[orcid=0009-0002-8751-8605, bioid=1] [orcid=0000-0002-2895-6544, bioid=2]
[orcid=0000-0003-0840-7350, bioid=3]

[cor1]Corresponding author:
Phone: +39 0776 2993381, Fax: +39 0776 2994355

# Gravity Network for end-to-end small lesion detection

Ciro Russo<EMAIL_ADDRESS>Alessandro Bria<EMAIL_ADDRESS>Claudio
Marrocco<EMAIL_ADDRESS>Department of Electrical and Information
Engineering, University of Cassino and L.M., Via G. Di Biasio 43, 03043
Cassino (FR), Italy

###### Abstract

This paper introduces a novel one-stage end-to-end detector specifically
designed to detect small lesions in medical images. Precise localization of
small lesions presents challenges due to their appearance and the diverse
contextual backgrounds in which they are found. To address this, our approach
introduces a new type of pixel-based anchor that dynamically moves towards the
targeted lesion for detection. We refer to this new architecture as
GravityNet, and the novel anchors as gravity points since they appear to be
“attracted" by the lesions. We conducted experiments on two well-established
medical problems involving small lesions to evaluate the performance of the
proposed approach: microcalcifications detection in digital mammograms and
microaneurysms detection in digital fundus images. Our method demonstrates
promising results in effectively detecting small lesions in these medical
imaging tasks.

###### keywords:

Small lesion detection Pixel-based anchor Convolutional neural networks
Mammograms Ocular fundus images

##  1 Introduction

Detection of small lesions in medical images has emerged as a compelling area
of research, which holds significant relevance in medicine, especially in
fields like Radiology and Oncology, when a timely disease diagnosis is
essential [43]. Small lesions are primarily characterized by a limited size
and can vary greatly in nature depending on their location and the involved
tissue. In numerous real-world scenarios, the identification and
classification of small lesions is a challenging and critical diagnostic
process. For example, retinal microaneurysms are the earliest sign of diabetic
retinopathy and are caused by small local expansion of capillaries in the
retina [14]. In ischemic stroke imaging, early identification of small
occlusion is crucial to initiate timely treatment [58]. In cancer diagnosis,
many forms of cancer originate as small lesions before they grow and spread,
such as breast calcifications, which are one of the most important diagnostic
markers of breast lesions [46], or pulmonary nodules, which can be the first
stage of a primary lung cancer [44]. The ability to early and accurately
detect small lesions can make a difference in the treatment and prognosis of
patients and have a substantial impact on patient health. Manual
interpretation of medical images can be time-consuming and susceptible to
human error, especially when the task involves of localization and
identification of small lesions within the full image space [13].

There is a long tradition of research on automatic lesion detection methods
[55]. Traditional image processing methods, such as thresholding, edge
detection, and morphological operations, can be effective for detecting small
lesions in images with clear and well-defined structures. However, these
methods are limited by the presence of noise and variability in medical
images. The use of Machine Learning, and in particular Deep Learning, helps to
enhance reliability, performance, and accuracy of diagnosing systems for
specific diseases [54]. Actually, the first lesion detection system based on
Convolutional Neural Networks (CNNs) was proposed back in 1995 to detect lung
nodules in X-ray images [40]. However, only in the last ten years, CNNs have
acquired great popularity thanks to their remarkable performance in computer
vision [32], rapidly becoming the preferable solution for automated medical
lesion detection [59, 17, 21, 22]. The reason for this success is due to the
ability of learning hierarchical representations directly from the images,
instead of using handcrafted features based on domain-specific knowledge. CNNs
are able to build features with increasing relevance, from texture to higher
order features like local and global shape [33].

A typical CNN architecture for medical image analysis is applied to subparts
of an image containing candidate lesions or background. This means that the
image is divided into patches of equal size and partially overlapping, and
each patch is processed individually. The output image is formed by
reassembling the individually processed patches [25]. Despite patch-based
methods being widely used, they suffer from several problems, especially in
the case of small lesion detection [7], where accurate detections requires
both local information about the appearance of the lesion and global
contextual information about its location. This combination is typically not
possible in patch-based learning architectures [37], even with a multi-scale
approach where the appearance of a small lesion can be missed. An alternative
is to use anchoring object detection methods of computer vision [26], like
RetinaNet [36], which can be adapted to be used in lesion detection problems
[42]. These methods face difficulties when the objects to be detected are very
small, mainly for two reasons: (i) lesions have an extremely small size
compared to natural objects; (ii) lesions and non-lesions have similar
appearances, making it difficult to detect them effectively [6].

