# H4VDM: H.264 Video Device Matching

Ziyue Xiang† Paolo Bestagini‡ Stefano Tubaro‡ Edward J. Delp†
†Video and Image Processing Lab (VIPER) School of Electrical and Computer
Engineering
Purdue University West Lafayette  Indiana  USA
‡Dipartimento di Elettronica Informazione e Bioingegneria  Politecnico di
Milano  Milano  Italy

###### Abstract

Methods that can determine if two given video sequences are captured by the
same device (e.g., mobile telephone or digital camera) can be used in many
forensics tasks. In this paper we refer to this as “video device matching”. In
open-set video forensics scenarios it is easier to determine if two video
sequences were captured with the same device than identifying the specific
device. In this paper, we propose a technique for open-set video device
matching. Given two H.264 compressed video sequences, our method can determine
if they are captured by the same device, even if our method has never
encountered the device in training. We denote our proposed technique as H.264
Video Device Matching (H4VDM). H4VDM uses H.264 compression information
extracted from video sequences to make decisions. It is more robust against
artifacts that alter camera sensor fingerprints, and it can be used to analyze
relatively small fragments of the H.264 sequence. We trained and tested our
method on a publicly available video forensics dataset consisting of 35
devices, where our proposed method demonstrated good performance.

Keywords: H.264 Video Compression, Video Device Matching, Digital Video
Forensics, Deep Learning

##  1 Introduction

Video Device Identification (VDI) is one of the most important tasks in
multimedia forensics [1, 2, 3]. A VDI method can associate a video with a
specific source device (e.g., a specific camera). VDI is valuable in forensics
investigations and court defense.

A large amount of VDI techniques rely on the analysis of video camera sensor
fingerprints. For example the Photo Response Non-uniformity (PRNU) pattern is
commonly used in image/video device identification techniques [4, 5, 6, 7].
Since PRNU patterns can capture the heterogeneity of the sensor response
caused by imperfections in the sensor manufacturing process, such sensor
fingerprints can attribute a video sequence to one source device uniquely.
Despite being powerful in many forensics tasks, the PRNU patterns can be
difficult to obtain. The estimation of PRNU patterns usually requires many
samples taken by the device under analysis [2]. Obtaining the PRNU patterns
from video sequences is more challenging due to the existence of video
compression and video stabilization [8]. These challenges limit the
application of PRNU patterns in VDI tasks.

H.264 is one of the most popular video compression techniques [9, 10]. It
offers a wide variety of compression configurations to balance data rate,
distortion, and computational complexity. Each configuration will induce a
distinct encoding response to the input video sequence. Even for the same
compression configuration, the behavior of H.264 encoder implementations used
by different device manufacturers may not be identical. Note that the H.264
compression standard only standardizes the decoding but not the encoding [9,
10]. In this paper we shall refer to the “encoding pattern” of a H.264
compressed video sequence as the parameters inserted into the compressed
bitstream by the encoder and used by the decoder to reconstruct the video
sequence. We will show in this paper that these encoding patterns along with
the video content can be used to determine the source device of an H.264 video
sequence. Since the H.264 encoding pattern tends to be the same for a specific
video camera model or firmware version, using it for VDI tasks may only result
in a model-level or firmware-level match, which is coarser compared to
techniques using video camera sensor fingerprints. However, these H.264
encoding patterns are closely related to the compressed digital video and are
less affected by operations that alter the sensor fingerprints such as video
stabilization. Compared to metadata forensic methods such as [11, 12, 13],
video forensics methods based on using the video content and encoding
parameters can still work even if the metadata information is modified (e.g.,
when an MP4 video file is converted to an AVI video file without transcoding
the video stream). Our proposed method only requires two H.264 Groups of
Pictures (GOPs) to make decisions, which allows our approach to work with
corrupted or fragmented H.264 data.

In this paper, we focus on open-set Video Device Matching (VDM), which is
related to VDI. Open-set classification is the problem of handling classes
that are not contained in the training dataset. Traditional classification
approaches assume that only known classes appear in the testing environment
[14]. In VDI, the video forensics method is asked to identify which device was
used to capture the video sequence. In VDM, the video forensics method is
asked to determine if two video sequences are captured by the same camera
model [15, 16, 17]. With VDM methods, VDI can be achieved by obtaining a video
sequence from a known device and then determining if the video sequence under
analysis is captured by the same device. Since VDI methods attribute a given
video sequence to a specific source device, these methods require prior
knowledge about the devices. As a result, VDI methods are often constrained to
closed-set problems, where the video sequence under test comes from a device
that is already known to the video forensics method [15]. This tends to have
limitations in practice, as forensic investigators are more likely to deal
with open-set problems where the video camera model has not been encountered
by the method before. Because VDM methods only analyze if the two video
sequences are from the same device, it is possible for these methods to work
in open-set scenarios where the devices have never been seen by the VDM
methods before [15]. We tested the open-set performance of our proposed VDM
technique and verified that it had good performance evaluation metrics for
unseen device models. The ability to attain open-set VDM allows our method to
be used in a wider range of forensic investigations.

The rest of the paper is organized as follows. In Section 2, we show the
related work and provide the background knowledge about the H.264 video
compression and the machine learning model used in our approach. In Section 3,
we describe the details of our proposed VDM method. In Section 4, we discuss
the details of our experiments and present the results. Section 5 concludes
the paper and gives insights on open problems and future challenges.

##  2 Background

In this section we show existing work related to Video Device Identification
(VDI), Video Device Matching (VDM), and H.264-based video forensics. Then, we
briefly introduce H.264 video compression and transformer neural networks,
which are two important concepts for understanding the mechanism of our
proposed open-set VDM method.

###  2.1 Related Work

Video Device Identification (VDI) is an important topic in video forensics. In
[18], the authors used the statistics of motion vectors from video codecs for
VDI. Yahaya et al. [19] used conditional probability features to achieve VDI.
In [11, 12, 13], the authors used metadata information stored in video
container formats for VDI. Most existing work on VDI are based on camera
sensor noise fingerprints, which was first studied in [1]. In the seminal work
proposed by Chen et al. [20], PRNU analysis was first used for VDI tasks. In
[21, 22, 7, 16, 23], the authors devised various strategies to improve the
performance of PRNU analysis of video, such as selecting key frames,
counteracting the effects of video stabilization, using the characteristics of
the video codecs, and weighting frames in terms of compression quality. Beyond
PRNU patterns, [24, 25, 26] used deep neural networks to extract features from
decoded video frames, which can be used in VDI tasks. Some VDI techniques use
multimodal data to improve the performance. Iuliani et al. [6] combined image
and video data from the same sensor for better VDI results. Dal Cortivo et al.
[27] proposed a VDI approach based on video and audio information.

As described in Section 1, Video Device Matching (VDM) is a concept derived
from VDI. It is also known as Video Device Verification. In [16, 17], the
authors improved PRNU analysis for VDM. Mayer et al. [15] addressed the
problem of open-set VDM using a deep neural network to extract features from
decoded video frames.

There have been a number of video forensics approaches that use the
characteristics of H.264 video compression. In [28, 29, 30, 31, 32, 33, 34,
35], the authors used information from the H.264 codec to determine if an
H.264 video is double compressed. Verde et al. [36] used deep neural networks
to extract H.264 codec information for video manipulation localization. We
believe our proposed technique is the first to use H.264 codec information for
open-set VDM problems.

###  2.2 H.264 Video Compression

The details of the H.264 video compression standard are extremely
sophisticated and beyond the scope of this paper. Therefore, we provide a
high-level overview of H.264 video compression that is sufficient for
understanding our proposed VDM method. More details about H.264 compression
can be obtained from [9, 10, 37].

One important concept that is a part of nearly all video compression standards
is the use of both spatial and temporal redundancy in a video sequence to
reduce its data rate, particularly the fact that consecutive frames in a small
temporal interval can be greatly correlated. The H.264 encoder examines the
video frames in a structure known as Group of Pictures (GOP), which is a
sequence of consecutive frames. Due to the high temporal correlation, the
frames in a GOP can be compressed using motion compensation [10]. The frames
in a GOP are divided into I-, P-, and B-frames that are used for the motion
compensation. Typically the first frame in a GOP is an I-frame that acts as
the reference frame for other frames in the GOP. The I-frame is compressed
using intra-frame compression similar to JPEG. The rest of the frames in an
H.264 GOP can be P-frames or B-frames, and are compressed with inter-frame
compression and motion compensation using the I-frame or a P-frame as a
reference frame [10].

