1 Reversible Data Hiding in Image with Invariance Dongdong Hou, Weiming Zhang, Kejiang Chen, Sian-Jheng Lin and Nenghai Yu

Abstract—Different from all the previous reversible data hiding Reversible data hiding (RDH) [3] is one special type of fragile schemes, a completely novel one for the color image is proposed, watermarking, and usually is used for integrity authentication, which reversibly embeds messages into the color host image by which the host signal as well as the embedded data can without modifying its corresponding gray version. The property of grayscale invariance is valuable, because many applications be both restored from the marked signal without any loss. and image processing algorithms for color images are based Many RDH algorithms have been proposed in the past decade, on the corresponding gray versions, such as and which can be roughly classified into three fundamental strate- printing, producing reading materials for color blind people, gies: lossless compression appending scheme [6], difference single- image processing, and so on. Thus, in terms of expansion [7] and histogram shift [8]. The state of the art of these applications and image processing algorithms, the presented scheme will make the generated color marked image be free RDH techniques combine these strategies to the residuals of for its further uses. In this paper, the unchanged gray version images such as prediction errors (PEs) [9]–[15] to achieve a is utilized efficiently in both the embedding processes and the better performance. extracting processes. Messages are embedded into the and All the RDH algorithms mentioned above are presented for channels of color image, and then the channel is gray images. Indeed, color images are much more popular than adjusted adaptively to remove the offsets from the gray version caused by modifying its red and blue channels. To return the the gray ones in practice, because color is a powerful visual adjusted green channel, error correcting bits guaranteeing the descriptor which simplifies the identification of an object from reversibility are regarded as one part of payloads to be recur- a scene. There are many watermarking algorithms in the color sively embedded. Therefore, the reversibility and the property of space, such as luminance-channel-based algorithms [16], [17], grayscale invariance are both achieved. -channel-based algorithms [18], [19], three-color- Index Terms—color image, grayscale invariance, free for image channel-based algorithms [20], [21] and blue-channel-based processing, free for applications, reversible data hiding. algorithms [22]. However, these watermarking algorithms will destroy color images irreversibly. To embed watermarking into I. Introduction color images and also restore images without any loss, some Ata hiding embeds messages into digital multimedia researchers begin to explore the characteristics of color images D such as image, audio, video etc through an imperceptible and design special RDH algorithms for color images. The way. Such technology plays the important role in protecting existing RDH algorithms for color images [23]–[26] mainly the privacy, which can be roundly classified as robust water- focus on generating the more shaper prediction error (PE) marking [1] having the ability to resist many attacks, fragile histogram and minimizing the total distortion by exploring the watermarking [2], [3] being fragile to any modifications, correlations within each color channel and among three color and steganography [4] with the strong undetectability. From channels (R, G, B). the application side, robust watermarking, steganography and RDH will cause distortion for the host image, and the fragile watermarking are mainly used for copyright protection, generated marked image will be deemed as a noisy image covert communication and integrity authentication respective- compared to the original host image. Therefore, after RDH on ly. A review of data hiding techniques and a few emerging the image, its further image processing will be interfered. For innovative solutions using data hiding are well introduced in example, in the system of object recognition, a marked image [5]. will affect the feature extraction and thus result in the decrease However, robust watermarking, steganography and many of recognition accuracy. For some important applications such fragile watermarking algorithms like [2] will destroy the as police system and military system, slight decrease of host signal irreversibly. Some special signals are so precious processing accuracy may involve fateful consequences. Of and cannot be damaged such as medical imagery, military course, as for RDH, the original copy can be restored from the imagery and law forensics. Therefore, recovering the host marked image before the further processing. However, besides signal from the marked signal completely is rather important. the computing cost of recovery algorithm, in many cases it is just rather difficult for the processors to recover each original This work was supported in part by the Natural Science Foundation of image due to that the embedding method and secret key are China under Grant 61572452, U1636201. D. Hou, W. Zhang, K. Chen, S. Lin and N. Yu are with the School mastered by image owners who perhaps do not want to share of Information Science and Technology, University of Science and Tech- this information, and the processors even do not know whether nology of China, Hefei, 230026, China. (Email: [email protected], the input image is a marked copy or not among the huge [email protected], [email protected], [email protected], yn- [email protected].) number of digital images on the Internet. Therefore, a novel Corresponding author: Weiming Zhang RDH scheme for images not interfering with the further image 2 processing is desired. rates the proposed embedding and extracting algorithms. The There are various of processing methods for color images, experimental results are given in Section IV to show the among which some methods only operate on luminance chan- feasibility of the proposed method, and finally this paper is nel (green version) and entirely ignore chromatic information. concluded with a discussion in Section V. For example, there are various of feature descriptors based on gray images such as the well known Haar-like descriptors II. Relative works [27], HOG (Histogram of Oriented Gradients) descriptors [28], Throughout this paper, we denote pure message as mp, SIFT (scale invariant feature transform) descriptors [29], and error correcting message as mc, and mathematical function so on. Taking SIFT descriptors as example, which can be as f with a subscript. Matrices and vectors are denoted by applied to image compression [30]–[32], pattern recognition boldface fonts, and a color {r, g, b} is denoted as a triplet [33], [34], image stitching [35], [36], and so on. As far as T = {r, g, b}. The scales at the position {i, j} from channels we know, almost all of SIFT or SIFT-extended descriptors R, G, B and gray version are denoted as r , , g , , b , and v , like SURF (Speeded Up Robust Features) [37] and their i j i j i j i j respectively, and the corresponding PEs from channels R and applications only operate on the luminance channel. That is B are denoted as eR and eB . to say as for many processing algorithms before processing i, j i, j Nearly all of the RDH algorithms consist of two steps. color images we usually convert them into gray versions. The The first step is to generate a host sequence with the small reasons are clearly, on the one hand, comparing to color image entropy, i.e., a host sequence has a sharp histogram. Then in processing, dealing with gray image reduces the computational the second step, the users reversibly embed messages into the costs greatly. On the other hand, gray version can well host sequence by modifying its histogram with methods like represent the structures and the contents of one image, thus for difference expansion and histogram shift. some processing methods operating on gray version is enough. PEs of image are the most commonly adopted host Therefore, as for these color image processing methods, to sequence because their entropy is small. Therefore, many avoid interfering with the further processing of color image scholars [9]–[15] devote to generating host PEs with sharp we only need to hold its corresponding gray version. histogram by exploring the correlations within neighboring Besides not interfering with the further image processing, pixels. Because one color image consists of three color chan- gray versions of color images are widely used in many nels and the correlations among three color channels are also other applications, including black and white printing (e-ink- high, researchers [24]–[26] begin to explore such correlations based book readers), producing reading materials for color to generate a much sharper PE histogram and correspondingly blind people, nonphotorealistic rendering with black-and-white reduce the total embedding distortion. media, and so on. For some devices even only gray images After generating host PEs, we will reversibly modify PEs are allowed, such as electron microscope, which measures the to accommodate messages. The commonly used embedding strength of an electron beam at various space points. Indeed, methods are difference expansion [7] and histogram shift [8]. due to the importance of gray images, how to better convert Taking difference expansion as example, we embed a binary color images to gray ones becomes a hot topic in academ- bit denoted as m ∈ {0, 1} into the PE e by expanding it to e′ ic circles [38]–[40]. All in all, RDH in color image with as grayscale invariance to avoid interfering with the further image e′ = e + m. processing and applications is a very meaningful scheme. 2 (1) In this paper, a novel RDH scheme in color image is Obviously, we can restore e and m from e′ by proposed. Different from all the previous color image RDH { = ′ algorithms [23]–[26], the corresponding gray version is kept m e mod 2 , = ⌊ ′/ ⌋ (2) unchanged by the presented scheme. Based on the unchanged e e 2 gray version, we present a novel causal predictor and pixel where ⌊x⌋ returns to the nearest integer less than or equal to selection strategy to generate PE sequence with a shape x. histogram for embedding. Messages are recursively embedded Although the existing color image RDH algorithms [23]– into red and blue channels, and then we adjust green channel [26] well reduce the embedding distortion by exploring corre- to eliminate the offsets on caused by embedding lations within color image, the gray version of host color image messages, thus grayscale invariance is achieved. To return is still polluted. As mentioned in the Introduction, the property the adjusted green channel, additional error correcting bits of grayscale invariance will introduce many conveniences in are necessary and regarded as one part of payloads to be the further color image applications. Next, we will elaborate recursively embedded, then the reversibility is also guaranteed. how to design color image RDH with grayscale invariance. The experimental results further show that, the proposed method as a novel RDH scheme for the color image not only III. Proposed scheme holds the gray version, but also well keeps the quality of color marked image. Taking SIFT feature extraction as example, we A. Algorithm overview demonstrate that the further processing is not interfered by the The fundamental difference between color image and gray presented method. image is that, a vector consisting of three components is The rest of this paper is organized as follows. We briefly assigned to a pixel for color image, while a scalar grayscale introduce the relative works in Section II. Section III elabo- is assigned to a pixel for gray image. As mentioned above, 3 there are various of algorithms [38]–[41] for converting color Pixel selection Color host fv( ) Gray and prediction Host PEs from images to gray versions, among which the most widely used image version channels R,B one is the following famous formula from [41]: v = ⌊0.299r + 0.587g + 0.114b⌉, (3) Modify channels Color marked Adjust channel G R,B to embed where ⌊x⌉ rounds the element x to its nearest integer, r, g, b image messages are scales from channels R, G, B respectively, and v is the generated grayscale. For convenience, we define Embedding processes