We propose a novel one-stage end-to-end detector based on a new type of
anchoring technique customised to small lesion detection in medical images.
Differently from classical anchor methods, which make use of anchor boxes to
capture scale and aspect ratio of specific classes of objects, the proposed
anchor is pixel-based and moves towards the lesion to be detected. We named
GravityNet this new architecture and gravity points the new anchors, because
they are distributed over the whole image and seem to be “attracted" by
hypothetical “gravitational masses" located in the centres of the lesions.
Such gravitational anchoring reveals to be particularly effective when small
lesions have to be detected in the whole image space. To evaluate the
performance of the proposed approach, we focused on two small lesions:
microcalcifications on digital mammograms and microaneurysms on digital fundus
images. In both cases, the lesions occupy only few pixels within an image,
resulting in limited features for them to be distinguished from the
surrounding tissues. Thus, their accurate localization becomes a main
challenge due to their appearance and to the heterogeneity of their contextual
backgrounds.

The paper is organized as follows. Section 2 is a brief overview of object
detection techniques in medical images and consequently of small lesion
detection methods. Section 3 introduces the proposed method. Section 4 reports
the experimental analysis, followed by results in Section 5. Finally, Sections
6 and 7 end the paper with discussion and conclusions.

Figure 1: Gravity-points distribution: on the left, the feature grid of size
$K\times K$; in the middle, the entire image $H\times W$; on the right, the
feature map $H_{FM}\times W_{FM}$.

##  2 Related work

###  2.1 Object detection in medical images

Object detectors can be divided into two categories: (i) two-stage detector,
the most representative is Faster R-CNN [15], (ii) one-stage detector, such as
YOLO [49], and SSD [39]. Two-stage detectors are characterized by high
localization and object recognition accuracy, whereas the one-stage detectors
achieve high inference speed [23, 67]. In a two-stage approach, the first
stage is responsible of generating candidates that should contain objects,
filtering out most of the negative proposals, whereas the second stage
performs the classification into foreground/background classes and regression
activities of the proposals from the previous stage.

Recently, the most popular object detection methods in computer vision have
been applied to medical imaging [68, 6, 21]. In [12], Faster R-CNN [15] is
applied with the VGG-16 [38] network as backbone for pulmonary nodule
detection. The YOLO architecture has been modified for lymphocyte detection in
immunochemistry [60, 50] and for pneumothorax detection on chest radiographs
[47]. In [48], a deep learning algorithm based on the YOLOv5 detection model
is proposed for automated detection of intracranial artery stenosis and
occlusion in magnetic resonance angiography. Other studies [29, 53, 42]
exploited architectures such as RetinaNet and Mask R-CNN for lung nodules and
breast masses localization. In [8], Mask R-CNN [19] is used by first assigning
bounding boxes for each tumor volume to perform detection and classification
of normal and abnormal breast tissue.

###  2.2 Small lesion detection

Although existing object detection models have been very successful with
natural images [23, 67], in medical images the high resolution makes the
problem particularly challenging to discover small lesions, requiring complex
architectures and the use of more than one stage for multi-resolution
detection. In [56], three CNN architectures, each at different scale, are
applied to lung nodule detection, whereas in [27] a multi-stream CNN is
designed to classify skin lesions, where each individual stream worked on a
different image resolution. In [64], a context-sensitive deep neural network
is developed to take into account both the local image features of a
microcalcification and its surrounding tissue background. In [52], a multi-
context architecture is proposed, based on the combination of different CNNs
with variable depth and individually trained on image patches of different
size. In [3], the problem of class imbalance between lesions and background is
addressed by proposing a two-stage deep learning approach where the first
stage is a cascade [2] of one-level decision trees, and the second is a CNN,
trained to focus on the most informative background configurations identified
by the first stage.

Recently, in [63] a hierarchical deep learning framework consisting of three
models each with a specific task is proposed for bone marrow nucleated
differential count. Some studies [10, 18] combined image processing techniques
and deep learning algorithms to evaluate lung tumor and liver tumor detection
respectively. In [66], the visibility of microcalcifications in mammographic
images is increased by difference filtering using the YOLOv4 model. A three-
stage multi-scale framework for the microaneurysms detection is designed in
[57], whereas multi-scale approach based on YOLOv5 is proposed for the
detection of stroke lesions [5].

