# Anabranch Network for Camouflaged Object Segmentation

Trung-Nghia Le Tam V. Nguyen Zhongliang Nie Minh-Triet Tran Akihiro Sugimoto
Department of Informatics, The Graduate University for Advanced Studies
(SOKENDAI), Tokyo, Japan  Department of Computer Science, University of
Dayton, Ohio, 45469, United States of America  University of Science, VNU-HCM,
Ho Chi Minh, Vietnam  National Institute of Informatics, Tokyo, Japan

###### Abstract

Camouflaged objects attempt to conceal their texture into the background and
discriminating them from the background is hard even for human beings. The
main objective of this paper is to explore the camouflaged object segmentation
problem, namely, segmenting the camouflaged object(s) for a given image. This
problem has not been well studied in spite of a wide range of potential
applications including the preservation of wild animals and the discovery of
new species, surveillance systems, search-and-rescue missions in the event of
natural disasters such as earthquakes, floods or hurricanes. This paper
addresses a new challenging problem of camouflaged object segmentation. To
address this problem, we provide a new image dataset of camouflaged objects
for benchmarking purposes. In addition, we propose a general end-to-end
network, called the Anabranch Network, that leverages both classification and
segmentation tasks. Different from existing networks for segmentation, our
proposed network possesses the second branch for classification to predict the
probability of containing camouflaged object(s) in an image, which is then
fused into the main branch for segmentation to boost up the segmentation
accuracy. Extensive experiments conducted on the newly built dataset
demonstrate the effectiveness of our network using various fully convolutional
networks.

###### keywords:

Camouflaged object segmentation, Anabranch Network

††journal: Computer Vision and Image Understanding

##  1 Introduction

Fig. 1: A few examples from our Camouflaged Object (CAMO) dataset with
corresponding pixel-level annotations. Camouflaged objects attempt to conceal
their texture into the background.

Camouflage is an attempt to conceal the texture of a foreground object into
the background (Singh et al., 2013). The term “camouflage” was first coined
from nature where animals used to hide themselves from predators by changing
their body pattern, texture, or color. Unlike salient objects, camouflaged
objects are hard to detect in their nature even for human beings. Autonomously
detecting/segmenting camouflaged objects is thus a challenging task where
discriminative features do not play an important role anymore since we have to
ignore objects that capture our attention. While detecting camouflaged objects
is technically difficult on one hand, it is beneficial in various practical
scenarios, on the other hand, which include surveillance systems, and search-
and-rescue missions. For example, detecting and segmenting camouflaged objects
in images definitely helps us search for camouflaged animals for the
preservation of wild animals and the discovery of new species. Nevertheless,
camouflaged object segmentation has not been well explored in the literature.

There exist two types of camouflaged objects in reality: naturally camouflaged
objects such as animals or insects hiding themselves from their predators, and
artificially camouflaged objects such as soldiers and weapons disguised by
artificial texture patterns. These objects first evaluate their surrounding
environment and then change to camouflaged textures via the colorization of
cloths or cover. For either type of camouflaged objects, it is not obvious to
identify them in images. Figure 1 shows a few examples of camouflaged objects
in real life. Based on this figure, we can easily see how challenging
camouflaged object segmentation is.

Camouflaged objects have naturally evolved to exploit weaknesses in the visual
system of their prey or predator, and thus understanding such a mechanism will
provide some insights for segmenting camouflaged objects. Without any prior,
we human beings easily miss detecting camouflaged objects; however, once we
are informed a camouflaged object exists in an image, we can carefully scan
the entire image to detect it. This also comes up in nature: once the predator
has an awareness of camouflaged animals in the scene, then it makes efforts to
localize them for hunting. Therefore, utilizing the awareness as a prior can
be an essential cue for camouflaged object segmentation. Then, the challenge
is what awareness should be incorporated in what way. We consider that
classification scheme can be used as the awareness of existing of camouflaged
objects in an image in order to combine with detection/segmentation scheme.

As reviewed in Singh et al. (Singh et al., 2013), there are a few methods that
study camouflaged object detection in simple contexts. These methods use hand-
crafted low-level features such as color, edge or texture to detect
camouflaged objects. Such hand-crafted low-level features are designed to be
as much discriminative as possible for detecting and segmenting objects, which
is the opposite way to camouflage. Therefore, their performances are limited
to camouflaged object segmentation. Furthermore, these methods can be applied
only to relatively low-resolution images with a uniform background. Deep
features, on the other hand, are known to outperform low-level features in
many tasks in computer vision such as image classification (Krizhevsky et al.,
2012), image semantic segmentation (Shelhamer et al., 2016), action
recognition (Tran et al., 2015), and face recognition (Wen et al., 2016). We
thus expect that deep features can replace low-level features even for
camouflaged object segmentation by incorporating them into a new network. To
use them, however, a large number of data of camouflaged objects are required.
Nevertheless, there is no public dataset (both for training data and testing
data) for the camouflaged object segmentation problem. This obviously poses
serious problems to (1) train deep learning models and (2) evaluate the
performance of the proposed network.

The overall contribution of this paper is two-fold:

  * •

We provide a new image dataset of camouflaged objects to promote new methods
for camouflaged object segmentation. Our newly constructed Camouflaged Object
(CAMO) dataset consists of 1250 images, each of which contains at least one
camouflaged object. Pixel-wise ground-truths are manually annotated to each
image. Furthermore, images in the CAMO dataset involve a variety of
challenging scenarios such as object appearance, background clutter, shape
complexity, small object, object occlusion, multiple objects, and distraction.
We emphasize that this is the very first dataset for camouflage segmentation.

  * •

We propose a novel end-to-end network, called the Anabranch111An anabranch is
a section of a river or stream that diverts from the main channel or stem of
the watercourse and rejoins the main stem downstream. Network (ANet), which is
general, conceptually simple, flexible, and efficient for camouflaged object
segmentation. Our proposed ANet leverages the advantages of different network
streams, namely, the classification stream and the segmentation stream, in
order to segment camouflaged objects. The classification stream is used as the
awareness of existing of camouflaged object(s) in an image. This design is
motivated by the observation that there is no guarantee that a camouflaged
object is always present in an image and thus we first identify whether a
camouflaged object is present in an image and then only if exists, the
object(s) should be accurately segmented; nothing is segmented otherwise.
Extensive experiments verify the effectiveness of our proposed ANet when the
network is applied on various fully convolutional networks (FCNs) for
camouflaged object segmentation.

The datasets, evaluation scripts, models, and results are publicly available
at our websites
222https://sites.google.com/view/ltnghia/research/camo,333https://sites.google.com/site/vantam/research

The remainder of this paper is organized as follows. Section 2 summarizes the
related work. Next, Section 3 and Section 4 introduce the constructed dataset
and the proposed network, respectively. Section 5 then reports extensive
experiments over the proposed method and baselines on the newly constructed
dataset. Section 5.5.2 discusses the joint training of ANet. Finally, Section
6 draws the conclusion and paves the way for the future work.

##  2 Related Work

This section first reviews existing works on camouflaged object segmentation
and salient object segmentation, which share some similarities. We also
clarify that the camouflaged object segmentation is more challenging than the
salient object segmentation. Then, we briefly introduce some advancements of
two-stream networks in several computer vision tasks.

###  2.1 Camouflaged Object Segmentation

As addressed in the intensive survey compiled by Singh et al. (Singh et al.,
2013), most of the related work (Kavitha et al., 2011; Pan et al., 2011;
Siricharoen et al., 2010; Song and Geng, 2010; Yin et al., 2011) uses
handcrafted, low-level features for camouflaged region detection/segmentation
where the similarity between the camouflaged region and the background is
evaluated using low-level features, e.g., color, shape, orientation, and
brightness (Galun et al., 2003; Song and Geng, 2010; Xue et al., 2016). We
note that there is little work that directly deals with camouflaged object
detection; most of the work is dedicated to detecting the foreground region
even when some of its texture is similar to the background. Pan et al. (Pan et
al., 2011) proposed a 3D convexity based method to detect camouflaged objects
in images. Liu et al. (Liu et al., 2012) integrated the top-down information
based on the expectation maximization framework for foreground object
detection. Sengottuvelan et al. (P. Sengottuvelan and Shanmugam, 2008) applied
a gray level co-occurrence matrix method to identify the camouflaged object in
images with a simple background. In the case where some parts of a moving
object and the background share the similar texture, optical flow is combined
with color to detect/segment the moving object (Yin et al., 2011; Gallego and
Bertolino, 2014). All these methods use handcrafted low-level features and
work for only a few cases where images/videos have the simple and non-uniform
background. Their performances are also unsatisfactory in camouflage
detection/segmentation when there is a strong similarity between the
foreground and the background.