fv(r, g, b) = ⌊0.299r + 0.587g + 0.114b⌉. (4) Pixel selection Through v = f (r, g, b) we convert a color pixel {r, g, b} to v Color marked fv( ) Gray and prediction Marked PEs from the gray one v. Based on the generated gray version, a shaper image version channels R,B host histogram can be generated from each color channel by a pixel selection and prediction strategy, and then we modify Recover channels host histogram to marked histogram to embed messages. In Color host Readjust R,B and extract both the embedding processes and the extracting processes the image channel G messages gray version is kept unchanged, thus marked histogram can be regenerated, from which we restore embedded messages and Extracting processes host histogram. The color pixel {r, g, b} is modified to {r′, g′, b′} after embed- Fig. 1: Overview of the proposed algorithm. ding messages, to achieve the property of grayscale invariance Eq. (5) needs to be held. $!"# '!"#$% '!"#$& ′ ′ ′ !"# !"#$% !"#$& fv(r, g, b) = fv(r , g , b ) (5) !$%"# !$%"#$% !$%"#$& '!$%"# '!$%"#$% '!$%"#$& The main idea to keep Eq. (5) is embedding messages into r and b, and then adjusting g. In detail, after {r, b} is modified !$&"# !$&"#$% !$&"#$& G!$&"# '!$&"#$% '!$&"#$& ′ ′ ′ to {r , b }, we generate g by solving the linear equation Eq. ".:JJVC ( ".:JJVC ) (3) with r′, g′ and v as the inputs. That is g′ = ⌊(v − 0.299r′ − 0.114b′)/0.587⌉. (6) 0!(%"#(% )!(%"# )!(%"#$% )!(%"#$& For convenience, we define )!"#(% %!"# )!"#$% )!"#$& ′ ′ ′ ′ fg(r , b , v) = ⌊(v − 0.299r − 0.114b )/0.587⌉. (7) )!$%"#(% )!$%"# )!$%"#$% )!$%"#$& At the receiver’s side, we firstly regenerate the gray version )!$&"#(% 0!$&"# )!$&"#$% )!$&"#$& from color marked image, based on which marked histogram is regenerated. Then we recover {r, b} from {r′, b′} after extracting *`:7 0V`-1QJ the embedded messages, with the help of r, b, v and one bit Fig. 2: Predicting ri, j or bi, j based on vi, j by exploring cor- c error correcting message denoted as m , g is restored. Error relations within {vi+1, j, vi, j+1, vi+1, j+1} and {ri+1, j, ri, j+1, ri+1, j+1} or { , , } correcting messages are essential for recovery and need to bi+1, j bi, j+1 bi+1, j+1 . be recursively embedded, which will be elaborated later. The overview of the proposed algorithm is shown as Fig. 1. To provide a good prediction for ri, j based on vi, j, the correlations within their nearby scales are explored. We B. Polynomial predictor based on gray version search the optimal polynomial coefficients which minimize As done in most of RDH algorithms, we should firstly the fitting errors with {vi+1, j, vi, j+1, vi+1, j+1} as inputs and generate a sequence having a sharp histogram as the host {ri+1, j, ri, j+1, ri+1, j+1} as outputs. That is sequence of RDH. Eq. (3) implies that the gray version has β∗ = arg min ∥y − Xβ∥2, (9) a great correlation with each color channel. By utilizing such β correlations, a shaper host histogram can be generated.   2 1 v + , v  In this subsection, based on the unchanged gray version,  i 1 j i+1, j  = T =  2  we propose a novel adaptive causal predictor which can where y [ri+1, j ri, j+1 ri+1, j+1] , X 1 vi, j+1 vi, j+1   2  provide satisfactory results for predicting red scales and blue 1 vi+1, j+1 vi+1, j+1 β = T scales. Taking the prediction of red scale ri, j as example, we and [a b c] . predict ri, j based on the input grayscale vi, j with a 3th degree By taking the partial derivatives of Eq. (9) with respect to polynomial formulated as the polynomial coefficients β and setting them equal to zero we get = + + 2 . ∗ − fp(vi, j) a bvi, j cvi, j (8) β = (XT X) 1XT y. (10) 4

