Since we deal with
high class imbalance, a high weight is not desirable. However, it is a hyper-
parameter that depends on the dataset and its class imbalance ratio, and needs
to be tuned. Various distance maps are depicted with different $w$, $R$, and
$k$ in Fig. 2.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Variation of the distance maps with different parameters when
applied on the ground truth. First image is (a) ground truth, and rest are
distance maps with (b) $w=5$, $R=4$, $k=0.1$, (c) $w=15$, $R=4$, $k=0.1$, (d)
$w=15$, $R=8$, $k=0.1$, (e) $w=5$, $R=4$, $k=0.5$, and (f) $w=5$, $R=4$,
$k=1$.

Finally, the distance map computed on the ground truth is used in the BCE loss
to create the distance map BCE loss. The BCE loss ($\text{J}_{\text{BCE}}$) is
given by:

$\text{J}_{\text{BCE}}(y,\hat{y})=-\frac{1}{N}\sum\limits_{i=1}^{N}(y_{i}\log\hat{y}_{i}+(1-y_{i})\log(1-\hat{y}_{i}))\enspace,$
(6)

where $\hat{y}$ is the prediction, $y$ is the ground truth, and $N$ is the
batch size during training. Incorporating the distance map into the BCE loss
($\text{J}_{\text{DBCE}}$) results in:

$\begin{split}\text{J}_{\text{DBCE}}(y,\hat{y})&=-\frac{1}{N}\sum\limits_{i=1}^{N}(y_{i}\log(\hat{y}_{i}\cdot
d_{i})\\\ &+(1-y_{i})\log(1-\hat{y}_{i}\cdot d_{i})\enspace.\end{split}$ (7)

##  IV Experimental Setup and Evaluation

###  IV-A Dataset

We perform our experiments on SAR images obtained from sites of the Antarctic
Peninsula (AP) and Greenland, depicted in Fig. 3. AP has been strongly
affected by changing climatic conditions [31]. The Dinsmoor-Bombardier-
Edgworth (DBE) and the Sjögren-Inlet (SI) glacier had undergone a strong
frontal retreat over the years [32, 33]. Jakobshavn Isbræ is one of the
largest tidewater glaciers in Greenland and went through a drastic retreat
over the last decades [34].

Figure 3: Maps of the study regions and the locations of the studied glaciers
at the Northern Antarctic Peninsula on the left, and Greenland Jakobshavn on
the right. Blue polygons indicate subsets used for the CNN analysis. Left
panels: Landsat LIMA Mosaic © USGS, NASA, BAS, NFS, Right panels: Bing Aerial
Maps © Microsoft.

The SAR images of the AP were calibrated and multi-looked to remove speckle
noise. Furthermore, the images were geo-coded and ortho-rectified by means of
the enhanced ASTER digital elevation model of the AP [35]. The ground truth of
the front lines is taken from Seehaus _et al. missing_ [32, 33]. The images
vary on the basis of their spatial resolution. High spatial resolution images
cover an area of $5\text{\times}5\text{\,}\mathrm{m}\text{\,}{\mathrm{}}^{2}$
per pixel and range until the lowest resolution of
$50\text{\times}50\text{\,}\mathrm{m}\text{\,}{\mathrm{}}^{2}$ per pixel. Out
of 244 available SAR images at the AP, randomly 50 images were kept for
testing, another 50 were used for validation, and the rest served for training
the model. We increase the AP training dataset three times by augmenting with
horizontally flipped image versions as well as by
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ rotated versions. The Jakobshavn
Isbræ glacier calving front locations were provided by Zhang et al _et al.
missing_ [12]. The SAR images of Jakobshavn Isbræ were preprocessed with a
median filter to remove speckle noise. All the images have a constant spatial
resolution of $6\text{\times}6\text{\,}\mathrm{m}\text{\,}{\mathrm{}}^{2}$ per
pixel. We use 25 random images for validation, 25 for testing, and the
remaining 119 images as the training set. The training data is augmented by
adding horizontally flipped image versions.