In I-, P-, and B-frames, H.264 uses $16\times 16$ frame patches known as
macroblocks in each video frame. Each macroblock is associated with a
macroblock type, which specifies how the information in the macroblock is
compressed. Macroblock-level compression can done by predicting patterns
within the macroblock (i.e., intra-coded macroblocks) or using difference
information from a similar macroblock in a motion compensated reference frame
(i.e., inter-coded macroblocks) [10]. This difference is known as the
prediction residual, which is more efficient to compress compared to the
original video frame. The pixels in the macroblocks are then transformed and
quantized before being entropy coded [10]. The quantization process is
controlled by a Quantization Parameter (QP). As the QP increases, the data
rate decreases and the quantization distortion increases; as QP decreases, the
rate-distortion trade-off goes towards the opposite direction [10]. For inter-
coded macroblocks the encoded prediction residuals and the motion vectors from
the motion compensation are placed in the compressed bitstream.

H.264 uses the YUV color space, which has one luma (luminance) channel and two
chroma (chrominance) channels [10]. Since the human visual system is more
sensitive to luminance than chrominance, H.264 prioritizes the compression
quality of the luma channel over the chroma channels to reduce data rate. Each
color channel has its own QP for rate-distortion control [10].

###  2.3 Vision Transformers

Our proposed VDM method is based on transformer neural networks [38]. These
networks demonstrated outstanding performance in a wide variety of tasks such
as language modeling [39], image classification [40], image segmentation [41],
video classification [42], audio signal processing [43], and protein structure
prediction [44].

Transformers can be used to process sequence data. They possess faster
computational speed, higher scalability, and better stability compared to
Recurrent Neural Networks such as LSTM [45] and GRU [46, 47]. Transformer
networks are made up of transformer layers. The output of a transformer layer
is used as the input to the next transformer layer. Denote the input to a
transformer layer by $\boldsymbol{Z}\in\mathbb{R}^{N\times D}$, where $N$ is
the sequence length and $D$ is the dimensionality of the vector at each time
step. Transformer layers estimate the relationship between the vectors at each
time step in $\boldsymbol{Z}$ based on the Self Attention (SA) mechanism. In
SA, the input $\boldsymbol{Z}$ is first linearly projected into three matrices
$\boldsymbol{q},\boldsymbol{k},\boldsymbol{v}\in\mathbb{R}^{N\times D_{h}}$
using a matrix $\boldsymbol{U}\in\mathbb{R}^{D\times 3D_{h}}$ such that
$[\hskip 0.5pt\boldsymbol{q}\hskip 0.5pt|\hskip 0.5pt\boldsymbol{k}\hskip
0.5pt|\hskip 0.5pt\boldsymbol{v}\hskip 0.5pt]=\boldsymbol{Z}\boldsymbol{U}$.
Then, the SA of $\boldsymbol{Z}$ is computed by
$\operatorname{SA}(\boldsymbol{Z})=\operatorname{softmax}(\nicefrac{{\boldsymbol{q}\boldsymbol{k}^{T}}}{{\sqrt{D_{h}}}})\boldsymbol{v}$.
An extended version of SA known as Multihead Self Attention (MSA) is used in
[38], where $h$ different SA values (known as “MSA heads”) are computed at the
same time. The $h$ different SA values are combined to form the result of MSA
using a matrix $\boldsymbol{V}\in\mathbb{R}^{D\times D}$ such that
$\operatorname{MSA}(\boldsymbol{Z})=[\operatorname{SA}_{1}(\boldsymbol{Z})\mid\cdots\mid\operatorname{SA}_{h}(\boldsymbol{Z})]\boldsymbol{V}$.
When choosing the number of MSA heads $h$, it must hold that $h$ divides $D$
and $D_{h}=\nicefrac{{D}}{{h}}$, which ensures the output of the transformer
layer $\operatorname{MSA}(\boldsymbol{Z})$ has the same dimensionality as the
input $\boldsymbol{Z}$. Since a transformer network consists of a series of
such layers, its input and output share the same dimensionality.

Vision Transformer (ViT) [40] uses transformer networks for computer vision
tasks. In order to convert image data into the format accepted by
transformers, the ViT splits the image into $K$ non-overlapping patches of
size $P\times P\times C$, where $P$ is the patch size and $C$ is the number of
channels. The patches are flattened to form the matrix
$\boldsymbol{Z}^{\prime}\in\mathbb{R}^{K\times P^{2}C}$. Since $P^{2}C$ is
usually large, the dimensionality of the flattened vectors from each patch is
reduced to $D_{\mathrm{ViT}}$ using a linear projection
$\boldsymbol{W}\in\mathbb{R}^{P^{2}C\times D_{\mathrm{ViT}}}$ such that
$\boldsymbol{Z}=\boldsymbol{Z}^{\prime}\boldsymbol{W}$. The original input
image can be represented by $\boldsymbol{Z}$, which is used as the input to
the transformer network.

##  3 Proposed Method

The H.264 Video Device Matching (VDM) problem can be formally defined as
follows: given two Groups of Pictures (GOPs) $\mathcal{G}_{1}$ and
$\mathcal{G}_{2}$ from two video sequences, determine if the two GOP s are
from the same device. Note the GOP information includes the decoded frame and
the coding parameters. We propose an H.264 Video Device Matching (H4VDM)
method for open-set VDM. The block diagram of H4VDM is shown in Figure 1. This
network architecture is commonly used in many open-set digital forensics
techniques [15, 5, 48]. The input to H4VDM is two H.264 GOP s
$\mathcal{G}_{1}$ and $\mathcal{G}_{2}$. They are then passed to the H4VDM
feature extractor separately.

Figure 1: The block diagram of our proposed H.264 Video Device Matching
(H4VDM) method. The details of the H4VDM feature extractor are described in
Section 3.1. Note the input GOP information includes the decoded frame and the
coding parameters.

The H4VDM feature extractor is the key component of our proposed method, which
is described extensively in Section 3.1. It extracts important information
from an H.264 GOP $\mathcal{G}$ and expresses this information in a
$D_{\boldsymbol{r}}$-dimensional vector representation $\boldsymbol{r}$ known
as the GOP feature vector. Let $\boldsymbol{r}_{1}$ and $\boldsymbol{r}_{2}$
be the corresponding GOP feature vectors of $\mathcal{G}_{1}$ and
$\mathcal{G}_{2}$, respectively. We compute a similarity score
$s(\boldsymbol{r}_{1},\boldsymbol{r}_{2})$ between the two GOP feature
vectors. The similarity score can be used for classification: a higher
similarity score indicates that $\mathcal{G}_{2}$ is more likely to be
captured by the same device as $\mathcal{G}_{1}$. The similarity score used in
our proposed method is described in Section 3.2.

###  3.1 H4VDM Feature Extractor

Figure 2: The block diagram of the H4VDM feature extractor.

The block diagram of the H4VDM feature extractor is shown in Figure 2. The
feature extractor computes GOP feature vectors using H.264 GOP s containing
$L$ frames, where the size of each frame is $H\times W$. If a GOP is longer
than $L$ frames, we extract features for the first $L$ frames only without
loss of generality. If the frame size of a GOP is larger than $H\times W$, we
extract features from an arbitrary $H\times W$ region in the GOP for our
analysis. Feature extraction is not possible when the length of GOP is less
than $L$ or when the GOP frame size is smaller than $H\times W$. Therefore, it
is important to set $L$, $H$, and $W$ appropriately so that the H4VDM feature
extractor can process GOP s from a wide range of H.264 video sequences.

The H4VDM feature extractor uses five types of data from an H.264 GOP to
generate GOP feature vectors, i.e., the I-frame, the frame differences, the
frame types, the macroblock types and the luma QPs. Each type of data is first
processed by a specific approach to generate an intermediate output denoted by
$\boldsymbol{o}\in\mathbb{R}^{D_{\boldsymbol{o}}}$. The five processing
methods for each data type are known as the I-, DF-, FT-, M-, and L-Proc. Many
of these processing methods use Vision Transformers (ViTs) to process image-
like information. In total, we use two ViT architectures in our proposed
method, i.e., ViT-1 and ViT-2 (as shown in Table 1). ViT-1 is a larger network
that processes more complicated data such as I-frames and residual frames.
ViT-2 is a smaller network that processes simpler data such as macroblock
types and luma Quantization Parameters (QPs). The details of each processing
method are described as below.

#### I-Proc.

This processing step extracts feature from the decoded I-frame in the GOP. The
I-Proc uses the ViT-1 architecture in Table 1. For each GOP, the input to the
I-Proc is an $H\times W\times 3$ vector consisting of the RGB pixel data from
the I-frame. The output of the I-Proc is $\boldsymbol{o}_{1}$.

