# A Lattice-Based Embedding Method for Reversible Audio Watermarking

Junren Qin, Shanxiang Lyu, Jiarui Deng, Xingyuan Liang, Shijun Xiang, Hao Chen
The authors are with the College of Information Science and Technology, and
the College of Cyber Security, Jinan University, Guangzhou 510632, China.
Corresponding author: Shanxiang Lyu (E-mail: lsx07@jnu.edu.cn).

###### Abstract

Reversible audio watermarking (RAW) is a promising technique in various
applications. To simultaneously meet the demand of achieving high
imperceptibility and robustness, this paper proposes a novel RAW scheme based
on lattices. The scheme is referred to as Meet-in-the-Middle Embedding (MME),
in which the lattice quantization errors are properly scaled and added back to
the quantized host signals. Simulations show that MME excels in a wide range
of metrics including signal-to-watermark ratio (SWR), objective difference
grade (ODG), and bit error rate (BER).

###### Index Terms:

Reversible Audio Watermarking (RAW), lattices, signal-to-watermark ratio
(SWR), bit error rate (BER).

##  1 Introduction

Reversible data hiding (RDH) is a technique where a payload is embedded into
host data such that the consistency of the host is perfectly preserved and the
host data are restored during the extraction of the payload [1]. Nowadays,
audio is widely spread on the Internet and has been one of the main digital
multimedia carriers. Its associated Reversible Audio Watermarking (RAW)
techniques are beneficial for archiving, transmitting, and authenticating
high-quality audio data that contains metadata, secret data, and so forth [2].

Similar to other types of multimedia, a critical step of RAW is to design a
proper reversible function. Varying on how the reversible function is
designed, conventional RDH methods include difference expansion (DE) [3],
prediction-error expansion (PEE) [4, 5], histogram shifting (HS) [6]. A common
feature of these methods is to embed some clues of the cover object into the
watermarked object, such that the cover object can be restored. Unfortunately,
these methods are mostly only suitable for a finite-alphabet cover domain
(e.g., the domain of $\\{0,...,255\\}$ in a gray-scale pixel), which implies
that they are more suitable for images rather than audio.

In recent years, the celebrated quantization index modulation (QIM) [7] has
also been adapted for RDH [8, 9, 10], in which the cover domain more generally
includes the real-valued domain where the digital audio resides. Ko et al. [8]
developed a reversible watermarking algorithm based on nested QIM via the
iterative process that reduces the normalized error. Peng _et al._ [9] made an
improvement in RDH for one dimensional scalar-QIM by maintaining the relative
distance, and extended it into $N$-dimensional region nesting model [10].
These methods are collectively referred to as improved-QIM (IQIM), as is named
in [9]. It is noteworthy that applying IQIM to digital audio is
straightforward, although they generally fail to meet the demand of high
robustness and imperceptibility. The reason of no robustness is that the
scaling factor of IQIM is fixed as $1-\frac{1}{2^{b}}$. The reason of high
distortion is more involved: i) The constructed difference vector is rather
sub-optimal. ii) The embedding method is based on the integer lattice
$\mathbb{Z}^{n}$, whose distortion performance is worse than other optimal
lattices.

The robustness of the embedded message is important in the sense that the
cover object may go through some attacks such as additive noises. In this
regard, robust RAW has been investigated in [11, 12, 13]. To be concise,
Nishimura et al. [11] proposed a robust RAW scheme by combining quantization
index modulation (QIM) and amplitude expansion, and also a scheme based on
error-expansion of linear prediction [13] for segmental audio and G.711
speech. Liang et al. [12] proposed a robust RAW scheme based on high-order
difference statistics. Nevertheless, these works overemphasize the robustness
of embedded messages, but ignore or even sacrifice its imperceptibility. Since
the sensibility of human auditory system (HAS), the watermarked cover object
should not cause perceivable changes.

To address the above issues, this paper makes a retrospective and prospective
revisit to RDH from the perspective of lattices [14, 15, 16], with the hope of
designing an efficient RAW scheme that simultaneously achieves good robustness
and imperceptibility. Lattices are discrete additive subgroups of the
Euclidean vector space, and QIM is in essence quantization based on nested
lattice codes [17, 14]. The reversibility, robustness and imperceptibility
features of QIM can then be better accommodated by using more advanced lattice
coding skills. Specifically, the contributions of this paper, along with the
highlights, are summarized as follows.

  * •

First, we propose a Meet-in-the-Middle Embedding (MME) RAW scheme. Within the
QIM framework, there exists a difference vector between the cover vector and
the quantized cover vector. By properly scaling the difference vector and
adding it back to the quantized cover vector, it becomes possible for the
receiver to estimate the cover vector with the aid of the scaled difference
vector. As the reversibility of MME hinges on the properly choosing the
scaling factor, we derive the feasible range of the scaling factor.

  * •

Second, we analyze the performance of MME in terms of distortion and
robustness. More specifically, we compare the signal-to-watermark ratio (SWR)
and generalized signal-to-noise ratio (GSNR) between MME and IQIM [9].

  * •

Finally, we make a comprehensive comparison between MME and other state-of-
the-art RAW methods, and the numerical simulations show that MME excels in the
robustness and imperceptibility metrics such as SWR, GSNR, objective
difference grade (ODG), and bit error rate (BER). The simulated results
ideally align with our theoretical analyses.

Notation: Matrices and column vectors are denoted by uppercase and lowercase
boldface letters. We write $\lfloor\cdot\rfloor$ for the floor function, and
$|\cdot|$ for the absolute value.

##  2 Preliminaries

###  2.1 QIM

The one-bit scalar-QIM is perhaps the most popular version of QIM in the data
hiding community [17]. Its rationale can be explained by a one-bit embedding
example in Fig. 1. Denote circle and cross positions in Fig. 1 as two sets
$\Lambda_{0}$ and $\Lambda_{1}$, respectively. Given a host/cover sample
$s\in\mathbb{R}$ and a one-bit message $m\in\\{0,1\\}$, the watermarked value
is simply moving $s$ to the nearest point in $\Lambda_{0}$ when $m=0$, and to
the nearest point in $\Lambda_{1}$ when $m=1$.

Figure 1: Embedding one bit into a sample with original QIM.