Figure 2: GravityNet architecture is composed of a backbone (blue) and two
subnetworks, attached to the backbone output, one for classification task
(orange) and one for regression task (green). The output is a representation
of the gravity points in the grid pattern at training time and the subsequent
attraction behavior towards the lesion at inference time. Gravity points in
light blue correspond to positive candidates trained to collapse toward the
ground truth in light green

##  3 GravityNet

This section explains the proposed network architecture and the concept of
gravity points, a new anchoring technique designed for small lesion detection.

The code is available at this link.

###  3.1 Gravity points

We define a gravity point as a pixel-based anchor, which inspects its
surroundings to detect lesions. The gravity-points distribution is generated
with a grid of points spaced by a user-defined step parameter. A base
configuration is generated in a squared reference window, named feature grid,
of size $K\times K$ where $K$ is equal to the upper integer of the ratio
between the dimensions of the image and the feature map:

$K\times K=\Biggl{\lceil}\frac{H}{H_{FM}}\Biggl{\rceil}\text{ }\times\text{
}\Biggl{\lceil}\frac{W}{W_{FM}}\Biggl{\rceil}$ (1)

Assuming that the first gravity point is located in the upper left corner of
the feature map, the number of gravity points in a feature grid is equal to:

$N^{FG}_{GP}=\Biggl{(}\text{
}\Biggl{\lfloor}\frac{K-2}{step}\Biggl{\rfloor}\text{ }+1\Biggr{)}^{2}$ (2)

where $0<step\leq K-2$. In cases where $K-2$ is multiple of step the
distribution will be equispaced in the feature grid.

Since each pixel in the feature map corresponds to a feature grid in the
image, the complete configuration is obtained by sliding the base
configuration over the whole image. The total number of gravity points
$N_{GP}$ in the image is equal to the base configuration times the number of
feature grids:

$N_{GP}=N^{FG}_{GP}\cdot H_{FM}\cdot W_{FM}$ (3)

Fig. 1 shows an example of gravity-points distribution.

###  3.2 Architecture

GravityNet is a one-stage end-to-end detector composed of a backbone network
and two specific subnetworks. The backbone is a convolutional network and
plays the role of feature extractor. The first subnet performs convolutional
object classification on the backbone output, whereas the second subnet
performs convolutional gravity-points regression. Fig. 2 shows the overall
architecture.

The backbone is the underlying network architecture of a detection model and
provides a feature map containing basic features and representations of input
data, which are then processed to perform a specific task. The bottom layers
of a backbone net usually extract simple features such as edges and corners,
while the top layers learn more complex features like parts of lesions. The
feature maps generated by these layers are used as a representation of the
input image and fed into two models for classification and regression tasks.

The classification subnet is a fully convolutional network that outputs the
probability of lesion presence at each gravity-point location. The subnetwork
applies four $3\times 3$ convolutional layers, each with $256$ filters, where
the first one maps the number of features output from the backbone, followed
by ReLU activations. The last layer applies a filter with $N_{AP}\cdot 2$
outputs and sigmoid activation to obtain the binary predictions for each
gravity point.

The regression subnetwork is connected to the output of the backbone with the
purpose of regressing the offset from each gravity point to the closest
lesion. The design of the regression subnet is the same of the classification
subnet. The last layer outputs $N_{AP}\cdot 2$ values, indicating the offsets
to move each gravity point towards a lesion. It is worth noting that the
classification and regression subnets, though sharing a common structure, use
separate parameters.

###  3.3 Gravity loss

Gravity Loss (GL) is a multi-task loss that contains two terms: one for
regression (denoted as $GL_{reg}$) and the other for classification (denoted
as $GL_{cls}$).

The multi-task loss can be written as:

$GL=GL_{cls}+\lambda GL_{reg}$ (4)

where $\lambda$ is an hyperparameter that controls the balance between the two
task losses.

Figure 3: Hooking process where gravity points (light blue) are hooked to a
lesion (light green)

####  3.3.1 Classification loss

Since significant class imbalance between lesion and background is usually
present in medical images [3], the classification loss is a variant of Focal
Loss [36]. This loss is designed to address the issue of class imbalance in
object detection tasks, where the majority of the examples belong to the
negative class (e.g., background) and only a few examples belong to the
positive class (e.g., lesion).