#### DF-Proc.

This processing step extracts features from differences of the decoded frames
in sequence. The differences we compute are between the decoded frames and the
decoded I-frame in the GOP (including the difference between the I-frame and
itself, which is all zeros). Using difference frames can better enable the DF-
Proc to learn about the characteristics of H.264 compression. The DF-Proc also
uses the ViT-1 architecture in Table 1. For each GOP, the input to the DF-Proc
is $L$ vectors of dimension $H\times W\times 3$, where each vector is the RGB
pixel difference. The $L$ frame difference vectors are processed one by one to
generate $L$ outputs $\boldsymbol{o}_{3},\ldots,\boldsymbol{o}_{L+2}$.

#### FT-Proc.

This step converts frames types into $D_{\boldsymbol{o}}$ dimensional vectors.
For each input GOP, the frame types are a sequence of $L$ positive integers,
each of which represents a valid frame type in H.264 (i.e., I, P, B). The FT-
Proc projects each integer to a $D_{\boldsymbol{o}}$-dimensional real-valued
vector with the widely used embedding technique [49]. In the original integer
representation, each integer is an index for a concept (e.g., frame types).
The distance between two integer indices is not meaningful. By converting the
integer indices into real-valued vectors using a projection learned in the
training phase, the vector representation of similar concepts can have smaller
distances, which makes learning easier for the machine learning method. The
final outputs are $L$ vectors
$\boldsymbol{o}_{L+4},\ldots,\boldsymbol{o}_{2L+3}$.

#### M-Proc.

Here we extract features from the macroblock types. In the original H.264 data
stream, a frame is subdivided into macroblocks of size $16\times 16$. Each
macroblock in the frame can be compressed with different methods. The
compression method used in a macroblock is stored as an integer known as the
macroblock type [34]. This macroblock type information is converted into a
$H\times W\times 1$ vector by “unpacking” the macroblocks. That is, every
pixel in the frame is associated with a macroblock type integer inherited from
the macroblock it belongs to. We use the embedding technique to project each
macrotype integer into a three dimensional real-valued vector, which is then
processed by a ViT-2 network described in Table 1. For the $L$ frames in the
GOP, the output is $L$ vectors
$\boldsymbol{o}_{2L+5},\ldots,\boldsymbol{o}_{3L+4}$.

#### L-Proc.

This step extracts features from the luma QP s. The input is an $H\times
W\times 1$ vector, where each element is an integer ranging from 0 to 51,
i.e., the luma QPs. As the luma QP increases, the H.264 quantization procedure
discards more details in the luma channel in exchange for lower data rate
[50]. Due to the ordered nature of luma QPs, we process them directly using a
ViT-2 network in Table 1. For the $L$ frames in the GOP, the output is $L$
vectors $\boldsymbol{o}_{3L+6},\ldots,\boldsymbol{o}_{4L+5}$.

Table 1: The hyperparameters of the vision transformers (ViT) [40] used in H4VDM, as discussed in Section 4.3.  Hyperparameters | ViT-1 | ViT-2
---|---|---
depth | 8 | 4
projection dimension | $D_{\mathrm{ViT}1}$ | $D_{\mathrm{ViT}2}$
number of MSA heads | 8 | 4
output dimension | $D_{\boldsymbol{o}}$ | $D_{\boldsymbol{o}}$
patch size | 16 | 16

The intermediate outputs from the five processing networks contain important
information about different data types. Similar to [51], we insert special
vectors (i.e., $\boldsymbol{o}_{2}$, $\boldsymbol{o}_{L+3}$,
$\boldsymbol{o}_{2L+4}$, $\boldsymbol{o}_{3L+5}$) to combine the information
acquired from various data types. These special vectors can be updated during
training. In total, there are $4L+5$ intermediate output vectors, which are
used as the input to the joint analysis network. This network is an 8-layer
transformer network [38]. The output of the joint analysis network is linearly
projected to a vector $\boldsymbol{r}\in\mathbb{R}^{D_{\boldsymbol{r}}}$. The
vector $\boldsymbol{r}$ is the output of the H4VDM feature extractor (i.e.,
the GOP feature vector).

Based on the five data types in the input GOP, the H4VDM feature extractor
characterizes macroblock type selection, luma QP selection, and other patterns
that are specific to the video capturing device. This information is contained
in a $D_{\boldsymbol{r}}$-dimensional GOP feature vector. By comparing the
similarity score between the corresponding GOP feature vectors from two video
sequences we can determine if the two video sequences were captured by the
same device. Since H4VDM only requires two H.264 GOP s to make decisions, it
is able to work in scenarios where data from a test device is scarce, e.g.,
when the H.264 video sequence is corrupted and only a few GOP s can be
recovered.

###  3.2 Similarity Score and Loss Function

The similarity score between two GOP feature vectors $\boldsymbol{r}_{1}$ and
$\boldsymbol{r}_{2}$ is a real number in the range $[0,1]$, where $1$
indicates the two vectors are the most similar and $0$ indicates the two
vectors are the most dissimilar. We compute the similarity score using the
following function

$\displaystyle
s(\boldsymbol{r}_{1},\boldsymbol{r}_{2})=1-\tanh\left(\lVert\boldsymbol{r}_{1}-\boldsymbol{r}_{2}\rVert_{2}\right),$
(1)

where $\lVert\cdot\rVert_{2}$ denotes $L_{2}$-norm. The function
$f(x)=1-\tanh(x)$ for $x\geq 0$ is shown in Figure 3. When
$\boldsymbol{r}_{1}$ and $\boldsymbol{r}_{2}$ are more similar,
$\lVert\boldsymbol{r}_{1}-\boldsymbol{r}_{2}\rVert_{2}\rightarrow 0$, which
indicates $s(\boldsymbol{r}_{1},\boldsymbol{r}_{2})\rightarrow 1$. Conversely,
$s(\boldsymbol{r}_{1},\boldsymbol{r}_{2})\rightarrow 0$ when
$\boldsymbol{r}_{1}$ and $\boldsymbol{r}_{2}$ are more dissimilar.

Figure 3: The similarity score function.

We use binary cross-entropy loss and the similarity score (Equation 1) to
compute the loss of the H4VDM method during training. Suppose the ground truth
label of the GOP feature vector pair $(\boldsymbol{r}_{1},\boldsymbol{r}_{2})$
is given by $y$, where $y=1$ indicates the two GOP features are from the same
video capturing device and $y=0$ indicates the opposite. The loss of the pair
is

$\displaystyle\ell(\boldsymbol{r}_{1},\boldsymbol{r}_{2},y)=$ $\displaystyle
y\log\left[s(\boldsymbol{r}_{1},\boldsymbol{r}_{2})\right]+(1-y)\log\left[1-s(\boldsymbol{r}_{1},\boldsymbol{r}_{2})\right].$
(2)

##  4 Experiments and Results

We describe our experiments in this section including the datasets used for
training and testing. More details about H4VDM such as hyperparameter
selection and training strategy are also discussed. Finally, we present the
results from the experiments. We used Area Under the Receiver Operating
Characteristic (AUC) score [52], $F_{1}$-score [53], and accuracy score to
evaluate the performance of H4VDM. Throughout the experiments, we selected the
dimensionality of the GOP feature vectors to be 1024, i.e.,
$D_{\boldsymbol{r}}=1024$.

###  4.1 Dataset Generation

The datasets used in our experiments are generated from the VISION dataset
[54]. The VISION dataset contains 648 H.264 video sequences from 35 different
video capturing devices. The list of devices in VISION is shown in Table 2.

