§ INTRODUCTION

The primary concern during the transmission of digital data over communication media is that anybody can access this data. Hence, to protect the data from being accessed by illegitimate users, the sender must employ some security mechanisms. In general, there are two main approaches used to protect secret data; cryptography [26] and steganography [17], with our focus on the latter.

Steganography is derived from the Greek words steganos for “covered” or “secret” and graphie for “writing”. In steganography, the secret data is hidden in some unsuspected cover media so that it is visually imperceptible. Here, both the secret data as well as the cover media may be text or multimedia. Recently, steganography schemes that use images as secret data as well as cover media have gained a lot of research interest due to their heavy use in World Wide Web applications. This is the focus of our work.

Next, we present some relevant previous studies in this domain. Secret data can be embedded in images in two ways; spatially or by using a transform. In the spatial domain based image steganography scheme, secret data is embedded directly into the image by some modification in the values of the image pixels. Some of the past works related to this are given in Table <ref>. In the transform domain based image steganography scheme, first, the image is transformed into frequency components, and then the secret data is embedded into these components. Some of the past works related to this are given in Table <ref>.

Spatial domain-based image steganography schemes.

Reference     Technique                                                              c]@c@Secret images         c]@c@Cover image
[7]     c]@c@A modified version of
Least Significant Bits
(LSB) with deep
neural networks     2 color     c]@c@color
[13]     c]@c@LSB      2 color      color
[12]     c]@c@LSB     3 binary     c]@c@grayscale and color

Transform domain-based image steganography schemes.

Reference                                     Technique                                                                                c]@c@Secret images         c]@c@Cover image
[24]                              c]@c@Discrete Wavelet
Transformation (DWT)
with Particle Swarm
Optimization (PSO)                                                  1 grayscale      color
[5]          c]@c@Redundant Integer Wavelet
Transform (RIWT) and QR
factorization                          1 binary         color
[19]     c]@c@Contourlet and Fresnelet
Transformations with
Genetic Algorithm (GA)
and PSO     c]@c@1 binary
(specifically, QR code)       grayscale
[6]     c]@c@RIWT, Singular Value
Decomposition (SVD),
and Discrete Cosine
Transformation (DCT)     1 binary       grayscale
[15]                              c]@c@DWT                                                       2 grayscale     color
[13]                                 DWT and SVD                                                                                2 color           color

Images are of three kinds; binary, grayscale, and color. A grayscale image has more information than a binary image. Similarly, a color image has more information than a grayscale image. Thus, hiding a color secret image is more challenging than hiding a grayscale secret image, which is more challenging than hiding a binary secret image. Similarly, applying this concept to the cover image, we see a reverse sequence; see Table <ref>. We focus on the middle case here, i.e., when both the secret images and the cover image are grayscale, which is considered challenging.

The difficulty in designing a good steganography scheme for embedding secret images into a cover image is increasing the embedding capacity of the scheme while preserving the quality of the resultant stego-image as well as making the scheme resistant to steganographic attacks. Usually, the more the number of secret images to be embedded (which often translates to heavier secret images), the lower the quality of the obtained stego-image. Hence, we need to balance these two competing requirements.  Until now, in most works, researchers have embedded two secret images in a cover image. Some people have looked at embedding three secret images but this is rare [12]. Here, not just the number of secret images but the total size of the secret images is also important. To capture this requirement of number as well as size, a metric of bits per pixel (bpp) is used.

In this work, we present a novel image steganography scheme wherein up to four images can be hidden in a single cover image. The size of a secret image is about half of that of the cover image, which results in a very high bpp capacity. No one has attempted embedding up to four secret images in a cover image until now, and those who have attempted embedding one, two, or three images have also not achieved the level of embedding capacity that we do. While enhancing the capacity as discussed above, the quality of our stego-image does not deteriorate much. Also, we do not need any cover image data to extract secret images on the receiver side, which is commonly required with other schemes. We do require some algorithmic settings on the receiver side, however, these can be communicated to the receiver separately. Thus, this makes our scheme more secure.

Our innovative scheme has three components, which we discuss next.
The first component, i.e., embedding of secret images, consists of the following parts:

(1) We perform sub-sampling on a cover image to obtain four sub-images of the cover image.

(2) We perform block-wise sparsification of each of these four sub-images using DCT (Discrete Cosine Transform) and form a vector.

(3) We represent each vector in two groups based upon large and small coefficients, and then project each of the resultant (or generated) sparse vector onto linear measurements by using a measurement matrix (random matrix whose columns are normalized). The oversampling at this stage leads to sparse approximation.

(4) We repeat the second step above for each of the secret images.

(5) We embed DCT coefficients from the four secret images into “a set” of linear measurements obtained from the four sub-images of the cover image using our new embedding rule.

Second, we generate the stego-image from these modified measurements by using the Alternating Direction Method of Multipliers (ADMM) to solve the Least Absolute Shrinkage and Selection Operator (LASSO) formulation of the underlying minimization problem. This method has fast convergence, is easy to implement, and also is extensively used in image processing. Here, the optimization problem is an $\ell_1$-norm minimization problem, and the constraints comprise an over-determined system of equations [25].

Third, we extract the secret images from the stego-image using our proposed extraction rule, which is the reverse of our embedding rule mentioned above. As mentioned earlier, we do not require any information about the cover image while doing this extraction, which makes the process blind. We call our scheme SABMIS (Sparse Approximation Blind Multi-Image Steganography).

For performance evaluation, we perform extensive experiments on a set of standard images. We first compute the embedding capacity of our scheme, which turns out to be very good. Next, we check the quality of the stego-images by comparing them with their corresponding cover images. We use both a visual measure and a set of numerical measures for this comparison. The numerical measures used are: Peak Signal-to-Noise Ratio (PSNR) value, Mean Structural Similarity (MSSIM) index, Normalized Cross-Correlation (NCC) coefficient, entropy, and Normalized Absolute Error (NAE). The results show very little deterioration in the quality of the stego-images.

Further, we visually demonstrate the high quality of the extracted secret images by comparing them with the corresponding original secret images. Also, via experiments, we support our conjecture that our scheme is resistant to steganographic attacks. Finally, we compare the embedding capacity of our scheme and PSNR values of our stego-images with the corresponding data from competitive schemes available in the literature[Most of the existing works do not compute the other numerical mesaures  of SSIM index, NCC coefficient, entropy and NAE.].  These last two checks (quality of the extracted secret images and experimentation for resistance to steganographic attacks) are not common in existing works, and hence, we are unable to perform these two comparisons. The superiority of our scheme over past works is summarized below.

Because of the lack of past work of embedding grayscale secret images into a grayscale cover image, we compare with the cases of embedding binary images into a grayscale image or embedding grayscale images into a color image. Both these problems are easier than our problem as discussed earlier; see Table <ref>.

Image types and levels of challenge.

Image Type     More Challenging     Medium Challenging     Less Challenging
Secret Image            Color                         Grayscale                       Binary

Cover Image             Binary                        Grayscale                       Color

For the case of embedding one secret image into a cover image, we compare with [6]. Here, a binary secret image is embedded into a grayscale cover image. The authors in [6] achieve an embedding capacity of 0.25bpp while we achieve an embedding capacity of 2bpp. When comparing the stego-image and the corresponding cover image, [6] achieve a PSNR value of 49.69dB while we achieve a PSNR value of 43.15dB. This is considered acceptable because we are compromising very little in quality while gaining a lot more in embedding capacity. Moreover, PSNR values over 30dB are considered good [13, 30, 18].

For the case of embedding two secret images in a cover image, we compare with [15]. Here, two grayscale images are embedded into a color cover image. The authors in [15] achieve an embedding capacity of 1.33bpp while we achieve an embedding capacity of 4bpp. When comparing the stego-image and the corresponding cover image, [15] achieve a PSNR value of 44.75dB while we achieve a PSNR value of 39.83dB. This is again considered acceptable because of the reason discussed above.

For the case of embedding three secret images in a cover image, we compare with [12]. Here, three binary images are embedded into a grayscale cover image. The authors in [12] achieve an embedding capacity of 2bpp while we achieve an embedding capacity of 6bpp. When comparing the stego-image and the corresponding cover image, [12] achieve a PSNR value of 46.36dB while we achieve a PSNR value of 38.12dB. Again, this is considered acceptable.

When embedding four secret images in a cover image, we achieve an embedding capacity of 8bpp and a PSNR value of 37.14dB, which no one else has done.

The remainder of this paper has three more sections. In Section <ref>, we present our proposed sparse approximation based blind multi-image steganography scheme. The experimental results are presented in Section <ref>. Finally, in Section <ref>, we discuss conclusions and future work.

§ PROPOSED APPROACH

Our sparse approximation based blind multi-image steganography scheme consists of the following components: (1) Embedding of secret images leading to the generation of the stego-data. (2) Construction of the stego-image. (3) Extraction of secret images from the stego-image. These parts are discussed in the respective subsections below.

§.§ Embedding Secret Images

First, we perform sub-sampling of the cover image to obtain four sub-images. This type of sampling is done because we are embedding up to four secret images. Let $CI$ be the cover image of size $r\times r$. Then, the four sub-images each of size $\frac{r}{2} \times \frac{r}{2}$ are obtained as follows [22]:

\begin{align}\label{eq:sampling}
& {CI}^1(n_1,n_2) = CI(2n_1-1,2n_2-1),   \refstepcounter{equation} \subeqn \\ %\nonumber
& {CI}^2(n_1,n_2) = CI(2n_1,2n_2-1),   \subeqn \\
& {CI}^3(n_1,n_2) = CI(2n_1-1,2n_2),   \subeqn \\
& {CI}^4(n_1,n_2) = CI(2n_1,2n_2),   \subeqn \label{eq:sampling_eq4}
\end{align}

where $CI^k$, $\text{for} \; k=\{1, 2, 3, 4\}$, are the four sub-images; $n_1,n_2 = 1,2,\hdots, \frac{r}{2}$ (in our case, $r$ is divisible by $2$); and $CI({}\cdot{},{}\cdot{})$ is the pixel value at $({}\cdot{},{}\cdot{})$. A cover image and the corresponding four sub-sampled images are shown in Figure <ref>.