4 We firstly initialize the prediction of ri, j as fp(vi, j) after x 10 ffi β∗ finding the optimal polynomial coe cients . However, there Proposed Predictor may exist some cases that ri, j cannot be well predicted by the 3.5 MED polynomial prediction because of over fitting or under fitting. 3 To solve this problem, we also utilize the correlations among red scales nearing ri, j to restrict such prediction. Inspired by 2.5 median-edge detector (MED) [42], the ultimate prediction of ri, j denoted byr ˆi, j is limited in the range of ri+1, j and ri, j+1. 2

That is Counts  1.5  , , ≤ ,  min(ri+1, j ri, j+1) if fp(vi, j) min(ri+1, j ri, j+1) rˆ , =  max(r + , , r , + ), if f (v , ) ≥ max(r + , , r , + ) . 1 i j  i 1 j i j 1 p i j i 1 j i j 1 fp(vi, j), otherwise (11) 0.5 By subtractingr ˆi, j from the original scale ri, j, we obtain its R 0 PE ei, j as −50 0 50 R = − . Prediction Errors ei, j ri, j rˆi, j (12) (a)

4 x 10 3 Proposed Predictor MED 2.5

(a) (b) (c) (d) 2 Fig. 3: Tested color images of size 510×510. (a) Airplane. (b) Barbara. (c) Lena. (d) Baboon. 1.5 Counts Empirically, we adopt 3th polynomial predictor and select 1 the nearest 3 pairs of scales labelled in dark in Fig. 2 for predicting ri, j. Obviously, the adopted 3th polynomial predictor can be easily extended to the higher order. Similar with MED, 0.5 the presented predictor moves prediction window in the raster 0 order, i.e., from left to right and from top to bottom, which is −50 0 50 also the half-enclosing causal one. The receiver will inversely Prediction Errors scan and process scales to get the same prediction values in (b) the same way, so there is no need to record these optimal polynomial coefficients. By the same prediction method, the Fig. 4: (a) Average PEs from channel R of Fig. 3. (b) Average PEs PEs of blue scales can be generated. from channel B of Fig. 3. To show that the presented polynomial predictor based on the unchanged gray version can generate shaper host histograms, we compare it with the classic casual MED The recipient will recalculate {rˆi, j, bˆi, j} by the same poly- predictor [42] operating on single channel. The average PEs { ′R, ′B} nomial predictor, and thus regain ei, j ei, j by subtracting the from airplane, barbara, lena and baboon are shown in Fig. 4, { , ˆ } { ′ , ′ } predicted scales rˆi, j bi, j from the marked scales ri, j bi, j . We from which we see that the presented predictor utilizing the { R , B } will extract the embedded two bits and restore ei, j ei, j from gray version outperforms MED a lot. The reason is that the ′R ′B {e , , e , } with Eq. (2), and then restore {ri, j, bi, j} as unchanged gray version consists of three color channels and i j i j { there exists high correlations among them. R r , = rˆ , + e i j i j i, j . = ˆ + B (14) bi, j bi, j ei, j C. Select host triplets for modification based on gray version After restoring {ri, j, bi, j}, gi, j can be roughly calculated with As for each triplet Ti, j = {ri, j, gi, j, bi, j}, we get the PEs { R , B } { ′R, ′B} ei, j ei, j and expand them to ei, j ei, j by Eq. (1) with carrying c = , , . { , } { ′ , ′ } gi, j fg(ri, j bi, j vi, j) (15) 2 bits. Accordingly, ri, j bi, j are modified to ri, j bi, j as { c ′ ′R Note that the calculated result denoted as g may not equal r = rˆ , + e i, j i, j i j i, j . ′ ′B (13) to gi, j due to the round function, but the magnitude of their b = bˆ , + e i, j i j i, j offset is 1 or 0, because the weight of color channel G is more ′ ′ = ′ , ′ , . Finally, we adjust gi, j to gi, j with gi, j fg(ri, j bi, j vi, j) to than 0 5. That is also the reason why we adjust channel G to ff c remove the o set from vi, j. hold the gray version. We need an error correcting bit mi, j to 5 record such offset, where D. Recursive embedding algorithm c = | − , , |, mi, j gi, j fg(ri, j bi, j vi, j) (16) We reserve some triplets for recording auxiliary parameters whose content will be elaborated at the end of this subsec- which needs to be recursively embedded into host triples. tion. These parameters are embedded into the LSBs (least To reduce the distortion, we usually select the PEs with the significant bits) of blue scales of reserved triplets, and these smaller magnitudes for modification firstly. As we all know, substituted LSBs will be regarded as a part of pure payloads the magnitude of each PE is proportional to its local variance, to be embedded. Note that, the reserved triplets cannot be and a small variance usually indicates a small magnitude of selected arbitrarily, because LSB substitutions of blue scales PE. In order to decrease the image distortion, we prefer to perhaps will also modify the corresponding grayscales. Our embed messages into triplets with the relatively smaller local objective is holding the gray version completely, thus we variances. To better represent the local variance of Ti, j denoted ρ ρ must reserve the triplets which can hold the corresponding by i, j, we calculate i, j based on the gray version (see Fig. grayscales after LSB substitutions of their blue