Table 2: The list of devices contained in the VISION dataset [54]. Device ID |  Device Name |  Device ID |  Device Name
---|---|---|---
1 | Samsung˙GalaxyS3Mini | 19 | Apple˙iPhone6Plus (iOS 10.2.1)
2 | Apple˙iPhone4s (iOS 7.1.2) | 20 | Apple˙iPadMini (iOS 8.4)
3 | Huawei˙P9 | 21 | Wiko˙Ridge4G
4 | LG˙D290 | 22 | Samsung˙GalaxyTrendPlus
5 | Apple˙iPhone5c (iOS 10.2.1) | 23 | Asus˙Zenfone2Laser
6 | Apple˙iPhone6 (iOS 8.4) | 24 | Xiaomi˙RedmiNote3
7 | Lenovo˙P70A | 25 | OnePlus˙A3000
8 | Samsung˙GalaxyTab3 | 26 | Samsung˙GalaxyS3Mini
9 | Apple˙iPhone4 (iOS 7.1.2) | 27 | Samsung˙GalaxyS5
10 | Apple˙iPhone4s (iOS 8.4.1) | 28 | Huawei˙P8
11 | Samsung˙GalaxyS3 | 29 | Apple˙iPhone5 (iOS 9.3.3)
12 | Sony˙XperiaZ1Compact | 30 | Huawei˙Honor5c
13 | Apple˙iPad2 (iOS 7.1.1) | 31 | Samsung˙GalaxyS4Mini
14 | Apple˙iPhone5c (iOS 7.0.3) | 32 | OnePlus˙A3003
15 | Apple˙iPhone6 (iOS 10.1.1) | 33 | Huawei˙Ascend
16 | Huawei˙P9Lite | 34 | Apple˙iPhone5 (iOS 8.3)
17 | Microsoft˙Lumia640LTE | 35 | Samsung˙GalaxyTabA
18 | Apple˙iPhone5c (iOS 8.4.1) |  |

We decoded the video frames and extracted the GOP s from video sequences in
VISION using a customized version of the
openh264111https://github.com/cisco/openh264 H.264 decoder, which allows us
access to the GOP information. In our analysis, we selected the length of GOP
$L=8$. That is, our method can analyze H.264 GOP with length greater than or
equal to 8. The height ($H$) and width ($W$) of the frames were set to
$H=W=224$, which is a common size used by popular image or video processing
techniques such as [55, 40, 41, 42]. With this choice of $H$ and $W$, the
frame sizes of all video sequences in the VISION dataset are larger than
$(H,W)$. Therefore, we cropped a $(H,W)$ region from the center of the frames
for analysis. We chose to crop the region at the center because the video
content at the center is more likely to change compared to those from the
edges or corners. From each video sequence in the VISION dataset, we randomly
sampled 15 GOP s whose length is greater than or equal to $L$. When the length
of a sampled GOP is greater than $L$, only the first $L$ frames in the GOP
were used.

We constructed data from the VISION dataset to train and test our method as
follows:

  1. 1.

Provide a set of device indices $\mathcal{S}$, selected from Table 2.

  2. 2.

Select all pairs of device indices $(i,j)\in\mathcal{S}\otimes\mathcal{S}$,
where $\otimes$ denotes Cartesian product. Denote the set of GOP s from device
$i$ and device $j$ by $\mathcal{A}_{i}$ and $\mathcal{A}_{j}$, respectively.
For each $(i,j)$ do the following:

    1. (a)

If $i\neq j$, randomly sample $n_{0}$ unique GOP pairs from the set
$\mathcal{A}_{i}\otimes\mathcal{A}_{j}$ that are not in the dataset. When
determining if a GOP pair is in the dataset, the GOP pair is considered to be
unordered. That is, if one swaps the first and the second element, the pair is
considered to be the same. These $n_{0}$ pairs are assigned label $0$ and
added to the dataset.

    2. (b)

If $i=j$, randomly sample $n_{1}$ unique GOP pairs from the set
$\mathcal{A}_{i}\otimes\mathcal{A}_{j}$ that are not in the dataset. When
determining if a GOP pair is in the dataset, the GOP pair is considered to be
unordered. These $n_{1}$ pairs are assigned label $1$ and added to the
dataset.

In our experiments, we chose $n_{0}=15$ and $n_{1}=120$. To better evaluate
the performance of H4VDM, we constructed 7 datasets, where each dataset
contained a training set and a testing set. For brevity, we refer to these
datasets as D1, D2, …, D7. In each dataset, the set of all device IDs in
VISION are split into two disjoint sets $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$. They are passed to the dataset construction step to
generate the training set and the testing set, respectively. For dataset
D1–D4, $\mathcal{S}_{2}$ contained half of the device IDs that were randomly
selected. For dataset D5–D7, the device IDs are split into three disjoint sets
and used as $\mathcal{S}_{2}$ of each dataset. The details of each dataset are
shown in Table 3. For the testing set of each dataset, we uniformly sampled 40
percent of the GOP pairs after dataset generation to reduce its size. In each
dataset, we uniformly removed $\nicefrac{{1}}{{8}}$ of the GOP pairs from the
testing set for validation during training. The training process is stopped
when the open-set validation performance no longer increases.

Table 3:  The details of each dataset. #0 and #1 indicates the number of GOP
pairs with class 0 (different device) and class 1 (same device), respectively.

Dataset | Test Device IDs ($\mathcal{S}_{2}$) | Training | Testing
---|---|---|---
#0 | #1 | #0 | #1
D1 |  {1, 2, 4, 5, 6, 14, 17, 18, 19, 21, 22, 23, 27, 28, 30, 32, 35} | 4590 | 2160 | 816 | 1632
D2 |  {1, 2, 4, 11, 14, 15, 17, 18, 19, 20, 21, 23, 26, 30, 32, 33, 35} | 4590 | 2160 | 816 | 1632
D3 |  {4, 5, 6, 10, 11, 13, 14, 16, 17, 19, 21, 22, 23, 30, 31, 32, 35} | 4590 | 2160 | 816 | 1632
D4 |  {3, 4, 6, 8, 13, 17, 19, 20, 21, 22, 23, 26, 29, 30, 31, 32, 34} | 4590 | 2160 | 816 | 1632
D5 |  {1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34} | 7590 | 2760 | 792 | 576
D6 |  {2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35} | 7590 | 2760 | 792 | 576
D7 |  {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33} | 8280 | 2880 | 660 | 528

We selected $L=8$ in our experiments based on several factors. Increasing $L$
is likely to improve the quality of GOP feature extraction, because the H4VDM
feature extractor is exposed to more information. However, it will increase
the complexity of the model and make the training process more computationally
intense. Since our method requires the length of the GOP under analysis to be
at least $L$, a larger $L$ value can also reduce the number of valid GOP
candidates in a video sequence. Note that it is computationally expensive to
acquire the decoded frames and GOP information from the H.264 bitstream. In
training, since the training data will be used repeatedly in each training
epoch, it is more efficient to store the decoded frames and GOP information on
disk. Therefore, another factor that affects the choice of $L$ is the storage
size of the uncompressed GOP s and decoded frames. With the current dataset
configurations, each dataset (training set and testing set) requires
approximately 176GB of disk space. Given such an enormous dataset size, the
training process is heavily bounded by the I/O performance of the storage
devices, which is much slower compared to other hardware components (e.g.,
GPUs). In the worst case, the training time can increase linearly with respect
to $L$, which is costly when $L$ is large. We used $L=8$ as a balance among
model performance, model complexity, the usability of our method, training
complexity, and training speed. Typical training times were 3-6 hours using
six 48GB GPUs.

###  4.2 Parameter Initialization and Training

From Figure 3 it can be seen that the gradients of the similarity score
function $f(x)$ will saturate when $x$ is large, which can significantly
reduce the efficiency of gradient-based learning. Therefore, when initializing
the model parameters, we limited the scale of the weights in the output layer
to be within $[-0.002,0.002]$. This reduced the initial scale of
$\lVert\boldsymbol{r}_{1}-\boldsymbol{r}_{2}\rVert_{2}$, which can facilitate
training.

We trained the H4VDM method using the Adam optimizer [56] with a minibatch
size of 72. We used 5 warm-up epochs, where the learning rate linearly
increased from $0$ to $8\times 10^{-6}$. After warm-up epochs, we trained the
method using an initial learning rate of $8\times 10^{-6}$. An exponential
learning rate decay was used with a decay factor of $0.97$. During training,
we monitored the AUC score on the validation set. The training was stopped
when the validation AUC score no longer increased.

###  4.3 Model Size Selection

We tuned the hyperparameters $D_{\mathrm{ViT1}}$ and $D_{\boldsymbol{o}}$ to
control the size of the model. We constructed 3 models of various sizes (i.e.,
S (small), B (baseline), and L (large)). For each model, we computed the best
AUC score on the testing set of Dataset D1. The results are shown in Table 4.
Since the H4VDM-B model achieved the best AUC score, further experiments were
conducted with this model.