A cover image and its 4 sub-sampled images [3, 1, 2].

Originally, these sub-images are not sparse; hence, next, we perform block-wise sparsification of each of these images. For this, we divide each sub-image into blocks of size $b\times b$ and obtain $\frac{r^2}{4\times b^2}$ blocks for each sub-image (in our case, $b$ divides $r$). Now, we sparsify each block using the discrete cosine transformation. That is,

\begin{equation}\label{eq:propose sparsification SABMIS}
%s_i = \Psi^T x_i,
s_i = DCT(x_i),
\end{equation}

where $i=1,2,\hdots,\frac{r^2}{4\times b^2}$, $x_i$ and $s_i$ are the $i^{th}$ original and sparse blocks of the same size, i.e, $b\times b$, respectively, and DCT is the Discrete Cosine Transform.
Further, we pick the final sparse blocks using a zig-zag scanning order as used in our earlier work [21], and obtain corresponding sparse vectors each of size $b^2 \times 1$. The zig-zag scanning order for a block of size $8 \times 8$ is shown in Figure <ref>. This order helps us to arrange the DCT coefficients with the set of large coefficients first, followed by the set of small coefficients, which assists in the preservation of a good quality stego-image.

Next, we represent each vector in two groups based upon large (say $\#p_1$) and small (say $\#p_2$) coefficients, i.e., $s_{i,u}\in \mathbb{R}^{p_1}$ and $s_{i,v}\in \mathbb{R}^{p_2}$, where $p_1 \leq p_2$. Each of these vectors is sparse and $p_1 + p_2 = b^2$. Further, we project each sparse vector onto linear measurements as

\begin{equation}\label{eq: propose measurements SABMIS}
y_i=\begin{bmatrix}y_{i,u}\\ y_{i,v} \end{bmatrix}=
\begin{bmatrix}
s_{i,u}\\ \Phi s_{i,v}
\end{bmatrix},
\end{equation}

where $y_i\in \mathbb{R}^{({p_1+p_3})\times 1}$ is the set of linear measurements, and $\Phi \in \mathbb{R}^{p_3\times p_2}$ is the column normalised measurement matrix consisting of normally distributed random numbers with $p_3>p_2$ and $p_3\in \mathbb{N}$ (i.e., the sparse coefficients are oversampled). This oversampling helps us to perform sparse approximation. By employing this approximation (along with our novel embedding rule discussed towards the end of this subsection), we achieve a higher embedding capacity. Moreover, our approach gains an extra layer of security because the linear measurements are measurement-matrix encoded small coefficients of the sparse vectors obtained after DCT. Since the distribution of coefficients of the generated sparse vectors is almost the same for all the blocks of an image, we use the same measurement matrix for all the blocks.

Zig-zag scanning order for a block of size $8 \times 8$.

Next, we perform processing of the secret images for embedding them into the cover image.
Let the size of each secret image be $m \times m$. Initially, we perform block-wise DCT of each of these images and obtain their corresponding DCT coefficients. Here, the size of each block taken is $l\times l$, and hence, we have $\frac{m^2}{l^2}$ blocks for each secret image. In our case, $l$ divides $m$, and we ensure that $\frac{m^2}{l^2}$ will be less than or equal to $\frac{r^2}{4\times b^2}$ so that the number of blocks of the secret image is less than or equal to the number of blocks of a sub-image. Thereafter, we arrange these DCT coefficients as a vector in the earlier discussed zig-zag scanning order. Let $t_{\hati} \in R^{l^2\times 1}$, for $\hati=1,2,\hdots,\frac{m^2}{l^2}$, be the vector representation of the DCT coefficients of one secret image. We pick the initial $p_4$  DCT coefficients with relatively larger values (out of the available $l^2$ coefficients) for embedding, where $p_4\in \mathbb{N}$.

Here, we show the embedding of only one secret image into one sub-image of the cover image. However, in our steganography scheme, we can embed a maximum of four secret images, one in each of the four sub-images of the cover image, which is demonstrated in the experimental results section. If we want to embed less than four secret images, we can randomly select the corresponding sub-images from the available four.

Next, using our novel embedding rule (discussed below), we embed the chosen $p_4$ DCT coefficients of the secret image into a selected set of $p_1 + p_3$ linear measurements obtained from the sub-image of the cover image, leading to the generation of the stego-data (we ensure that $p_4$ is less than $p_1 + p_3$). The selected linear measurements are chosen to give the best results, i.e., satisfy all the three goals of higher embedding capacity, less deterioration in the quality of the stego-image and more security.

Initially, we embed the first coefficient of the secret image into the $(p_1 - 2c)^\text{th}$ index of the original linear measurements vector to obtain the $p_1^\text{th}$ index of the resultant modified linear measurements vector (where $c$ is some user chosen constant). Further, we embed the next $c-1$ coefficients  from the secret image into $p_1 -  2c + 1$ to $p_1 - c  - 1$ indices of the original linear measurements vector to obtain the $p_1 -  c + 1$ to $p_1 - 1$ indices of the resultant modified linear measurements vector.

Finally, we embed the remaining $p_4 - c$ coefficients from the secret image into $p_1 + c + 1$ to $p_1 + p_4 $ indices of the original linear measurements vector to obtain $p_1 + p_4 + 1$ to $p_1 + 2\times p_4 - c$ indices of the resultant modified linear measurements vector. The whole process in given in Algorithm <ref>. Specifically, the embedding rules discussed above are given on line <ref>,  lines <ref> – <ref>, and lines <ref> – <ref> of this algorithm, respectively.

Embedding Rule
[1]

* $y_i$: Sequence of linear measurements of the cover image with $i=1,2,\hdots,\frac{r^2}{4\times b^2}$.

* $t_{\hati}$: Sequence of transform coefficients of the secret image with $\hati=1,2,\hdots,\frac{m^2}{l^2}$.

* The choice of our $r$, $b$, $m$, and $l$ is such that $\frac{m^2}{l^2}$ is less than or equal to $\frac{r^2}{4\times b^2}$.

* $p_1$ and $p_4$ are lengths of certain vectors defined on pages p1_page_ref_section and p4_page_ref_section, respectively.

* $\alpha$, $\beta$, $\gamma$, and $c$ are algorithmic constants that are chosen based upon experience. The choices of these constants are discussed in the experimental results sections.

* $y'_i$: The modified version of the linear measurements with $i=1,2,\hdots,\frac{r^2}{4\times b^2}$.

Initialize ${y'_i}$ to $y_i$, where $i=1,2,\hdots,\frac{r^2}{4\times b^2}$.
$i = 1$ to $\frac{r^2}{4\times b^2}$

// Embedding of the first coefficient.

$y'_i(p_1) = y_i(p_1-2c) + \alpha \times t_i(1)$.

$j = p_1 - c + 1$ to $p_1 - 1$

// Embedding of the next $c-1$ coefficients.

$y'_i(j) = y_i(j-c) + \beta \times t_i(j-p_1+c+1)$.

$k = p_1 + p_4 + 1$ to $p_1 + 2\times p_4 - c$

// Embedding of the remaining $p_4 - c$ coefficients.

$y'_i(k) = y_i(k - p_4 + c) + \gamma \times t_i(k - p_1 - p_4 + c)$.

§.§ Construction of the Stego-Image

As mentioned earlier, the next step in our scheme is the construction of the stego-image.
Since we can embed a maximum of four secret images into four sub-images of a single cover image, we first construct four sub-stego-images and then perform inverse sampling to obtain a single stego-image. Let $s'_i$ be the sparse vector of the $i^{th}$ block of a sub-stego-image. The sparse vector $s'_i$ is the concatenation of $s'_{i,u}$ and $s'_{i,v}$. Here, the size of ${s}'_{i,u}$, ${s}'_{i,v}$, and ${s}'$ is the same as that of $s_{i,u}$, $s_{i,v}$, and $s$, respectively. Then, we have

\begin{align}
\label{eq:stego reconstruction SABMIS}
s'_{i,u} &= y'_{i,u}, \refstepcounter{equation} \subeqn \\
s'_{i,v} &= \argmin_{s'_{i,v}\in \mathbb{R}^{p_2}} \|{s'_{i,v}} \|_{1}
\quad\text{subject to}\quad
\Phi s'_{i,v} = y'_{i,v}. \subeqn
\label{eq:stego reconstruction SABMIS_eq2}
\end{align}

\begin{align}\label{eq:stego reconstruction SABMIS}
& s'_{i,u} = y'_{i,u} \: \text{and} \refstepcounter{equation} \subeqn \\
& s'_{i,v} = \argmin_{s'_{i,v}\in \mathbb{R}^{p_2}} \left \|{s'_{i,v}} \right \|_{1} \subeqn \label{eq:stego reconstruction SABMIS_eq2}\\
& \text{Subject to} \hspace{0.1cm} \Phi {s'_{i,v}} = y'_{i,v}. \label{eq:stego reconstruction SABMIS_eq3} \subeqn
\end{align}

The second part (<ref>) (i.e., obtaining $s'_{i,v}$), is an $\ell_1$-norm minimization problem. Here, we can observe that in the above optimization problem, the constraints are oversampled. As earlier, this oversampling helps us to do sparsification, which leads to increased embedding capacity without degradation of the quality of both the stego-image and the secret image. For the solution of the minimization problem (<ref>)–(<ref>), we use ADMM [8, 10] to solve the LASSO [16, 20] formulation of this minimization problem. The reason is that this method has a fast convergence, is easy to implement, and also is extensively used in image processing [8, 16].

Next, we convert each vector ${s}'_{i}$ into a block of size $b\times b$. After that, we perform inverse sparsification (i.e., we apply the two-dimensional Inverse DCT) to each of these blocks to generate blocks ${x}'_{i}$ of the image. That is,

\begin{equation}\label{eq:SABMIS_IDCT}
{x}'_{i}=IDCT\left( {s}'_{i}\right).
\end{equation}

Next, we construct the sub-stego-image of size $\frac{r}{2} \times \frac{r}{2}$ by arranging all these blocks ${x}'_{i}$. We repeat the above steps to construct all four sub-stego-images. At last, we perform inverse sampling and obtain a single constructed stego-image from these four sub-stego-images. In the experiments section, we show that the quality of the stego-image is also very good.