scales. The 2) as triplet with ability resisting disturbance is called invariant ∆ 2 + ∆ 2 + ∆ 2 + ∆ 2 + ∆ 2 vi−1, j vi, j−1 vi, j vi+1, j vi, j+1 triplet, and how to select these invariant triplets is the first ρ , = , (17) i j 4 step we need to do. Indeed, the modified magnitude by LSB substitution is 1 where ∆v − , = v − , −v¯, , ∆v , − = v , − −v¯, , ∆v , = v , −v¯, , i 1 j i 1 j i j i j 1 i j 1 i j i j i j i j or 0. As long as Eq. (19) is held by one triplet, after LSB ∆v + , = v + , − v¯, , ∆v , + = v , + − v¯, , andv ¯, = (v − , + i 1 j i 1 j i j i j 1 i j 1 i j i j i 1 j substitution of its blue scale Eq. (19) still holds, and then such vi, j−1 +vi, j +vi+1, j +vi, j+1)/5. Of course, at the receiver’s side we can recalculate these local variances based on the unchanged triplet is invariant. As for an invariant triplet, it is still invariant gray version. at the receiver’s side. Therefore, if we reserve some invariant Some selected triplets may yield serious distortion by the triplets in a fixed region such as the corner or the edges embedding method Eq. (1) because of the large magnitudes of of image, the receiver will relocate these reserved invariant PEs. Traditional RDH algorithms usually group PEs into the triplets to extract auxiliary parameters. The reason why we inner region and the outer region by a threshold, and the PEs select blue scales for LSB substitution is that the weight of in outer region are shifted without carrying messages to avoid blue channel is the least and so the ratio of invariant triplet is causing large distortion. Different from the traditional RDH the maximum. algorithms, it will be rather wasteful to shift the PEs in outer b b f (r, g, 2⌊ ⌋) = f (r, g, 2⌊ ⌋ + 1) (19) region for the presented novel scheme. Because as long as one v 2 v 2 triplet is modified, not only distortion is caused, but also it will { , , ..., } yield one bit error correcting message for recovery, even such Assume the messages to be embedded are m1 m2 mL , triplet does not accommodate any pure messages. Therefore, and the number of reserved invariant triplets in a fixed region ρ we need to skip these triplets yielding large distortion. is Lr. The local variance of each triplet apart from these The distortion caused by modifying Ti, j = {ri, j, gi, j, bi, j} to border triplets and reserved invariant triplets is calculated with ′ ′ ′ ′ = { , , } Eq. (17) firstly, and then we select a threshold denoted as ρT to Ti, j ri, j gi, j bi, j is √ determine the positions of selected host triples. ρT is initialized ′ ′ ′ = − 2 + − 2 + − 2. as ρ0, where ρ0 is the minimum integer value with L + Lr Di, j (ri, j ri, j) (gi, j gi, j) (bi, j bi, j) (18) triplets’ local variances less than it. If the triplets with local We give a threshold denoted by DT to control the distortion. variances less than ρT cannot provide enough capacity, then ′ In detail, we modify Ti, j to Ti, j by embedding one bit pure we increase ρT by 1. And so on, until selecting N host triplets, message and one bit error correcting message. If the distortion which can include L+Lr legal triplets and some skipped illegal ′ Di, j between Ti, j and Ti, j is larger than DT , we just skip this triplets. triplet for modification. Clearly that, the number of skipped After ρT is determined, we scan host image in the raster triplets will decrease with the increase of DT . Note that, order starting from the upper left corner, and collect N triplets usually D is initialized as 20 empirically. As for some images T {T1, T2, ..., TN } with variances less than ρT . The prediction with rich textures, we can enlarge DT to expand the number window with 2 × 2 size is also moved in the raster order from of legal triplets. After embedding messages, some triplets the upper left corner, and PEs of red scales and blue scales perhaps will yield overflow/underflow scales which are less of collected triplets are calculated with the proposed causal than 0 or more than 255, and these triplets are also skipped polynomial predictor. for modification. That is, as long as current triplet meets one As shown in Fig. 5, for each host triplet Ti = {ri, gi, bi}, with of the following two illegal conditions, we label such illegal the method of Eq. (1) we embed one bit pure message denoted triplet with “tag = 0”; otherwise, current triplet is legal and p R ′R as mi into ri by expanding its PE ei to ei . At the same time, labelled with “tag = 1”. These tags are recorded as auxiliary c one error correcting bit denoted as mi−1 is embedded into bi parameters, which are easy to be compressed due to that the B ′B c by expanding ei to ei , where mi−1 is utilized for recovering number of skipped illegal triplets is usually fewer. p c { , } { ′, ′} gi−1. After embedding mi and mi− we modify ri bi to ri bi . Two illegal conditions: 1 ′ To hold the corresponding grayscale vi, we generate gi with ′ ′ ′ ′ • D , > D ; = , , = { , , } i j T gi fg(ri bi vi). Then gi is modified to gi , and Ti ri gi bi • Overflow/underflow scale exists. ′ = { ′, ′, ′} becomes Ti ri gi bi . 6