Table 4: The best AUC scores of various models on the testing set of Dataset D1. Model | $D_{\mathrm{ViT}1}$ | $D_{\mathrm{ViT}2}$ | $D_{\boldsymbol{o}}$ | Parameters | AUC
---|---|---|---|---|---
H4VDM-S | 192 | 64 | 192 | 48.93M | 79.8
H4VDM-B | 256 | 64 | 256 | 80.10M | 80.2
H4VDM-L | 320 | 64 | 320 | 118.95M | 69.4

###  4.4 Results

#### Results on Datasets D1–D4.

In Table 5, we show the performance of the H4VDM-B model on datasets D1–D4.
When computing the $F_{1}$-score, we selected the threshold such that the sum
of True Positive Rate (TPR) and True Negative Rate (TNR) is maximized. Our
method achieved an overall $F_{1}$-score of 67.2 and an average AUC score of
77.4 on datasets D1–D4. Overall, for class 0 (GOP pairs from different
devices) the precision is high, which means most retrieved GOP pairs are
relevant. For class 1 (GOP pairs from the same device) the recall is high,
which means most relevant GOP pairs are retrieved. In Figure 4, we show the
accuracy score matrix of device index pairs of each dataset in testing. From
this figure, it can be seen that H4VDM can match devices at a firmware level.
For example, device 29 and 34 in dataset D4 are the same device (Apple
iPhone5) with different operating system versions. H4VDM was able to
distinguish them with high accuracy.

Table 5: The testing performance of the H4VDM-B model on datasets D1–D4. Dataset | Class 0 | Class 1 | All Classes
---|---|---|---
Pre. | Rec. | $F_{1}$ | Pre. | Rec. | $F_{1}$ | Pre. | Rec. | $F_{1}$ | AUC
D1 | 95.9 | 53.6 | 68.8 | 50.6 | 95.4 | 66.1 | 80.8 | 67.5 | 67.9 | 80.2
D2 | 91.7 | 49.7 | 64.5 | 47.6 | 91.1 | 62.5 | 77.0 | 63.5 | 63.8 | 74.1
D3 | 89.9 | 55.4 | 68.5 | 49.4 | 87.5 | 63.2 | 76.4 | 66.1 | 66.8 | 78.0
D4 | 86.0 | 64.0 | 73.4 | 52.3 | 79.1 | 63.0 | 74.8 | 69.0 | 69.9 | 77.3
Overall | 90.5 | 55.7 | 68.9 | 49.9 | 88.3 | 63.7 | 76.9 | 66.5 | 67.2 | 77.4
Pre.=Precision Rec.=Recall
Figure 4: The testing accuracy score matrix of device index pairs on datasets
D1–D4.

#### Results on Datasets D5–D7.

From the results of H4VDM on datasets D1–D4 (Figure 4), it can be seen that
the performance of H4VDM was low for specific device pairs. This may be caused
by the fact that datasets D1–D4 contain only a small number of devices in the
training set, which makes it difficult for the method to generalize to a
broader scope of devices. To test the performance of H4VDM on datasets with
more training devices, we trained and tested H4VDM on datasets D5–D7, whose
training sets contain more devices in the VISION dataset. The testing
performance of the H4VDM-B model on datasets D5–D7 is shown in Table 6. The
accuracy score matrix of device index pairs is shown in Figure 5. On datasets
D5–D7, our method achieved an overall $F_{1}$-score of 78.6 and an average AUC
score of 85.2. It can be seen that as the number of devices in the training
set increases, the performance of H4VDM becomes better. However, since the
number of devices in the testing set decreases, the testing performance can
vary significantly depending on the choice of the testing devices. Overall,
the results on datasets D1–D4 show that H4VDM can learn to achieve open-set
VDM given a small number of training devices. The results on datasets D5–D7
show that the performance of H4VDM can improve quickly as more devices are
available in training.

Table 6:  The testing performance of the H4VDM-B model on datasets D5–D7.  Dataset | Class 0 | Class 1 | All Classes
---|---|---|---
Pre. | Rec. | $F_{1}$ | Pre. | Rec. | $F_{1}$ | Pre. | Rec. | $F_{1}$ | AUC
D5 | 85.9 | 74.5 | 79.8 | 75.5 | 86.5 | 80.6 | 80.9 | 80.2 | 80.2 | 87.0
D6 | 96.1 | 73.4 | 83.2 | 76.9 | 96.7 | 85.7 | 86.9 | 84.5 | 84.4 | 90.0
D7 | 94.8 | 48.8 | 64.4 | 65.5 | 97.3 | 78.3 | 80.1 | 73.0 | 71.3 | 78.5
Overall | 92.3 | 65.6 | 75.8 | 72.6 | 93.5 | 81.5 | 82.6 | 79.2 | 78.6 | 85.2
Pre.=Precision Rec.=Recall
Figure 5: The testing accuracy score matrix of device index pairs on datasets
D5–D7.

##  5 Conclusion

In this paper we proposed an H.264-based open-set VDM method known as H4VDM.
H4VDM uses transformer neural networks to process five types of data from the
H.264 decoded frames and the GOP s. We trained and tested the H4VDM-B model on
datasets generated from the VISION dataset [54]. The experimental results
showed that H4VDM demonstrated good VDM performance on unseen devices.

Despite the good performance, H4VDM has still room for improvement. When
selecting hyperparameters, we greedily used the model that had the best
performance on Dataset D1. The selected model may not have the best overall
performance across all datasets. Some important H.264 codec information such
as motion vectors and true prediction residuals were not used in H4VDM. The
small number of devices in the VISION dataset also limited the performance
evaluation of H4VDM. On datasets D1–D4, less dataset bias is introduced in
training/testing split, but the performance of H4VDM is relatively low due to
small number of devices in the training set. On datasets D5–D7, the performane
of H4VDM is higher, but the influence of dataset bias is stronger due to the
small number of devices in the testing set, which resulted in fluctuating
testing perforamnce. A dataset with more devices is required to evaluate the
performance of H4VDM more comprehensively.

In future work, we will examine the use of motion vectors and prediction
residuals. We will develop methods that use popular deep learning frameworks
efficiently so that the training speed is less constrained by hardware I/O
speed. We will collect video data from more video capturing devices for future
video forensics research. We are investigating other video compression
techniques including H.265, H.266, VP9, and AV1.

## Acknowledgments

This material is based on research sponsored by the Defense Advanced Research
Projects Agency (DARPA) and Air Force Research Laboratory (AFRL) under
agreement number FA8750-20-2-1004. The U.S. Government is authorized to
reproduce and distribute reprints for Governmental purposes notwithstanding
any copyright notation thereon. The views and conclusions contained herein are
those of the authors and should not be interpreted as necessarily representing
the official policies or endorsements, either expressed or implied, of DARPA,
AFRL or the U.S. Government. Address all correspondence to Edward J. Delp,
<EMAIL_ADDRESS>
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# Interacting holographic dark energy in Brans-Dicke theory

Ahmad Sheykhi<EMAIL_ADDRESS>Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran

###### Abstract

We study cosmological application of interacting holographic energy density in
the framework of Brans-Dicke cosmology. We obtain the equation of state and
the deceleration parameter of the holographic dark energy in a non-flat
universe. As system’s IR cutoff we choose the radius of the event horizon
measured on the sphere of the horizon, defined as $L=ar(t)$. We find that the
combination of Brans-Dicke field and holographic dark energy can accommodate
$w_{D}=-1$ crossing for the equation of state of noninteracting holographic
dark energy. When an interaction between dark energy and dark matter is taken
into account, the transition of $w_{D}$ to phantom regime can be more easily
accounted for than when resort to the Einstein field equations is made.

##  I Introduction

Recent data from type Ia supernova, cosmic microwave background (CMB)
radiation, and other cosmological observations suggest that our universe is
currently experiencing a phase of accelerated expansion and nearly three
quarters of the universe consists of dark energy with negative pressure Rie .
Nevertheless, the nature of such a dark energy is still the source of much
debate. Despite the theoretical difficulties in understanding dark energy,
independent observational evidence for its existence is impressively robust.
Explanations have been sought within a wide range of physical phenomena,
including a cosmological constant, exotic fields, a new form of the
gravitational equation, new geometric structures of spacetime, etc, see Pad
for a recent review. One of the dramatic candidate for dark energy, that arose
a lot of enthusiasm recently, is the so-called “Holographic Dark Energy” (HDE)
proposal. This model is based on the holographic principle which states that
the number of degrees of freedom of a physical system should scale with its
bounding area rather than with its volume Suss1  and it should be constrained
by an infrared cutoff Coh . On these basis, Li Li  suggested the following
constraint on its energy density $\rho_{D}\leq 3c^{2}m^{2}_{p}/L^{2}$, the
equality sign holding only when the holographic bound is saturated. In this
expression $c^{2}$ is a dimensionless constant, $L$ denotes the IR cutoff
radius and $m^{2}_{p}=(8\pi G)^{-1}$ stands for the reduced Plank mass. Based
on cosmological state of holographic principle, proposed by Fischler and
Susskind Suss2 , the HDE models have been proposed and studied widely in the
literature Huang ; Hsu ; HDE ; Setare ; Seta1 ; Setare1 . The HDE model has
also been tested and constrained by various astronomical observations Xin ;
Feng  as well as by the anthropic principle Huang1 . It is fair to claim that
simplicity and reasonability of HDE model provides more reliable frame to
investigate the problem of dark energy rather than other models proposed in
the literature. For example, the coincidence problem can be easily solved in
some models of HDE based on the fundamental assumption that matter and HDE do
not conserve separately Pav1 .