Define $Q(s)=\Delta\lfloor s/\Delta\rfloor$ with $\Delta$ being a step-size
parameter. Then the embedding process can be described as

$s_{\mathrm{QIM}}\triangleq Q_{m}(s)=Q(s-d_{m})+d_{m},~{}m\in\\{0,1\\},$ (1)

where $d_{0}=-(\Delta/4)$, $d_{1}=\Delta/4$,
$\Lambda_{0}=d_{0}+\Delta\mathbb{Z}$ and $\Lambda_{1}=d_{1}+\Delta\mathbb{Z}$.

Assume that the transmitted $s_{\mathrm{QIM}}$ has undergone the contamination
of an additive noise term $n$, then at the receiver’s side we have:
$y=s_{\mathrm{QIM}}+n$. A minimum distance decoder is therefore given by

$\hat{m}=\mathop{arg\min}_{m\in\\{0,1\\}}\left[\mathop{\min}_{s\in\Lambda_{m}}|y-s|\right].$
(2)

If $|n|<\Delta/4$, the $\hat{m}$ is correct.

Regarding the embedding distortion, as shown in Fig. 2, the maximum error
caused by embedding is $\Delta/2$. If the quantization errors are distributed
uniformly over $[-(\Delta/2),(\Delta/2)]$, then the mean-squared embedding
distortion is $D=\Delta^{2}/12$.

Figure 2: Selection of watermarked signal $x$ with given $s$ and
$m\in\\{0,1\\}$ in one-bit scalar-QIM. (a) Prototype symmetric function. (b)
$x=Q_{m}(s)$ with $m=0$. (c) $x=Q_{m}(s)$ with $m=1$.

###  2.2 IQIM

Inspired by conventional QIM, a reversible version of QIM called improved-QIM
(IQIM) has been proposed in [9]. Its embedding and extraction processes can be
described as follows.

####  2.2.1 Embedding

Step i) Calculate the number of quantization interval $\gamma$ and relative
distance of $s$ in the $\gamma^{th}$ quantization interval $r$ by:

$\left\\{\begin{aligned}
\gamma&=\left\lfloor\frac{s}{2^{b}\Delta}\right\rfloor\\\
r&=s-2^{b}\gamma\Delta\end{aligned},\right.$ (3)

where $b$ is the bit length of a watermark $m$, and $\Delta$ is again the step
size of quantization.

Step ii) To embed the watermark $m$, the final embedded point
$s_{\mathrm{IQIM}}$ is set as

$s_{\mathrm{IQIM}}=2^{b}\gamma\Delta+m\Delta+\frac{r}{2^{b}}.$ (4)

####  2.2.2 Extraction

Step i) With either a clean or a noisy version of $s_{\mathrm{IQIM}}$ denoted
as $y$, calculate the number of quantization interval $\gamma_{w}$ and the
relative distance $r_{w}$ by

$\left\\{\begin{aligned}
\gamma_{w}&=\left\lfloor\frac{y}{\Delta}\right\rfloor\\\
r_{w}&=y-\gamma_{w}\Delta\end{aligned}.\right.$ (5)

Step ii) Estimate the watermark $m$ by

$\hat{m}=\gamma_{w}-2^{b}\lfloor\frac{\gamma_{w}}{2^{b}}\rfloor.$ (6)

Step iii) Estimate the cover $s$ by

$\hat{s}=2^{b}\lfloor\frac{\gamma_{w}}{2^{b}}\rfloor\Delta+2^{b}r_{w}.$ (7)

If $y$ is noiseless, one may verify that $\hat{m}=m$ and $\hat{s}=s$. For
noisy $y$, although $\hat{s}$ becomes inaccurate, $\hat{m}$ may still be
reliable if the noise pollution is small enough. In addition, IQIM can also be
described by an $N$-dimension integer set $\mathbb{Z}^{N}$, which is no more
than independently performing embedding and extraction in each dimension. For
instance, the embedding function w.r.t. the watermark $\mathbf{m}$ is given by

$\displaystyle\mathbf{s}_{\mathrm{IQIM}}=\beta\left[2^{b}\Delta\left[\gamma_{1}~{}\cdots~{}\gamma_{N}\right]^{\top}+\frac{\Delta\mathbf{m}}{\beta}\right]+(1-\beta)\mathbf{s},$
(8)

where $\beta=1-\frac{1}{2^{b}}$ denotes the scaling factor in IQIM.

###  2.3 Lattices

This paper will use some concepts from lattices to arrive at simpler
description of the algorithm and more elegant analysis. Hereby we review some
basic concepts of lattices and nested lattice codes.

An $N$-dimensional lattice in $\mathbb{R}^{N}$ is defined as
$\Lambda=\left\\{\mathbf{Gz}|\mathbf{z}\in\mathbb{Z}^{N}\right\\}$, where the
full-ranked matrix $\mathbf{G}\in\mathbb{R}^{N\times N}$ represents the
generator matrix of $\Lambda$. For any $\mathbf{x}\in\mathbb{R}^{N}$, finding
its closest vector in a lattice $\Lambda$ is referred to as the closest vector
problem (CVP). Based on efficient algorithms to solve CVP [14], a nearest
neighbor quantizer $Q_{\Lambda}(\cdot)$ is defined as

$\displaystyle
Q_{\Lambda}(\mathbf{x})=\mathop{\arg\min}_{\mathbf{\lambda}\in\Lambda}\|\mathbf{x}-\mathbf{\lambda}\|.$
(9)

Details of some optimal low-dimensional lattices for quantization are listed
in Table I. Based on the CVP quantizer we also define
$\rm{mod}(\mathbf{x},\Lambda)=\mathbf{x}-Q_{\Lambda}(\mathbf{x})$.