′ ′ ρ Next, we will check the validity of Ti . If Ti is illegal, collected triplets N, the threshold of local variance T , triplets = c then we label it with tagi 0 and skip it for modification. tags and the error correcting bit of the last triplet mN , which c = c ′ = Correspondingly, mi mi−1 and Ti Ti; If Ti is legal, one are compressed and embedded into the LSBs of blue scales c = | − , , | p error correcting bit mi gi fg(ri bi vi) together with mi+1 of reserved invariant triplets. c p are embedded into Ti+1. That is we embed mi−1 and mi into T , and generate T′, mc and tag , which is formulated as i i i i E. Recursive extracting algorithm ′, c, = , p, c , = , , ..., . (Ti mi tagi) femb(Ti mi mi−1) i 1 2 N (20) At the decoder’s side, we firstly relocate invariant triplets with Eq. (19) in a fixed region. Then the parameters, including ρ c N, T , tags and mN are extracted, from the blue scales of $ $ reserved invariant triplets. The local variance ρ of each triplet % % apart from these border triplets and reserved invariant triplets is calculated with Eq. (17), and then according to N and ρ , ! ! T we scan the image in the raster order to collect N marked triplets {T′ , T′ , ..., T′ }. " " 1 2 N

# # & & & c % g figiii r= b v ( , , ) 888 % ! % 888

!  !   "  "  c mi = 0?

'(" '(" g g c c ii = vfrgbiviii=+ (,1,) ? Fig. 5: Recursive embedding processes.

gg=+ c 1 gg=- c 1 ii ii '(( !( '  ! # ! '  " "( ! 

 Fig. 7: The processes of recovering green scale. % $

! ! !  " %& " # " "( $ !( %  ! # ! %  " "( !  & $ $ % ! # ! # !  ' +  !  " ! ! )%)%$  $  ) II   '(" + LL+ +  ! ! ! !  " JJ; " "( " # & & ) *%*%  !  ! + KKLL+  +  # ! #  ! # # ! # !

I L"( "(t  " !  !( I &&' J * ! !" *  +  I!! " ! & ! LL KL%( • # "  ! ! !( • JJ; *(() *  + ## ! ! KL !" # KL Fig. 6: Example of the proposed embedding algorithm.

"%) "(  ! ! !( We give a simple example depicted in Fig. 6 to show the " R = embedding processes of one legal triplet. We expand ei 1 ′R = p = B = ′B = to ei 2 by embedding mi 0, and ei 0 to ei 1 by CI %  ! # ! !( S ′ ′ ) $ c = = = = DT+ J ; " # embedding mi−1 1, then ri 113, bi 121. To keep vi 129 DT%  ! "' ! !' ′ = ′ = { , , } EUK $ unchanged, gi is adjust to gi 139, thus Ti 113 139 121 . c = The error correcting bit mi 1 for restoring gi is calculated Fig. 8: Example of the proposed extracting algorithm. c = | − , , | with mi 140 fg(112 120 129) . ′ { ′ , ′ , ..., ′ } The receiver needs some auxiliary parameters to inversely Beginning with TN we decode T1 T2 TN in the inverse decode marked triplets to restore host triplets and embedded order. The prediction window with 2 × 2 size is also moved in messages. The auxiliary parameters include the number of the inverse raster order, and we get marked PEs of red scale 7