§.§ Extraction of the Secret Images

In this subsection, we discuss the process of extracting secret images from the stego-image. Initially, we perform sampling (as done in Section <ref> using (<ref>)–(<ref>)) of the stego-image to obtain four sub-stego-images. Since the extraction of all the secret images is similar, here, we discuss the extraction of only one secret image from one sub-stego-image. First, we perform block-wise sparsification of the chosen sub-stego-image. For this, we divide the sub-stego-image into blocks of size $b\times b$. We obtain a total of $\frac{r^2}{4\times b^2}$ blocks. Further, we sparsify each block (say $x''_i$) by computing the corresponding sparse vector (say $s''_i$). That is,

\begin{equation}
%s''_i = \Psi^T x_i,
s''_i = DCT(x''_i).
\end{equation}

Next, as earlier, we arrange these sparse blocks in a zig-zag scanning order, obtain the corresponding sparse vectors each of size $b^2 \times 1$, and then categorize each of them into two groups $s''_{i,u}\in \mathbb{R}^{p_1}$ and $s''_{i,v}\in \mathbb{R}^{p_2}$. Here, as before, $p_1$ and $p_2$ are the numbers of coefficients having large values and small values (or zero values), respectively. After that, we project each sparse vector onto linear measurements (say $y''_i\in \mathbb{R}^{(p_1+p_3)\times 1}$),

\begin{equation}
y''_i=\begin{bmatrix}y''_{i,u}\\ y''_{i,v} \end{bmatrix}=
\begin{bmatrix}
s''_{i,u}\\ \Phi s''_{i,v}
\end{bmatrix}.
\end{equation}

From $y''_i$, we extract the DCT coefficients of the embedded secret image using Algorithm <ref>. This extraction rule is the reverse of the embedding rule given in Algorithm <ref>.

Extraction Rule
[1]

* ${y''_i}$: Sequence of linear measurements of the stego-image with $i=1,2,\hdots,\frac{r^2}{4\times b^2}$.

* $p_1$ and $p_4$ are lengths of certain vectors defined on pages p1_page_ref_section and p4_page_ref_section, respectively.

* $\alpha$, $\beta$, $\gamma$, and $c$ are algorithmic constants that are chosen based upon experience. The choices of these constants are discussed in the experimental results section.

* $t'_{\hati}$: Sequence of transform coefficients of the secret image with $\hati=1,2,\hdots,\frac{m^2}{l^2}$.

Initialize $t'_{\hati}$ to zeros, where $\hati=1,2,\hdots,\frac{m^2}{l^2}$.
$i = 1$ to $\frac{r^2}{4\times b^2}$

// Extraction of the first coefficient.

${t'}_i(1)  = \frac{y''_i(p_1) - y''_i(p_1-2c)}{\alpha}$.

$j = p_1 - c + 1$ to $p_1 - 1$

// Extraction of the next $c-1$ coefficients.

${t'_i}(j-p_1+c+1)  = \frac{y''_i(j) - y''_i(j-c)}{\beta}$.

$k = p_1 + p_4 + 1$ to $p_1 + 2\times p_4 - c$

// Extraction of the remaining $p_4 - c$ coefficients.

${t'_i}(k - p_1 - p_4 + c)  = \frac{y''_i(k) - y''_i(k - p_4 + c)}{\gamma}$.

In Algorithm <ref>, $t'_{\hati} \in \mathbb{R}^{l^2\times 1}$, for ${\hati}=1,2,\hdots,\frac{m^2}{l^2}$, are the vector representations of the DCT coefficients of the blocks of one extracted secret image. Next, we convert each vector $t'_{\hati}$ into blocks of size $l\times l$, and then perform a block-wise Inverse DCT (IDCT) (using (<ref>)) to obtain the secret image pixels. Finally, we obtain the extracted secret image of size $m\times m$ by arranging all these blocks column wise. As mentioned earlier, this steganography scheme is a blind multi-image steganography scheme because it does not require any cover image data at the receiver side for the extraction of secret images.

Here, the process of hiding (and extracting) secret images is not fully lossless[This is common in transform-based image steganography.], resulting in the degradation of the quality of extracted secret images. This is because we first oversample the original image using (<ref>), and then we construct the stego-image by solving the optimization problem (<ref>), which leads to a loss of information. However, our algorithm is designed in such a way that we are able to extract high-quality secret images. We support this fact with examples in the experiments section (specifically, Section <ref>). We term our algorithm Sparse Approximation Blind Multi-Image Steganography (SABMIS) scheme due to the involved sparse approximation and the blind multi-image steganography.

§ EXPERIMENTAL RESULTS

Experiments are carried out in MATLAB on a machine with an Intel Core i3 processor @2.30 GHz and 4GB RAM. We use a set of 31 standard grayscale images available from miscellaneous categories of the USC-SIPI image database [3] and two other public domain databases [1, 2]. In this work, we report results for 10 images (shown in Figure <ref>). However, our SABMIS scheme is applicable to other images as well. Our selection is justified by the fact that the image processing literature has frequently used these 10 images or a subset of them.

Living room

Camera man

Test images used in our experiments.

Here, we take all ten images shown in Figure <ref> as the cover images, and four images; Figures <ref>, <ref>, <ref> and <ref> as the secret images for our experiments. However, we can use any of the ten images as the secret images.

Although the images shown in Figure <ref> look to be of the same dimension, they are of varying sizes. For our experiments, each cover image is converted to $1024\times 1024$ size (i.e., $r\times r$). We take blocks of size $8\times 8$ for the cover images (i.e., $b\times b$). Recall from subsection <ref> that the size of the DCT sparsified vectors is $(p_1+p_2) \times 1$ with $p_1+p_2=b^2$ (here, $b^2 = 64$). Applying DCT on images results in a sparse vector where more than half of the coefficients have values that are either very small or zero [4, 21]. This is the case here as well. Hence, in our experiments, we take $p_1 = p_2 = 32$. Recall, the size of the measurement matrix $\Phi$ is $p_3\times p_2$ with $p_3 > p_2$. We take $p_3 = 50\times p_2$. Without loss of generality, the element values of the column-normalized measurement matrix are taken as random numbers with mean $0$ and standard deviation $1$, which is a common standard.

Each of the secret image is converted to $512\times 512$ size (i.e., $m\times m$). This choice is also motivated by the fact that we chose the size of the secret image to be half of that of the cover image ($1024\times 1024$). We take blocks of size $8\times 8$ for the secret images as well (i.e., $l\times l$). In general, the DCT coefficients can be divided into three sets; low frequencies, middle frequencies, and high frequencies. Low frequencies are associated with the illumination, middle frequencies are associated with the structure, and high frequencies are associated with the noise or small variation details. Thus, these high-frequency coefficients are of very little importance for the to-be embedded secret images. Since the number of high-frequency coefficients is usually half of the total number of coefficients, we take $p_4 = 32$ (using $8\times 8$ divided by $2$).

The values of the constants in Algorithm <ref> and Algorithm <ref> are taken as follows[The values of these constants do not affect the convergence of ADMM much. Determining the range of values that work best here is part of our future work.] (based upon experience): $\alpha=0.01$, $\beta=0.1$, $\gamma=1$, and $c=6$. For ADMM, we set the maximum number of iterations as $500$, the absolute stopping tolerance as $1\times 10^{-4}$, and the relative stopping tolerance as $1\times 10^{-2}$. These values are again taken based upon our experience with a similar algorithm [4]. Eventually, our ADMM almost always converges in $10$ to $15$ iterations.

As mentioned earlier, in the five sections below we experimentally demonstrate the usefulness of our steganography scheme. First, in Section <ref>, we show analytically that our SABMIS scheme gives excellent embedding capacities. Second, in Section <ref>, we show that the quality of the constructed stego-images, when compared with the corresponding cover images, is of high. Third, in Section <ref>, we demonstrate the good quality of the extracted secret images when compared with the original secret images. Fourth, in Section <ref>, we show that our SABMIS scheme is resistant to steganographic attacks. Finally and fifth, in Section <ref>, we perform a comparison of our scheme with various other steganography schemes.

§.§ Embedding Capacity Analysis

The embedding capacity (or embedding rate) is the number (or length) of secret bits that can be embedded in each pixel of the cover image. It is measured in bits per pixel[Since in the transform domain-based steganography schemes, some specific transform coefficients are embedded into the cover image (along with the secret bits), a more appropriate term that can be used for embedding capacity is “bits of information per pixel” (bipp). However, to avoid confusion, we use the term bpp in this paper, which is commonly used.] (bpp) and is calculated as follows:

\begin{equation}
\text{EC in \si{bpp}} = \frac{\text{Total number of secret bits embedded}}{\text{Total number of pixels in the cover image}}.
\end{equation}

In this scheme, we embed a maximum of four secret images each of size $512\times 512$ in a cover image of size $1024\times 1024$. Thus, we obtain embedding capacities of 2bpp, 4bpp, 6bpp, and 8bpp while embedding one, two, three, and four secret images, respectively.

§.§ Stego-Image Quality Assessment

In general, the visual quality of the stego-image degrades as the embedding capacity increases. Hence, preserving the visual quality becomes increasingly important. There is no universal criterion to determine the quality of the constructed stego-image. However, we evaluate it by visual and numerical measures.
We use Peak Signal-to-Noise Ratio (PSNR), Mean Structural Similarity (MSSIM) index, Normalized Cross-Correlation (NCC) coefficient, entropy, and Normalized Absolute Error (NAE) numerical measures.

When using the visual measures, we construct the stego-images corresponding to the different cover images used in our experiments and then check their distortion visually. We also check their corresponding histograms and edge map diagrams. Here, we present the visual comparison only for `Zelda' as the cover image with `Peppers' secret image and the corresponding stego-image. We get similar results for the other images as well. The comparison is given in Figure <ref>. The cover image and its corresponding histogram and edge map are shown in parts (a), (b) and (c) of this figure. The stego-image and its corresponding histogram and edge map are given in parts (d), (e) and (f) of the same figure. When we compare each figure with its counterpart, we find that they are very similar.