The SAR images of AP also vary in size. The smallest of the images has
dimensions of $377\times 458$, and the largest one contains $3770\times 4581$
pixels. Conversely, Jakobshavn Isbræ images are all approximately $1600\times
3500$ pixels in size. For both datasets, two ground truth images are
available: The calving front lines, and ice/non-ice zones. Figure 4 depicts a
sample SAR image in our dataset, along with its corresponding ice/non-ice
zones ground truth and its glacier calving front ground truth images. The
zones have a training class-imbalance of 1:3 in both datasets. The front line
ground truth is severely imbalanced with a ratio of 1:2097 in AP training data
and a ratio of 1:1124 in Jakobshavn Isbræ training data.

(a)

(b)

(c)

Figure 4: 4(a) Example of a SAR image with its corresponding ground truth
images: 4(b) ice/non-ice zones in which black represents the ice and white
represents the non-ice zones, and 4(c) glacier calving front line ground
truth.

###  IV-B Experimental Setup

####  IV-B1 Architecture and Parameters

We followed Zhang _et al. missing_ [12] as our baseline architecture. Both the
convolution and transposed convolution layers have kernel sizes of $(5,5)$. We
use leaky ReLUs as the activation functions with a negative slope of
$\alpha=0.1$, and the pooling layers have a pooling size of $(2,2)$. For
regularization, batch normalization layers have been exploited after every
convolutional layer. Furthermore, we use Adam as the optimizer [36] and a
cyclic learning rate for optimization, which is beneficial to the learning of
the network [37]. Due to the constant changes in the network’s performance by
the cyclic training during training, we set a rather high patience value (30
epochs) for early stopping. Note that we also tried various other patience
values for a subset of the experiments and the same behavior has been
observed. Hence, we decided to use 30 for all experiments so that we give BCE
early stopping criterion enough time to bring the validation loss down, too.
We use a patch size of $256\times 256$ pixels for patch extraction. The
prediction of zones differs from the prediction of lines in the context of
class-imbalance. Hence, we use slightly different parameters for these two
different ground truths, which are explained in the following experiments.

####  IV-B2 Experiment 1: Early Stopping Criterion: MCC vs. BCE

To demonstrate the importance of the metric used as the early stopping
criterion, we perform the segmentation on (a) the zones and (b) on the lines,
and compare the performance of BCE and MCC as the early stopping criteria. For
segmenting zones, a batch size of $20$ is used. The maximum learning rate of
$1e-2$ and a minimum of $1e-7$ is used in the cyclic learning rate with a step
size of $5\times$ epochs. To extract the calving front lines from the
predicted zones, we post-process our predictions as follows: First, the false
alarms are removed by extracting the largest connected component in the image.
Then, we use canny edge detection [38] to delineate the calving front lines
from the rectified zones.

In the second variant, where we train our model using the calving front lines
directly, the front lines are one-pixel wide. This contributes to the major
reason for the extreme class-imbalance. To facilitate the network’s training
on such a highly imbalanced dataset, the lines were thickened using
morphological dilation with a rectangular structuring element of size $5\times
5$ pixels. This reduces the class-imbalance of AP Dataset lines to $1:420$ and
of Jakobshavn Isbræ to $1:225$. We use a batch size of $15$ for the front
lines. The cyclic learning rates parameters for the lines are $1e-4$ and
$1e-8$ as maximum and minimum, respectively.

####  IV-B3 Experiment 2: Improved Distance Map BCE Loss vs. BCE loss

After proving the fact that MCC as an early stopping criterion accomplishes
better results, we perform our second experiment using MCC as an early
stopping criterion. In the second experiment, we compare the performance of
the improved distance map BCE loss (i.e., with optimum $k$) with the
conventional BCE loss, inverse class-frequency weighted BCE, and the
conventional distance map BCE loss (i.e., with $k=1$). This experiment is
designed only to tackle high class-imbalance. Hence, the models are trained
only using the front line ground truth images of both the datasets. We use the
same parameters for predicting front line masks as in our first experiment. We
search for the best parameters for the modified distance map formulation using
grid search and thereby, select $w=3$ and $R=1$ for both datasets. The best
value of $k$ was found to be $0.1$ for AP and $0.25$ for Jakobshavn Isbræ.