′ and blue scale of TN with the proposed causal polynomial pre- auxiliary messages. We cannot enlarge DT without limits, once dictor firstly. After restoring TN we further calculate marked DT ≥ 50, we keep DT unchanged, and enlarge the number ′ PEs of red scale and blue scale of TN−1, and so on. of reserved triplets Lr. Note that, there is no need to record ′ ′ According to tagi we get whether Ti is legal or not. If Ti DT , because the receiver will relocate legal triplets according c = c = ′ p is illegal, then mi−1 mi , Ti Ti and mi is empty. For to triplet tags, the number of collected triplets N and the ′ { ′R ′B} { ′ ′} ρ legal Ti , ei , ei are calculated from ri , bi by the proposed threshold of local variance T . We initialize the threshold of p c { R B} ρ ρ polynomial predictor, from which mi , mi−1 and ei , ei are local variance T as 0, which is the minimum integer value restored with Eq. (2). Further, ri and bi can be restored. Since with L + Lr triplets’ local variances less than it. If the triplets = ′, ′, ′ c = ρ + vi fv(ri gi bi ), we will restore gi from the calculated gi with local variances less than T cannot provide L Lr legal , , c c = ρ fg(ri bi vi) (Eq. 6) according to mi . To be detailed, if mi triplets, we increase T by 1. And so on, until selecting N = c = , , + − − 0, then gi gi fg(ri bi vi); Otherwise, we get that the host triplets, which include L Lr legal triplets and N L Lr ff c = c+ ρ magnitude of the o set between gi and gi is 1, that is gi gi 1 skipped illegal triplets. The pseudo codes of selecting Lr, T = c − . or gi gi 1. Since the weight of gi to generate vi is 0 587, and DT are shown as Algorithm 1. which is larger than 0.5, the wrong one must will modify vi , c + , , Algorithm 1 L ρ D through fv( ) while the right one will not, i.e., fv(ri gi 1 bi) Pseudo code of selecting r, T and T , c − , , vi or fv(ri gi 1 bi) vi. Therefore, it is easy to eliminate 1: Lr = 0.02L, DT = 20, f lag1 = true and f lag2 = true; the wrong one and get the right one. The processes to get gi 2: Initialize ρT = ρ0; c according to mi are described in Fig 7. From each marked 3: while f lag1 do ′ c c p triplet Ti , with the help of mi and tagi we restore mi−1, mi 4: while f lag2 do and Ti, that is 5: Locate the triplets whose local variances are less ρ , p, c = ′, c, , = , − , ..., . than T ; (Ti mi mi−1) fext(Ti mi tagi) i N N 1 1 (21) 6: Embed L+Lr bits messages into the located triplets As depicted in Fig. 8, we continue the above example to in the raster order; ′ = { , , } show how Ti 113 139 121 is decoded with the help of 7: if Legal triplets are enough then c = { R = B = } = the error correcting bit mi 1. We restore ei 1, ei 0 8: f lag2 f alse; ′ ′ p { R = B = } = c = 9: from ei 2, ei 1 after extracting mi 0 and mi−1 1, else and then further restore {ri = 112, bi = 120}. vi = 129 can 10: ρT = ρT + 1; = , , c = = be got with vi fv(113 139 121), thus we get gi 139 11: end if , , c = , + , = fg(112 120 129). Since mi 1, and fv(112 139 1 120) 12: end while 129 while fv(112, 139 − 1, 120) = 128 , 129, we get gi = 140. 13: Compress auxiliary parameters and generate La bits = { , , } p = c = Therefore, Ti 112 140 120 , mi 0 and mi−1 1 are all auxiliary messages; restored. 14: if La ≤ Lr then After extracting the embedded messages and restoring the 15: f lag1 = f alse; host triplets, we further reconstruct the blue scales of reserved 16: else invariant triplets with the extracted LSBs. 17: if DT < 50 then 18: DT = DT + 5; 19: else F. Set parameters 20: Lr = Lr + 0.01L; There are some auxiliary messages for restoring host image 21: end if and embedded messages from marked image. Invariant triplets 22: end if are located and reserved to record auxiliary messages by LSB 23: end while substitution, and the substituted LSBs will be embedded as 24: Record the number of scanned triplets N, including legal one part of pure payloads. The number of reserved invariant triplets and illegal triplets; triplets L needs to be larger than the length of auxiliary r 25: return Lr, ρT and N; messages denoted by La. Indeed, triplet tags account for the vast majority of auxiliary messages. Apart from triplet tags, 50 bits messages are used to record the rest of auxiliary IV. Experimental results parameters, including the number of collected triplets N, the A. Payload-distortion performance threshold of local variance ρT and the error correcting bit of the c last triplet mN . In this subsection, we discuss how to properly Firstly, we need to determine the threshold of embedding select the parameters ρT , Lr, and the threshold of embedding distortion (see Eq. (18)) DT , the threshold of local variance ρT distortion (see Eq. (18)) DT . and the number of reserved invariant triplets Lr with Algorithm Empirically, DT is initialized as 20, and Lr is initialized as 1. After DT , ρT and Lr are determined, the length of auxiliary 0.02L, where L is the length of given embedded messages. For messages La and the number of scanned triplets N will be also some cases, 0.02L bits may not accommodate the auxiliary determined. With typical images (Fig. 3) as test images, the messages. If that happens we firstly enlarge DT to reduce selected DT , ρT , Lr and the corresponding generated La, N the number of illegal triplets, thus making triplet tags to under different payloads are shown in TABLE I, from which be compressed more efficiently and reducing the length of we see that for general images like airplane and barbara, 8 setting L = 0.02L and D = 20 can usually accommodate r T 60 auxiliary messages. For lena, we can reduce the length of Airplane auxiliary messages through enlarging DT . But for some images 55 Barbara like baboon, due to its rich textures, enlarging DT cannot Lena ffi Baboon reduce the length of auxiliary messages e ciently. In such 50 case, the step of enlarging the number of reserved invariant triplets is executed. 45 The presented RDH algorithm is the first one for color image with grayscale invariance. A good algorithm should PSNR(dB) 40 not only hold the gray version, but also keep the quality of color marked image as far as possible. Therefore, one key 35 assessment indicator to evaluate such novel RDH scheme is the visual quality of color marked image, and peak signal to noise 30 ratio (PSNR) is the generally adopted assessment indicator. 2 4 6 8 10 12 Payloads(bits) 4 PSNR is defined as x 10 2552 Fig. 9: Payload-distortion performances on color image of the PS NR = 10log ( ), (22) 10 MSE proposed algorithm. and MSE is computed by ∑C ∑M ∑N 1 B. Application example: SIFT feature extraction MSE = (s(i, j, k) − s′(i, j, k))2, (23) M × N × C k=1 i=1 j=1 As mentioned in Introduction, there are various of applica- where M and N are the width and the height of host image, C tions for the gray versions of color images. In this subsection, means the number of color channel which will be 3 for color we just take SIFT feature extraction as example to show that image and 1 for gray image, s(i, j, k) represents the scale value how previous RDH schemes in color image [23]–[26] interfere at the position (i, j) in the kth color channel of host image and with the feature extraction while the presented method does s′(i, j, k) is the scale value after modification. not. SIFT descriptors are widely applied to image compression The payload-distortion performances of the proposed [30]–[32], pattern recognition [33], [34], image stitching [35], method with Fig. 3 as test images are shown in Fig. 9. Of [36], and so on. To the best of our knowledge, almost all of course, all the generated color marked images will have the SIFT or SIFT-extended descriptors and their applications for same gray versions as original color host images. Fig. 9 shows color images only operate on the corresponding gray versions. that by the proposed method not only the gray versions are The previous RDH schemes in color image [23]–[26] mainly unchanged, but also the quality of color marked images are focus on minimizing the total distortion for the yielded color well kept. From Fig. 9 we also see that baboon’s performance marked image, among which Ou et al.’s method [26] is the is relatively poorer than the others’. The reasons are mainly state-of-the-art one. By Ou et al.’s method the total payload is about two aspects: one is that the PE histograms generated adaptively allocated to three color channels according to their from baboon are not sharp enough due to its rich texture, and PEs so that well keeping image quality. However, as shown the other is that the number of illegal triplets are rather large, in Fig. 10, by Ou et al.’s method the gray versions of color which causes much more auxiliary messages for embedding marked images will be polluted. Naturally, after modifications (see TABLE I). the earlier RDH schemes [23]–[25] will also change the gray By the proposed method, one triplet accommodates one versions greatly. Once the corresponding gray versions of bit pure message but also will yield one bit error correcting color images are polluted, there is no doubt that the further message. Obviously, the large number of error correcting applications based on gray versions will be disturbed. bits limit the embedding capacity. However, in this paper To extract SIFT features from color image, we usually the error correcting messages are difficult to be compressed. convert the color image to the gray version, and then extract For example, the ratios of “0” in the binary error correcting features from the converted gray one. The first step for messages for Airplane, Barbara, Lena and Baboon are all about SIFT feature extraction is keypoint location. Response is an 0.587. That is because the error correcting messages imply indicator of how good a point is, and a bigger response means whether green scales are completely restored or not and the a stronger SIFT point. For lena image, we locate the top weight of green channel is 0.587. Of course, compressing the eight points with the strongest responses as signed in Fig. error correcting messages is a potential solution to enlarge the 11. Since the presented method will hold the corresponding embedding capacity in the future. gray version, SIFT features from the generated marked image The design of invariant triplets is important for grayscale will be the same as those from the host image. However, by invariance. We embed auxiliary parameters into blue scales of Ou et al.’s method [26], SIFT features from the generated invariant triplets by LSB substitution. Since the weight of blue marked image will be modified due to the modification on channel is 0.114, the ratio of invariant triplets is about 0.886. the corresponding grayscales. For example, after embedding Therefore, we think the number of invariant triplets is enough 140000 bits messages, 200000 bits messages and 260000 to accommodate these auxiliary parameters. bits messages into lena by Ou et al.’s method respectively, 9