Cover Image (CI)

CI Histogram

CI Edge Map

Stego-Image (SI)

SI Histogram

SI Edge Map

Visual quality analysis between `Zelda' cover image (CI) and its corresponding stego-image (SI).

Next, when using the numerical measures to assess the quality of the stego-image with respect to the cover image, we first evaluate the most common measure of PSNR value in Section <ref>. Subsequently, we evaluate the other more rarely used numerical measures of MSSIM index, NCC coefficient, entropy, and NAE in Section <ref>.

§.§.§ Peak Signal-to-Noise Ratio (PSNR) Value

We compute the PSNR values to evaluate the imperceptibility of stego-images (SI) with respect to the corresponding cover images (I) as follows [9]:

\begin{equation}%\label{eq:PSNR}
PSNR(I, SI)=10\log_{10}\frac{R^{2}}{MSE(I, SI)}\: \si{dB},
\end{equation}

where $MSE(I, SI)$ represents the mean square error between the cover image $I$ and the stego-image $SI$, $R$ is the maximum intensity of the pixels, which is $255$ for grayscale images, and dB refers to decibel. This error is calculated as

\begin{equation}%\label{eq:MSE}
MSE(I, SI)=\frac{\sum_{i=1}^{r1}\sum_{j=1}^{r2}\left ( I\left ( i,\, j \right )-SI\left ( i,\, j \right ) \right )^2}{r1\times r2},
\end{equation}

where $r1$ and $r2$ represent the row and column numbers of the image (for us either cover or stego), respectively, and $I(i,j)$ and  $SI(i,j)$ represent the pixel values of the cover image and the stego-image, respectively.

A higher PSNR value indicates a higher imperceptibility of the stego-image with respect to the corresponding cover image. In general, a value higher than 30dB is considered to be good since human eyes can hardly distinguish the distortion in the image [13, 30, 18].

The PSNR values of the stego-images corresponding to the ten cover images are given in Figure <ref> and Figure <ref>. In Figure <ref>, we show the PSNR values of all the stego-images when separately all the four secret images (mentioned above in Figure <ref>) are embedded. In this figure, we obtain the highest PSNR value (47.32dB) when the `Zelda' secret image is hidden in the `Zelda' cover image, while the lowest PSNR value (40.21dB) is obtained when the `Peppers' secret image is hidden in the `Boat' cover image.

In Figure <ref>, we show the PSNR values for the four cases of embedding one, two, three, and four secret images in the ten cover images. As we have four secret images, when embedding one secret image, we have a choice of embedding any one of them and present the resulting PSNR values. However, we separately embed all four images, obtain their PSNR values, and then present the average results. Similarly, the average PSNR values are presented for the cases when we embed two and three images. In this figure, we obtain the highest average PSNR value (45.67dB) when one secret image is hidden in the `Zelda' cover image, while the lowest PSNR value (34.98dB) is obtained when all four secret images are hidden in the `Livingroom' cover image. Also, we observe that for all test cases, we obtain PSNR values higher than 30dB which, as earlier, are considered good.

PSNR values of the stego-images when only one secret image is hidden.

PSNR values of the stego-images when different numbers of images are hidden.

§.§.§ Other Numerical Measures

Mean Structural Similarity (MSSIM) Index
This is an image quality assessment metric used to measure the structural similarity between two images, which is most noticeable to humans [14, 9]. MSSIM between the cover image $I$ and the stego-image $SI$ is given as

\begin{align}%\label{eq:SSIM}
& MSSIM(I, SI)=\frac{1}{M}\sum_{j=1}^{M}SSIM(i_{j}, si_{j}),
\end{align}

where $i_{j}$ and $si_{j}$ are the content of the cover image and the stego-image, respectively, at the $j^{th}$ local window with $M$ being the number of local windows [14, 27], and

\begin{align}%\label{eq:SSIM}
& SSIM(x, y)=\frac{(2\mu _{x}\mu _{y}+C_{1})(2\sigma _{xy}+C_{2})}{(\mu _{x}^{2}+\mu _{y}^{2}+C_{1})(\sigma _{x}^{2}+\sigma _{y}^{2}+C_{2})},\end{align}

where for vectors $x$ and $y$, $\mu_{x}$ is the weighted mean of $x$, $\mu_{y}$ is the weighted mean of $y$, $\sigma_{x}$ is the weighted standard deviation of $x$, $\sigma_{y}$ is the weighted standard deviation of $y$, $\sigma_{xy}$ is the weighted covariance between $x$ and $y$, $C_{1}$ and $C_{2}$ are positive constants.

We take $M$ = $1069156$, $C_1 = (0.01\times255)^2$, and $C_2 = (0.03\times255)^2$ based upon the recommendations from [14, 27].
The value of the MSSIM index lies between $0$ and $1$, where the value $0$ indicates that there is no structural similarity between the cover image and the corresponding stego-image, and the value $1$ indicates that the images are identical.

Normalized Cross-Correlation (NCC) Coefficient:
This metric measures the amount of correlation between two images [23]. The NCC coefficient between the cover image $I$ and the stego-image $SI$ is given as

\begin{align}%\label{eq:NC}
& NCC(I,SI)=\frac{\sum_{i=1}^{r1}\sum_{j=1}^{r2}I(i,j)SI(i,j)}{\sum_{i=1}^{r1}\sum_{j=1}^{r2}I^{2}(i,j)},
\end{align}

where $r1$ and $r2$ represent the row and column numbers of the image (for us either cover or stego), respectively, and $I(i,j)$ and $SI(i,j)$ represent the pixel values of the cover image and the stego-image, respectively. The NCC coefficient value of $1$ indicates that the cover image and the stego-image are highly correlated while a value of $0$ indicates that the two are not correlated.

In general, entropy is defined as the measure of average uncertainty of a random variable.

In the context of an image, it is a statistical measure of randomness that can be used to characterize the texture of the image [11]. For a grayscale image (either a cover image or a stego-image in our case), entropy is given as

\begin{align}%\label{eq:entropy}
E=-\sum_{i=0}^{255}(p_{i}\log _{2}p_{i}), % MCS: better?
%Entropy=-\sum_{i=0}^{255}(p_{i}\log _{2}p_{i}),
\end{align}

where $p_{i} \in [0, 1]$ is the fraction of image pixels that have the value $i$. If the stego-image is similar to its corresponding cover image, then the two should have similar entropy values (due to similar textures).

Normalized Absolute Error (NAE):
This metric is a distance measure that captures pixel-wise differences between two images [6]. NAE between the cover image $I$ and the stego-image $SI$ is given as

\begin{equation}%\label{eq:NAE}
NAE(I, SI)=\frac{\sum_{i=1}^{r1}\sum_{j=1}^{r2}\left ( | CI\left ( i,\, j \right )-SI\left ( i,\, j \right ) | \right )}{\sum_{i=1}^{r1}\sum_{j=1}^{r2} CI\left ( i,\, j \right )},
\end{equation}

where $r1$ and $r2$ represent the row and the column numbers of the image (for us either cover or stego), respectively, and $CI(i,j)$ and $SI(i,j)$ represent the pixel values of the cover image and the stego-image, respectively. NAE has values in the range $0$ to $1$. A value close to $0$ indicates that the cover image is very close to its corresponding stego-image, and a value close to $1$ indicates that the two are substantially far apart.

In Table <ref>, we present the values of MSSIM index, NCC coefficient, entropy and NAE for our SABMIS scheme when hiding all four secret images. We do not present the values for the cases of embedding less than four secret images as their results will be better than those given in Table <ref>. Hence, our reported results are for the worst case. From this table, we observe that all values of the MSSIM index are nearly equal to $1$ (different in the sixth place of decimal), the values of NCC coefficients are close to 1, and values of NAE are close to $0$. The entropy values of the cover and the stego-images are almost identical. All these values indicate that the cover images and their corresponding stego-images are almost identical.

MSSIM index, NCC coefficient, entropy, and NAE of the stego-images when compared with the corresponding cover images.

Image     1c|2*MSSIM     1c|2*NCC      2c|Entropy                                                                                1c|2*NAE

1|c|      1c|      1c|                                   1c|c]@c@Cover
Image     1c|c]@c@Stego-
Image     1c|

1|c|Lena                                                                                1c|1                                   1c|0.9992                                                    1c|7.443                             1c|7.463     1c|0.009
1|c|Pepper                                                                              1c|1                                   1c|0.9997                                                    1c|7.573                                                                 1c|7.598     1c|0.012
1|c|Boat                                                                                1c|1                                   1c|0.9998                                                    1c|7.121                                                                 1c|7.146          1c|0.012
1|c|Goldhill                                                                            1c|1                                   1c|0.9998                                                     1c|7.471                                                                  1c|7.486       1c|0.013
1|c|Zelda                                                                               1c|1                                   1c|0.9964                                                    1c|7.263                                                                 1c|7.272                      1c|0.011
1|c|Tiffany                                                                             1c|1                                   1c|0.9999                                                    1c|6.602                                                                 1c|6.628                 1c|0.006
1|c|Livingroom                                                                          1c|1                                   1c|0.9996                                                    1c|7.431                                                                 1c|7.438                    1c|0.013
1|c|Tank                                                                                1c|1                                   1c|0.9998                                                    1c|6.372                                                                 1c|6.405                  1c|0.014
1|c|Airplane                                                                            1c|1                                   1c|0.9972                                                    1c|6.714                                                                  1c|6.786         1c|0.015
1|c|Cameraman                                                                           1c|1                                   1c|1.0000                                                        1c|7.055                                                                 1c|7.123                      1c|0.009
1|c|Average                                                                    1c|1                          1c|0.9991                                 1c|7.104       1c|7.1343                                                     1c|0.011

§.§ Secret Image Quality Assessment

Since human observers are considered the final arbiter to assess the quality of the extracted secret images, we compare one such original secret image and its corresponding extracted secret image. The results of all other combinations are almost the same. In Figures <ref> and <ref>, we show the original `Peppers' secret image and the extracted `Peppers' secret image (from the `Zelda' stego-image). From these figures, we observe that there is very little distortion in the extracted image. Besides this, for these two images, we also present their corresponding histograms (in Figures <ref> and <ref>, respectively), and edge map diagrams (in Figures <ref> and <ref>, respectively). Again, we observe minimal variations between the original and the extracted secret images.