The classification loss is defined as:

$GL_{cls}=-\alpha_{t}\cdot(1-p_{t})^{\varphi}\cdot\log(p_{t})$ (5)

where $p_{t}$ is the predicted probability of the true class (lesion),
$\varphi$ is a focusing parameter that controls the rate at which the
modulating factor $\alpha_{t}$ decreases as the predicted probability $p_{t}$
increases.

To evaluate $p_{t}$ with gravity points, we introduce a criterion based on the
Euclidean distance between the gravity points and the ground-truth
lesions111Without loss of generality, we consider as ground truth the center
of the smallest bounding box containing the lesion.. We consider as belonging
to the positive class those gravity points with a distance from the closest
ground-truth lesion lower than a threshold distance that we named hooking
distance $d_{h}$. All the gravity points within that distance are hooked to
the lesion and trained to move towards it. Fig. 3 shows an example of gravity-
points hooking process.

Figure 4: An example of NMS: on the left, gravity points and corresponding
boxes (light blue) hooked to a lesion (green); on the right, the final
candidate corresponding to the gravity point with the highest score (blue)

####  3.3.2 Regression loss

Let us indicate as $(d_{x},d_{y})$ the distance between a gravity point and
the relative hooked lesion, and as $(o_{x},o_{y})$ the output of the
regression subnetwork, which represents the offset to move each gravity point
towards the hooked lesion.

We evaluate the regression loss as:

$GL_{reg}=\sum_{\forall\text{ }hooked\text{
}GP}\sum_{i\in\\{x,y\\}}smooth_{L1}(d_{i}-o_{i})$ (6)

where $smooth_{L1}(t)$ is the Smooth L1 loss [15], defined as:

$smooth_{L1}(t)=\begin{cases}0.5t^{2},&\text{if }|t|<1\\\
|t|-0.5,&\text{otherwise}\end{cases}$ (7)

###  3.4 Inference time

The model produces two output predictions for each gravity point for each
subnetwork. Non-Maxima-Suppression (NMS) is applied to reduce the number of
false candidates (see Fig. 4): (i) an $L\times L$ box is built for each hooked
gravity points, where $L$ is chosen equal to the average size of the lesions
to be detected; (ii) all boxes with an Intersection over Union (IoU) greater
than 0.5 are merged; and (iii) for each merger, the gravity point with the
highest score is considered as final candidate. After NMS, we determine the
lesion class with a threshold $\gamma$ on the classification score: all
predictions with scores above $\gamma$ belong to the positive class (lesion),
the remaining ones to the negative class (no lesion).

##  4 Experiments

We proved the effectiveness of GravityNet on two detection problems in medical
image analysis: (i) microcalcifications on full field digital mammograms and
(ii) microaneurysms on digital ocular fundus images.

Microcalcifications (MCs) are calcium deposits and are considered as robust
markers of breast cancer [41]. MCs appear as fine, white specks, similar to
grains of salt, with size between $0.1$ $mm$ and $1$ $mm$. Due their small
dimensions and the inhomogeneity of the surrounding breast tissue, identifying
MCs is a very challenging task. Moreover, mammograms contain a variety of
linear structures (such as vessels, ducts, etc.) that are very similar to MCs
in size and shape, making detection even more complex.

Microaneurysms (MAs) are the earliest visible manifestation of Diabetic
Retinopathy, one of the leading causes of vision loss globally [34]. MAs are
described as isolated small red dots of $10$-$100$ $\mu m$ of diameter sparse
in retinal fundus images, but sometimes they appear in combination with
vessels. Retinal vessels, together with dot-hemorrhages and other objects like
the small and round spots resulting from the crossing of thin blood vessels,
make MAs hard to distinguish.

###  4.1 Dataset

####  4.1.1 Microcalcifications dataset

We used the publicly available INBreast database [45], acquired at the Breast
Centre in Centro Hospitalar de S. João (CHSJ) in Porto, Portugal. The
acquisition equipment was the MammoNovation Siemens FFDM, with a solid-state
detector of amorphous selenium, pixel size of $70$ $\mu m$ (microns) and
14-bit contrast resolution. The image matrix was $4,084\times 3,328$ ($243$
images) or $3,328\times 2,560$ ($167$ images), depending on the compression
plate used in the acquisition and according to the breast size of the patient.

The database has a total of $410$ images, amounting to 115 cases, from which
$90$ cases are from women with both breasts, and $25$ are from mastectomy
patients. Calcifications can be found in $313$ images for a total of $7,142$
individual calcifications. In this work, only calcifications with a radius of
less than $7$ pixels were considered for testing, for a total of $5,657$
microcalcifications identified in $296$ images.