TABLE I Parameters under different payloads.

Payloads (bits) 10000 30000 50000 70000 90000 110000 130000

DT = 20 20 20 20 20 20 20 ρT = 1 1 2 2 3 5 7 airplane N= 10199 30599 50999 71399 91800 112201 132608 Lr= 200 600 1000 1400 1800 2200 2600 La= 51 51 51 51 73 89 190

DT = 20 20 20 20 20 20 25 ρT = 1 3 4 6 9 15 30 barbara N= 10203 30616 51033 71446 91886 112348 132775 Lr= 200 600 1000 1400 1800 2200 2600 La= 101 258 457 614 1049 1688 1985

DT = 40 25 25 25 25 25 25 ρT = 2 3 4 5 7 9 13 lena N= 10212 30630 51039 71451 91866 112302 132754 Lr= 200 600 1000 1400 1800 2200 2600 La= 193 403 520 661 844 1236 1783

DT = 50 50 50 50 50 50 50 ρT = 9 21 36 59 94 160 313 baboon N= 12043 34402 56199 78602 101123 126164 159783 Lr= 1800 3900 5500 7700 9900 14300 26000 La= 1770 3836 5482 7163 9598 14058 25877

72 Lena 70 Barbara Baboon Airplane 68

66

64 PSNR(dB)

62

60

58 2 4 6 8 10 12 payloads(bits) 4 x 10 Fig. 10: Payload-distortion performances on gray versions of color marked images generated from Ou et al.’s method.