`Peppers' Original Secret Image (OSI)

OSI Histogram

OSI Edge Map

`Peppers' Extracted Secret Image (ESI)

ESI Histogram

ESI Edge Map

Visual quality analysis between the `Peppers' original secret image and the `Peppers' extracted secret image (from the `Zelda' stego-image).

§.§ Security Analysis

The SABMIS scheme is a transform domain based technique which employs an indirect embedding strategy, i.e., it does not follow the Least Significant Bit (LSB) flipping method, and hence, it is immune to statistical attacks [28, 29].
Moreover, in the SABMIS scheme, the measurement matrix $\Phi$, and the embedding/ extraction algorithmic settings are considered as secret-keys, which are shared between the sender and the legitimate receiver. It is assumed that these keys are not known to the eavesdropper. Hence, we achieve increased security in our proposed system.

To justify this, we extract the secret image in two ways, i.e., by using correct secret-keys and by using wrong secret-keys. Since the measurement matrix, which we use (random matrix having numbers with mean $0$ and standard deviation $1$) is one of the most commonly used measurement matrix and the eavesdropper can easily guess it, we use this same measurement matrix while building wrong secret-keys. Here, we use the same dimension of this matrix as well, i.e., $p3 \times p2$. In reality, the guessed matrix size would be different from the original matrix size, which would make the extraction task of the eavesdropper more difficult.

The algorithmic settings that we use will be completely unknown to the eavesdropper. These involve using few constants ($\alpha=0.01$, $\beta=0.1$, $\gamma=1$ and $c=6$) and a set of cover image coefficients where secret image coefficients are embedded (using $p1$ and $p4$). While building wrong secret-keys, we take the common guess of one for all constants (i.e., $\alpha=1$, $\beta=1$, $\gamma=1$ and $c=1$) without changing the rest of the settings (i.e., same $p1$ and $p4$). In reality, the eavesdropper would not be able to correctly guess these other settings as well resulting in a further challenges during extraction.

We compare the average NAE values (between the original secret images and the corresponding extracted ones) for both these cases. These values are presented in Figure <ref>. In this figure, we see that for correct secret-keys, the NAE values are close to 0, and the for wrong secret-keys, the NAE values are very high. Hence, in the SABMIS scheme, a change in secret-keys will lead to a shift in the accuracy between the original secret images and their corresponding extracted ones, in turn, making our scheme secure.

Moreover, in Figure <ref> (a) and (b), we compare the `Peppers' secret image when extracted using correct and wrong secret-keys (from the `Zelda' stego-image), respectively. From this figure, we see that when using correct secret-keys, the visual distortion in the extracted secret image is negligible, and when using the wrong secret-keys, the distortion in the extracted secret image is very high (it is almost black).

Average NAE value between the four original secret images and their corresponding extracted secret images when using correct and wrong secret-keys.

`Peppers' extracted secret image (using correct secret-keys)

`Peppers' extracted secret image (using wrong secret-keys)

Visual quality analysis between the `Peppers' extracted secret image using correct and wrong secret-keys (from the `Zelda' stego-image).

§.§ Performance Comparison

We compare our SABMIS scheme with the existing steganography schemes (discussed in the Introduction section) for the embedding capacity, the quality of stego-images, and resistance to steganographic attacks. For the stego-image quality comparison, since most works have computed PSNR values only, we use only this metric for analysis. Further, resistance to steganographic attacks is not experimented by most of the past works. Rather, they have just stated whether their scheme has such a property or not. Hence, we use this information for comparison. Finally, although we check the quality of the extracted secret images by comparing them with the corresponding original secret images (as earlier), this check is not common in the existing works. Hence, we do not perform this comparison.

As mentioned in the introduction, because of the lack of past work of embedding grayscale secret images into a grayscale cover image, we compare with the cases of embedding binary images into a grayscale image or embedding grayscale images into a color image. Both these problems are easier than our problem.

This comparison is given in Table <ref>. As evident from this table, our scheme outperforms the competitive schemes in embedding capacity for the three cases of embedding one, two, and three secret images. We do have a slight loss in the quality of the stego-images. However, as earlier, this is considered acceptable because we are compromising very little in quality while gaining a lot more in embedding capacity. Moreover, PSNR values over 30dB are considered good [13, 30, 18]. Further, our scheme, like most of the other competitive schemes (not all; please see [12]), is resistant to steganographic attacks. Finally, we are the first ones to embed four secret images in a cover image.

Performance comparison of our SABMIS scheme with various other steganography schemes.

images     Steganography Scheme             c]@c@Type of
secret image     c]@c@Type of
cover images     c]@c@Embedding
(in bpp)     c]@c@PSNR
(in dB)     c]@c@Resistant
to stegan-
2*1                                                                     [6]     Binary                                                                   Grayscale                                                                        0.25                                                                                49.69                                                               Yes
SABMIS                           Grayscale                                                                   Grayscale                                                                     2                                                                                43.15                                                               Yes

2*2                                                                     [15]                              Grayscale                                                                       Color                                                                     1.33                                                                                44.75                                                               Yes
SABMIS                           Grayscale                                                                   Grayscale                                                                     4                                                                                39.83                                                               Yes

2*3                                                                     [12]                              Binary                                                                   Grayscale                                                                         2                                                                                46.36                                                               No
SABMIS                           Grayscale                                                                   Grayscale                                                                     6                                                                                38.12                                                               Yes

4                                                                                SABMIS                           Grayscale                                                                   Grayscale                                                                     8                                                                                37.14                                                               Yes

§ CONCLUSIONS AND FUTURE WORK

Securing secret information over Internet-based applications becomes important as illegitimate users try to access them. One way to secure this data is to hide them into an image as the cover media, known as image steganography. Here, the challenges are increasing the embedding capacity of the scheme, maintaining the quality of the stego-image, and ensuring that the scheme is resistant to steganographic attacks. We propose a blind multi-image steganography scheme for securing secret images in cover images to substantially overcome all the three challenges. All our images are grayscale, which is a hard problem.

First, we perform sampling of the cover image to obtain four sub-images. Further, we sparsify (using DCT) each sub-image and obtain its sparse linear measurements using a random measurement matrix. Then, we embed DCT coefficients of the secret images into these linear measurements using our novel embedding rule. Second, we construct the stego-image from these modified measurements by solving the resulting LASSO problem using ADMM. Third, using our proposed extraction rule, we extract the secret images from the stego-image. We call this scheme SABMIS (Sparse Approximation Blind Multi-Image Steganography). Using these components together helps us to embed up to four secret images into one cover image (instead of the common embedding of two secret images).

We perform experiments on several standard grayscale images. We obtain embedding capacities of 2bpp (bits per pixel), 4bpp, 6bpp, and 8bpp while embedding one, two, three, and four secret images, respectively. These embedding capacities are higher than all the embedding capacities obtained in the literature until now.  Further, there is with very little deterioration in the quality of the stego-image as compared to its corresponding cover image (measured by the above metrics). The quality of the original secret images and their corresponding extracted secret images is also almost the same. Further, due to our algorithmic design, our scheme is resistant to steganographic attacks as well.

Next, we discuss the future work in this context. First, since we embed secret images in images, in the future we plan to embed the secret data into text, audio, and video. Second, we plan to apply optimization techniques to improve the values of parameters $\alpha, \beta, \gamma$, etc. used in the embedding and the extraction algorithms of the SABMIS scheme. Third, we plan to extend this work to real-time applications such as hiding fingerprint data, iris data, medical information of patients, and personal signatures.

§ ACKNOWLEDGEMENT

This work has been supported by the DAAD in the project
`A new passage to India' between IIT Indore and the Leibniz Universität Hannover.

[1]
(Accessed: 22-10-2019a).
Images Database.
<https://homepages.cae.wisc.edu/ ece533/images/>.

[2]
(Accessed: 22-10-2019b).
Images Database.

[3]
(Accessed: 22-10-2019).
The USC-SIPI Images Database.

[4]
Agrawal, R. and Ahuja, K. (2021).
CSIS: compressed sensing-based enhanced-embedding capacity image
steganography scheme.
IET Image Processing, 15(9):1909–1925.

[5]
Arunkumar, S., Subramaniyaswamy, V., Ravichandran, K. S., and Logesh, R.
RIWT and QR factorization based hybrid robust image steganography
using block selection algorithm for IoT devices.
Journal of Intelligent & Fuzzy Systems, 35(5):4265–4276.

[6]
Arunkumar, S., Subramaniyaswamy, V., Vijayakumar, V., Chilamkurti, N., and
Logesh, R. (2019b).
SVD-based robust image steganographic scheme using RIWT and DCT
for secure transmission of medical images.
Measurement, 139:426–437.

[7]
Baluja, S. (2019).
Hiding images within images.
IEEE transactions on pattern analysis and machine intelligence,

[8]
Boyd, S., Parikh, N., Chu, E., Peleato, B., and Eckstein, J. (2010).
Distributed optimization and statistical learning via the alternating
direction method of multipliers.
Foundations and Trends in Machine Learning, 3(1):1–122.

[9]
Elzeki, O. M., Elfattah, M. A., Salem, H., Hassanien, A. E., and Shams, M.
A novel perceptual two layer image fusion using deep learning for
imbalanced covid-19 dataset.
PeerJ Computer Science, 7:e364.

[10]
Gabay, D. (1976).
A dual algorithm for the solution of nonlinear variational problems
via finite element approximation.
Computers & Mathematics with Applications, 2(1):17–40.

[11]
Gonzalez, R. C., Woods, R. E., and Eddins, S. L. (2004).
Digital image processing using MATLAB.
Pearson Education India.

[12]
Guttikonda, P., Cherukuri, H., and Mundukur, N. B. (2018).
Hiding encrypted multiple secret images in a cover image.
In Proceedings of International Conference on Computational
Intelligence and Data Engineering, pages 95–104. Springer.

[13]
Gutub, A. and Shaarani, F. A. (2020).
Efficient implementation of multi-image secret hiding based on LSB
and DWT steganography comparisons.
Arabian Journal for Science and Engineering, 45(4):2631–2644.