One of the main reasons why there has been very little work on camouflaged
object segmentation is the lack of the standard dataset for this problem. A
benchmarking dataset is thus mandatory to develop advanced techniques for
camouflaged object segmentation. Note that the dataset has to include both
training data and testing data. To our best of knowledge, there is so far no
standard dataset for camouflage object segmentation, which has the sufficient
number of data for training and evaluating deep networks. A few of datasets
related to camouflaged animals have been proposed, but they have a limited
number of samples. Bideau et al. (Pia Bideau, 2016) proposed a camouflaged
animal video dataset but the dataset has only 9 videos, and camouflaged
animals really exist in a third of videos. The unpublished dataset proposed by
Skurowski et.al.444http://zgwisk.aei.polsl.pl/index.php/en/research/other-
research/63-animal-camouflage-analysis has only 76 images of camouflaged
animals. Therefore, we collect a new camouflaged object dataset. Note that our
work is the first work solve the real camouflage segmentation problem in
diverse images. Different from the two above datasets, our constructed dataset
consists of both camouflaged animals and human.

###  2.2 Salient Object Segmentation

Salient object predictors aim to detect and segment salient objects in images.
Though “saliency” is opposed to “camouflage”, techniques developed for salient
object segmentation may be useful for camouflaged object segmentation. This is
because the two tasks highlight image regions with certain characteristics.

Early work on salient object segmentation is based on biologically-inspired
approaches (Itti et al., 1998; Koch and Ullman, 1987) where used features are
the contrast of low-level features such as orientation of edges, or direction
of movement. Since human vision is sensitive to color, different approaches
using color were proposed where the contrast of color features is locally or
globally analyzed (Achanta et al., 2009; Cheng et al., 2011). Local analysis
methods estimate the saliency of a particular image region against its
neighborhoods based on color histogram comparison (Cheng et al., 2011); global
analysis methods achieve globally consistent results by computing color
dissimilarities to the mean image color (Achanta et al., 2009). Various patch-
based methods were also proposed that estimate dissimilarity between image
patches (Cheng et al., 2011; Margolin et al., 2013; Zhang et al., 2015). To
leverage the advantage of deep learning, recent methods first conduct
superpixel segmentation and then feed segmented regions into a convolutional
neural network (CNN) individually to obtain saliency scores (Le and Sugimoto,
2018; Li and Yu, 2015; Wang et al., 2015). More recent methods modified fully
convolutional networks (FCNs) to compute a pixel-wise saliency map (Hou et
al., 2017; Le and Sugimoto, 2017; Li et al., 2017a; Liu and Han, 2016; Wang et
al., 2017b, a). Skip-layer structures are usually employed in FCNs to obtain
multi-scale feature maps, which are critically needed to segment a salient
object (Le and Sugimoto, 2017; Liu and Han, 2016; Hou et al., 2017). Li et al.
(Li and Yu, 2016) modified the FCN to combine pixel-wise and segment-wise
feature maps for saliency value prediction. Wang et al. (Wang et al., 2017b)
integrated a pyramid pooling module and a multi-stage refinement mechanism
into an FCN for salient object segmentation. Wang et al. (Wang et al., 2017a)
proposed a two-stage training method to compute saliency maps using image-
level tags. Object proposal based networks, which are originally developed for
category-specific object detection, are also utilized to identify salient
objects (Han et al., 2018b; Le and Sugimoto, 2019). In addition, salient
objects from common categories can be co-segmented by exploiting
correspondence relationship among multiple relevant images (Han et al., 2018a;
Yao et al., 2017).

|  |
---|---|---
(a) Camouflaged object |  (b) Ambiguous |  (c) Salient object
Fig. 2: Ambiguous differences between camouflaged objects and salient objects.

It is worth noting that camouflaged object segmentation is more challenging
than salient object segmentation. In fact, salient objects tend to be
outstanding and discriminative from the background. We thus only need to focus
on identifying such outstanding and discriminative regions to segment salient
objects. Camouflaged objects, however, tend to conceal themselves into the
background environment by decreasing discriminativeness as much as possible.
This makes it hard to identify boundaries of camouflaged objects. In
particular, when the background is cluttered, camouflaged objects blend with
the background too much, and discriminating them from the background becomes
even harder.

We remark that the same object can be either a camouflaged object or a salient
object, meaning that the differences between camouflaged objects and salient
objects are ambiguous (cf. Fig. 2). This suggests that some salient object
detection methods may work for the camouflaged object segmentation task;
however, this is not the case. Indeed, existing methods on salient object
segmentation are designed to segment objects for any reason because a salient
object is assumed to always exist in a training/testing image. On the
contrary, there is no guarantee that a camouflaged object is always present in
an image. The approach to camouflaged object segmentation should be completely
different so that it allows us to handle practical scenarios where we do not
know whether or not a target object exists.

###  2.3 Two-Stream Network

There are several works that use two-stream architectures (Simonyan and
Zisserman, 2014a; Feichtenhofer et al., 2016; Wang et al., 2016; Jain et al.,
2017; He et al., 2017; Li et al., 2017b). Fusion networks were proposed for
action recognition (Simonyan and Zisserman, 2014a; Feichtenhofer et al., 2016;
Wang et al., 2016) and object segmentation (Jain et al., 2017) in videos. The
network consists of an appearance stream and a motion stream, and the final
result is fused from the outputs of the two streams. Two streams, however,
have the same architecture for the same task despite the fact that they
receive different inputs (i.e., video frame and optical flow). Two streams
(i.e., a classification stream and a segmentation stream) of region-based
networks for instance segmentation (He et al., 2017; Li et al., 2017b) have
different architectures for different tasks. The two streams are built up from
the region proposal network (Ren et al., 2015) and then work on specific
regions.

In contrast with the above mentioned two-stream networks, the two streams (the
classification stream and the segmentation stream) of our proposed ANet have
different architectures mutually reinforce each other on the whole image for
different tasks. The output of the classification stream is fused with that of
the segmentation stream to enhance segmentation result. In this sense, the
role of the classification stream is complementary to that of the segmentation
stream.

##  3 Camouflaged Object Dataset

###  3.1 Motivation

In the literature, no dataset is available for camouflaged object
segmentation. To promote camouflaged object segmentation, a publicly available
dataset with pixel-wise ground-truth annotation is mandatory. Note that the
dataset has to include both training data and testing data. The training data
is suitable for training a deep neural network whereas the testing data is
used for evaluation. Therefore, we aim to construct a dataset, Camouflaged
Object (CAMO) dataset, to promote advancements in camouflaged object
segmentation and its evaluation. We stress that our CAMO dataset is the very
first dataset for camouflaged object segmentation.

###  3.2 Dataset Construction

To build the dataset, we initially collected 3000 images in which at least one
camouflaged object exists. We collected the images of camouflaged objects from
the Internet with keywords: “camouflaged objects”, “camouflaged animals”,
“concealed objects”, “hidden animals”, “camouflaged soldier”, “human body
painting”, etc. Note that we consider both types of camouflaged objects,
namely, natural and artificial camouflage as mentioned in Section 1. Then, we
manually discarded images with low resolution or duplication. Finally, we
ended up with 1250 images. To avoid inconsistency in labeling ground-truth, we
asked three people to annotate camouflaged objects in all 1250 images
individually using a custom designed interactive segmentation tool. On
average, each person takes 2-5 minutes to annotate one image depending on its
complexity. The annotation stage spanned about one month.