Mammograms have been cropped to the size $3,328\times 2,560$ to have all
images in the dataset with equal size. We ensured that no MC was missed after
cropping.

####  4.1.2 Microaneurysms dataset

We used the publicly available E-ophtha database [11], designed for scientific
research in Diabetic Retinopathy. The acquired images have dimensions ranging
from $960\times 1,440$ to $1,696\times 2,544$ with a $45^{\circ}$ field of
view (FOV) and a pixel size of 7-15 $\mu m$. The database has a total of $381$
images: $148$ images from unhealthy patients containing $1,306$
microaneurysms, and $233$ images from healthy patients.

The original retinal fundus images are RGB, but in this work, green channel
has been extracted due to its rich information and high contrast in comparison
with the other two color channels [62]. We also evaluated the average
dimensions of all the retinas in the dataset and resized all the images to an
average dimensions of $1,216\times 1,408$.

Table 1: Data overview

INbreast Images Unhealthy MCs 1-fold 2-fold 1-fold 2-fold 1-fold 2-fold Train
143 143 108 117 2,408 2,051 Validation 62 62 39 45 516 724 Test 205 205 154
142 2,756 2,901 E-ophtha-MA Images Unhealthy MAs 1-fold 2-fold 1-fold 2-fold
1-fold 2-fold Train 154 151 60 60 542 552 Validation 38 38 14 14 105 107 Test
189 192 74 74 659 647

###  4.2 Data preparation

For all experiments we applied 2-fold image-based cross-validation. The
dataset is divided into two equal-sized folds, where one fold is used as
training set and the other as test set and vice versa. A subset of the
training set fold is used as validation set for parameter optimisation. See
Tab. LABEL:tab:DataOverview for more details.

In both datasets, data augmentation techniques are used to address class
imbalance and enhance the model robustness and accuracy [31]. Three more
samples for each image are generated by using horizontal and vertical
flipping. All data are normalized with min-max transformation.

|  |  |  |
---|---|---|---|---
(a) | (b) | (c) | (d) | (e)
Figure 5: Examples of initial gravity-points configurations represented in a
reference window $K\times K$, where: (a) step = 5 (b) step = 6 (c) step = 10
(d) step = 15 (e) step = 30

###  4.3 Architecture parameters

GravityNet uses ResNet [20] as its backbone, in order to solve the well-known
vanishing/exploding gradient problems [1] by using residual connections.
ResNet is composed of $5$ max-pooling layers, each halving the dimensions of
the feature map. According to the input dimensions, the feature map size is
$104\times 80$ for mammograms and $38\times 44$ for retina fundus images. As a
consequence, according to Eq. 1, we obtain $K=32$ and a feature grid of
$32\times 32$. We generate gravity-points configurations with step multiple of
$K-2$ to ensure equi-spatiality. To take into account the computational cost,
we chose to use configurations that did not exceed $300,000$ gravity points.
Fig. 5 shows some examples of the initial configurations used in this work. To
ensure that at least one gravity point hooks a lesion, the hooking distance
$d_{h}$ was always chosen equal to the step. At inference time, we use NMS
with $L=7$ for MCs and $L=3$ for MAs, which correspond to the average size of
the lesions to be detected.

We train ResNet in transfer learning by using a model pretrained on natural
images [65]. Both subnetworks are initialised with Xavier technique [16].
During training we apply Adam optimization algorithm [30]. The learning rate
was set to an initial value of $10^{-4}$ and decreased with a factor of $0.1$
with patience equal to $3$. The balance between the two task losses (see Eq.
4) is handled by $\lambda$ equal to $10$. The batch size is by default equal
to $8$. All training parameters were optimized on the validation set. The
training was stopped after $50$ epochs. Experiments were conducted on a GPU
NVIDIA A100 80GB.

###  4.4 FROC analysis

The detection quality was evaluated in terms of lesion-based Free-Response
Operating Characteristic (FROC) curve by plotting True Positive Rate (TPR)
against the average number of False Positives per Image (FPpI) for a series of
thresholds $\gamma$ on the classification score associated to each sample.

A prediction with a value higher than $\theta$ is considered as True Positive
(TP) when its distance from the center of a lesion is no larger than the
largest side of the bounding box containing the ground-truth lesion.
Otherwise, it is considered as False Positive. Notably, (i) if multiple
predictions are associated to the same lesion, only the one with the highest
classification score is selected as TP, and (ii) all predictions for gravity
points outside the tissue were ignored.