[14]
Habib, W., Sarwar, T., Siddiqui, A. M., and Touqir, I. (2016).
Wavelet denoising of multiframe optical coherence tomography data
using similarity measures.
IET Image Processing, 11(1):64–79.

[15]
Hemalatha, S., Acharya, U. D., Renuka, A., and Kamath, P. R. (2013).
A secure image steganography technique to hide multiple secret
In Computer Networks & Communications (NetCom), pages
613–620. Springer.

[16]
Hwang, H. J., Kim, S., and Kim, H. J. (2016).
Reversible data hiding using least square predictor via the LASSO.
EURASIP Journal on Image and Video Processing, 2016(1):42.

[17]
Kordov, K. and Zhelezov, S. (2021).
Steganography in color images with random order of pixel selection
and encrypted text message embedding.
PeerJ Computer Science, 7:e380.

[18]
Liu, C. L. and Liao, S. R. (2008).
High-performance JPEG steganography using complementary embedding
Pattern Recognition, 41(9):2945–2955.

[19]
Maheswari, S. U. and Hemanth, D. J. (2017).
Performance enhanced image steganography systems using transforms and
optimization techniques.
Multimedia Tools and Applications, 76(1):415–436.

[20]
Nardone, D., Ciaramella, A., and Staiano, A. (2019).
A sparse-modeling based approach for class specific feature
PeerJ Computer Science, 5:e237.

[21]
Pal, A. K., Naik, K., and Agrawal, R. (2019).
A steganography scheme on JPEG compressed cover image with high
embedding capacity.
The International Arab Journal of Information Technology,

[22]
Pan, J. S., Li, W., Yang, C. S., and Yan, L. J. (2015).
Image steganography based on subsampling and compressive sensing.
Multimedia Tools and Applications, 74(21):9191–9205.

[23]
Parah, S. A., Sheikh, J. A., Loan, N. A., and Bhat, G. M. (2016).
Robust and blind watermarking technique in DCT domain using
inter-block coefficient differencing.
Digital Signal Processing, 53:11–24.

[24]
Sanjutha, M. K. (2018).
An image steganography using particle swarm optimization and
transform domain.
International Journal of Engineering & Technology,

[25]
Srinivas, M. and Naidu, R. M. (2015).
Sparse approximation of overdetermined systems for image retrieval
In Agrawal, P., Mohapatra, R., Singh, U., and Srivastava, H.,
editors, Mathematical Analysis and its Applications, volume 143, pages
219–227. Springer.

[26]
Stallings, W. (2019).
Cryptography and network security: principles and practice.
Prentice Hall.

[27]
Wang, Z., Bovik, A., Sheikh, H., and Simoncelli, P. (2004).
Image quality assessment: From error visibility to structural
IEEE Transactions on Image Processing, 13(4):600–613.

[28]
Westfeld, A. and Pfitzmann, A. (2000).
Attacks on steganographic systems.
In Pfitzmann, A., editor, Information Hiding, volume 1768,
pages 61–76. Springer.

[29]
Yu, L., Zhao, Y., Ni, R., and Zhu, Z. (2009).
PM1 steganography in JPEG images using genetic algorithm.
Soft Computing, 13(4):393–400.

[30]
Zhang, Y., Jiang, J., Zha, Y., Zhang, H., and Zhao, S. (2013).
Research on embedding capacity and efficiency of information hiding
based on digital images.
International Journal of Intelligence Science, 3(02):77–85.


# The mechanism of twin thickening and the elastic strain state of TWIP steel
nanotwins

TWJ Kwok  TP McAuliffe  AK Ackerman  BH Savitzky  M Danaie  C Ophus  D Dye
<EMAIL_ADDRESS>Department of Materials, Royal School of Mines,
Imperial College London, Prince Consort Road, London, UK  National Center for
Electron Microscopy (NCEM), Molecular Foundry, Lawrence Berkeley Lab, USA
Electron Physical Science Imaging Centre (ePSIC), Diamond Light Source,
Oxford, UK

###### Abstract

A Twinning Induced Plasticity (TWIP) steel with a nominal composition of
Fe-16.4Mn-0.9C-0.5Si-0.05Nb-0.05V was deformed to an engineering strain of 6%.
The strain around the deformation twins were mapped using the 4D-STEM
technique. Strain mapping showed a large average elastic strain of
approximately 6% in the directions parallel and perpendicular to the twinning
direction. However, the large average strain comprised of several hot spots of
even larger strains of up to 12%. These hot spots could be attributed to a
high density of sessile Frank dislocations on the twin boundary and correspond
to shear stresses of 1–1.5 GPa. The strain and therefore stress fields are
significantly larger than other materials known to twin and are speculated to
be responsible for the early thickness saturation of TWIP steel nanotwins. The
ability to keep twins extremely thin helps improve grain fragmentation, i.e.
the dynamic Hall-Petch effect, and underpins the large elongations and strain
hardening rates in TWIP steels.

††journal: arXiv

Twinning has become increasingly significant in modern metallurgy [1].
Deformation or mechanical twinning occurs when a region of the crystal is
sheared, with each atom moving less than one lattice spacing, such that the
twinned region forms a mirror image of the parent crystal about the twinning
or habit plane [2]. Many modern materials such as TWIP steels [3, 4], medium
Mn steels [5], TWIP Ti [6, 7], Mg alloys and Zr alloys [8] all capitalise on
twinning to enable high strain hardening rates and improve elongation.
However, as Qin and Bhadeshia [9] point out, the maximum contribution of
twinning to total elongation (true strain) in a random polycrystal is $<0.3$.
This is compared to total elongation of 0.5–0.6 true strain in TWIP steels
[10, 11]. It is therefore the interaction between twins and dislocations which
are of importance. In TWIP steels, long and thin twins are formed during
deformation, acting as barriers to dislocation motion. The volume fraction of
twins increase with increasing strain and continuously subdivides the grain,
resulting in a high strain hardening rate and also elongation. This mechanism
has been termed the dynamic Hall-Petch effect [12, 13].

Figure 1: Schematic ball and stick model of twinning in FCC materials. (a)
Perfect FCC lattice with ABCABC stacking. (b) Passage of one, (c) two and (d)
three Shockley partial dislocations on successive (111) planes building up the
twin thickness. Grey regions indicate the growing twin. (e) Schematic of the
interaction between dislocation and twin boundaries described by Idrissi and
Schryvers [14], drawn after De Cooman et al. [10].

In the case of Face Centred Cubic (FCC) crystals, the origin of twinning is
still a matter of academic debate with many different mechanisms being
proposed although the three layer twin nucleus model by Mahajan and Chin [15]
is increasingly being used in computational models [16, 10]. Nevertheless, the
fundamentals of FCC twinning are well understood and first involves the
dissociation of a full $\frac{a}{2}\\{111\\}\langle 110\rangle$ into two
$\frac{a}{6}\\{111\\}\langle 112\rangle$ Shockley partial dislocations, where
$a$ is the FCC lattice spacing. In TWIP steels, the formation of partial
dislocations is favoured due to the low Stacking Fault Energy (SFE). An FCC
twin is then created, as shown in Figure 1, when partial dislocations
propagate on successive $\\{111\\}$ planes and increasing the twin thickness.

In TWIP steels, the deformation twins are often between 20–30 nm in thickness,
corresponding to the glide of approximately 100 partial dislocations [10].
Interestingly, these twins generally do not continue to thicken with strain
once this thickness is reached, rather choosing to form new twins instead.
This is compared to a larger twin thickness of $\leq 100$ nm in Cu-Al alloys
[17] and 1–2 µm in $\alpha$-brass [18]. Therefore, while there is a general
mechanism for twin thickening, there also appears to be mechanism for twin
thickness saturation based on some material property such as SFE [17] and
grain size [19, 20]. Unfortunately, the reasons for twin thickness saturation
are not very well studied, even though the ability of TWIP steels to form a
high density of very fine twins is paramount to a high degree of grain
fragmentation, underpinning the dynamic Hall-Petch effect and overall tensile
properties.

Nevertheless, Idrissi and Schryvers [14] studied the interaction between
coherent twin boundaries and dislocations and proposed a “pole $+$ deviation”
mechanism that might help to control twin thickness. Illustrated in Figure 1e,
the first step involves two Shockley partials (D$\gamma$ and $\gamma$A)
gliding on the K1 primary slip plane which constrict and recombine under
applied stress to form a full dislocation (DA). In the second step, the full
dislocation meets the coherent twin boundary and dissociates again under
stress into a sessile Frank partial dislocation (D$\delta$) and a Shockley
partial dislocation ($\delta$A) according to the reaction:

$\frac{a}{2}\langle 110\rangle\rightarrow\frac{a}{3}\langle
111\rangle_{sessile}+\frac{a}{6}\langle 11\mathrm{bar}{2}\rangle$ (1)

where the sessile Frank partial dislocation (D$\delta$) remains along the twin
boundary and the Shockley partial dislocation ($\delta$A) glides off on the K2
conjugate plane, increasing the twin thickness by one layer. Idrissi and
Schryvers [14] then propose that the high density of sessile Frank partials on
the twin boundary will affect the mobility of Shockley partials on the
twinning plane (K1), limiting the thickening of the twin. Recent computational
work by Grilli et al. [21] confirmed the observations by Idrissi and Schryvers
[14] that residual dislocations on the twin boundary can completely stop twin
thickening and add that materials with a larger SFE produces larger twins.

To further investigate the reasons behind the early twin thickness saturation
in TWIP steels, Four Dimensional Scanning Transmission Electron Microscopy
(4D-STEM) can used to map the strain state in and around a mechanical twin.
4D-STEM is a relatively new technique where an electron diffraction pattern is
acquired at every point in a scan grid where sub-nm spatial resolution can be
achieved [22]. The term “4D” refers to the collection of 2D diffraction
patterns over a 2D grid. A comprehensive review of 4D-STEM and its
applications in strain mapping, imaging and ptychography is available in a
review by Ophus [22].