###  IV-C Evaluation

We use the performance metrics, that are commonly used for segmentation
problems [39, 7], i. e., Intersection Over Union (IOU) and Dice coefficient,
to evaluate our experiments. We also report the MCC. IOU is the area of
overlap between the predicted masks $p$ and the ground truth masks $g$ divided
by the area of union between the predicted masks and the ground truth masks.
The Dice coefficient is twice the area of overlap divided by the total number
of pixels from predictions and ground truth.

$\text{IOU}=\frac{p\cap g}{p\cup g}$ (8)
$\text{Dice}=\frac{2\sum\limits_{i=1}^{N}p_{i}g_{i}}{\sum\limits_{i=1}^{N}p_{i}+\sum\limits_{i=1}^{N}g_{i}}$
(9)

These performance metrics solely represent how well the predictions overlap
with the ground truth masks. This makes it hard to evaluate the calving front
lines because even if they are very close to the ground truth locations, they
will be considered as a miss by these metrics. Therefore, to make these
metrics more comprehensible, we further dilate the predictions and ground
truth masks. The dilated pixels around the ground truth are considered as the
relaxation or tolerance. Since our datasets also come with spatial resolution,
we calculate the tolerance in meters. The spatial resolution $s$ denotes the
actual distance one pixel covers on earth. Therefore, the tolerance $t$ is
calculated as $t=s\times p/2$. The tolerance scheme is depicted in Fig. 5. The
metrics are reported in our tables along with the tolerance (in meters) for
the predicted calving front lines.

Figure 5: Tolerance scheme based on dilation of lines and the spatial
resolution of image

Table I shows the results when using BCE and MCC as an early stopping
criterion. Monitoring MCC clearly outperforms BCE in all cases. Prediction of
zones in Jakobshavn Isbræ does not show much improvement as BCE as early
stopping performed very well, having IOU and Dice score of around
$97\text{\,}\mathrm{\char 37\relax}$. The reason is that the use of MCC
continues training to roughly 150 epochs, while BCE loss already stops at
around 50 epochs. Validation BCE decides that the network is overfitting too
early. On the other hand, MCC is more comprehensive about the validation error
and correctly allows the network to be trained further. Table I also
illustrates that monitoring MCC shows improvement not only when dealing with
high class-imbalance, but also is quite reliable for less imbalanced datasets
(AP and Jakobshavn Isbræ zones).

TABLE I: Performance when using BCE and MCC as early stopping criterion. Datasets |  Metrics |  Tolerance |  Early Stopping Criterion
---|---|---|---
|  |  | BCE | MCC
AP |  IOU |  | 0.8264 | 0.8912
zones |  Dice Coeff |  | 0.8984 | 0.9389
|  MCC |  | 0.8764 | 0.9254
| IOU |  65 m | 0.1524 | 0.2817
|  110 m | 0.2201 | 0.3957
|  155 m | 0.2712 | 0.4561
| Dice Coeff |  65 m | 0.2617 | 0.4421
AP |  110 m | 0.3549 | 0.5385
lines |  155 m | 0.4196 | 0.5997
| MCC |  65 m | 0.3328 | 0.4496
|  110 m | 0.4112 | 0.5456
|  155 m | 0.4657 | 0.6062
Jakobshavn |  IOU |  | 0.9613 | 0.9666
Isbræ |  Dice Coeff |  | 0.9801 | 0.9828
zones |  MCC |  | 0.9725 | 0.9763
| IOU |  60 m | 0.3755 | 0.5681
|  105 m | 0.4424 | 0.6451
|  150 m | 0.4801 | 0.6928
Jakobshavn | Dice Coeff |  60 m | 0.5437 | 0.7223
Isbræ |  105 m | 0.6108 | 0.7824
lines |  150 m | 0.6459 | 0.8167
| MCC |  60 m | 0.5712 | 0.7208
|  105 m | 0.6273 | 0.7783
|  150 m | 0.6553 | 0.8103