To analyze and compare FROC curves, we chose the non-parametric approach
suggested in [4]. The figure-of-merit is the Partial Area under the FROC curve
($AUFC_{\gamma}$) to the left of $FPpI=\gamma$ calculated by trapezoidal
integration. We normalized $AUFC_{\gamma}$ by dividing with $\gamma$ to obtain
an index in the range $[0,1]$. In particular, for both MCs and MAs detection,
we selected $\gamma=10$, a commonly used value in the literature of the
respective fields [9, 52]. All results are presented in percentage values.

##  5 Results

###  5.1 Model analysis

To verify the effectiveness of the model, for both small lesion detection
problems, experiments were conducted using different gravity-points
configurations for all different depths of ResNet222It is worth noting that,
due to memory constraints, for MCs detection, we use in training a batch size
equal to $4$ for ResNet-50 and $2$ for ResNet-101 and ResNet-152. Results are
reported in Tab. LABEL:tab:ResultsMCs for MCs, and in Tab.
LABEL:tab:ResultsMAs for MAs together with the parameters of the gravity-
points configurations. The best result for each backbone is shown in bold,
whereas the best of all in italic. FROC curves of the best ResNet
configurations are shown in Fig. LABEL:fig:FROC_Results.

For MCs, the best result is a $AUFC_{\gamma}$ equal to $72.25\%$ by using
ResNet-34 and step $10$. Configuration with step $10$ turns to be the best for
all backbones, except ResNet-50, which achieves a $AUFC_{\gamma}$ equal to
$71.25\%$ with step $6$. Dense configurations present better results with
shallower backbones, e.g. with a ResNet-18 we obtain a $AUFC_{\gamma}$ equal
to $70.89\%$ and $71.47\%$ respectively with step $6$ and $10$ as opposed to
$65.58\%$ and $55.90\%$ respectively with step $15$ and $30$.

For MAs, the highest $AUFC_{\gamma}$ ($71.53\%$) is obtained with a ResNet-50
and step $6$. Configuration with step $6$ turns to be the best for all
backbones, except ResNet-18, which achieves a $AUFC_{\gamma}$ equal to
$65.36\%$ with step $10$. Sparse configurations decrease the performance, even
with deeper backbones, e.g. with a ResNet-152 we obtain a $AUFC_{\gamma}$ of
$67.51\%$ and $54.18\%$ respectively with step $15$ and $30$ as opposed to
$65.81\%$ and $69.86\%$ respectively with step $5$ and $6$.

Through result analysis, it becomes evident that we need to find the
appropriate density configuration for addressing the detection problem at
hand. A sparse configuration might fail to identify all lesions, particularly
in the case of small ones, whereas a dense configuration could potentially
generate a high number of lesion candidates.

###  5.2 Comparison with the literature

We compare our best models, that are ResNet-34 with step $10$ for MCs
detection and ResNet-50 with step $6$ for MAs detection, with methods proposed
in the scientific literature for the detection problems at hand:

  * -

Context-Sensitive CNN (CSNet) [64]: the architecture comprises two
convolutional subnetworks: one for processing the large image context with a
window of size $96\times 96$ pixels and another for processing the small
microcalcification texture with a window of size $9\times 9$ pixels. The
features extracted from both subnetworks are subsequently merged and fed into
a fully connected network.

  * -

Deep Cascade (DC) [2]: a cascade of decision stumps able to learn effectively
from heavily class-unbalanced datasets. It builds on Haar features computed in
a small detection window of $12\times 12$ pixels, which can contain
diagnostically relevant lesions, while limiting the exponential growth of the
number of features that are extracted during training.

  * -

MCNet with DC hard mining (DC-MCNet) [3]: a two-stage patch-based deep
learning framework, which comprises a DC for hard mining the background
samples, followed by a second stage represented by a CNN that discriminates
between lesions and the more challenging background configurations.

  * -

Multicontext Ensemble of MCNets (ME-MCNet) [52]: a multi-context ensemble of
CNNs aiming to learn different levels of image spatial context by training
multiple-depth networks on image patches of different dimensions ($12\times
12$, $24\times 24$, $48\times 48$, and $96\times 96$).

To evaluate the behavior of the proposed anchoring technique, we also compared
with RetinaNet [36], a well-known one-stage object detector based on anchoring
technique. We slightly modified the original anchors configuration by using an
anchor box size ranging from $8^{2}$ to $128^{2}$ in order to be more suitable
for small lesion detection. We applied it to the whole image without any kind
of rescale or patching.