On the other side, scalar-tensor theories of gravity have been widely applied
in cosmology Fara . Scalar-tensor theories are not new and have a long
history. The pioneering study on scalar-tensor theories was done by Brans and
Dicke several decades ago who sought to incorporate Mach’s principle into
gravity BD . In recent years this theory got a new impetus as it arises
naturally as the low energy limit of many theories of quantum gravity such as
superstring theory or Kaluza-Klein theory. Because the holographic energy
density belongs to a dynamical cosmological constant, we need a dynamical
frame to accommodate it instead of general relativity. Therefore it is
worthwhile to investigate the HDE model in the framework of the Brans-Dicke
theory. The studies on the HDE model in the framework of Brans-Dicke cosmology
have been carried out in Pavon2 ; Setare2 ; other . The purpose of the present
paper is to construct a cosmological model of late acceleration based on the
Brans-Dicke theory of gravity and on the assumption that the pressureless dark
matter and HDE do not conserve separately but interact with each other. Given
the unknown nature of both dark matter and dark energy there is nothing in
principle against their mutual interaction and it seems very special that
these two major components in the universe are entirely independent. Indeed,
this possibility is receiving growing attention in the literature Ame ; Zim ;
wang1 ; wang2 ; wang3  and appears to be compatible with SNIa and CMB data Oli
. On the other hand, although it is believed that our universe is spatially
flat, a contribution to the Friedmann equation from spatial curvature is still
possible if the number of e-foldings is not very large Huang . Besides, some
experimental data has implied that our universe is not a perfectly flat
universe and recent papers have favored the universe with spatial curvature
spe .

In the light of all mentioned above, it becomes obvious that the investigation
on the interacting HED in the framework of non-flat Brans-Dicke cosmology is
well motivated. We will show that the equation of state of dark energy can
accommodate $w_{D}=-1$ crossing. As systems’s IR cutoff we shall choose the
radius of the event horizon measured on the sphere of the horizon, defined as
$L=ar(t)$. Our work differs from that of Ref. Pavon2  in that we take
$L=ar(t)$ as the IR cutoff not the Hubble radius $L=H^{-1}$. It also differs
from that of Ref. Setare2 , in that we assume the pressureless dark matter and
HDE do not conserve separately but interact with each other, while the author
of Setare2  assumes that the dark components do not interact with each other.

This paper is outlined as follows: In section II, we consider noninteracting
HDE model in the framework of Brans-Dicke cosmology in a non-flat universe. In
section III, we extend our study to the case where there is an interaction
term between dark energy and dark matter. We summarize our results in section
IV.

##  II HDE in Branse-Dicke cosmology

The action of Brans-Dicke theory is given by

$S=\int{d^{4}x\sqrt{g}\left(-\varphi{R}+\frac{\omega}{\varphi}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+L_{M}\right)}.$
(1)

The above action can be transformed into the standard canonical form by re-
defining the scalar field $\varphi$ and introducing a new field $\phi$, in
such a way that

$\varphi=\frac{\phi^{2}}{8\omega}.$ (2)

Therefore, in the canonical form, the action of Brans-Dicke theory can be
written Arik

$S=\int{d^{4}x\sqrt{g}\left(-\frac{1}{8\omega}\phi^{2}{R}+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+L_{M}\right)},$
(3)

where ${R}$ is the scalar curvature and $\phi$ is the Brans-Dicke scalar
field. The non-minimal coupling term $\phi^{2}R$ replaces with the Einstein-
Hilbert term ${R}/{G}$ in such a way that
$G^{-1}_{\mathrm{eff}}={2\pi\phi^{2}}/{\omega}$, where $G_{\mathrm{eff}}$ is
the effective gravitational constant as long as the dynamical scalar field
$\phi$ varies slowly. The signs of the non-minimal coupling term and the
kinetic energy term are properly adopted to $(+---)$ metric signature. The HDE
model will be accommodated in the non-flat Friedmann-Robertson-Walker (FRW)
universe which is described by the line element

$\displaystyle
ds^{2}=dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(4)

where $a(t)$ is the scale factor, and $k$ is the curvature parameter with
$k=-1,0,1$ corresponding to open, flat, and closed universes, respectively. A
closed universe with a small positive curvature ($\Omega_{k}\simeq 0.01$) is
compatible with observations spe . Varying action (3) with respect to metric
(4) for a universe filled with dust and HDE yields the following field
equations

$\displaystyle\frac{3}{4\omega}\phi^{2}\left(H^{2}+\frac{k}{a^{2}}\right)-\frac{1}{2}\dot{\phi}^{2}+\frac{3}{2\omega}H\dot{\phi}\phi=\rho_{M}+\rho_{D},$
(5)
$\displaystyle\frac{-1}{4\omega}\phi^{2}\left(2\frac{{\ddot{a}}}{a}+H^{2}+\frac{k}{a^{2}}\right)-\frac{1}{\omega}H\dot{\phi}\phi-\frac{1}{2\omega}\ddot{\phi}\phi-\frac{1}{2}\left(1+\frac{1}{\omega}\right)\dot{\phi}^{2}=p_{D},$
(6)
$\displaystyle\ddot{\phi}+3H\dot{\phi}-\frac{3}{2\omega}\left(\frac{{\ddot{a}}}{a}+H^{2}+\frac{k}{a^{2}}\right)\phi=0,$
(7)

where $H=\dot{a}/a$ is the Hubble parameter, $\rho_{D}$ and $p_{D}$ are,
respectively, the energy density and pressure of dark energy. We further
assume the energy density of pressureless matter can be separated as
$\rho_{M}=\rho_{BM}+\rho_{DM}$, where $\rho_{BM}$ and $\rho_{DM}$ are the
energy density of baryonic and dark matter, respectively. We also assume the
holographic energy density has the following form

$\rho_{D}=\frac{3c^{2}\phi^{2}}{4\omega L^{2}},$ (8)

where $\phi^{2}={\omega}/({2\pi G_{\mathrm{eff}}})$. In the limit of Einstein
gravity where $G_{\mathrm{eff}}\rightarrow G$, the above expression reduces to
the holographic energy density in standard cosmology

$\rho_{D}=\frac{3c^{2}}{8\pi GL^{2}}=\frac{3c^{2}m^{2}_{p}}{L^{2}}.$ (9)

The radius $L$ is defined as

$L=ar(t),$ (10)

where the function $r(t)$ can be obtained from the following relation

$\int_{0}^{r(t)}{\frac{dr}{\sqrt{1-kr^{2}}}}=\int_{0}^{\infty}{\frac{dt}{a}}=\frac{R_{h}}{a}.$
(11)

It is important to note that in the non-flat universe the characteristic
length which plays the role of the IR-cutoff is the radius $L$ of the event
horizon measured on the sphere of the horizon and not the radial size $R_{h}$
of the horizon. Solving the above equation for the general case of the non-
flat FRW universe, we have

$r(t)=\frac{1}{\sqrt{k}}\sin y,$ (12)

where $y=\sqrt{k}R_{h}/a$. Now we define the critical energy density,
$\rho_{\mathrm{cr}}$, and the energy density of the curvature, $\rho_{k}$, as

$\displaystyle\rho_{\mathrm{cr}}=\frac{3\phi^{2}H^{2}}{4\omega},\hskip
22.76228pt\rho_{k}=\frac{3k\phi^{2}}{4\omega a^{2}}.$ (13)

We also introduce, as usual, the fractional energy densities such as

$\displaystyle\Omega_{M}$ $\displaystyle=$
$\displaystyle\frac{\rho_{M}}{\rho_{\mathrm{cr}}}=\frac{4\omega\rho_{M}}{3\phi^{2}H^{2}},$
(14) $\displaystyle\Omega_{k}$ $\displaystyle=$
$\displaystyle\frac{\rho_{k}}{\rho_{\mathrm{cr}}}=\frac{k}{H^{2}a^{2}},$ (15)
$\displaystyle\Omega_{D}$ $\displaystyle=$
$\displaystyle\frac{\rho_{D}}{\rho_{\mathrm{cr}}}=\frac{c^{2}}{H^{2}L^{2}}.$
(16)