###  3.3 Dataset Description

Fig. 3: Categories in Camouflaged Object Dataset. Natural camouflaged objects
are shown in red while artificial camouflaged objects are shown in blue.

(a) Examples of challenging attributes

(b) Attribute distribution over the CAMO dataset

Fig. 4: Some examples of challenging attributes and the attribute distribution
over the CAMO dataset.

We describe in this section the Camouflaged Object (CAMO) dataset specifically
designed for the task of camouflaged object segmentation. Some examples are
shown in Fig. 1 with the corresponding ground-truth label annotations. We
focus on two categories, i.e., naturally camouflaged objects and artificially
camouflaged objects, which usually correspond to animals and humans in the
real world, respectively. Camouflaged animals consist of amphibians, birds,
insects, mammals, reptiles, and underwater animals in various environments,
i.e., ground, underwater, desert, forest, mountain, and snow. Camouflaged
human falls into soldiers on the battlefields and human body painting arts.
The ratios for each category are shown in Fig. 3.

In our CAMO dataset, it is also noteworthy that multiple objects, including
separate single objects and spatially connected/overlapping objects, possibly
exist in some images ($8\%$). This also makes our dataset more challenging for
camouflaged object segmentation. The challenge of the CAMO dataset is also
enhanced due to some attributes, i.e. object appearance, background clutter,
shape complexity, small object, object occlusion, and distraction (cf. Fig.
4):

  * •

Object appearance: The object has a similar color appearance with the
background, causing large ambiguity in segmentation.

  * •

Background clutter: The background is not uniform but contains small-scale
structures or is composed of several complex parts.

  * •

Shape complexity: The object has complex boundaries such as thin parts and
holes, which is usually the case for the legs of insects.

  * •

Small object: The ratio between the camouflaged object area and the whole
image area is smaller than 0.1.

  * •

Object occlusion: The object is occluded, resulting in disconnected parts or
in touching the image border.

  * •

Distraction: The image contains distracted objects, resulting in losing
attention to camouflaged objects. This makes camouflaged objects more
difficult to discover even by human beings.

##  4 Anabranch Network

In this section, we give a detailed description of our Anabranch Network
(ANet). Specifically, we first discuss the motivation of network design. Then,
we introduce the architecture of ANet.

###  4.1 Network Design

As mentioned above, there is no guarantee that a camouflaged object is always
present in the scene. Therefore, a method that systematically segments objects
for any image will not work. Moreover, directly applying discriminative
features from segmentation models (i.e., semantic segmentation and salient
object segmentation, etc.) to camouflaged object segmentation is not effective
because camouflaged objects conceal their texture into the surrounding
environment. In order to segment camouflaged objects, we need an additional
function that identifies whether a camouflaged object exists in an image. To
do so, each pixel needs to be classified as a part of camouflaged objects or
not. Such classification can be used not only to enhance segmentation accuracy
but also to segment multiple camouflaged objects. Indeed, this classification
only strengthens features extracted from the camouflaged part and weakens
features extracted from a non-camouflaged part. Accordingly, segmentation and
classification tasks should be closely combined in the network with different
architectures for camouflaged object segmentation.

###  4.2 Network Architecture

Fig. 5: Overview of our proposed Anabranch Network (ANet). The proposed
network leverages the strength of both image classification and semantic
segmentation tasks for camouflaged object segmentation.

Figure 5 depicts the overview of our proposed ANet, a general network for
camouflaged object segmentation. The ANet leverages the advantages of the two
different network models, namely, the classification network model based on a
convolutional neural network (CNN) and the segmentation network model based on
a fully convolutional network (FCN). The CNN reveals object hypotheses in the
image (Zhou et al., 2014), yielding a cue on whether or not the input image
contains camouflaged object(s). We thus aim to train the CNN model so that it
classifies two classes: camouflaged image class and non-camouflaged image one.
The FCN, on the other hand, provides us with the pixel-wise semantic
information in the image. We consider the output of the FCN as the semantic
information of different objects in the image. The two outputs from the two
network streams are finally fused to produce a pixel-wisely accurate
camouflage map which uniformly covers camouflaged objects. We note that our
classification stream output, i.e., the probability (scalar), is multiplied
($\otimes$) with each pixel value of 2D map produced by the segmentation
stream.

Segmentation Stream: The input and the output of the segmentation stream are
both images, and, thus, any end-to-end FCN for segmentation can be employed.
In the sense that we can employ any FCN for segmentation, ANet can be
considered as a general network.

Though, as addressed above, methods for salient object detection alone may not
work for accurate camouflaged object segmentation, effective compensation by
the classification stream can be expected to boost up the performance in the
ANet. Recent state-of-the-art methods for salient object detection have
demonstrated their good performances. In addition, as we observed in Fig. 2,
we have ambiguity in difference between camouflaged objects and salient
objects. Based on these reasons, we consider a saliency model is suitable to
use into ANet. We can easily swap the model with any other saliency model. We
remark that it is also interesting to utilize saliency models in a similar but
different domain such as camouflaged object segmentation.

Fig. 6: Examples of non-camouflaged objects from MS-COCO.

Classification Stream: We build the classification stream on top of
convolution layers, which play as feature extraction module, of FCNs. In
particular, we use three fully connected layers with 4096, 4096, and 2
filters, respectively (cf. Fig. 5 and Table 1). We note that each of the first
two fully connected layers is followed by a Rectified Linear Unit (ReLU)
activation layer (Krizhevsky et al., 2012). The last layer is the soft-max
layer. To train this stream, we use soft-max with the cross-entropy loss.
During the training process, we use dropout layers with a dropout rate of
$50\%$ after the first two fully connected layers to avoid over-fitting.

Table 1: Architecture of the classification stream. No | Layer | Output | Dropping Rate
---|---|---|---
1 | Fully connected | 2048 |
2 | ReLU | 2048 |
3 | Dropout | 2048 | 50%
4 | Fully connected | 2048 |
5 | ReLU | 2048 |
6 | Dropout | 2048 | 50%
7 | Fully connected | 2 |
8 | Soft-max | 2 |

As seen above, our introduced ANet is conceptually simple. Each stream in ANet
has its own task and; thus, each stream is expected to be individually trained
using data suitable for the task to enhance its ability. Then, the fusion of
the two streams into one boosts up segmentation accuracy of camouflaged
objects. Furthermore, ANet is flexible in the sense that we can employ any
end-to-end FCN and easily switch it with another. As we see in our
experimental results, ANet is robust since it maintains good segmentation
accuracy in both cases of with and without a camouflaged object in an input
image. We also see its computational efficiency.

##  5 Experiments

In this section, we first introduce evaluation datasets and evaluation
criteria used in experiments. Then we describe implementing details as well as
network training of our proposed ANet and other baselines. In experiments, we
show the potential ability of transforming domain from salient object
segmentation to camouflaged object segmentation. We next compare instances of
our ANet with the FCN models fine-tuned on two evaluation datasets, to
demonstrate that classifying camouflaged objects and non-camouflaged objects
can boost up the camouflaged object segmentation. We also present the
efficiency of our general ANet through short network training and fast running
time. These results can be considered as the first baselines for the
camouflaged object segmentation problem. Finally, we discuss the challenge of
the proposed CAMO dataset to show rooms for further research on this problem.

Table 2: CAMO-COCO dataset used in our experiments. | Training | Testing | Total
---|---|---|---
Camouflaged object images (CAMO) | 1000 | 250 | 1250
Non-Camouflaged object images (MS-COCO) | 1000 | 250 | 1250
Total | 2000 | 500 | 2500

###  5.1 Datasets and Experimentation Setup

Any image in the CAMO dataset contains at least one camouflaged object. There
is thus prior information that camouflaged objects are always present. In a
realistic scenario, however, there is no guarantee that a camouflaged object
is always present in an image. Therefore, we set up another dataset, called
CAMO-COCO555The link to the dataset will be available along with the
publication of this paper., consisting of camouflaged object images and non-
camouflaged object images. We used the entire images in the CAMO dataset for
the camouflaged object images and collected additional 1250 images from the
MS-COCO dataset (Lin et al., 2014) for the non-camouflaged object images (cf.
Fig. 6). We note that we created zero-mask ground-truth labels (all pixels
have zero values) for the non-camouflaged object images. For each of the
camouflaged image and non-camouflaged object image sets, we randomly chose 80%
for training (1000 images) and used the remaining 20% for testing (250
images). Table 2 shows the number of images in the CAMO-COCO dataset used for
experiments.