The set of vectors in $\mathbb{R}^{N}$ that are closer to $\lambda\in\Lambda$
than other vectors in $\Lambda$ is called the Voronoi region
$\mathcal{V}_{\lambda}$, i.e.,

$\displaystyle\mathcal{V}_{\lambda}=\\{\mathbf{x}:Q_{\Lambda}(\mathbf{x})=\lambda\\}.$
(10)

The fundamental Voronoi region is represented as

$\displaystyle\mathcal{V}_{\Lambda}=\\{\mathbf{x}:Q_{\Lambda}(\mathbf{x})=\mathbf{0}\\},$
(11)

and the volume of the fundamental region is

$\displaystyle\rm{Vol}(\mathcal{V}_{\Lambda})=\int_{\mathcal{V}_{\Lambda}}d\mathbf{x}=|\rm{det}\mathbf{G}|.$
(12)

The packing radius $r_{\rm{pack}\left(\Lambda\right)}$, represents the radius
of the largest sphere contained in $\mathcal{V}_{\Lambda}$:

$\displaystyle
r_{\rm{pack}(\Lambda)}=\frac{1}{2}\rm{min}_{\lambda\in\Lambda\backslash\\{\mathbf{0}\\}}\|\lambda\|.$
(13)

The covering radius $r_{\rm{cov}\left(\Lambda\right)}$ refers to the radius of
the smallest sphere containing $\mathcal{V}_{\Lambda}$:

$r_{\rm{cov}\left(\Lambda\right)}=\mathop{\max}_{\mathbf{x}\in\mathbb{R}^{N}}\mathop{\min}_{\lambda\in\Lambda}\|\mathbf{x}-\mathbf{\lambda}\|.$
(14)

In the lattice quantization, a typical and efficient method is using the
nested lattices based shaping, called coset coding [14, 17]. Two lattices
$\Lambda_{f}$ and $\Lambda_{c}$ are regarded as nested when these two lattices
have a inclusion relation, i.e., $\Lambda_{c}\subset\Lambda_{f}$. The lattice
$\Lambda_{f}$ is called the _fine/coding_ lattice, and its subset
$\Lambda_{c}$ is called the _coarse/shaping_ lattice. The generator matrices
of $\Lambda_{c}$ and $\Lambda_{f}$ which are expressed by $\mathbf{G}_{c}$ and
$\mathbf{G}_{f}$, respectively, have a relation

$\displaystyle\mathbf{G}_{c}=\mathbf{G}_{f}\mathbf{J},$ (15)

where the sub-sampling matrix $\mathbf{J}$ represents _nesting matrix_. When
$\mathbf{J}$ is an identity matrix, it denotes a self-similar shaping [18]
operation. When $\mathbf{J}$ is a diagonal matrix with different elements, it
denotes a rectangular shaping [19] operation.

TABLE I: The generator matrices of some optimal low-dimensional quantizers [21]. N | Quantizer | Generator Matrix | $r_{pack}(\Lambda)$ | $r_{cov}(\Lambda)$ | $G(\Lambda)$
---|---|---|---|---|---
1 | $\mathbb{Z}$ | 1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0.083333
2 | $A_{2}$ | $\left[\begin{matrix}\frac{\sqrt{3}}{2}&0\\\ \frac{1}{2}&1\end{matrix}\right]$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{3}$ | 0.080188
3 | $A_{3}\cong D_{3}$ | $\left[\begin{matrix}-1&1&0\\\ -1&-1&1\\\ 0&0&-1\end{matrix}\right]$ | $\frac{\sqrt{2}}{2}$ | $1$ | 0.078745
4 | $D_{4}$ | $\left[\begin{matrix}2&1&1&1\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{matrix}\right]$ | $\frac{\sqrt{2}}{2}$ | $1$ | 0.076603
8 | $E_{8}$ | $\left[\begin{matrix}2&-1&0&0&0&0&0&\frac{1}{2}\\\ 0&1&-1&0&0&0&0&\frac{1}{2}\\\ 0&0&1&-1&0&0&0&\frac{1}{2}\\\ 0&0&0&1&-1&0&0&\frac{1}{2}\\\ 0&0&0&0&1&-1&0&\frac{1}{2}\\\ 0&0&0&0&0&1&-1&\frac{1}{2}\\\ 0&0&0&0&0&0&1&\frac{1}{2}\\\ 0&0&0&0&0&0&0&\frac{1}{2}\\\ \end{matrix}\right]$ | $\frac{\sqrt{2}}{2}$ | $1$ | 0.071682

##  3 The Proposed Method

###  3.1 Lattice-QIM reformulation

Following [17, 20, 21], we reformulate QIM for arbitrary lattices and payloads
from the perspective of lattice quantization and coset coding, based on which
the description of the proposed RAW scheme can also be simpler.

####  3.1.1 Embedding

Step i) Choose a pair of nested lattices $\Lambda_{f}$ and
$\Lambda_{c}\subset\Lambda_{f}$ in $\mathbb{R}^{N}$. The fine lattice
$\Lambda_{f}$ can be decomposed as the union of $|\rm{det}\mathbf{J}|$ cosets
of the coarse lattice $\Lambda_{c}$:

$\displaystyle\Lambda_{f}=\bigcup\limits_{i=0}^{|\rm{det}\mathbf{J}|-1}\Lambda_{i}=\bigcup\limits_{\mathbf{d}_{i}\in\Lambda_{f}\backslash\Lambda_{c}}(\mathbf{d}_{i}+\Lambda_{c})$
(16)

where each coset $\Lambda_{i}=\Lambda_{c}+\mathbf{d}_{i}$ is a translated
coarse lattice and $\mathbf{d}_{i}$ is called the coset representative of
$\Lambda_{i}$.

Step ii) Perform labeling to associate the
$i\in\left\\{0,1,\ldots,|\rm{det}\mathbf{J}|-1\right\\}$ to the $i$th message
vector $\mathbf{m}_{i}\in\mathcal{M}$.

Step iii) Consider a cover/host vector denoted as $\mathbf{s}$. To embed a
message vector $\mathbf{m}_{i}$ over $\mathbf{s}$, the watermarked cover
vector is set as

$\mathbf{s}_{\mathrm{QIM}}\triangleq
Q_{\Lambda_{i}}(\mathbf{s})=Q_{\Lambda_{c}}(\mathbf{s}-\mathbf{d}_{i})+\mathbf{d}_{i}.$
(17)

####  3.1.2 Decoding

Let the (possibly) noisy observation of $\mathbf{s}_{\mathrm{QIM}}$ be
$\mathbf{y}$.