We applied a statistical comparison by means of bootstrap method [51] to test
the significance of observed performances. Cases were sampled with replacement
$1,000$ times, with each bootstrap containing the same number of cases as the
original set. At each bootstrapping iteration, FROC curves were recalculated,
and differences in figures-of-merit $\Delta AUFC_{\gamma}$ between GravityNet
and each of the compared methods were evaluated. Finally, the obtained FROC
curves were averaged along the TPR axis, and $p$-values were computed as the
fraction of $\Delta AUF_{\gamma}$ populations that were negative or zero. The
statistical significance level was chosen as $\alpha=0.05$. Average FROC curve
are shown in Fig. LABEL:fig:FROC_ResultsComparison.

The statistical comparison results for MCs and MAs detection are shown in Tab.
LABEL:tab:ComparisonResults where significant performances are indicated in
bold. Results of the proposed architecture were statistically significantly
better than all the other considered approaches. The highest improvement in
terms of $AUFC_{\gamma}$ is of $+50.04\%$ with RetinaNet for MAs and of
$+42.25\%$ with CSNet for MCs. Compared to patch-based methods such as DC, DC-
MCNet and ME-MCNet the improvement is respectively $+41.00\%$, $+19.52\%$,
$+11.90\%$ for MCs and $+15.72\%$, $+10.15\%$, $+5.90\%$ for MAs.

##  6 Discussion

###  6.1 Gravity points configuration

The gravity points configuration depends directly on the size of the input
image and is managed by the step parameter. This implies a higher number of
gravity points for images with larger dimensions. In the cases studied,
mammograms have a larger size than retina images and consequently have a
higher $N_{GP}$, so requiring much more computational efforts.

Depending on the chosen configuration, gravity-points will behave differently.
We chose to train all the configurations with a $d_{h}$ equal to the step to
measure the capacity of gravity-points to move towards ground-truth lesions. A
small $d_{h}$ will have less impact on the movement of gravity-points,
compared to a large $d_{h}$ that let them move more widely, always within the
specified distance value. For MCs detection, the best configurations are those
with step $10$ and thus $d_{h}$ $10$ because these values are more
representative of the size and distribution of MCs in mammographies. On the
other hand, for MAs detection, where lesions are usually isolated,
configurations with a higher density, such as step $5$ and step $6$, are
needed. Fig. LABEL:fig:output shows two detection outputs of the best
GravityNet models for MCs with step $10$ and ResNet-34 and for MAs with step
$6$ and ResNet-50. We can see the gravitational behaviour towards the centres
of the lesions in Fig. LABEL:subfig:output_MCs for MCs and Fig.
LABEL:subfig:output_MAs for MAs. Hooked gravity-points that are, at inference
time, within the radius of the lesion to be detected are shown in light blue
and are defined as predictions of possible TP. The NMS, whose output can be
seen in the right panel of the same figures, merges all hooked gravity-points
in a single detection so as to obtain a single prediction (in blue) for each
small lesion (in green).

###  6.2 Comparison with anchoring methods

We compared the proposed one-stage detector with a widespread exponent of one-
stage object detection methods, i.e. RetinaNet, which has also been usefully
applied to medical detection problems [24, 42]. Small lesions such as MCs and
MAs are often less than 10 pixels in diameter and, in this case, anchoring
methods face two main obstacles: (i) the number and size of anchor boxes, and
(ii) the pyramidal approach for multi-scale resolution.

Regarding the first issue, we tried to train RetinaNet with the original range
of anchor boxes size (from $32^{2}$ to $512^{2}$ according to the Feature
Pyramid Network (FPN) level), but due to the small size of the lesions, the
train failed; thus, we reduced the size in the range $8^{2}$ to $128^{2}$. The
proposed anchoring technique is based on pixel-shaped gravity points, which
only require an initial configuration setting without specifying a box size.
This is advantageous, especially in the case of small lesions with variable
sizes, as demonstrated in the MA results. In addition, considering all FPN
resolution levels, RetinaNet generates a number of anchor boxes more than 10
times the number of gravity points. This is a considerable advantage in
computational and temporal terms (see Section 6.4).

As to the second issue, RetinaNet adopts a multi-scale architecture. However,
this approach proves to be ineffective because positive anchors (those
containing a lesion) only belong to the first level of FPN, which corresponds
to the highest resolution level. In GravityNet, we decided to not use a multi-
scale approach given the shape of the lesions to be detected. For the sake of
comparison, we tried to use RetinaNet without FPN, considering only the
outputs of the first level, but this did not improve the performance.