###  5.2 Evaluation Criteria

We used the F-measure ($F_{\beta}$) (Achanta et al., 2009), Intersection Over
Union (IOU) (Long et al., 2015), and Mean Absolute Error (MAE) as the metrics
to evaluate obtained results. The first metric, F-measure, is a balanced
measurement between precision and recall as follows:

${F_{\beta}}=\frac{{\left({1+{\beta^{2}}}\right)Precision\times
Recall}}{{{\beta^{2}}\times Precision+Recall}}.$ (1)

Note that we set $\beta^{2}=0.3$ as used in (Achanta et al., 2009) to put an
emphasis on precision. IOU is the area ratio of the overlapping against the
union between the predicted camouflage map and the ground-truth map.
Meanwhile, MAE is the average of the pixel-wise absolute differences between
the predicted camouflage map and the ground-truth.

For MAE, we used the raw grayscale camouflage map. For the other metrics, we
binarized the results depending on two contexts. In the first context, we
assume that camouflaged objects are always present in every image like salient
objects; we used an adaptive threshold (Jia and Han, 2013) $\theta=\mu+\eta$
where $\mu$ and $\eta$ are the mean value and the standard deviation of the
map, respectively. In the second context which is much closer to a real-world
scenario, we assume that the existence of camouflaged objects is not
guaranteed in each image; we used the fixed threshold $\theta=0.5$.

###  5.3 Implementation Details

To demonstrate the generality and flexibility of our proposed ANet, we employ
various recent state-of-the-art FCN-based salient object segmentation models
for the segmentation stream. They are DHS (Liu and Han, 2016), DSS (Hou et
al., 2017), SRM (Wang et al., 2017b), and WSS(Wang et al., 2017a).

For each employed model, we train two streams sequentially. We first fine-
tuned the segmentation stream using the CAMO dataset from the publicly
available pre-trained model and then trained the classification stream using
CAMO-COCO with fixed parameters in the segmentation stream. The final model is
considered as our baseline for camouflaged object segmentation.

In both training steps, we set the size of each mini-batch to 2, and used the
Stochastic Gradient Descent (SGD) optimization (Rumelhart et al., 1988) with a
moment $\beta=0.9$ and a weight decay of 0.0005. We trained the segmentation
stream for 10 epochs (corresponding to 5k iterations) with the learning rate
of $10^{-4}$ and trained the classification stream for 3 epochs (3k
iterations) with the learning rate of $10^{-6}$. During the fine-tuning
process, a simple data augmentation technique was used to avoid over-fitting.
Images were rescaled to the resolution of the specific FCN employed in the
segmentation stream and randomly flipped horizontally.

We note that for comparison, our employed FCN models (original) were trained
on CAMO (or CAMO-COCO) dataset for 10 epochs with the learning rate of
$10^{-4}$ and the other parameters were set similarly to our ANet. We also
remark that we implemented our method in C/C++, using Caffe (Jia et al., 2014)
toolbox and conducted all the experiments on a computer with a Core i7 3.6 GHz
processor, 32 GB of RAM, and two GTX 1080 GPUs.

###  5.4 Experimental Results

####  5.4.1 Baseline Evaluation

Table 3: Experimental results on two datasets: CAMO dataset (the left part),
and CAMO-COCO dataset (the right part). The evaluation is based on F-measure
(Achanta et al., 2009) (the higher the better), IOU (Long et al., 2015) (the
higher the better), and MAE (the smaller the better). The 1st and 2nd places
are shown in blue and red, respectively.

Dataset in test | CAMO |  | CAMO-COCO
---|---|---|---
Method |  | Adaptive Threshold | Fixed Threshold |  |  | Adaptive Threshold | Fixed Threshold
| MAE $\Downarrow$ | Fβ $\Uparrow$ | IOU $\Uparrow$ | Fβ $\Uparrow$ | IOU $\Uparrow$ |  | MAE $\Downarrow$ | Fβ $\Uparrow$ | IOU $\Uparrow$ | Fβ $\Uparrow$ | IOU $\Uparrow$
DHS (Liu and Han, 2016) (pre-trained) | 0.173 | 0.548 | 0.351 | 0.562 | 0.332 |  | 0.204 | 0.746 | 0.571 | 0.750 | 0.549
DHS (fine-tuned with CAMO) | 0.129 | 0.640 | 0.459 | 0.643 | 0.444 |  | 0.169 | 0.794 | 0.633 | 0.793 | 0.618
DHS (fine-tuned with CAMO-COCO) | 0.138 | 0.596 | 0.388 | 0.614 | 0.367 |  | 0.072 | 0.796 | 0.679 | 0.808 | 0.681
ANet-DHS (baseline) | 0.130 | 0.626 | 0.437 | 0.631 | 0.423 |  | 0.072 | 0.812 | 0.712 | 0.814 | 0.705
DSS (Hou et al., 2017) (pre-trained) | 0.157 | 0.564 | 0.320 | 0.563 | 0.333 |  | 0.176 | 0.757 | 0.559 | 0.756 | 0.570
DSS (fine-tuned with CAMO) | 0.141 | 0.622 | 0.425 | 0.631 | 0.420 |  | 0.152 | 0.791 | 0.633 | 0.795 | 0.630
DSS (fine-tuned with CAMO-COCO) | 0.145 | 0.582 | 0.385 | 0.584 | 0.381 |  | 0.076 | 0.790 | 0.686 | 0.792 | 0.687
ANet-DSS (baseline) | 0.132 | 0.587 | 0.404 | 0.607 | 0.390 |  | 0.067 | 0.795 | 0.701 | 0.804 | 0.694
SRM (Wang et al., 2017b) (pre-trained) | 0.171 | 0.448 | 0.258 | 0.425 | 0.213 |  | 0.191 | 0.699 | 0.535 | 0.685 | 0.502
SRM (fine-tuned with CAMO) | 0.120 | 0.683 | 0.507 | 0.688 | 0.498 |  | 0.176 | 0.815 | 0.651 | 0.812 | 0.634
SRM (fine-tuned with CAMO-COCO) | 0.127 | 0.663 | 0.454 | 0.656 | 0.421 |  | 0.067 | 0.830 | 0.717 | 0.831 | 0.708
ANet-SRM (baseline) | 0.126 | 0.654 | 0.475 | 0.662 | 0.466 |  | 0.069 | 0.826 | 0.732 | 0.830 | 0.727
WSS (Wang et al., 2017a) (pre-trained) | 0.178 | 0.559 | 0.323 | 0.531 | 0.265 |  | 0.197 | 0.754 | 0.567 | 0.740 | 0.528
WSS (fine-tuned with CAMO) | 0.145 | 0.658 | 0.477 | 0.661 | 0.465 |  | 0.174 | 0.807 | 0.649 | 0.810 | 0.637
WSS (fine-tuned with CAMO-COCO) | 0.149 | 0.642 | 0.439 | 0.638 | 0.382 |  | 0.085 | 0.811 | 0.678 | 0.820 | 0.687
ANet-WSS (baseline) | 0.140 | 0.661 | 0.459 | 0.643 | 0.407 |  | 0.078 | 0.826 | 0.710 | 0.820 | 0.697

We evaluated our model (baseline) on two datasets: CAMO (camouflaged object
images only) and CAMO-COCO (both camouflaged and non-camouflaged object
images) where only test images in each dataset were used for this evaluation.
We employed an FCN (DHS (Liu and Han, 2016), DSS (Hou et al., 2017), SRM (Wang
et al., 2017b), or WSS(Wang et al., 2017a)) as the segmentation stream. For
comparison, we also evaluated each of used FCNs to see how it works for
camouflaged object segmentation. Each was evaluated with three different
models: the pre-trained models on saliency datasets (e.g. MSRA (Cheng et al.,
2015), DUT-OMRON (Yang et al., 2013), and DUTS (Wang et al., 2017a)), the
models that were fine-tuned with CAMO or CAMO-COCO (the model is specified by
the word inside the parentheses). Figures 7, 8, and Table 3 illustrate the
obtained results of our experiments. We note that ANet-DHS (baseline), for
example, denotes our proposed model employing DHS as the segmentation stream.