Step i) Estimate the coset representative $\mathbf{d}_{i}$ of $\Lambda_{i}$ by

$\hat{\mathbf{d}}_{i}={\rm mod}(Q_{\Lambda_{f}}(\mathbf{y}),\Lambda_{c}).$
(18)

Step ii) Perform inverse labeling to turn $\hat{\mathbf{d}}_{i}$ into the
estimated message vector $\hat{\mathbf{m}}_{i}$.

The extracted message vector is correct as long as
$Q_{\Lambda_{f}}(\mathbf{y})=\mathbf{s}_{\mathrm{MME}}$. Moreover, the
information transmission rate per dimension of Lattice-QIM is simply

$R=\frac{1}{N}\log{|\rm{det}\mathbf{J}|}.$ (19)

###  3.2 Meet-in-the-middle embedding

Based on the above, we observe that there exists a difference vector
$\mathbf{e}$ between the cover vector $\mathbf{s}$ and its quantized
watermarked vector $\mathbf{s}_{\mathrm{QIM}}$, i.e.,

$\displaystyle\mathbf{e}=\mathbf{s}-Q_{\Lambda_{i}}(\mathbf{s}).$ (20)

Obviously the information about $\mathbf{e}$ is lost if we only use
$Q_{\Lambda_{i}}(\mathbf{s})$ as the watermarked vector.

Notice that QIM has certain error tolerance capability. If we treat
$\mathbf{e}$ as the “beneficial noise” and add it back to
$Q_{\Lambda_{i}}(\mathbf{s})$, then the information about the cover
$\mathbf{s}$ can be maintained, and the scheme becomes reversible. The tricky
part about adding “beneficial noise” is that, they should be properly scaled
to meet several demands. First, the scaled $\mathbf{e}$ should be small enough
such that it does not go beyond the Voronoi region of the fine lattice.
Second, the scaled $\mathbf{e}$ should not be too small to avoid exceeding the
used representation accuracy of numbers. Details of the proposed scheme is
explained as follows.

######  Definition 1 (MME Embedding).

The watermarked cover vector is set as

$\displaystyle\mathbf{s}_{\mathrm{MME}}=\mathcal{W}(\mathbf{s})$
$\displaystyle=Q_{\Lambda_{i}}\left(\mathbf{s}\right)+\left(1-\alpha\right)(\mathbf{s}-Q_{\Lambda_{i}}\left(\mathbf{s}\right))$
$\displaystyle=\alpha
Q_{\Lambda_{i}}\left(\mathbf{s}\right)+\left(1-\alpha\right)\mathbf{s},$ (21)

in which $\alpha$ is a chosen scaling factor such that
$(1-\alpha)(\mathbf{s}-Q_{\Lambda_{i}}\left(\mathbf{s}\right))\in\mathcal{V}_{\Lambda_{f}}$.

######  Definition 2 (Noiseless MME Extraction).

If the receiver’s side has $\mathbf{s}_{\mathrm{MME}}$, the estimated cover
vector can be accurately restored by

$\displaystyle\hat{\mathbf{s}}=\mathcal{W}^{-1}\left(\mathbf{s}_{\mathrm{MME}}\right)$
$\displaystyle=\frac{\mathbf{s}_{\mathrm{MME}}-Q_{\Lambda_{f}}\left(\mathbf{s}_{\mathrm{MME}}\right)}{1-\alpha}+Q_{\Lambda_{f}}\left(\mathbf{s}_{\mathrm{MME}}\right)$
$\displaystyle=\frac{1}{1-\alpha}\mathbf{s}_{\mathrm{MME}}-\frac{\alpha}{1-\alpha}Q_{\Lambda_{f}}\left(\mathbf{s}_{\mathrm{MME}}\right).$
(22)

In this case, the message vector can also be accurately decoded by using the
decoder of QIM.

In the noisy setting, consider an additive white Gaussian noise (AWGN) channel
in the form of $\mathbf{y}=\mathbf{s}_{\mathrm{MME}}+\mathbf{n}$, where the
entries of $\mathbf{n}$ admit a standard Gaussian distribution
$\mathcal{N}(0,\sigma_{\mathbf{n}}^{2})$. In this case, accurately restring
the cover vector is impossible, but the message vector may be accurately
decoded by using the decoder of QIM. Compared with IQIM, MME is advantageous
in the following aspects.

  * •

As shown in Figs. 3 and 4, the difference vector of MME can be confined to a
small Voronoi cell (hence the name “meet-in-the-middle”) while IQIM cannot. In
the distortion analysis we will shown that this brings certain performance
gains.

  * •

MME can be applied to general lattices while IQIM is only feasible for the
standard integer lattice.

Figure 3: Comparison of embedding process between IQIM and MME in one-
dimension with different host vector $\mathbf{s}$. (a) $\mathbf{s}=0.5$. (b)
$\mathbf{s}=1.5$. Figure 4: Comparison of embedding process between IQIM and
MME in two-dimension with $\mathbf{s}=(1.61,0.73)$.

###  3.3 Correctness analysis

####  3.3.1 Reversibility

For the embedding function Eq. (1), its definition field $\mathcal{A}$ is
noted as

$\mathcal{A}=\\{\mathbf{s}|\mathbf{s}\in\mathbb{R}^{N}\\}.$ (23)

And the value field $\mathcal{B}$ is defined as

$\mathcal{B}=\\{\mathbf{s}_{\mathrm{MME}}|\mathbf{s}_{\mathrm{MME}}=\mathcal{W}\left(\mathbf{s}\right),\mathbf{s}\in\mathcal{A}\\}.$
(24)

Hereby we introduce a theorem about the one-to-once correspondence between
about the cover and the watermarked cover.

######  Theorem 1 (Noiseless Recovery).

The function $\mathcal{W}$ in MME is bijective.

###### Proof.

It suffices to prove that $\mathcal{W}$ is both injective and surjective.
“Injective” means no two elements in the domain of the function gets mapped to
the same image, i.e., for any cover vectors
$\mathbf{s}_{a},\mathbf{s}_{b}\in\mathcal{A}$,

$\mathbf{s}_{a}\neq\mathbf{s}_{b}\rightarrow\mathcal{W}\left(\mathbf{s}_{a}\right)\neq\mathcal{W}\left(\mathbf{s}_{b}\right).$
(25)

With $\mathbf{s}_{a}\neq\mathbf{s}_{b}$, we analyze the relationship between
their output $\mathcal{W}\left(\mathbf{s}_{a}\right)$ and
$\mathcal{W}\left(\mathbf{s}_{b}\right)$ in three cases:

  * 1.