In this study, a 400 g ingot of steel with nominal composition of
Fe-16.4Mn-0.9C-0.5Si-0.05Nb-0.05V (SFE $\approx$ 23 mJ m-2 [23]) based on the
composition by Scott et al. [24] was produced by arc melting pure elements in
a vacuum arc melter. The steel was cast into a copper crucible and homogenised
at 1300 °C for 24 h in Ar. The steel ingot was then hot rolled from
$10\rightarrow 5$ mm (50% reduction) in 2 passes at 1000 °C, cold rolled from
$5\rightarrow 1.5$ mm (67% reduction) and finally annealed at 1000 °C for 5
min. Sub-sized tensile samples with gauge dimensions of 19 $\times$ 1.5
$\times$ 1.5 mm were obtained from the rolled strip via electrical discharge
machining. Tensile samples were then tested to 6% engineering strain
(approximately 5% plastic strain) and to failure, both at a nominal strain
rate of $10^{-3}$ s-1.

Electron Backscattered Diffraction (EBSD) was conducted using a Sigma 300 FEG-
SEM on the as-annealed and 6% strained conditions. A strain of 6% was chosen
to produce a sufficient twin density in order to locate the twins using EBSD
while keeping background dislocation density to a minimum. In the 6% strained
sample, a grain with a $\langle 111\rangle$ crystallographic direction in the
vertical axis of the microscope field of view was selected and a thin foil was
extracted using the Focussed Ion Beam (FIB) lift-out technique, producing a
foil with its $\langle 111\rangle$ direction parallel to the foil normal.

4D-STEM was performed on the foil using the probe corrected JEOL ARM200CF
Transmission Electron Microscope (TEM) at ePSIC (Oxford, United Kingdom).
Diffraction patterns were collected with a Merlin (MediPix) direct electron
detector. An accelerating voltage of 200 kV and a camera length of 40 cm were
employed with a 10 µm CL1 aperture. A $68.2\times 83.1$ nm area of interest
was scanned in $188\times 229$ real space pixels and with a $256\times 256$
pixel diffraction pattern captured at each of these scan positions with a 1 ms
dwell time per pattern.

The in-plane elastic strain was calculated from the electron diffraction
pattern at each scan location. Bragg peak identification, dataset calibration
including elliptical distortion correction and diffraction shift correction
and elastic strain calculation were performed with the open source py4DSTEM
analysis package [25]. For locating Bragg peaks, a correlation power of 1 was
used, corresponding to cross-correlation [26], and achieve subpixel precision
with local Fourier up-sampling by a factor of 16 [27]. For the rest of the
paper, an orthogonal frame of reference with directions 1, 2, 3 will be
adopted, where 3 refers to the $[111]$ direction perpendicular to the surface
of the foil, 1 as $[1\mathrm{bar}{2}1]$ and 2 as $[10\mathrm{bar}1]$.

Figure 2: EBSD IPF-X (right) and image quality maps of (a) as-annealed sample
where black lines indicate high angle grain boundaries and white lines
indicate austenite $\Sigma 3$ boundaries (rolling direction parallel to the
horizontal axis and transverse direction pointing out of the page), (b) sample
deformed to 6% strain, demonstrating fine deformation twins. (c) Engineering
tensile curves of the investigated steel when tested to failure and to 6%
strain.

The microstructure of the as-annealed TWIP steel is shown in Figure 2a with an
area weighted average grain size of 12 µm. An EBSD map of the 6% strained
sample is shown in Figure 2b. Unfortunately, due to the fine twin thickness,
it is not easy to index deformation twins using EBSD. However, they can be
identified as darker lines in the Image Quality (IQ) map. The tensile
behaviour of the investigated steel is shown in Figure 2c and has a yield
strength of 360 MPa, tensile strength of 1050 MPa and total elongation of 69%.

Figure 3: (a,b) Virtual bright and (c,d) dark field images of the steel
nanotwin using different virtual apertures indicated by the open red circles.
(e) Average deconvolution maps, sums over real space of the identified Bragg
peak locations weighted by intensity.

Figure 3a–d shows the Virtual Bright and Dark Field (VBF and VDF respectively)
images reconstructed from the intensity collected from the highlighted digital
apertures. VBF images in Figure 3a–b clearly distinguish the twin from the
matrix. Given the axis system we have adopted with [111] out of the plane and
the diffraction vectors indexed in Figure 3e, it is inferred that the twins
have a habit plane of $(11\mathrm{bar}{1})$.

In conventional TEM, the direct beam contains a large amount of structure-
dependent information. Conventional bright field imaging when used with an
objective aperture to isolate the direct beam, uses electron wave phase
information as well as intensity to re-interfere and reconstruct an image
[28]. In VBF, only access to electron intensity in the diffraction plane is
available. Therefore it is likely that the observed contrast between twin and
matrix is derived from local strain, lattice rotation or dynamical effects
which will alter the ratio of diffracted to direct intensity. The high angle
VDF image in Figure 3c is similar to convention High-Angle Annular Dark Field
(HAADF) imaging which detects electrons which are scattered to high angles and
also suggests that there was no detectable variation in local chemistry
between the twin and the matrix. The twin is explicitly highlighted in Figure
3d by reconstructing the spatial image from the $00\mathrm{bar}{2}$ reciprocal
lattice point for the twinned region only, similar to a conventional TEM dark
field image. The action of twinning rotates the the diffraction pattern about
the direct beam, such that the twinned and untwinned $02\mathrm{bar}{2}$ peaks
are separated. In Figure 3e, a Bragg vector map is presented (after Savitzky
et al. [25]), showing the distribution of all measured Bragg peak locations
for the untwinned region. It is acknowledged that a small amount of intensity
is observed at the twin reciprocal lattice points even for the untwinned
material. This is possibly due to the geometry of the sample, i.e. twinning
habit plane not perfectly parallel to beam direction, and through thickness
sampling of both untwinned and twinned material near the interface.

Figure 4: Elastic strain maps resolved in the (a) 11, (b) 22, (c) 12 and (d)
33 directions. The average 1-direction profile, perpendicular to the twin’s
length, was calculated by integrating all points along the 2-direction within
the highlighted region.

The strain components $\varepsilon_{11}$, $\varepsilon_{22}$,
$\varepsilon_{12}$ were calculated from the relative movements of the
diffraction spots. This operation was performed independently for twinned and
untwinned regions. A set of reference reciprocal lattice vectors were obtained
by averaging the untwinned regions’s reciprocal lattice basis vectors. The
magnitude of the twin basis vectors were normalised to this unstrained length.
As such, elastic strains are given with reference to this “unstrained” state.
The measurement could alternatively be considered as the elastic strain
variation across the area of interest.

Maps of the measured (11, 22, 12) and inferred strain components across the
area of interest are shown in Figure 4. An average line profile in the
1-direction was obtained by integrating points along the 2-direction within
the highlighted area. From the average profiles in Figure 4, it is striking
that the average elastic strains in the 11 and 22 directions reach values as
high as 6% at locations along the twin boundaries. Hot spots of up to 12% were
also observed at certain points along the twin boundaries. These values far
exceed other elastic strain measurements observed in the literature (Table 1),
even between other 4D-STEM experiments with the same methodology, e.g. $\sim
4\%$ observed by Pekin et al. along twin boundaries in stainless steel 304
[26]. However, it should be noted that the averaged strains have been
disproportionately raised above the baseline of approximately 4% by the
presence of hot spots.

Table 1: Comparison of strain, $\varepsilon$, in % obtained with different
strain mapping methods on various materials. *SS–Stainless Steel,
CP–Commercially Pure, N-PED–Nano-Precession Electron Diffraction.

Material Method Feature $\varepsilon$ Reference AISI 321 (SS) 4D-STEM Planar
dislocations 0.5 [29] Boron Carbide N-PED Grain boundary 0.75 [30] Pd HR-TEM
Twins 2.5 [31] SS 304 4D-STEM Twins 4 [26] CP Mg 4D-STEM Twins 4 [32] TWIP
steel 4D-STEM Twins 6 Current

The presence of hot spots was also observed by Rösner et al. [31] in a cold
rolled Pd foil. These hot spots of approximately $\leq 25\%$ strain were
attributed to dislocation debris when mapping the strain of dislocation cores
of stacking faults or partial dislocations in the foil. It should therefore
also follow, that the hot spots observed at the twin boundaries in Figure 4
were the result of a highly dislocated structure. Following the “pole $+$
deviation” mechanism proposed by Idrissi and Schryvers [14], the hot spots are
likely to be the locations where one slip system impinges onto the twin
boundary, generating a high density of sessile Frank partial dislocations at
the twin boundary and also generating a very large stress field. Faulted twin
boundaries as a result of impingement from another slip system were also shown
by Dao et al. [33] in nanotwinned Cu using HR-TEM. Furthermore, since the
sample was only deformed to 5% strain, it is likely that only a few slip
systems would have been activated, hence the localisation of strain only at
certain points along the entire twin boundary.

In the study by Grilli et al. [21], it was found that an additional shear
stress of 400 MPa to the critical resolved shear stress for twinning in
$\alpha$-uranium (orthorhombic) was sufficient to completely retard twinning
and favour slip instead. Using the analytical model from Müllner et al. [34,
35, 36], it was estimated that the strain hot spots at the twin boundaries in
the investigated steel resulted in a shear stress of approximately 1–1.5 GPa
(shear modulus of austenite being 65 GPa [12]), significantly larger than the
shear stress needed to retard twinning in $\alpha$-uranium. This large shear
stress may help to explain why deformation twins in TWIP steels remain very
thin.

However, it should be noted that the twins imaged in the current sample were
only 4 nm thick, whereas the “saturation” twin thickness in TWIP steels is
approximately 30 nm [10], suggesting that these twins will continue to thicken
with increasing strain. Based on the conclusions by Grilli et al. [21] that
twin growth is stopped when a critical density of residual dislocations on the
twin boundary is reached, it is speculated that by the time the twin has grown
to approximately 30 nm, the strain field along the twin boundary would have
become more uniform, i.e. no more hot spots. The high density of dislocations
along the twin boundary would then be able to completely stop the motion of
Shockley partial dislocations along the twin boundary and the twin would then
reach its maximum thickness.