Fig. 11: The top eight SIFT points with the strongest responses. the Euclidean distances between SIFT features from marked image and host image at the same signed positions of Fig. 11 are shown in TABLE II. Once the extracted SIFT features are modified, the further SIFT applications will be interfered. Continuing with the example of lena image, SIFT points with responses larger than messages are embedded into the color host image by the 0.5 in both host image and yielded marked image are located, presented method, the matching results will be the same as Fig. and these located SIFT points in host image and marked image 12(d), which carrying 260000 bits messages. That is because are matched. The results of SIFT matching for Ou et al.’s Ou et al.’s method will destroy both color host images and method [26] and the proposed method are shown in Fig. 12. corresponding gray versions more seriously with the increase Note that, the horizontal line between two SIFT points means of embedded messages, and thus correspondingly interfere such two SIFT points are matched correctly, while the oblique with SIFT feature extraction more seriously, but the proposed line means a false match between two SIFT points. From Fig. method will keep the gray versions unchanged throughout. 12(a)-(c) we can see that the number of false matches will Therefore, a novel RDH scheme in color image not interfering increase with the increase of embedded messages. However, with its further image processing is achieved by the presented as long as the given payload is allowed no matter how many method. 10

TABLE II Euclidean distances of SIFT features from host image and marked image.

keypoints (in scan order) 1 2 3 4 5 6 7 8 After embedding 140000 bits 4.795832 2.449490 2.236068 3.162278 7.745967 1.732051 3.162278 5.916080 After embedding 200000 bits 5.916080 3.000000 2.000000 3.316625 4.690416 1.414214 4.123106 3.741657 After embedding 260000 bits 18.681541 3.741657 2.236068 7.483315 7.000000 2.000000 5.916080 4.472136

(a) (b)

(c) (d)

Fig. 12: (a) SIFT matching after embedding 140000 bits messages by Ou et al.’s method. (b) SIFT matching after embedding 200000 bits messages by Ou et al.’s method. (c) SIFT matching after embedding 260000 bits messages by Ou et al.’s method. (d) SIFT matching after embedding 260000 bits messages by the presented method.

V. Conclusion code in the paper [26]. To help readers to use the proposed method, we will post the Matlab implementation of this paper Different from all the previous RDH algorithms, we propose on our website: http://home.ustc.edu.cn/%7Ehoudd. a completely novel one for color image, which can hold the corresponding gray version of color host image. Grayscale References invariance for RDH in color image is valuable in practice, [1] X. Zhu, J. Ding, H. Dong, K. Hu and X. Zhang, “Normalized because it does not affect the further applications and image Correlation-Based Quantization Modulation for Robust Watermarking,” processing of the yielded marked image in terms of many IEEE Trans. Multimedia, vol. 16, no. 7, pp. 1888-1904, Nov. 2014. [2] A. Ahmad, M. Pant, and C. Ahn, “SVD based fragile watermarking cases. scheme for tamper localization and self-recovery,” International Journal In the proposed method, the large number of error correcting of Machine Learning and Cybernetics, vol. 7, no. 6, pp. 1225-1239, bits limit the embedding capacity, and thus compressing the 2016. [3] A. Khan, A. Siddiqa, S. Munib, and S. A. Malik, “A recent survey error correcting messages is a potential solution to improve of reversible watermarking techniques,” Information sciences, vol. 279, the proposed scheme in the future. On the other hand, beside pp. 251–272, 2014. the famous formula Eq. (3), there are still many other methods [4] B. Li, M. Wang, J. Huang, and X. Li, “A new cost function for spatial image steganography,” in 2014 IEEE International Conference on Image [38]–[40] for converting color images to the gray ones. When Processing (ICIP), pp. 4206-4210, 2014. adopting these advanced converting algorithms, how to hold [5] M. Asikuzzaman and M. R. Pickering, “An Overview of Digital Video corresponding gray version for the color image after RDH is Watermarking,” IEEE Trans. Circuits System and Video Technology, vol., no. 99, pp. 1-1. doi: 10.1109/TCSVT.2017.2712162. a more challenging but interesting problem. [6] J. Fridrich and M. Goljan, “Lossless Data Embedding for All Image Formats,” in SPIE Proceedings of Photonics West, Electronic Imaging, cknowledgements Security and Watermarking of Multimedia Contents, vol. 4675, pp. 572- VI. A 583, San Jose, Jan. 2002. [7] J. Tian, “Reversible Data Embedding Using a Difference Expansion,” The authors thank the anonymous reviewers for their valu- IEEE Trans. Circuits System and Video Technology, vol. 13, no. 8, pp. able comments and thank Ou et al. for offering the source 890-896, Aug. 2003. 11

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Sian-Jheng Lin received the B.Sc., M.Sc., and Ph.D. degrees in computer science from National Chiao Tung University, Hsinchu, Taiwan, in 2004, 2006, and 2010, respectively. From 2010 to 2014, he was a postdoc with the Research Center for In- formation Technology Innovation, Academia Sinica. From 2014 to 2016, he was a postdoc with the Electrial Engineering Department at King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. He was a part-time lecturer at Yuanpei University from 2007 to 2008, and at Hsuan Chuang University From 2008 to 2010. He is currently a project researcher with the School of Information Science and Technology at University of Science and Technology of China (USTC), Hefei, China. In recent years, his research focuses on the algorithms for MDS codes and its applications to storage systems.

Nenghai Yu received his B.S. degree in 1987 from Nanjing University of Posts and Telecommunica- tions, M.E. degree in 1992 from Tsinghua University and Ph.D. degree in 2004 from University of Science and Technology of China, where he is currently a professor. His research interests include multimedi- a security, multimedia information retrieval, video processing, information hiding and security, privacy and reliability in cloud computing.