$\mathbf{s}_{a}$ and $\mathbf{s}_{b}$ belong to two different Voronoi regions
$\mathcal{V}_{\lambda_{a}}$ and $\mathcal{V}_{\lambda_{b}}$ of the coarse
lattice $\Lambda_{c}$. It follows from
$\mathcal{V}_{\lambda_{a}}\cap\mathcal{V}_{\lambda_{b}}=\emptyset$ that
$\mathcal{W}\left(\mathbf{s}_{a}\right)\neq\mathcal{W}\left(\mathbf{s}_{b}\right)$.

  * 2.

$\mathbf{s}_{a}$ and $\mathbf{s}_{b}$ belong to the same Voronoi region in
$\Lambda_{c}$ but different watermarks
($Q_{\Lambda_{i}}\left(\mathbf{s}_{a}\right)\neq
Q_{\Lambda_{i}}\left(\mathbf{s}_{b}\right)$). Since the “self-noise” does not
go beyond the Voronoi cell of a fine lattice, i.e.,
$(1-\alpha)(\mathbf{s}-Q_{\Lambda_{i}}\left(\mathbf{s}\right))\in\mathcal{V}_{\Lambda_{f}}$,
then obviously
$\mathcal{W}\left(\mathbf{s}_{a}\right)\neq\mathcal{W}\left(\mathbf{s}_{b}\right)$.

  * 3.

$\mathbf{s}_{a}$ and $\mathbf{s}_{b}$ belong to the same Voronoi region in
$\Lambda_{c}$ and watermark
($Q_{\Lambda_{i}}\left(\mathbf{s}_{a}\right)=Q_{\Lambda_{i}}\left(\mathbf{s}_{b}\right)$).
Since $\mathbf{s}_{a}\neq\mathbf{s}_{b}$, one has
$(1-\alpha)\mathbf{s}_{a}\neq(1-\alpha)\mathbf{s}_{b}$, and they are still
unequal after both adding $Q_{\Lambda_{i}}\left(\mathbf{s}\right)$.

Combining the above three cases, the injection is proved.

“Surjective” means that any element in the range of the function is hit by the
function. We prove this by using contradiction. If $\mathcal{W}$ is not
surjective, i.e., $\exists\mathbf{s}_{\mathrm{MME}}\in\mathcal{B}$,
$\forall\mathbf{s}\in\mathcal{A}$ s.t.
$\mathcal{W}(\mathbf{s})\neq\mathbf{s}_{\mathrm{MME}}$, then we obtain
$\mathbf{s}_{\mathrm{MME}}\notin\mathcal{B}$ which contradicts Eq. (24). ∎

The above theorem says that $\mathcal{W}$ is a reversible function, where each
element of one set is paired with exactly one element of the other set. Thus
the correctness of the estimated covers is guaranteed.

In the noisy scenario of $\mathbf{n}\neq\mathbf{0}$ in
$\mathbf{y}=\mathbf{s}_{\mathrm{MME}}+\mathbf{n}$, we argue that the cover
vector is also approximately reversible if
$Q_{\Lambda_{f}}\left(\hat{\mathbf{v}}\right)=\mathbf{0}$, where

$\hat{\mathbf{v}}=(1-\alpha)\mathbf{e}+\mathbf{n}$ (26)

is referred to as the composite noise vector at the receiver’s side. To be
concise, by using the estimation function

$\displaystyle\hat{\mathbf{s}}_{\mathrm{noisy}}=\mathcal{W}^{-1}\left(\mathbf{y}\right)$
$\displaystyle=\frac{\mathbf{y}-Q_{\Lambda_{f}}\left(\mathbf{y}\right)}{1-\alpha}+Q_{\Lambda_{f}}\left(\mathbf{y}\right),$
(27)

we have

$\hat{\mathbf{s}}_{\mathrm{noisy}}-\mathbf{s}=\frac{1}{1-\alpha}\mathbf{n}.$
(28)

It says that, if the composite noise $\hat{\mathbf{v}}$ is small enough to
ensure the correctness of the message, then the distance from the noisy
estimated cover and the clean cover is only $||{1}/{(1-\alpha)}\mathbf{n}||$.

####  3.3.2 Correctness of the estimated messages

We have the following theorem about the robustness of the watermarking scheme.
Since the “self-noise” is within the boundary of the Voronoi region, the
estimated messages in the noiseless setting is correct for sure. More
generally in the noisy setting, if the composite noise is small, the
correctness of the estimated messages is also guaranteed.

######  Theorem 2.

If $\hat{\mathbf{v}}$ satisfies

$Q_{\Lambda_{f}}(\hat{\mathbf{v}})=\mathbf{0},$ (29)

then the estimated message by using MME is correct.

###### Proof.

At the receiver’s side, the noisy observation can be written as
$\mathbf{y}=Q_{\Lambda_{i}}\left(\mathbf{s}\right)+\hat{\mathbf{v}}$. Since
$Q_{\Lambda_{i}}\left(\mathbf{s}\right)\in\Lambda_{f}=\cup_{\mathbf{d}_{i}\in\Lambda_{f}\backslash\Lambda_{c}}(\mathbf{d}_{i}+\Lambda_{c})$,
$Q_{\Lambda_{f}}(\mathbf{y})$ can be written as

$Q_{\Lambda_{f}}(\mathbf{y})=Q_{\Lambda_{f}}(Q_{\Lambda_{i}}\left(\mathbf{s}\right))+Q_{\Lambda_{f}}(\hat{\mathbf{v}}).$
(30)

Then we have

$\displaystyle{\rm mod}(Q_{\Lambda_{f}}(\mathbf{y}),\Lambda_{c})$
$\displaystyle=$ $\displaystyle{\rm
mod}(Q_{\Lambda_{f}}(Q_{\Lambda_{i}}\left(\mathbf{s}\right))+Q_{\Lambda_{f}}(\hat{\mathbf{v}}),\Lambda_{c})$
$\displaystyle=$ $\displaystyle{\rm mod}({\rm
mod}(Q_{\Lambda_{f}}(Q_{\Lambda_{i}}\left(\mathbf{s}\right)),\Lambda_{c})+{\rm
mod}(Q_{\Lambda_{f}}(\hat{\mathbf{v}}),\Lambda_{c}),\Lambda_{c})$
$\displaystyle=$ $\displaystyle{\rm
mod}(\mathbf{d}_{i}+Q_{\Lambda_{f}}(\hat{\mathbf{v}}),\Lambda_{c}).$ (31)

Thus ${\rm mod}(Q_{\Lambda_{f}}(\mathbf{y}),\Lambda_{c})=\mathbf{d}_{i}$ if
$Q_{\Lambda_{f}}(\hat{\mathbf{v}})=\mathbf{0}$. Since each coset
representative $\mathbf{d}_{i}$ corresponds a unique message $\mathbf{m}_{i}$,
the lemma is proved. ∎

###  3.4 Setting the scaling factor

In this subsection, we will discuss the feasible range of $\alpha$.