The existence of hot spots may also help to explain the variability of twin
thickness. De Cooman et al. [10] showed that while twins are commonly
considered to be plate-like in morphology, TEM observations have shown twin
thicknesses to be highly variable and twin boundaries to be highly faulted
[33]. The formation of faulted twin boundaries appear to be caused by the
disorderly overlapping of Shockley partial dislocations [10]. Therefore, it is
speculated that while the hot spots may not have been sufficient to completely
stop twin thickening at early strains, they will certainly disrupt the
propagation of Shockley partials along the twin boundary leading to twin
growth with irregular thicknesses.

In summary, the 4D-STEM technique was used to experimentally observe the
strain fields in and around several deformation twins in a TWIP steel with
sub-nm resolution. The results show several strain hot spots of up to 12%
along the twin boundaries which very likely correspond to sessile Frank
dislocations along the twin boundaries according to the “pole $+$ deviation”
mechanism proposed by Idrissi and Schryvers [14]. The observed average strain
of 6% corresponds to a large shear stress of 1–1.5 GPa along the twin
boundaries compared to other materials and is likely to be responsible for
early retardation of twin growth and thickness saturation. This mechanism for
twin thickness saturation helps to keep deformation twins in TWIP steels
extremely fine, resulting in increased grain fragmentation with applied strain
which is central to the dynamic Hall-Petch effect and underpins the impressive
tensile properties of TWIP steels.

Acknowledgements

TPM and DD would like to acknowledge support from the Rolls-Royce plc - EPSRC
Strategic Partnership in Structural Metallic Systems for Gas Turbines
(EP/M005607/1), and the Centre for Doctoral Training in Advanced
Characterisation of Materials (EP/L015277/1) at Imperial College London. TWJK
is grateful for a studentship from A*STAR. AA acknowledges EPSRC grant IAA
EP/R511547/1. We thank Diamond Light source for access and support in use of
the ePSIC instrument E02 and proposal no. EM18770 that contributed to the
results presented here. BHS and CO acknowledge funding from the Toyota
Research Institute, and from by the Office of Science, Office of Basic Energy
Sciences, of the U.S. Department of Energy under contract no. DE-
AC02-05CH11231. We are grateful to TB Britton for insight into electron
diffraction and micromechanics and B Poole for guidance with basis rotation.
We also thank VA Vorontsov and AJ Knowles for helpful conversations when
planning the project.

References

## References

  * [1] Lu K, Lu L, Suresh S.  Strengthening materials by engineering coherent internal boundaries at the nanoscale.  Science, 2009; vol. 324 (5925), pp. 349–352.
  * [2] Cottrell A.  An Introduction to Metallurgy, The Institute of Materials, London1995, 2nd ed.
  * [3] Rahman KM, Vorontsov VA, Dye D.  The dynamic behaviour of a twinning induced plasticity steel.  Materials Science and Engineering A, 2014; vol. 589, pp. 252–261.
  * [4] Rahman KM, Jones NG, Dye D.  Micromechanics of twinning in a TWIP steel.  Materials Science and Engineering A, 2015; vol. 635, pp. 133–142.
  * [5] Kwok TWJ, Gong P, Rose R, Dye D.  The relative contributions of TWIP and TRIP to strength in fine grained medium-Mn steels.  Materials Science & Engineering A, 2022; vol. 855 (143864).
  * [6] Zhao GH, Xu X, Dye D, Rivera-Díaz-del Castillo PE.  Microstructural evolution and strain-hardening in TWIP Ti alloys.  Acta Materialia, 2020; vol. 183, pp. 155–164.
  * [7] Gao J, Huang Y, Guan D, Knowles AJ, Ma L, Dye D, Rainforth WM.  Deformation mechanisms in a metastable beta titanium twinning induced plasticity alloy with high yield strength and high strain hardening rate.  Acta Materialia, 2018; vol. 152, pp. 301–314.
  * [8] Abdolvand H, Wilkinson AJ.  Assessment of residual stress fields at deformation twin tips and the surrounding environments.  Acta Materialia, 2016; vol. 105, pp. 219–231.
  * [9] Qin B, Bhadeshia HK.  Plastic strain due to twinning in austenitic TWIP steels.  Materials Science and Technology, 2008; vol. 24 (8), pp. 969–973.
  * [10] De Cooman BC, Estrin Y, Kim SK.  Twinning-induced plasticity (TWIP) steels.  Acta Materialia, 2018; vol. 142, pp. 283–362.
  * [11] De Cooman BC, Chin Kg, Kim J.  High Mn TWIP Steels for Automotive Applications.  In M Chiaberge (Ed.), New Trends and Developments in Automotive System Engineering, chap. 6, pp. 101–128, IntechOpen2011.
  * [12] Allain S, Chateau JP, Bouaziz O.  A physical model of the twinning-induced plasticity effect in a high manganese austenitic steel.  Materials Science and Engineering A, 2004; vol. 387-389, pp. 143–147.
  * [13] Bouaziz O, Guelton N.  Modelling of TWIP effect on work-hardening.  Materials Science and Engineering A, 2001; vol. 319-321, pp. 246–249.
  * [14] Idrissi H, Schryvers D.  Investigation of the elementary mechanisms controlling dislocation/twin boundary interactions in fcc metals and alloys: from conventional to advanced TEM characterization.  In A Méndez-Vilas (Ed.), Current microscopy contributions to advances in science and technology, pp. 1213–1224, Badajoz, Spain2012.
  * [15] Mahajan S, Chin GY.  Formation of Deformation Twins in F.C.C. Crystals.  Acta Metallurgica, 1973; vol. 21, pp. 1353–1363.
  * [16] Steinmetz DR, Jäpel T, Wietbrock B, Eisenlohr P, Gutierrez-Urrutia I, Saeed-Akbari A, Hickel T, Roters F, Raabe D.  Revealing the strain-hardening behavior of twinning-induced plasticity steels: Theory, simulations, experiments.  Acta Materialia, 2013; vol. 61 (2), pp. 494–510.
  * [17] Zhang Y, Tao NR, Lu K.  Effect of stacking-fault energy on deformation twin thickness in Cu-Al alloys.  Scripta Materialia, 2009; vol. 60 (4), pp. 211–213.
  * [18] Roy B, Kumar NK, Nambissan PMG, Das J.  Evolution and interaction of twins, dislocations and stacking faults in rolled $\alpha$-brass during nanostructuring at sub-zero temperature.  AIP Advances, 2014; vol. 4 (6).
  * [19] Rahman KM, Vorontsov VA, Dye D.  The effect of grain size on the twin initiation stress in a TWIP steel.  Acta Materialia, 2015; vol. 89, pp. 247–257.
  * [20] Ghaderi A, Barnett MR.  Sensitivity of deformation twinning to grain size in titanium and magnesium.  Acta Materialia, 2011; vol. 59 (20), pp. 7824–7839.
  * [21] Grilli N, Cocks AC, Tarleton E.  A phase field model for the growth and characteristic thickness of deformation-induced twins.  Journal of the Mechanics and Physics of Solids, 2020; vol. 143\.
  * [22] Ophus C.  Four-Dimensional Scanning Transmission Electron Microscopy (4D-STEM): From Scanning Nanodiffraction to Ptychography and Beyond.  Microscopy and Microanalysis, 2019; pp. 563–582.
  * [23] Sun B, Fazeli F, Scott C, Brodusch N, Gauvin R, Yue S.  The influence of silicon additions on the deformation behavior of austenite-ferrite duplex medium manganese steels.  Acta Materialia, 2018; vol. 148, pp. 249–262.
  * [24] Scott C, Remy B, Collet JL, Cael A, Bao C, Danoix F, Malard B, Curfs C.  Precipitation strengthening in high manganese austenitic TWIP steels.  International Journal of Materials Research, 2011; vol. 102 (5), pp. 538–549.
  * [25] Savitzky BH, Hughes L, Bustillo KC, Deng HD, Jin NL, Lomeli EG, Chueh WC, Herring P, Minor A, Ophus C.  py4DSTEM: Open Source Software for 4D-STEM Data Analysis.  Microscopy and Microanalysis, 2019; vol. 25 (S2), pp. 124–125.
  * [26] Pekin TC, Gammer C, Ciston J, Minor AM, Ophus C.  Optimizing disk registration algorithms for nanobeam electron diffraction strain mapping.  Ultramicroscopy, 2017; vol. 176 (December 2016), pp. 170–176.
  * [27] Guizar-Sicairos M, Thurman ST, Fienup JR.  Efficient subpixel image registration algorithms.  Optics Letters, 2008; vol. 33 (2), p. 156.
  * [28] Williams DB, Carter CB.  Transmission electron microscopy: A textbook for materials science.  Springer, 2nd ed.
  * [29] Pekin TC, Gammer C, Ciston J, Ophus C, Minor AM.  In situ nanobeam electron diffraction strain mapping of planar slip in stainless steel.  Scripta Materialia, 2018; vol. 146, pp. 87–90.
  * [30] Rottmann PF, Hemker KJ.  Nanoscale elastic strain mapping of polycrystalline materials.  Materials Research Letters, 2018; vol. 6 (4), pp. 249–254.
  * [31] Rösner H, Boucharat N, Padmanabhan KA, Markmann J, Wilde G.  Strain mapping in a deformation-twinned nanocrystalline Pd grain.  Acta Materialia, 2010; vol. 58 (7), pp. 2610–2620.
  * [32] Chen JS, Liu Y, McCabe RJ, Wang J, Tomé CN.  Quantifying elastic strain near coherent twin interface in magnesium with nanometric resolution.  Materials Characterization, 2020; vol. 160 (December 2019), p. 110082.
  * [33] Dao M, Lu L, Shen YF, Suresh S.  Strength, strain-rate sensitivity and ductility of copper with nanoscale twins.  Acta Materialia, 2006; vol. 54 (20), pp. 5421–5432.
  * [34] Müllner P, Solenthaler C, Speidel MO.  Second order twinning in austenitic steel.  Acta Metallurgica Et Materialia, 1994; vol. 42 (5), pp. 1727–1732.
  * [35] Müllner P.  On the ductile to brittle transition in austenitic steel.  Materials Science and Engineering A, 1997; vol. 234-236, pp. 94–97.
  * [36] Müllner P, Solenthaler C, Uggowitzer PJ, Speidel MO.  Brittle fracture in austenitic steel.  Acta Metallurgica Et Materialia, 1994; vol. 42 (7), pp. 2211–2217.