For latter convenience we rewrite Eq. (16) in the form

$\displaystyle HL=\frac{c}{\sqrt{\Omega_{D}}}.$ (17)

Taking derivative with respect to the cosmic time $t$ from Eq. (10) and using
Eqs. (12) and (17) we obtain

$\displaystyle\dot{L}=HL+a\dot{r}(t)=\frac{c}{\sqrt{\Omega_{D}}}-\cos y.$ (18)

Consider the FRW universe filled with dark energy and pressureless matter
which evolves according to their conservation laws

$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=0,$ (19)
$\displaystyle\dot{\rho}_{M}+3H\rho_{M}=0,$ (20)

where $w_{D}$ is the equation of state parameter of dark energy. We shall
assume that Brans-Dicke field can be described as a power law of the scale
factor, $\phi\propto a^{\alpha}$. A case of particular interest is that when
$\alpha$ is small whereas $\omega$ is high so that the product $\alpha\omega$
results of order unity Pavon2 . This is interesting because local astronomical
experiments set a very high lower bound on $\omega$ Will ; in particular, the
Cassini experiment implies that $\omega>10^{4}$ Bert ; Aca . Taking the
derivative with respect to time of relation $\phi\propto a^{\alpha}$ we get

$\displaystyle\dot{\phi}=\alpha H\phi,$ (21)
$\displaystyle\ddot{\phi}=\alpha^{2}H^{2}\phi+\alpha\phi\dot{H}.$ (22)

Taking the derivative of Eq. (8) with respect to time and using Eqs. (18) and
(21) we reach

$\displaystyle\dot{\rho}_{D}=2H\rho_{D}\left(\alpha-1+\frac{\sqrt{\Omega_{D}}}{c}\cos
y\right).$ (23)

Inserting this equation in conservation law (19), we obtain the equation of
state parameter

$\displaystyle
w_{D}=-\frac{1}{3}-\frac{2\alpha}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}\cos y.$
(24)

It is important to note that in the limiting case $\alpha=0$
($\omega\rightarrow\infty$), the Brans-Dicke scalar field becomes trivial and
Eq. (24) reduces to its respective expression in non-flat standard cosmology
Huang

$\displaystyle w_{D}=-\frac{1}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}\cos y.$ (25)

We will see that the combination of the Brans-Dicke field and HDE brings rich
physics. For $\alpha\geq 0$, $w_{D}$ is bounded from below by

$\displaystyle
w_{D}=-\frac{1}{3}-\frac{2\alpha}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}.$ (26)

If we take $\Omega_{D}=0.73$ for the present time and choosing $c=1$ c , the
lower bound becomes $w_{D}=-\frac{2\alpha}{3}-0.9$. Thus for $\alpha=0.15$ we
have $w_{D}=-1$. The cases with $\alpha>0.15$ and $\alpha<0.15$ should be
considered separately. In the first case where $\alpha>0.15$ we have
$w_{D}<-1$. This is an interesting result and shows that, theoretically, the
combination of Brans-Dicke scalar field and HDE can accommodate $w_{D}=-1$
crossing for the equation of state of dark energy. Therefore one can generate
phantom-like equation of state from a noninteracting HDE model in the Brans-
Dicke cosmology framework. This is in contrast to the general relativity where
the equation of state of a noninteracting HDE cannot cross the phantom divide
Li . In the second case where $0\leq\alpha<0.15$ we have $-1<w_{D}\leq-0.9$.
Since $\alpha\approx 1/{\omega}$ and for $\omega\geq 10^{4}$ the Brans-Dicke
theory is consistent with solar system observations Bert , thus practically
$\alpha\simeq 10^{-4}$ is compatible with recent cosmological observations
which implies $w_{D}\simeq-0.903$ for the present time in this model. In both
cases discussed above $w_{D}<-1/3$ and the universe undergoing a phase of
accelerated expansion. It is worthwhile to note that since $\alpha\approx
1/\omega$ and $\omega>10^{4}$, therefore for all practical purposes Brans-
Dicke theory reduces to Einstein gravity as one can see from the above
discussion.

For completeness, we give the deceleration parameter

$\displaystyle q=-\frac{\ddot{a}}{aH^{2}}=-1-\frac{\dot{H}}{H^{2}},$ (27)

which combined with the Hubble parameter and the dimensionless density
parameters form a set of useful parameters for the description of the
astrophysical observations. Dividing Eq. (6) by $H^{2}$, and using Eqs. (8),
(17), (21) and (22), we find

$\displaystyle
q=\frac{1}{2\alpha+2}\left[(2\alpha+1)^{2}+2\alpha(\alpha\omega-1)+\Omega_{k}+3\Omega_{D}w_{D}\right].$
(28)

Substituting $w_{D}$ from Eq. (24), we get

$\displaystyle
q=\frac{1}{2\alpha+2}\left[(2\alpha+1)^{2}+2\alpha(\alpha\omega-1)+\Omega_{k}-(2\alpha+1)\Omega_{D}-\frac{2}{c}{\Omega^{3/2}_{D}}\cos
y\right].$ (29)

If we take $\Omega_{D}=0.73$ and $\Omega_{k}\approx 0.01$ for the present time
and choosing $c=1$, $\alpha\omega\approx 1$, $\omega=10^{4}$ and $\cos y\simeq
1$, we obtain $q=-0.48$ for the present value of the deceleration parameter
which is in good agreement with recent observational results Daly . When
$\alpha\rightarrow 0$, Eq. (29) restores the deceleration parameter for HDE
model in Einstein gravity wang2

$\displaystyle
q=\frac{1}{2}(1+\Omega_{k})-\frac{\Omega_{D}}{2}-\frac{\Omega^{3/2}_{D}}{c}\cos
y.$ (30)

##  III Interacting HDE in Branse-Dicke cosmology

In this section we extend our study to the case where both dark components-
the pressureless dark matter and the HDE- do not conserve separately but
interact with each other. Although at this point the interaction may look
purely phenomenological but different Lagrangians have been proposed in
support of it Tsu . Besides, in the absence of a symmetry that forbids the
interaction there is nothing, in principle, against it. Further, the
interacting dark energy has been investigated at one quantum loop with the
result that the coupling leaves the dark energy potential stable if the former
is of exponential type but it renders it unstable otherwise Dor . Therefore,
microphysics seems to allow enough room for the coupling. With the interaction
between the two dark constituents of the universe, we explore the evolution of
the universe. The total energy density satisfies a conservation law

$\dot{\rho}+3H(\rho+p)=0.$ (31)

where $\rho=\rho_{M}+\rho_{D}$ and $p=p_{D}$. However, since we consider the
interaction between dark energy and dark matter, $\rho_{DM}$ and $\rho_{D}$ do
not conserve separately. They must rather enter the energy balances Pav1

$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q,$ (32)
$\displaystyle\dot{\rho}_{DM}+3H\rho_{DM}=Q,$ (33)
$\displaystyle\dot{\rho}_{BM}+3H\rho_{BM}=0,$ (34)

where we have assumed the baryonic matter does not interact with dark energy.
Here $Q$ denotes the interaction term and we take it as
$Q=3b^{2}H(\rho_{DM}+\rho_{D})$ with $b^{2}$ is a coupling constant. This
expression for the interaction term was first introduced in the study of the
suitable coupling between a quintessence scalar field and a pressureless cold
dark matter field Ame ; Zim . The choice of the interaction between both
components was meant to get a scaling solution to the coincidence problem such
that the universe approaches a stationary stage in which the ratio of dark
energy and dark matter becomes a constant. In the context of HDE models, this
form of interaction was derived from the choice of Hubble scale as the IR
cutoff Pav1 .