####  3.4.1 Lower bound

The lower bound of $\alpha$ is determined by ensuring the correctness of
estimated messages. As mentioned in Definition 1, one should have
$(1-\alpha)(\mathbf{s}-Q_{\Lambda_{i}}\left(\mathbf{s}\right))\in\mathcal{V}_{\Lambda_{f}}$.
This implies that

$(1-\alpha)\mathcal{V}_{\Lambda_{c}}\subseteq\mathcal{V}_{\Lambda_{f}}.$ (32)

It suffices to have

$(1-\alpha)r_{cov}(\Lambda_{c})\leq r_{pack}(\Lambda_{f}),$ (33)

which means

$\alpha\geq 1-\frac{r_{pack}(\Lambda_{f})}{r_{cov}(\Lambda_{c})}.$ (34)

For the special case of self similar shaping, a tighter bound can be derived.
To be concise, let $\Lambda_{c}=\Gamma\Lambda_{f}$. It follows from
$(1-\alpha)\Gamma\mathcal{V}_{\Lambda_{f}}\subseteq\mathcal{V}_{\Lambda_{f}}$
that

$\alpha\geq 1-1/\Gamma.$ (35)

Figure 5: Distribution of the watermarked cover vector
$\mathbf{s}_{\mathrm{MME}}$ with $\Lambda_{f}$ being the $A_{2}$ lattice and
$\Lambda_{c}=4\Lambda_{f}$. (a) $\alpha=1-\frac{\sqrt{3}}{6}$ based on Eq.
(34). (b) $\alpha=3/4$ based on Eq. (35).

Fig. 5 depicts the distribution of the watermarked cover vector
$\mathbf{s}_{\mathrm{MME}}$ based on the general bound in Eq. (34) and the
special-case bound in Eq. (35).

####  3.4.2 Upper bound

The upper bound of $\alpha$ is related to the correct extraction of the
covers. Recall that the audio data is stored as single or double precision
floating-point numbers (as shown in Table II), and it cannot represent
arbitrary real-valued numbers without loss of accuracy. Overflow occurs when
the magnitude of a number exceeds the range allowed by the size of the bit
field. To avoid the effect of overflow, we need to have

$\alpha\leq 1-2^{-(\mathcal{L}+1)},$ (36)

where $\mathcal{L}$ denotes the mantissa bit length. For example, we have
$\alpha\leq 1-2^{-53}$ when using double precision floating-point numbers.

TABLE II: The storage structure of floating-point numbers. Data Type | Total length (bit) | Sign length (bit) | Order length (bit) | Mantissa length (bit)
---|---|---|---|---
Float | 32 | 1 | 8 | 23
Double | 64 | 1 | 11 | 52

##  4 Performance metrics

###  4.1 Embedding distortion

To evaluate the embedding distortion, we adopt the signal to watermark ratio
(SWR) metric defined as

$\displaystyle{SWR}~{}(\mathrm{dB})=10\times
lg\left(\frac{\sigma^{2}_{\mathbf{s}}}{\sigma^{2}_{\mathbf{w}}}\right),$ (37)

where $\sigma^{2}_{\mathbf{s}}$, $\sigma^{2}_{\mathbf{w}}$ represent the power
of the host and the additive watermark, respectively. For MME, we have

$\displaystyle\sigma^{2}_{\mathbf{w}}$
$\displaystyle=E(\|\alpha\mathbf{e}\|^{2})$
$\displaystyle=\alpha^{2}E(\|\mathbf{e}\|^{2})$
$\displaystyle=\alpha^{2}NG({\Lambda_{c}})\rm{Vol}(\mathcal{V}_{\Lambda_{c}})^{2/N},$
(38)

in which $G(\Lambda_{c})$ represents the normalized second moment of the
chosen coarse lattice $\Lambda_{c}$, and Eq. (38) is derived from the widely
adopted flat-host assumption [22]. For the simplest setting of
$\Lambda_{f}=\Delta\mathbb{Z}^{N}$ and $\Lambda_{c}=\Delta
2^{b}\mathbb{Z}^{N}$, since $G(\Delta 2^{b}\mathbb{Z}^{N})=1/12$, we have

$\sigma_{\mathbf{w}}^{2}=\frac{N\alpha^{2}2^{2b}\Delta^{2}}{12}.$ (39)

######  Proposition 1.

By setting $\Lambda_{f}=\Delta\mathbb{Z}^{N}$, $\Lambda_{c}=\Delta
2^{b}\mathbb{Z}^{N}$, and $\alpha=\beta$, MME has a larger SWR than IQIM.