Figures 7 and 8 show that ANet yields better results than its employed
original FCN (pre-trained) independently of the choice of the FCN. In
particular, for non-camouflaged object images, the superiority of performances
of ANet against its original FCN is distinguished. This suggests that the
fusion with the classification stream indeed works effectively to boost the
segmentation accuracy.

Fig. 7: Visual comparison of our ANet employing different FCNs on natural
camouflaged object images (CAMO dataset). From left to right, input image and
ground-truth are followed by outputs obtained using ANet with DHS (Liu and
Han, 2016), DHS (pre-trained), ANet with DSS (Hou et al., 2017), DSS (pre-
trained), ANet with SRM (Wang et al., 2017b), SRM (pre-trained), ANet with WSS
(Wang et al., 2017a), and WSS (pre-trained) in this order. The results
obtained by ANet is surrounded with red rectangles. Fig. 8: Visual comparison
of our ANet employing different FCNs. From left to right, input image and
ground-truth are followed by outputs obtained using ANet with DHS (Liu and
Han, 2016), DHS (pre-trained), ANet with DSS (Hou et al., 2017), DSS (pre-
trained), ANet with SRM (Wang et al., 2017b), SRM (pre-trained), ANet with WSS
(Wang et al., 2017a), and WSS (pre-trained) in this order. The first eight
rows are images with artificial camouflaged objects (CAMO dataset), and the
last seven rows are images without camouflaged objects (COCO dataset). The
results obtained by ANet is surrounded with red rectangles.

####  5.4.2 Performance of Camouflaged Object Segmentation

It is interesting for the saliency community to see the segmentation
capabilities of saliency systems to segment camouflaged objects (which should
actually not be possible due to different properties of those objects).
Therefore, we show results of FCNs using pre-trained models for salient object
segmentation in Table 3. Although results are moderate (the worst results
among compared methods for each FCN), it also illustrates the potential
ability to transfer from salient object segmentation to camouflaged object
segmentation due to the ambiguous border of these kinds of object. This shows
the possibility of using existing saliency models to tackle camouflaged object
segmentation problem.

Table 3 shows that the fine-tuned FCN models achieve around 60$+$% accuracy in
$F_{\beta}$ and IOU (around 0.1 in MAE). Naturally, the model fine-tuned with
the dataset used for testing performs better than the pre-trained model or the
model fine-tuned with the other dataset. We also observe the model fine-tuned
with CAMO-COCO performs better than the one fine-tuned with CAMO. This is
because CAMO is a subset of CAMO-COCO. These mean that salient object
segmentation methods are, to some extent, capable of segmenting even a not-
target object such as a camouflaged object. This may come from the ambiguity
of the differences between salient objects and camouflaged objects. This,
simultaneously, also supports the use of an FCN developed for salient object
segmentation as the segmentation stream in our ANet.

We then draw our attention to the performances of ANet using an FCN. Since the
segmentation stream of ANet is trained using CAMO and its output is multiplied
by the output of the classification stream, the segmentation ability of ANet
is limited by that of the employed FCN fine-tuned with CAMO. In other words,
the performance of ANet on CAMO is expected to be comparable with that of the
employed FCN (fine-tuned with CAMO), which is confirmed in Table 3. When we
evaluate ANet on CAMO-COCO, however, the performance of ANet can be better
than that of the employed FCN (fine-tuned with CAMO). This can be explained by
the nature of CAMO-COCO where an image may not contain any camouflaged object,
and thus the role of the classification stream becomes more crucial; in other
words, the accuracy of the classification stream boosts up the segmentation
accuracy of ANet. The segmentation accuracy of the FCN, in contrast, is
degraded because it is not trained on CAMO-COCO. A similar conclusion can be
derived from the comparison between ANet and the employed FCN fine-tuned with
CAMO-COCO. Accordingly, ANet works with comparable accuracy on both CAMO and
CAMO-COCO while each FCN works well only on one of the two datasets. Indeed,
this can be observed in Table 3. This clearly demonstrates that ANet can
maintain with flexibility a good segmentation accuracy, regardless whether a
camouflaged object is present in the input image or not.

We also statistically evaluated the significance of the differences using the
t-test with the significance level of $90\%$. For CAMO-COCO dataset, we
confirmed that on all the metrics, ANet significantly outperforms the other
methods mostly or is at least similar to the best methods. For CAMO dataset,
we also confirmed ANet is significantly better than FCNs fine-tuned on CAMO-
COCO, and FCNs fine-tuned on CAMO are significantly better than ANet. This
means ANet’s ability for CAMO dataset is between FCNs fine-tuned on CAMO and
FCNs fine-tune on CAMO-COCO. This is reasonable because the performance of the
classification stream is not perfect, leading to slightly reducing the
performance on CAMO.

To summarize, ANet has good performance for both CAMO and CAMO-COCO datasets.
On the other hand, FCNs fine-tuned with CAMO-COCO have good performance for
only CAMO-COCO but not for CAMO; whereas FCNs fine-tuned with CAMO have good
performance for only CAMO but not for CAMO-COCO (not for both).

####  5.4.3 Performance of Camouflaged Object Classification

In order to evaluate the level of accuracy achieved by our classification
stream, we compared the performance of our classification stream with that of
the SVM-based classification using the Bag-of-Words model (Fei-Fei and Perona,
2005) with SIFT features (Lowe, 1999) (denoted by SVM-BoW) and CNNs, including
AlexNet (Krizhevsky et al., 2012) and VGG-16 (Simonyan and Zisserman, 2014b).
For CNNs, we fine-tuned them on our CAMO-COCO dataset from pre-trained models
on the ImageNet dataset (Russakovsky et al., 2015). The results are shown in
Table 4, indicating that the accuracy of our classification stream achieves
around 90%, which outperforms SVM-BoW. ANet baseline models are also
significantly better than both AlexNet and VGG-16. As a closer look, the
classification stream achieves high accuracy on few training iterations (3
epochs).

Table 4: The accuracy (%) of camouflaged object classification on the CAMO-COCO dataset. Method | Accuracy
---|---
SVM-BoW (Fei-Fei and Perona, 2005) | 54.0
AlexNet (Krizhevsky et al., 2012) | 76.0
VGG-16 (Simonyan and Zisserman, 2014b) | 88.6
ANet-DHS | 91.4
ANet-DSS | 89.2
ANet-SRM | 90.6
ANet-WSS | 89.6

####  5.4.4 Computational Efficiency

We further evaluated the computational efficiency of all the baseline models.
We also evaluated the average running time of the original FCN models employed
in our ANet. The results are shown in Table 5. We observe that stacking the
classification stream does not incur much more processing time, indicating
that the ANet is able to keep the running time computationally efficient.

Table 5: The average wall-clock time in millisecond for each image. Method | FCN | ANet (baseline)
---|---|---
DHS (Liu and Han, 2016) | 63 | 73
DSS (Hou et al., 2017) | 76 | 79
SRM (Wang et al., 2017b) | 81 | 86
WSS (Wang et al., 2017a) | 61 | 65

###  5.5 Discussion

####  5.5.1 Failure Cases

The accuracy of our baselines on CAMO for camouflaged object segmentation is
far lower than the state-of-the-art accuracy for salient object segmentation,
which can achieve up to round 90% in Fβ. This is mainly caused by the
insufficient ability of the segmentation stream when we transfer domain from
salient object segmentation to camouflaged object segmentation. Figure 9 shows
some failure segmentation results obtained by our used FCNs where input images
involve challenging scenarios in CAMO (object appearance and background
clutter in the first row; the shape complexity of insect legs in the second
row; distraction in the third row).