###  6.3 Comparison with patch-based methods

Patch-based methods have the computational disadvantage of assembling all the
individual results to obtain the final one, as opposed to end-to-end systems
like GravityNet that obtain the final result directly.

The class imbalance between lesions and background is another issue that
affects small lesion detection. We compared our approach with two existing
methods, DC and DC-MCNet, which are designed to manage this problem. DC
discards the majority of easily detectable background samples early in the
process, while DC-MCNet utilizes a CNN on the output of DC to enhance
detection performance. In this work, we propose the Gravity Loss, a variant of
Focal Loss typically applied in deep learning methods to address class
imbalance issues.

Since small lesions do not have a clear appearance and are similar to the
surrounding background, CSNet and ME-MCNet propose two context-sensitive
patch-based approaches, where the model is trained with patches of different
sizes and then combined. In contrast, our proposal works with the full image
without patches and is able to identify small lesions thanks to the new
anchoring technique and the regression subnet, which focus more on the
distance to the lesion rather than its appearance.

###  6.4 Computational and inference time

Computational and inference time are an important aspect in medical imaging
systems, improving interactivity and the time taken to formulate a diagnosis.
We evaluated the computational time for all the compared methods by measuring
the average Time per Epoch (TpE) in training, and the Time per Image (TpI) and
the Throughput333Throughput is defined as the maximum number of input
instances that the method can process in one second in test. Tab.
LABEL:tab:TimeComparisonResults shows the results. We can see how patch-based
methods are computationally time-consuming, whereas our proposal has a very
high Throughput and an average TpI below one second.

Table 2: Computational times compared in terms of Time per Epoch (TpE) in
training, and Time per Image (TpI) and Throughput in test

Method TpE (s) TpI (s) Throughput MCs detection RetinaNet 1254 0.121 14.70
CSNet 6959 822 $1.2\times 10^{-3}$ DC n.a. 1.1 0.9 DC-MCNet 32 1.2 0.8 ME-
MCNet 5360 386 $3.8\times 10^{-3}$ GravityNet 607 0.061 19.25 MAs detection
RetinaNet 137 0.057 34.04 CSNet 1260 266 $3.8\times 10^{-3}$ DC n.a. 1.3 0.7
DC-MCNet 6 1.4 0.7 ME-MCNet 1564 203 $4.9\times 10^{-3}$ GravityNet 184 0.045
37.49

###  6.5 Limitations

Although our method achieves excellent results in the detection of small
lesions, there are some limitations to be considered:

  * -

Clinical applicability: we require a dataset with individually annotated
lesions for the training phase, and this can be difficult to meet in a real
clinical scenario. In addition, further post-processing (e.g. benign vs.
malignant lesion classification) is needed to build a full CAD system.

  * -

Configuration limit: by employing an equispaced grid configuration, the
distribution of gravity points becomes uniform, even in areas of the image
where there is no tissue. In training this might not be advantageous.
Different approaches to generate the initial configuration can be
investigated.

  * -

Computational requirements: the number of gravity points directly increases
with the size of the image. In case of large images, GravityNet can require
considerable computational resources. A solution can be to limit the number of
gravity points by using sparse initial configuration, but this can affect the
detection performance of the method.

  * -

Memory constraints: the use of a backbone in the proposed model necessitates
remarkable resource requirements. As the backbone architecture becomes more
complex and deeper, it requires a larger memory allocation, which can be a
significant limitation for training the model.

##  7 Conclusions and future work

In this work, we introduced GravityNet, a new one-stage end-to-end detector
specifically designed to detect small lesions in medical images. The accurate
localization of small lesions, given their appearance and diverse contextual
backgrounds, is a challenge in several medical applications. To address this
point, our approach employed a novel pixel-based anchor that dynamically moves
towards the targeted lesion during detection. Through a comparative evaluation
with state-of-the-art anchoring and patch-based methods, our proposed approach
demonstrated promising results in effectively detecting small lesions.

Our primary future direction will involve testing GravityNet in various
detection problems, particularly those where the target object is point-like,
such as nuclei localization in whole-slide images [22]. We will also explore
the possibility of extending the proposed architecture to address other tasks
or image dimensionality involving small lesions, such as segmentation [35] or
three-dimensional images [28, 61].

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