###### Proof.

Based on the inequality that

$x-1<\left\lfloor x\right\rfloor\leq x,$ (40)

the embedding distortion of IQIM can be bounded as

$\displaystyle\sigma_{\mathbf{w},\rm{IQIM}}^{2}$
$\displaystyle=E\left(\beta^{2}\left\lVert\mathbf{s}-\left[2^{b}\Delta\left[\gamma_{1}~{}\cdots~{}\gamma_{N}\right]^{\top}+\frac{\Delta\mathbf{m}}{\beta}\right]\right\rVert^{2}\right)$
$\displaystyle=\frac{\beta^{2}}{2^{b}}\sum_{q=1}^{N}\sum_{m_{i,q}=0}^{2^{b}-1}\int_{0}^{2^{b}\Delta}\left[2^{b}\Delta\gamma_{q}+\frac{m_{i,q}\Delta}{\beta}-s_{q}\right]^{2}$
$\displaystyle\cdot f(s_{q})ds_{q}$
$\displaystyle>\frac{\beta^{2}}{2^{b}}\sum_{q=1}^{N}\sum_{m_{i,q}=0}^{2^{b}-1}\int_{0}^{2^{b}\Delta}\left[\frac{m_{i,q}\Delta}{\beta}-2^{b}\Delta\right]^{2}f(s_{q})ds_{q}$
$\displaystyle=\frac{\left(2^{b}\right)^{2}\left(2\cdot
2^{b}-1\right)}{6\left(2^{b}-1\right)}N\beta^{2}\Delta^{2}.$ (41)

Letting $\alpha=\beta$ in Eq. (39), we have

$\sigma_{\mathbf{w},\rm{IQIM}}^{2}-\sigma_{\mathbf{w}}^{2}>\frac{\beta^{2}2^{2b}\left(3\cdot
2^{b}-1\right)N\Delta^{2}}{12\left(2^{b}-1\right)}>0.$ (42)

As IQIM has a larger denominator in SWR, the proposition is proved. ∎

Proposition 1 justifies the advantage of MME over IQIM: by using a better
meet-in-the-middle approach to construct the “beneficial noise”, it enjoys a
larger SWR.

###  4.2 Robustness of messages

In addition to the distortion/imperceptibility metric, it is also necessary to
evaluate the robustness of MME against additive noises. Following [17], we
define the generalized signal-to-noise ratio (GSNR) as

$\displaystyle{\rm GSNR}$
$\displaystyle\triangleq\frac{\min_{i,j\in\mathcal{I},i\neq
j}\|\mathbf{d}_{i}-\mathbf{d}_{j}\|^{2}}{E(\|\hat{\mathbf{v}}\|^{2})}$
$\displaystyle=\frac{4r_{{\rm{pack}}(\Lambda_{f})}^{2}}{E(\|\hat{\mathbf{v}}\|^{2})},$
(43)

where $\hat{\mathbf{v}}$ is the composite noise vector defined in Eq. (26).

Due to the independence of $\mathbf{e}$ and $\mathbf{n}$, we have

$\displaystyle E(\|\hat{\mathbf{v}}\|^{2})$
$\displaystyle=E(\|(1-\alpha)\mathbf{e}+\mathbf{n}\|^{2})$
$\displaystyle=(1-\alpha)^{2}E(\|{\mathbf{e}}\|^{2})+E(\|{\mathbf{n}}\|^{2})$
$\displaystyle=(1-\alpha)^{2}NG({\Lambda_{c}}){\rm{Vol}}(\mathcal{V}_{\Lambda_{c}})^{2/N}+N\sigma_{\mathbf{n}}^{2}.$
(44)

By substituting (44) into (43), the GSNR of MME can be expressed as

${\rm
GSNR}=\frac{4r_{{\rm{pack}}(\Lambda_{f})}^{2}}{(1-\alpha)^{2}NG({\Lambda_{c}}){\rm{Vol}}(\mathcal{V}_{\Lambda_{c}})^{2/N}+N\sigma_{\mathbf{n}}^{2}}.$
(45)

Since the energy of the self-noise $(1-\alpha)\mathbf{e}$ is smaller than that
of IQIM, its composite-noise also features a smaller energy. Therefore, MME
has a larger GSNR than IQIM.

##  5 Simulations

We carry out some simulations in this section to verify the effectiveness of
MME. The simulation setups are summarized as follows.

TABLE III: The level of objective difference grade scores [23]. ODG | Quality | Impairment
---|---|---
0 | Excellent | Imperceptible
-1 | Good | Perceptible but not annoying
-2 | Fair | Slightly annoying
-3 | Poor | Annoying
-4 | Bad | Very annoying

Datasets: In the experiments, two online datasets [24] and [25] are adopted,
which contain ringtone, natural sound, absolute music, songs, and speeches.
They consist of $5$ and $70$ bipolar $16$-bits Waveform Audio File Format
(WAV) files, respectively, which are labeled as No. 1001-1005 and 2001-2070.
The cover vectors are generated from these datasets, and the messages are
generated from uniform random distributions.

Distortion metrics: In addition to the SWR metric, the objective difference
grade (ODG) [23] is also introduced in the experiments, which is a popular
metric to assess accurately the objective auditory perception and is produced
by the Perceptual Evaluation of Audio Quality (PEAQ) measure standardized in
the ITU-R BS. 1387 [26]. The ODG scores are computed by the software
downloaded from [27], which is the Matlab version of Evaluation of Audio
Quality (EAQUAL) [28]. ODG gives a score in the range of $[-4.5,0.5]$. As
shown in Table III, the audio quality is said to be excellent, good, fair,
poor, and bad based on the respective ODG scores.

Robustness metrics: In addition to the GSNR metric, we also introduce the bit
error rate (BER) metric to exactly show the decoding error rates. BER is
defined as

$BER=\frac{\sum_{k=1}^{M}\sum_{q=1}^{N}\left(\mathbf{x}_{k,q}\oplus\mathbf{x}^{\prime}_{k,q}\right)}{M*N},$
(46)

where $\mathbf{x}$ and $\mathbf{x}^{\prime}$ separately represent the original
and estimated messages, while $q$ and $k$ are dimensional and sample
identifiers respectively.

Benchmarks and settings: The state-of-the-art methods in [9, 12, 11] are
adopted as the benchmarks, which are respectively named as “IQIM”, “Liang” and
“Nishimura”. For IQIM, Nishimura and the proposed MME, unless stated
otherwise, we set $\alpha=0.6569$, $\Delta=2000$,
$\Lambda_{f}=\Delta\mathbb{Z}^{2}$, and $\Lambda_{c}=\Delta
2^{R}\mathbb{Z}^{2}$. In addition, “Liang” is configured with parameters
$S=300,~{}T=7500,~{}G=20000$.

###  5.1 Imperceptibility performance

Figure 6: The comparisons between methods in [9, 12, 11] and the proposed MME
with different metrics (a) SWR. (b)ODG.