####  5.5.2 Joint Training

The classification stream is expected to boost the segmentation accuracy in
ANet, however, fusing the two streams by the multiplication is still
insufficient. Further effective fusion of the two streams should be explored
to improve the segmentation accuracy.

Fig. 9: Failure cases of FCNs on CAMO. Each case is shown in a horizontal
triplet of images, from left to right: input image, ground-truth, and result
by the best FCN (fine-tuned with CAMO) in this order.

For a two-stream network, jointly training two streams after training each
stream separately is known as a strategy to improve the performance (Jain et
al., 2017). Following this strategy, we jointly trained the ANet (baseline)
using CAMO-COCO where we used 10 epochs (10k iterations) with the learning-
rates of $10^{-4}$ (for the segmentation stream) and $10^{-6}$ (for the
classification stream). Other parameters are set similarly to training two
streams individually. The loss function for joint-training is defined as the
summation of the loss of the classification stream and that of the
segmentation stream:

$\mathcal{L}=\mathcal{L}_{\rm seg}(\hat{y}_{\rm seg},y_{\rm
seg})+\mathcal{L}_{\rm cls}(\hat{y}_{\rm cls},y_{\rm cls}),$ (2)

where $\hat{y}_{\rm seg}$, $y_{\rm seg}$, $\hat{y}_{\rm cls}$, and $y_{\rm
cls}$ denote outputs and ground-truth labels of the segmentation stream and
the classification stream, respectively. $\mathcal{L}_{\rm cls}$ is the soft-
max loss of the classification stream and $\mathcal{L}_{\rm seg}$ is the
specific loss of the corresponding FCN implemented in the segmentation stream.

Table 6 illustrates the improvement of performances by introducing the joint-
training to ANet. We see that differently from other two-stream network cases,
jointly-training ANet does not lead to better performance for all methods. DHS
and WSS reduce accuracy while DSS and SRM slightly increase accuracy (see
Table 6). This can be explained as follows. Each stream of ANet has its own
task and thus using training data suitable for the task is mandatory to
enhance its ability. During the joint-training of ANet, however, data used for
training cannot be suitable for each of the tasks of the two streams,
resulting in interfering with each other stream. This is also facilitated by
the branch structure of ANet. Accordingly, joint-training potentially brings
to ANet some collapse in improving training of each stream.

Therefore, jointly training two streams may not bring gain to ANet due to
opposite attributes of camouflaged objects and non-camouflaged objects. The
two-step training approach is sufficient for ANet for the task of camouflaged
object segmentation on the CAMO-COCO dataset. Further investigation on this
issue is left for future work.

Table 6: Performance of joint-training on CAMO dataset and CAMO-COCO dataset.
The best results are shown in blue.

Dataset in test | CAMO |  | CAMO-COCO
---|---|---|---
Method |  | Adaptive Threshold | Fixed Threshold |  |  | Adaptive Threshold | Fixed Threshold
| MAE $\Downarrow$ | Fβ $\Uparrow$ | IOU $\Uparrow$ | Fβ $\Uparrow$ | IOU $\Uparrow$ |  | MAE $\Downarrow$ | Fβ $\Uparrow$ | IOU $\Uparrow$ | Fβ $\Uparrow$ | IOU $\Uparrow$
ANet-DHS (baseline) | 0.130 | 0.626 | 0.437 | 0.631 | 0.423 |  | 0.072 | 0.812 | 0.712 | 0.814 | 0.705
ANet-DHS (+joint training) | 0.201 | 0.509 | 0.314 | 0.490 | 0.130 |  | 0.109 | 0.749 | 0.628 | 0.758 | 0.564
ANet-DSS (baseline) | 0.132 | 0.587 | 0.404 | 0.607 | 0.390 |  | 0.067 | 0.795 | 0.701 | 0.804 | 0.694
ANet-DSS (+joint training) | 0.126 | 0.644 | 0.441 | 0.651 | 0.417 |  | 0.064 | 0.822 | 0.719 | 0.825 | 0.707
ANet-SRM (baseline) | 0.126 | 0.654 | 0.475 | 0.662 | 0.466 |  | 0.069 | 0.826 | 0.732 | 0.830 | 0.727
ANet-SRM (+joint training) | 0.123 | 0.680 | 0.481 | 0.682 | 0.454 |  | 0.063 | 0.839 | 0.733 | 0.841 | 0.726
ANet-WSS (baseline) | 0.140 | 0.661 | 0.459 | 0.643 | 0.407 |  | 0.078 | 0.826 | 0.710 | 0.820 | 0.697
ANet-WSS (+joint training) | 0.145 | 0.626 | 0.394 | 0.628 | 0.324 |  | 0.074 | 0.813 | 0.692 | 0.816 | 0.662

##  6 Conclusion

In this paper, we addressed an interesting yet challenging problem of
camouflaged object segmentation by providing a new image dataset of
camouflaged object segmentation where each image is manually annotated with
pixel-wise ground-truth. We believe that our novel dataset will promote new
advancements on camouflaged object segmentation. We also aim to explore
camouflaged object segmentation on videos in the near future.

We proposed a simple and flexible end-to-end network, namely Anabranch
Network, for camouflaged object segmentation where the classification stream
and the segmentation stream are effectively combined to show the baseline
performance. To show effectiveness of our proposed framework, which can boot
the segmentation using classification, we applied it to different FCNs.
Extensive experiments conducted on the newly built dataset demonstrate the
superiority of the proposed network. In addition, our method is
computationally efficient.

This paper focused on only regions, but the potential idea of utilizing
classification into segmentation can be useful when applying for instance
segmentation in appropriate ways. For examples, instance classification can be
combined with instance segmentation, where each instance is classified whether
it is camouflaged or salient. This is left for the future work.

## Acknowledgment

This work is in part granted by University of Dayton SEED Grant (US), and by
SOKENDAI Short-Stay Study Abroad Program (Japan). We also thank Vamshi Krishna
Madaram and Zhe Huang (University of Dayton) for their support in the
annotation of the CAMO dataset. We gratefully acknowledge NVIDIA for the
support of GPU.

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# Boundedness and compactness of weighted composition operators and monomial
operators

I. Chalendar  Isabelle Chalendar, Université Gustave Eiffel, LAMA, (UMR 8050),
UPEM, UPEC, CNRS, F-77454, Marne-la-Vallée (France)  isabelle.chalendar@univ-
eiffel.fr and  J.R. Partington  Jonathan R. Partington, School of Mathematics,
University of Leeds, Leeds LS2 9JT, UK<EMAIL_ADDRESS>
Abstract: This paper characterises the boundedness and compactness of
Agler–McCarthy monomial operators by reducing them to weighted composition
operators and deriving explicit Carleson measure criteria on the half-plane.
The results are illustrated by examples.

Keywords: Monomial operator, Volterra operator, weighted composition operator,
Carleson measure, compactness

MSC class: 30H10, 47B33

##  1\. Introduction

Motivated by questions of polynomial approximation, Agler and McCarthy [1]
defined a monomial operator as a bounded linear operator $T:L^{2}(0,1)\to
L^{2}(0,1)$ mapping monomials to monomials; that is, such that for each
$n\in\mathbb{N}$ we have (with a slight abuse of notation)
$T(x^{n})=c_{n}x^{p_{n}}$ for some $p_{n}\in\mathbb{C}$ with $\mathop{\rm
Re}\nolimits p_{n}>-1/2$. Such an operator is flat if there is a constant $b$
with $p_{n}=n+b$ for each $n$. We shall also say that it has affine index if
$p_{n}=an+b$ for real constants $a,b\geq 0$.

Well-known examples of monomial operators include the Volterra operator $V$
with

$Vf(x)=\int_{0}^{x}f(t)\,dt,\qquad Vx^{n}=\frac{x^{n+1}}{n+1},$

and the Hardy operator $H$ (also known as the continuous Cesàro operator [2])
with

$Hf(x)=\frac{1}{x}\int_{0}^{x}f(t)\,dt,\qquad Hx^{n}=\frac{x^{n}}{n+1},$

which are clearly flat operators. Agler and McCarthy gave conditions for flat
operators to be bounded, and showed that they were never compact except in
trivial cases, but the general case was left open.