The fact that the BCE-monitored network is not trained enough can be clearly
observed in the qualitative results that are depicted in Fig. 6. The MCC-
monitored network performs significantly better for both zones and lines.
Apart from the fact that the calving front lines generated from the MCC
monitored network are more accurate, the predictions also seem to have fewer
numbers of false positives (incorrect white lines on the black regions).

(a)

(b)

(c)

Figure 6: Comparing the segmentation results using BCE and MCC as the early
stopping criterion on AP glaciers: 6(a) ground truth, 6(b) segmentation maps
predicted using BCE as the early stopping criterion, 6(c) segmentation maps
predicted using MCC as the early stopping criterion.

The results of BCE Loss and improved distance map BCE Loss are shown in Table
II. The improved distance map BCE loss is also compared with the distance map
using $(k=1)$. It is interesting to notice that the distance map BCE loss with
$k=1$ did not improve the performance of BCE loss in our task of glacier front
line segmentation. However, it does show slight improvement with low tolerance
($65\text{\,}\mathrm{m}$) for the AP glaciers. In contrast, the optimum $k$
helps to deal with the class-imbalance better. The qualitative results of the
predicted lines on Jakobshavn Isbræ regions are depicted in Fig. 7. In this
figure, the green color represents ground truth, the red color shows the
incorrect prediction, and the yellow color depicts the correct predictions, i.
e., the overlap between the prediction and the ground truth. The line
predictions of Jakobshavn Isbræ glaciers in both losses are disconnected.
However, the distance map loss manages to identify more regions of the calving
fronts. BCE loss also incorrectly predicts severe outliers (false positives)
that are far away from the real locations as depicted in Fig. 7. Such false
positives are less numerous in the case of the distance map BCE predictions.

TABLE II: Performance of the improved distance map BCE Loss (i. e., with optimum $k$) vs. the BCE loss, inverse class frequency weighted BCE, and the conventional distance map BCE Loss. Datasets | Metrics | Tolerance (meters) | Loss function
---|---|---|---
|  |  | BCE |  BCE (with weights) |  Dmap BCE, $k=1$ |  Dmap BCE, optimum $k$
AP lines | IOU | 65 | 0.2817 | 0.2798 | 0.3078 | 0.3085
110 | 0.3957 | 0.3728 | 0.3991 | 0.4072
155 | 0.4561 | 0.4355 | 0.4591 | 0.4734
Dice Coeff. | 65 | 0.4421 | 0.4211 | 0.4456 | 0.4463
110 | 0.5385 | 0.5242 | 0.5442 | 0.5506
155 | 0.5997 | 0.5873 | 0.6046 | 0.6152
MCC | 65 | 0.4496 | 0.4302 | 0.4502 | 0.4544
110 | 0.5456 | 0.5295 | 0.5467 | 0.5566
155 | 0.6062 | 0.5913 | 0.6059 | 0.6204
Jakobshavn Isbræ lines | IOU | 60 | 0.5681 | 0.5535 | 0.5675 | 0.5908
105 | 0.6451 | 0.6272 | 0.6417 | 0.6714
150 | 0.6928 | 0.6717 | 0.6872 | 0.7182
Dice Coeff. | 60 | 0.7223 | 0.7101 | 0.7217 | 0.7401
105 | 0.7824 | 0.7684 | 0.7794 | 0.801
150 | 0.8167 | 0.8012 | 0.8123 | 0.8338
MCC | 60 | 0.7208 | 0.707 | 0.7204 | 0.7364
105 | 0.7783 | 0.7031 | 0.7754 | 0.7957
150 | 0.8103 | 0.7937 | 0.8058 | 0.8271

Figure 7: Comparison of predictions of 7 BCE Loss, and 7 DMap BCE Loss. 7 and
7 show magnified versions of the marked areas from the BCE Loss prediction
image, and 7 and 7 show magnified areas from the DMap BCE loss prediction
image.