Fig. 6(a) and Fig. 6(b) have respectively depicted the average value of SWR
and ODG for the whole dataset. The blue and black bars represent the
confidence interval (CI) and the range of standard deviation (SD). Fig. 6-(a)
shows that the proposed method attains the highest SWR of $32.9891$dB, which
is better than the $31.6769$dB of Liang, and the $28.3758$dB of IQIM.
Moreover, the Nishimura method fails to work in this setting. Regarding ODG,
the proposed method, IQIM and Nishimura work well, and they all belong the the
class of QIM-based methods.

Figure 7: The impact of code rate $R$ of the performance of (a) SWR. (b)ODG.

More in-depth comparisons are made with the most related IQIM method hereby.
Fig. 7 shows the performance of SWR and ODG with different code rate $R$. We
have the following observations: i) For the SWR performance, MME is strictly
better than IQIM regardless of the code rate $R$. ii) In terms of ODG, when
the performance is above the bar of “fair”, MME generally outperforms IQIM,
but falls short when both methods become unreliable.

The advantages of using more general lattices for MME are further examined in
Table IV. In addition to the simple integer lattice $\mathbb{Z}$, the optimal
lattices in $2$, $4$, and $8$ dimensions are respectively $A_{2}$, $D_{4}$ and
$E_{8}$. The Conway-Sloane type labeling technique [21] can be used for these
lattices. Specifically, we have the following observations from the table.
i)MME outperforms IQIM in all the metrics (SWR and ODG), regardless of the
chosen type of lattices. For instance, the SWR of proposed lower bound in
$A_{2}$ $(R=1)$ is $34.3771$dB, which is greater than the $28.3758$ SWR in
IQIM. ii) Choosing larger-dimensional optimal lattices is more desirable.
Steady improvements can be observed by increasing the dimensions of the
lattices from $\mathbb{Z}$ to $A_{2}$, $D_{4}$ and $E_{8}$. iii) Since the
fine lattices are fixed to feature unit volumes while the coarse lattices grow
as the code rate $R$ increases, the SWR/ODG decreases with $R$.

TABLE IV: The performance of MME with different lattices. Rate $R$ | Metrics | MME with $\alpha=1-\frac{r_{pack}(\Lambda_{f})}{r_{cov}(\Lambda_{c})}$ |  IQIM [9]
---|---|---|---
$\mathbb{Z}$ | $A_{2}$ | $D_{4}$ | $E_{8}$
1 | $\alpha$ | 0.5 | 0.5670 | 0.6464 | 0.6464 | 0.5
SWR(dB) | 34.5727 | 34.3771 | 34.2464 | 36.0888 | 28.3758
ODG | -1.3185 | -1.3237 | -1.3316 | -1.2840 | -1.4659
2 | $\alpha$ | 0.75 | 0.7835 | 0.8232 | 0.8232 | 0.75
SWR (dB) | 25.2209 | 25.75211 | 26.3638 | 28.3057 | 20.2369
ODG | -1.5459 | -1.5290 | -1.5132 | -1.4723 | -1.6339

###  5.2 Robustness performance

With the same setting as in Fig. 6-(a), we further investigate the robustness
performance of the watermarking algorithms against AWGN attacks. Fig. 8 has
depicted the BER comparison for MME, Liang and IQIM. The Nishimura method has
been excluded as its robustness performance is weak. The reason for MME’s
better BER performance is that, by adjusting the scaling factor $\alpha$, it’s
watermarked vector can be tuned further away from the boarder of the decoding
regions.

Fig. 9-(a) and Fig. 9-(c) have drawn the GSNR for MME and IQIM for various SNR
and $\alpha$. The GSNR of MME can go well beyond $10^{12}$, while IQIM
flattens at about $3\times 10^{11}$. Fig. 9-(b) and Fig. 9-(d) are also
attached to show the better GSNR performance of MME is not generated by
scarifying the SWR performance. The simulation results above are remarkable:
compared with other QIM-variants, MME can simultaneously achieve smaller
distortion and better robustness.

Since both the BER and SWR metrics are related to the $\Delta$ in MME, we plot
the trade-off relations of BER versus SWR in Fig. 10 by changing $\Delta$ and
letting SNR=$25$dB. We have the following observations: i) The trade-off
capability becomes better when $\Delta$ increases, where the curve rises
smoothly when $\Delta=500$ and become sharper when $\Delta$ increases. ii) The
ratio of SWR/BER decreases when $\Delta$ increases, which means larger
distortion and higher robustness is achieved with a larger $\Delta$.

Figure 8: BER versus SNR.

Figure 9: GSNR and SWR comparisons between MME and IQIM with different
parameters. (a), (b) SNR ($\alpha=0.6569$, $R=1$). (c), (d) $\alpha$
(SNR$=25$, $R=1$). Figure 10: BER versus SWR.

###  5.3 Reversibility

Figure 11: The signal waveforms at different stages of MME. (a) Original
signals. (b) Watermarked signals by using MME, whose SWR are $38.0775$dB,
$33.2487$dB, $34.0998$dB and $35.1670$dB. (c) Restored signals. (d) The
differences between the original signals and the restored signals.

Due to the sensibility of HAS, the human awareness of sounds from different
sources is also diverse, which may cause a distinct feeling for heterogeneous
sounds. To vividly demonstrate the actual effect of embedding, we draw the
original, watermarked and restored signal of four typical audio host signals
in Fig. 11. They are absolute music, songs, speeches and white noises,
respectively. These host signals are labeled as No. 2002, 2003, 2040 and 2050.
Judging by the naked eye, the wave forms of the original and the restored
signals are basically the same. In terms of watermarked signals, MME achieves
the SWRs of $38.0775$dB, $33.2487$dB, $34.0998$dB and $35.1670$dB for these
host signals. Lastly, Fig. 11-(d) shows that are basically no difference
between the original cover signals and the restored cover signals.

##  6 Conclusions

In this paper, a novel reversible audio watermarking method referred to as
Meet-in-the-Middle Embedding (MME) has been proposed. We have justified its
correctness, and make comparisons with existing methods, which show that MME
exhibits less distortion and better robustness. Theoretical analyses on the
SWR and GSNR have been presented. Simulations show that the proposed method
features a smaller amount of distortion and probability of decoding errors
compared with IQIM and other representative methods in RAW.

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