Their work was based on a transformation to the Hardy space of the half-plane
$\SS=\\{s\in\mathbb{C}:\mathop{\rm Re}\nolimits s>-1/2\\}$ and showed that the
adjoint of a bounded monomial operator can be expressed as an operator on
$H^{2}(\SS)$. By means of a closely-related transformation we shall work on
the Hardy space $H^{2}(\mathbb{C}_{+})$ on the standard right half-plane, and
derive results in the theory of weighted composition operators, which will be
applied to characterise bounded and compact operators. The results are
particularly simple for flat and affine-index operators, as will be
illustrated by examples.

##  2\. Boundedness and Compactness

The approach here is very similar to Agler–McCarthy [1], but adapted for own
purposes.

We transfer the operator $T$ to $L^{2}(0,\infty)$. Writing $x=e^{-t}$ we have

$\int_{0}^{1}|f(x)|^{2}\,dx=\int_{0}^{\infty}|f(e^{-t})|^{2}e^{-t}\,dt,$

giving an isometry by $Jf(t)=e^{-t/2}f(e^{-t})$. So
$J^{-1}g(x)=\frac{1}{\sqrt{x}}g(-\log x)$. The unitarily equivalent operator
$\tilde{T}$ on $L^{2}(0,\infty)$ maps $e^{-nt-t/2}$ to $c_{n}e^{-p_{n}t-t/2}$.

Then by taking Laplace transforms and using the Paley–Wiener theorem we have a
unitarily equivalent operator $T_{0}$ on $H^{2}(\mathbb{C}_{+})$ which maps
$1/(s+n+1/2)$ to $c_{n}/(s+p_{n}+1/2)$.

Since the reproducing kernel for $H^{2}(\mathbb{C}_{+})$ is
$k_{w}(s)=1/(s+\overline{w})$ we have

$T_{0}^{*}G(n+1/2)=\overline{c_{n}}G(p_{n}+1/2).$

The Blaschke sequences (zero sequences) $(z_{n})$ for $H^{2}(\mathbb{C}_{+})$
are those with $\displaystyle\sum_{n}\frac{\mathop{\rm Re}\nolimits
z_{n}}{1+|z_{n}|^{2}}<\infty$ [4], and so the sequence $(n+1/2)$ is not a
Blaschke sequence for the Hardy space of the right half-plane. Thus we have a
unique formula for $T_{0}^{*}$, namely the weighted composition operator
$W_{h,\phi}$ with

(2.1) $\phi(n+1/2)=p_{n}+1/2\qquad\hbox{and}\qquad h(n+1/2)=\overline{c_{n}},$

since $(T_{0}^{*}-W_{h,\phi})(G)=0$ for all $G\in H^{2}(\mathbb{C}_{+})$.

For affine-index operators with $p_{n}=an+b$ we have
$T_{0}^{*}G=h(G\circ\phi)$ with $h(n+1/2)=\overline{c}_{n}$ and
$\phi(s)=as+b$. This is consistent with Agler–McCarthy [1] except that we use
the “usual” right half-plane not the shifted version.

######  Remark 2.1.

The classical Müntz–Szász theorem says that the functions $\\{x^{n}:n\in S\\}$
span a dense subspace of $L^{2}(0,1)$ if and only if $\displaystyle\sum_{n\in
S}\frac{1}{n+1}=\infty$. Thus under this condition a bounded monomial operator
is completely defined by $\\{Tx^{n}:n\in S\\}$. We see this from the above
discussion, since such an $S$ does not constitute a Blaschke sequence, and the
interpolation problem has a unique solution.

Boundedness of the weighted composition operator $W_{h,\phi}:f\mapsto
h(f\circ\phi)$ can be characterised using a Carleson measure criterion by
proving a result similar to those in [3, 6]. Namely, for Borel subsets
$E\subseteq\overline{\mathbb{C}_{+}}$ define a measure by

(2.2) $\mu(E)=\int_{\phi^{-1}(E)\cap i\mathbb{R}}|h(s)|^{2}\,|ds|.$

######  Theorem 2.2.

The weighted composition operator $W_{h,\phi}$ is bounded if and only if $\mu$
is a Carleson measure; that is, $\sup_{I}\mu(Q_{I})/|I|<\infty$, where
$Q_{I}=[0,|I|]\times I$ is a Carleson square based on an interval $I\subset
i\mathbb{R}$. Hence the monomial operator $T$ is bounded if and only if the
same condition holds for the associated weighted composition operator.

###### Proof.

In [3] the analogous result is proved for weighted composition operators on
the disc and in [6] a similar result is derived for the Bergman space. For the
half-plane an identical proof works with obvious modifications, as follows:

We claim that for $g\geq 0$ Borel measurable we have

(2.3)
$\int_{\overline{\mathbb{C}_{+}}}g\,d\mu=\int_{i\mathbb{R}}|h|^{2}(g\circ\phi)|ds|.$

As in [3] we can easily verify this result for simple functions, and then for
arbitrary $g$ take an increasing sequence of positive simple functions
$(g_{n})$ with $g_{n}\to g$ pointwise in $\overline{\mathbb{C}_{+}}$.

From (2.3) we deduce that
$\|W_{h,\phi}f\|=\|f\|_{L^{2}(\overline{\mathbb{C}_{+}},\mu)}$. Now, the
Carleson measure condition that for some fixed $C>0$

$\int_{\overline{\mathbb{C}_{+}}}|f|^{2}\,d\mu\leq C\|f\|^{2}\qquad(f\in
H^{2}(\mathbb{C}_{+}))$

is equivalent to the condition that $W_{h,\phi}$ is bounded.

In the context of a monomial operator $T$ the adjoint operator is unitarily
equivalent to $W_{h,\phi}$ and the result follows. ∎

######  Remark 2.3.

It is interesting to consider the case $h=1$, an unweighted composition
operator $C_{\phi}$ on the right half-plane. We see that $C_{\phi}$ is bounded
if and only if there is a constant $K>0$ such that
$\mu(Q_{I})=m(\phi^{-1}(E)\cap i\mathbb{R})\leq K|I|$ for all Carleson squares
(here $m$ is Lebesgue measure). This may be compared with the alternative
condition given by Elliott and Jury [5]; namely, that there is an angular
derivative at $\infty$, the non-tangential limit
$\phi^{\prime}(\infty)=\lim_{z\to\infty\ {\rm n.t.}}z/\phi(z)$ satisfying
$0<\phi^{\prime}(\infty)<\infty$.

######  Corollary 2.4.

Let $T$ be an affine-index monomial operator with $p_{n}=an+b$ ($a,b>0$). Then
$T$ is bounded if and only if the associated function $h$ satisfies

(2.4) $\sup_{t\in\mathbb{R}}\int_{t}^{t+1}|h(iy)|^{2}\,dy<\infty.$

###### Proof.

By the simple structure of $\phi$, we have $\mu$ concentrated on
$b+i\mathbb{R}$ and the Carleson condition is

$\int_{u/a}^{v/a}|h(iy)|^{2}\,dy\leq C(v-u)$

but only for $v-u\geq b$; that is, equivalently, (2.4). ∎

######  Example 2.5.

We can have $c_{n}\to 0$ and the monomial operator not even bounded. For
example, take $c_{n}=(n+1/2)e^{-(n+1/2)}$, for which the associated function
$h$ can only be $se^{-s}$, which does not satisfy the Carleson criterion. This
fact may also be proved using the fact that the monomials do not form an
unconditional basis, but this is more explicit.

We know from [1] that there are no compact monomial operators of the form
$x_{n}\mapsto c_{n}x^{n}$, except the zero operator, and we also know that
there are no compact composition operators on the half-plane [7].