Referring to Fig. 8, which depicts the front line predictions of the AP
glaciers, we see that the BCE loss predicts several images quite inaccurately.
These images are well handled by the distance map loss.

(a)

(b)

(c)

Figure 8: Predicted lines achieved by 8(b) BCE loss, and 8(b) distance map BCE
loss. Image 8(a) depicts the ground truth. The two samples are from the AP
glaciers dataset.

(a)

(b)

Figure 9: Comparison of the predictions achieved by models trained on 9(a)
front lines, and 9(b) zones and then post-processed. The sample is from the AP
glaciers dataset.

When working on the problem of automatic detection of CFPs in the SAR images,
the available ground truth data for the training could be the ice zones or the
calving front lines. The semantic segmentation of zones on the SAR images
needs post-processing to find the calving front locations. Unlike semantically
segmenting the zones, a direct semantic segmentation of the calving front
locations is a harder problem due to the more severe class-imbalance. Figure 9
qualitatively compares the segmented front lines using the two approaches for
the AP glaciers dataset: one approach predicts the front lines directly and
the other post-processes the predicted zones. Direct predictions of calving
front lines bring minor disjoints in the predictions, whereas the post-
processed zone predictions are continuous. However, the result of the post-
processed zone is not entirely the glacier calving front lines. It shows the
boundary between ice zones and non-ice zones, which may contain some bedrock
boundaries as well. This is also observable on the post-processed predictions
for the Jakobshavn glacier in Fig. 10. Moreover, some regions are better
predicted by direct front line predictions.

Figure 10: Calving front prediction of Jakobshavn Isbræ computed by
postprocessing the predicted ice/non-ice zones.

##  V Conclusion

This paper proposes methods to improve deep learning-based semantic
segmentation of the glacier calving fronts. The detection of calving fronts is
challenged by its severe class inequality. We have tackled the class-imbalance
problem by two approaches. First, the use of Mathews Correlation Coefficient
as an early stopping criteria showed a great performance gain. Then, to help
the binary cross-entropy loss function to handle the imbalanced data, we
incorporated a novel distance map loss. The distance map can be used in
conjunction with any other loss function. The MCC-monitored network
outperforms the BCE-monitored network by about $15\text{\,}\mathrm{\char
37\relax}$ dice coefficient and IOU. Using the modified distance map BCE loss
results in a further improvement of approximately $2\text{\,}\mathrm{\char
37\relax}$ dice coefficient.

Finally, two approaches of predicting front lines were compared, i. e., either
to directly predict the front lines or to predict ice zones and then post-
process the areas to delineate the calving fronts. The presented results in
this work are encouraging and we believe that other image segmentation
applications with severe-class-imbalance can benefit from the proposed
approaches in this work.