Combining Eqs. (13) and (21) with the first Friedmann equation (5), we can
rewrite this equation as

$\displaystyle\rho_{\mathrm{cr}}+\rho_{k}=\rho_{BM}+\rho_{DM}+\rho_{D}+\rho_{\phi},$
(35)

where we have defined

$\displaystyle\rho_{\phi}\equiv\frac{1}{2}\alpha
H^{2}\phi^{2}\left(\alpha-\frac{3}{\omega}\right).$ (36)

Dividing Eq. (35) by $\rho_{\mathrm{cr}}$, this equation can be written as

$\displaystyle\Omega_{BM}+\Omega_{DM}+\Omega_{D}+\Omega_{\phi}=1+\Omega_{k},$
(37)

where

$\displaystyle\Omega_{\phi}=\frac{\rho_{\phi}}{\rho_{\mathrm{cr}}}=-2\alpha\left(1-\frac{\alpha\omega}{3}\right).$
(38)

Thus, we can rewrite the interaction term $Q$ as

$\displaystyle Q=3b^{2}H(\rho_{DM}+\rho_{D})=3b^{2}H\rho_{D}(1+r),$ (39)

where $r={\rho_{DM}}/{\rho_{D}}$ is the ratio of the energy densities of two
dark components,

$\displaystyle
r=\frac{\Omega_{DM}}{\Omega_{D}}=-1+\frac{1}{\Omega_{D}}\left[1+\Omega_{k}-\Omega_{BM}+2\alpha\left(1-\frac{\alpha\omega}{3}\right)\right].$
(40)

Using the continuity equation (34), it is easy to show that

$\displaystyle\Omega_{BM}=\Omega_{BM0}a^{-3}=\Omega_{BM0}(1+z)^{3},$ (41)

where $\Omega_{BM0}\approx 0.04$ is the present value of the fractional energy
density of the baryonic matter and $z=a^{-1}-1$ is the red shift parameter.
Inserting Eqs. (23), (39) and (40) in Eq. (32) we obtain the equation of state
parameter

$\displaystyle
w_{D}=-\frac{1}{3}-\frac{2\alpha}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}\cos
y-b^{2}{\Omega^{-1}_{D}}\left[1+\Omega_{k}-\Omega_{BM}+2\alpha\left(1-\frac{\alpha\omega}{3}\right)\right].$
(42)

If we define, following Setare1 , the effective equation of state as

$\displaystyle w^{\mathrm{eff}}_{D}=w_{D}+\frac{\Gamma}{3H},$ (43)

where $\Gamma=3b^{2}(1+r)H$. Then, the continuity equation (32) for the dark
energy can be written in the standard form

$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w^{\mathrm{eff}}_{D})=0.$ (44)

Substituting Eq. (42) in Eq. (43), we find

$\displaystyle
w^{\mathrm{eff}}_{D}=-\frac{1}{3}-\frac{2\alpha}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}\cos
y,$ (45)

From Eq. (45) we see that with the combination of Brans-Dicke field and HDE,
the effective equation of state, $w^{\mathrm{eff}}_{D}$, can cross the phantom
divide. For instance, taking $\Omega_{D}=0.73$ for the present time and $c=1$,
the lower bound of Eq. (45) is $w^{\mathrm{eff}}_{D}=-\frac{2\alpha}{3}-0.9$.
Thus for $\alpha>0.15$ we have $w^{\mathrm{eff}}_{D}<-1$. Therefore, the
Brans- Dicke field plays a crucial role in determining the behaviour of the
effective equation of state of interacting HDE. It is important to note that
in standard HDE ($\alpha=0$) it is impossible to have $w^{\mathrm{eff}}_{D}$
crossing $-1$ Setare1 . Returning to the general case (42), we see that when
the interacting HDE is combined with the Brans-Dicke scalar field the
transition from normal state where $w_{D}>-1$ to the phantom regime where
$w_{D}<-1$ for the equation of state of interacting dark energy can be more
easily achieved for than when resort to the Einstein field equations is made.
In the absence of the Brans- Dicke field ($\alpha=0$), Eq. (42) restores its
respective expression in non-flat standard cosmology wang2

$\displaystyle w_{D}=-\frac{1}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}\cos
y-b^{2}{\Omega_{D}}^{-1}\left(1+\Omega_{k}-\Omega_{BM}\right).$ (46)

Next, we examine the deceleration parameter, $q=-\ddot{a}/(aH^{2})$.
Substituting $w_{D}$ from Eq. (42) in Eq. (28), one can easily show

$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha+2}\left[(2\alpha+1)^{2}+2\alpha(\alpha\omega-1)+\Omega_{k}-(2\alpha+1)\Omega_{D}-\frac{2}{c}{\Omega^{3/2}_{D}}\cos
y\right.\ $ (47)
$\displaystyle\left.-3b^{2}\left(1+\Omega_{k}-\Omega_{BM}+2\alpha\left(1-\frac{\alpha\omega}{3}\right)\right)\right].$

If we take $\Omega_{D}=0.73$ and $\Omega_{k}\approx 0.01$ for the present time
and $c=1$, $\alpha\approx 1/\omega$, $\omega=10^{4}$, $\cos y\simeq 1$,
$\Omega_{BM}\approx 0.04$ and $b=0.1$, we obtain $q=-0.5$ which is again
compatible with recent observational data Daly . When $\alpha=0$, Eq. (47)
reduces to the deceleration parameter of the interacting HDE in Einstein
gravity wang2

$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\Omega_{k})-\frac{\Omega_{D}}{2}-\frac{{\Omega^{3/2}_{D}}}{c}\cos
y-\frac{3b^{2}}{2}\left(1+\Omega_{k}-\Omega_{BM}\right).$ (48)

We can also obtain the evolution behavior of the dark energy. Taking the
derivative of Eq. (16) and using Eq. (18) and relation
${\dot{\Omega}_{D}}=H{\Omega^{\prime}_{D}}$, we find

$\displaystyle{\Omega^{\prime}_{D}}=2\Omega_{D}\left(-\frac{\dot{H}}{H^{2}}-1+\frac{\sqrt{\Omega_{D}}}{c}\cos
y\right),$ (49)

where the dot is the derivative with respect to time and the prime denotes the
derivative with respect to $x=\ln{a}$. Using relation
$q=-1-\frac{\dot{H}}{H^{2}}$, we have

$\displaystyle{\Omega^{\prime}_{D}}=2\Omega_{D}\left(q+\frac{\sqrt{\Omega_{D}}}{c}\cos
y\right),$ (50)

where $q$ is given by Eq. (47). This equation describes the evolution behavior
of the interacting HDE in Brans-Dicke cosmology framework. In the limit of
standard cosmology ($\alpha=0$), Eq. (50) reduces to its respective expression
in HDE model wang2

$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left[(1-\Omega_{D})\left(1+\frac{2\sqrt{\Omega_{D}}}{c}\cos
y\right)-3b^{2}(1+\Omega_{k}-\Omega_{BM})+\Omega_{k}\right].$ (51)

For flat universe, $\Omega_{k}=0$, and Eq. (51) recovers exactly the result of
wang1 .

##  IV Summary and discussion

In summary, we studied the interacting holographic model of dark energy in the
framework of Brans-Dicke cosmology where the HDE density
$\rho_{D}={3c^{2}}/(8\pi GL^{2})$ is replaced with
$\rho_{D}={3c^{2}\phi^{2}}/({4\omega L^{2}})$. Here $\phi^{2}={\omega}/({2\pi
G_{\mathrm{eff}}})$, where $G_{\mathrm{eff}}$ is the time variable Newtonian
constant. In the limit of Einstein gravity, $G_{\mathrm{eff}}\rightarrow G$.
With this replacement in Brans-Dicke theory, we found that the accelerated
expansion will be more easily achieved for than when the standard HDE is
considered. We obtained the equation of state and the deceleration parameter
of the HDE in a non-flat universe enclosed by the event horizon measured on
the sphere of the horizon with radius $L=ar(t)$. Interestingly enough, we
found that the combination of Brans-Dicke and HDE can accommodate $w_{D}=-1$
crossing for the equation of state of noninteracting dark energy. For
instance, taking $\Omega_{D}=0.73$ for the present time and $c=1$, the lower
bound of $w_{D}$ becomes $w_{D}=-\frac{2\alpha}{3}-0.9$. Thus for
$\alpha>0.15$ we have $w_{D}<-1$. This is in contrast to Einstein gravity
where the equation of state of noninteracting HDE cannot cross the phantom
divide $w_{D}=-1$ Li . When the interaction between dark energy and dark
matter is taken into account, the transition from normal state where
$w_{D}>-1$ to the phantom regime where $w_{D}<-1$ for the equation of state of
HDE can be more easily accounted for than when resort to the Einstein field
equations is made.

In Brans-Dicke theory of interacting HDE, the properties of HDE is determined
by the parameters $c$, $b$ and $\alpha$ together. These parameters would be
obtained by confronting with cosmic observational data. In this work we just
restricted our numerical fitting to limited observational data. Giving the
wide range of cosmological data available, in the future we expect to further
constrain our model parameter space and test the viability of our model. The
issue is now under investigation and will be addressed elsewhere.

###### Acknowledgements.

I thank the anonymous referee for constructive comments. This work has been
supported by Research Institute for Astronomy and Astrophysics of Maragha,
Iran.

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