With $T(x^{n})=c_{n}x^{p_{n}}$, if $T$ is bounded then the sequence

(2.5) $\|Tx^{n}\|/\|x^{n}\|=|c_{n}|\sqrt{\frac{2n+1}{2\mathop{\rm Re}\nolimits
p_{n}+1}}$

is bounded. So for affine-index operators if $T$ is bounded we clearly have
$(c_{n})$ bounded.

If $T$ is compact, then observe first that the normalized functions
$e_{n}:x\mapsto\sqrt{2n+1}\,x^{n}$ tend weakly to zero. For they form a
bounded sequence tending weakly to zero on the dense subspace consisting of
polynomials, since

$\langle e_{m},e_{n}\rangle=\frac{\sqrt{2m+1}\sqrt{2n+1}}{m+n+1}\to
0\qquad\hbox{as}\quad n\to\infty.$

Hence compactness implies that $\|Te_{n}\|\to 0$ and this is given by (2.5).
That is, for compact affine-index operators, $c_{n}\to 0$.

Using the completely continuous characterization of compactness in reflexive
Banach spaces, namely, that if $f_{n}\to 0$ weakly then
$\|W_{h,\phi}f_{n}\|\to 0$, we see that a necessary and sufficient condition
on compactness will be compactness of the mapping $J:H^{2}(\mathbb{C}_{+})\to
L^{2}(\overline{\mathbb{C}_{+}},\mu)$. This is characterised for the disc in
[3].

On the disc this is the familiar vanishing Carleson measure criterion for
$\mu$. On the half-plane the condition that

(2.6) $\mu(Q_{I})/|I|\to 0\qquad\hbox{as}\qquad|I|\to 0,$

where $Q_{I}=[0,|I|]\times I$ is a Carleson square based on an interval
$I\subset i\mathbb{R}$, is not sufficient, as we now show.

Consider the mapping $x^{n}\mapsto x^{n+1}$, which is bounded but not compact.
We have $h(s)=1$ and $\phi(s)=s+1$. Now according to (2.2) we have
$\mu(E)=m(E\cap(1+i\mathbb{R}))$, for $E\subset\mathbb{C}_{+}$, and thus

$\|C_{\phi}f\|^{2}=\int_{\overline{\mathbb{C}_{+}}}|f|^{2}\,d\mu=\int_{1+i\mathbb{R}}|f|^{2}\,|ds|.$

The measure $\mu$ of a square $Q_{I}$ is $0$ if $|I|<1$ and so this $\mu$
satisfies (2.6), but nonetheless we do not have compactness.

In fact the correct test for compactness for embeddings from
$H^{2}(\mathbb{C}_{+})$ into $L^{2}(\overline{\mathbb{C}_{+}},\mu)$ is given
in [8] in the more general context of Zen spaces, namely, it can be written

(2.7) $\sup_{I:C(I)\in S_{r}}\mu(Q_{I})/|I|\to 0\qquad\hbox{as}\quad r\to 0,$

where $C(I)$ is the centre of a Carleson square $Q_{I}$ and for $0<r<1$ we
define

$S_{r}=\\{z\in\mathbb{C}_{+}:0<\mathop{\rm Re}\nolimits z<r\hbox{ or
}|z|>1/r\\},$

which is the union of the complement of a large semi-circle and a narrow strip
near the axis.

Equivalently, for each $\epsilon>0$ there is a compact subset
$K\subset\mathbb{C}_{+}$ such that $\mu(Q_{I})/|I|<\epsilon$ for every $Q_{I}$
with centre lying outside $K$.

We therefore have the following result.

######  Theorem 2.6.

The monomial operator $T$ is compact if and only if the associated measure
$\mu$ satisfies the condition (2.7).

This can be made more explicit for operators with affine index.

######  Corollary 2.7.

Let $T$ be an affine-index monomial operator with $p_{n}=an+b$ ($a,b>0$). Then
$T$ is compact if and only if the associated function $h$ satisfies the
condition that for each $\epsilon>0$ there is an $M>0$ such that

(2.8) $\int_{(t-L)/a}^{(t+L)/a}|h(iy)|^{2}\,dy<\epsilon L$

for all $|t|\geq M$ and $L\geq b/2$. This holds whenever $h\in
L^{2}(i\mathbb{R})$.

###### Proof.

Equation (2.8) is a direct rewriting of the vanishing condition (2.7), using
the fact that a square $[0,2L]\times[t-L,t+L]$ with centre $L+it$ meets the
line $\\{s\in\mathbb{C}:\mathop{\rm Re}\nolimits s=b\\}$ if and only if $L\geq
b/2$. Now if $h\in L^{2}(i\mathbb{R})$, the inequality (2.8) holds for all $t$
if $L>L_{0}:=\|h\|_{2}^{2}/\epsilon$. For $b/2\leq L\leq L_{0}$, we may make
the integral less than $\epsilon b/2$ by choosing $|t|$ sufficiently large. ∎

##  3\. Examples

For the Volterra operator, we have from (2.1) that $h(s)=(s+1/2)^{-1}$ and
$\phi(s)=s+1$. Now we can apply Corollary 2.7 and since

$\int_{-\infty}^{\infty}\frac{1}{|1/2+iy|^{2}}\,dy<\infty$

we can deduce that the operator is compact, as is well known.

For the bounded but non-compact Hardy operator we have $h(s)=(s+1/2)^{-1}$ and
$\phi(s)=s$. Corollary 2.7 does not apply since $a=0$, and we must consider
all Carleson squares, since they all meet the imaginary axis. We have

$\int_{u}^{v}|h(iy)|^{2}\,dy=O(v-u),\qquad\hbox{but not}\qquad o(v-u)$

as $v-u\to 0$, and so the operator is bounded but not compact, as is again
well known.

For some non-flat examples consider the following operators defined on
$L^{2}(0,1)$:

$\displaystyle T_{1}f(x)$ $\displaystyle=$ $\displaystyle f(x^{2}),$
$\displaystyle T_{2}f(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{x}f(t^{2})\,dt,$ $\displaystyle T_{3}f(x)$
$\displaystyle=$ $\displaystyle\int_{0}^{x}tf(t^{2})\,dt.$

We have $T_{1}(x^{n})=x^{2n}$, $T_{2}(x^{n})=\dfrac{x^{2n+1}}{2n+1}$,
$T_{3}(x^{n})=\dfrac{x^{2n+2}}{2n+2}$. These operators are not flat, but they
are affine-index.

It is elementary to see that $T_{1}$ is unbounded, since it sends $x^{-1/3}$
to $x^{-2/3}$. We shall see that $T_{2}$ is unbounded, but $T_{3}$ is bounded
(even compact).

For $T_{1}$ we have $\phi(n+1/2)=2n+1/2$, giving $\phi(s)=2s-1/2$, which is
not a self-map of $\mathbb{C}_{+}$.

For $T_{2}$ take $f(x)=1/(\sqrt{t}\log t)$ so that $f\in L^{2}(0,1)$. Then

$T_{2}f(x)=\int_{0}^{x}\frac{1}{t(2\log t)},$

and since this integral diverges we see that $T_{2}$ is unbounded. Now we have
$\phi(n+1/2)=2n+3/2$, giving $\phi(s)=2s+1/2$. Also $h(n+1/2)=1/(2n+1)$,
giving $h(s)=1/(2s)$.

We may also use the Carleson test so that $\phi^{-1}(E)\cap
i\mathbb{R}=\phi^{-1}(E\cap(1/2+i\mathbb{R}))$. However since $|h|^{2}$ is not
locally integrable on the imaginary axis we clearly do not get a Carleson
measure.

For $T_{3}$ we have $\phi(n+1/2)=2n+5/2$, and $\phi(s)=2s+3/2$. Then
$h(n+1/2)=1/(2n+2)$, giving $h(s)=1/(2s+1)$. It is now easy to see that $\mu$
is a Carleson measure, concentrated on the line $\\{s:\mathop{\rm Re}\nolimits
s=3/2\\}$, and the operator is bounded. Indeed, since $h\in
L^{2}(i\mathbb{R})$ we see from Corollary 2.7 that $T_{3}$ is indeed compact.

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