## References

  * [1] M. van den Broeke, J. Bamber, J. Ettema, E. Rignot, E. Schrama, W. J. van de Berg, E. van Meijgaard, I. Velicogna, and B. Wouters, “Partitioning recent greenland mass loss,” science, vol. 326, no. 5955, pp. 984–986, 2009.
  * [2] J. J. Fürst, G. Durand, F. Gillet-Chaulet, L. Tavard, M. Rankl, M. Braun, and O. Gagliardini, “The safety band of antarctic ice shelves,” Nature Climate Change, vol. 6, no. 5, pp. 479–482, 2016.
  * [3] P. Friedl, T. C. Seehaus, A. Wendt, M. H. Braun, and K. Höppner, “Recent dynamic changes on fleming glacier after the disintegration of wordie ice shelf, antarctic peninsula,” The Cryosphere, vol. 12, no. 4, p. 1347, 2018\.
  * [4] H.-G. Sohn and K. Jezek, “Mapping ice sheet margins from ers-1 sar and spot imagery,” International Journal of Remote Sensing, vol. 20, no. 15-16, pp. 3201–3216, 1999.
  * [5] A. Seale, P. Christoffersen, R. I. Mugford, and M. O’Leary, “Ocean forcing of the greenland ice sheet: Calving fronts and patterns of retreat identified by automatic satellite monitoring of eastern outlet glaciers,” Journal of Geophysical Research: Earth Surface, vol. 116, no. F3, 2011.
  * [6] C. A. Baumhoer, A. J. Dietz, C. Kneisel, and C. Kuenzer, “Automated extraction of antarctic glacier and ice shelf fronts from sentinel-1 imagery using deep learning,” Remote Sensing, vol. 11, no. 21, p. 2529, 2019.
  * [7] S. Minaee, Y. Boykov, F. Porikli, A. Plaza, N. Kehtarnavaz, and D. Terzopoulos, “Image segmentation using deep learning: A survey,” arXiv preprint arXiv:2001.05566, 2020.
  * [8] N. O’Mahony, S. Campbell, A. Carvalho, S. Harapanahalli, G. V. Hernandez, L. Krpalkova, D. Riordan, and J. Walsh, “Deep learning vs. traditional computer vision,” in Science and Information Conference, pp. 128–144, Springer, 2019.
  * [9] V. Badrinarayanan, A. Kendall, and R. Cipolla, “Segnet: A deep convolutional encoder-decoder architecture for image segmentation,” IEEE transactions on pattern analysis and machine intelligence, vol. 39, no. 12, pp. 2481–2495, 2017.
  * [10] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in International Conference on Medical image computing and computer-assisted intervention, pp. 234–241, Springer, 2015\.
  * [11] Y. Mohajerani, M. Wood, I. Velicogna, and E. Rignot, “Detection of glacier calving margins with convolutional neural networks: A case study,” Remote Sensing, vol. 11, no. 1, p. 74, 2019.
  * [12] E. Zhang, L. Liu, and L. Huang, “Automatically delineating the calving front of jakobshavn isbræ from multitemporal terrasar-x images: a deep learning approach,” The Cryosphere, vol. 13, no. 6, pp. 1729–1741, 2019.
  * [13] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” The journal of machine learning research, vol. 15, no. 1, pp. 1929–1958, 2014\.
  * [14] X. Ying, “An overview of overfitting and its solutions,” Journal of Physics: Conference Series, vol. 1168, p. 022022, 02 2019.
  * [15] L. Prechelt, “Early stopping-but when?,” in Neural Networks: Tricks of the trade, pp. 55–69, Springer, 1998.
  * [16] A. Luque, A. Carrasco, A. Martín, and A. de las Heras, “The impact of class imbalance in classification performance metrics based on the binary confusion matrix,” Pattern Recognition, vol. 91, pp. 216–231, 2019.
  * [17] G. Jurman, S. Riccadonna, and C. Furlanello, “A comparison of mcc and cen error measures in multi-class prediction,” PloS one, vol. 7, no. 8, p. e41882, 2012.
  * [18] D. Chicco and G. Jurman, “The advantages of the matthews correlation coefficient (mcc) over f1 score and accuracy in binary classification evaluation,” BMC genomics, vol. 21, no. 1, p. 6, 2020.
  * [19] L. Shi, G. Campbell, W. Jones, F. Campagne, Z. Wen, S. Walker, Z. Su, T. Chu, F. Goodsaid, L. Pusztai, et al., “The maqc-ii project: a comprehensive study of common practices for the development and validation of microarray-based predictive models.,” 2010.
  * [20] A. Davari, H. C. Özkan, A. Maier, and C. Riess, “Fast and efficient limited data hyperspectral remote sensing image classification via gmm-based synthetic samples,” IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 12, no. 7, pp. 2107–2120, 2019.
  * [21] X. Guo, Y. Yin, C. Dong, G. Yang, and G. Zhou, “On the class imbalance problem,” Fourth International Conference on Natural Computation, ICNC ’08, vol. Vol. 4, 10 2008.
  * [22] P. Pompey and I. Solomos, “Kaggle human protein atlas,” tech. rep., Standford University.
  * [23] H. Kervadec, J. Bouchtiba, C. Desrosiers, E. Granger, J. Dolz, and I. B. Ayed, “Boundary loss for highly unbalanced segmentation,” in International conference on medical imaging with deep learning, pp. 285–296, 2019.
  * [24] F. Caliva, C. Iriondo, A. M. Martinez, S. Majumdar, and V. Pedoia, “Distance map loss penalty term for semantic segmentation,” 2019.
  * [25] S. P. Adhikari and H. Kim, “Distance weighted loss for forest trail detection using semantic line,” in International Conference on Advanced Concepts for Intelligent Vision Systems, pp. 302–311, Springer, 2020.
  * [26] I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning.  MIT Press, 2016.  http://www.deeplearningbook.org.
  * [27] S. Jadon, “A survey of loss functions for semantic segmentation,” 2020.
  * [28] P. Baldi, S. Brunak, Y. Chauvin, C. A. Andersen, and H. Nielsen, “Assessing the accuracy of prediction algorithms for classification: an overview,” Bioinformatics, vol. 16, no. 5, pp. 412–424, 2000.
  * [29] S. F. Frisken, R. N. Perry, A. P. Rockwood, and T. R. Jones, “Adaptively sampled distance fields: A general representation of shape for computer graphics,” in Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pp. 249–254, 2000.
  * [30] P. Soille, Morphological image analysis: principles and applications.  Springer-Verlag New York, Inc., 2003.
  * [31] J. Turner, H. Lu, I. White, J. C. King, T. Phillips, J. S. Hosking, T. J. Bracegirdle, G. J. Marshall, R. Mulvaney, and P. Deb, “Absence of 21st century warming on antarctic peninsula consistent with natural variability,” Nature, vol. 535, no. 7612, pp. 411–415, 2016.
  * [32] T. Seehaus, S. Marinsek, V. Helm, P. Skvarca, and M. Braun, “Changes in ice dynamics, elevation and mass discharge of dinsmoor–bombardier–edgeworth glacier system, antarctic peninsula,” Earth and Planetary Science Letters, vol. 427, pp. 125–135, 2015.
  * [33] T. C. Seehaus, S. Marinsek, P. Skvarca, J. M. van Wessem, C. H. Reijmer, J. L. Seco, and M. H. Braun, “Dynamic response of sjögren inlet glaciers, antarctic peninsula, to ice shelf breakup derived from multi-mission remote sensing time series,” Frontiers in Earth Science, vol. 4, p. 66, 2016.
  * [34] I. Joughin, I. M. Howat, M. Fahnestock, B. Smith, W. Krabill, R. B. Alley, H. Stern, and M. Truffer, “Continued evolution of jakobshavn isbrae following its rapid speedup,” Journal of Geophysical Research: Earth Surface, vol. 113, no. F4, 2008.
  * [35] A. J. Cook, T. Murray, A. Luckman, D. G. Vaughan, and N. E. Barrand, “A new 100-m digital elevation model of the antarctic peninsula derived from aster global dem: methods and accuracy assessment.,” Earth system science data., vol. 4, no. 1, pp. 129–142, 2012.
  * [36] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv preprint arXiv:1412.6980, 2014.
  * [37] L. N. Smith, “Cyclical learning rates for training neural networks,” in 2017 IEEE Winter Conference on Applications of Computer Vision (WACV), pp. 464–472, IEEE, 2017.
  * [38] J. Canny, “A computational approach to edge detection,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-8, no. 6, pp. 679–698, 1986.
  * [39] A. A. Taha and A. Hanbury, “Metrics for evaluating 3d medical image segmentation: analysis, selection, and tool,” BMC medical imaging, vol. 15, no. 1, p. 29, 2015.