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Fault-tolerant and progressive transmission of images
Shang-Kuan Chen, Ja-Chen Lin
∗Department of Computer and Information Science, National Chiao Tung University, Hsinchu 30050, Taiwan, ROC Received 1 July 2004; received in revised form 4 April 2005; accepted 4 April 2005
Abstract
A fault-tolerant progressive image transmission method is proposed. The advantages include the following: (1) Unlike most progressive methods, the image is divided into n parts with equal importance to avoid worrying about which part is lost or transmitted first. (2) If the image is a secret image, then the transmission can use n distinct channels (one shared result per channel), and intercepting up tor1− 1 channels by the enemy (r1· · ·rkn are all pre-set constants) will not reveal any secret. Meanwhile, the disconnection up ton − rk channels will not affect the lossless recovery of the secret image. 䉷2005 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved.
Keywords: Progressive; Image sharing; Image hiding; Quality-control
1. Introduction
Traditionally, when an image is transmitted in a progres-sive way, the information contained in the image is par-titioned into several parts and the most significant part is transmitted first while the least significant part is transmit-ted last. If the most significant part is damaged or lost, the recovered image will be significantly degraded. Therefore, a “fault-tolerant” progressive image transmission system is more useful. In most researches[1–3]about progressive im-age transmission, the receiver can immediately stop trans-mitting an image if the received rough version of the im-age shows that the imim-age being received (for example, a jet airplane) is not the one the receiver desired (for example, Lena). However, none of these methods is fault-tolerant. The goal of the current paper is therefore a fault-tolerant pro-gressive image transmission method. This goal is achieved through the careful use of an image sharing technique[4] derived from Ref.[5,6].
∗Corresponding author. Fax: +886 3 572 1490. E-mail address:[email protected](J.-C. Lin).
0031-3203/$30.00䉷2005 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2005.04.002
Blakley [5] and Shamir [6] first proposed the idea of secret sharing, known as the(r, n) threshold scheme. In their
(r, n) threshold scheme, the system consists of a dealer and n participants; the dealer distributes a secret number into n shares and each participant holds one share. Later, if r shares are received, then the secret number can be revealed. If less than r shares are received, then no information about the secret number can be revealed. This secret sharing scheme is fault-tolerant in the sense thatn−r shares can be lost during the reconstruction (because only r shares are needed).
Thien and Lin[4]applied the idea in Ref.[6]to share a secret image and generated n shadow images (the detail is described in Section 2). The size of each shadow image was only 1/r of the original secret image. The secret image can be recovered if r of the n shadow images were received. Due to the smaller size(1/r) of the shadow images, the total transmission time needed to recover an image (by receiving r shadow images) would not increase.
In the scheme of Thien and Lin[4], if less than r shadow images were received, people could not get any information about the secret image. In the current paper, a progressive version of image sharing is proposed. The user can set sev-eral thresholds, namely, the k thresholds:r1r2· · · rk=r. If less thanr1shared results are received, nothing can be
revealed. However, ifr1shared results are received, a rough version of the original image can be revealed; then, for each s > 1, if rs shared results are received andrs> rs− 1, the quality of the recovered image is better than the one using rs−1shared results. Finally, ifrkshared results are received, the image can be recovered without any loss. Therefore, the absence ofn − rk shared results cannot affect the lossless recovery.
Notably, since the content of each shared result of an original (secret) image always looks noisy, an attack from hackers is quite likely. Therefore, a data-hiding method[7] is utilized to hide the shared results in some host images to form stego images, which look ordinary instead of being noisy, to avoid attracting the hackers’ attention[8–10].
Section 2 introduces the secret image sharing method proposed by Thien and Lin [4]. Section 3 describes the details of our scheme. Section 4 discusses the decoding phase. Section 5 provides the experimental result. Finally, Section 6 discusses the design to control quality.
2. Thien and Lin’s secret image sharing method Thien and Lin[4]proposed a method to share a secret image based on Shamir’s[6]polynomial threshold scheme with a prime number p. The image is divided into several non-overlapping sectors, and each sector has r pixels. For each sector j, the r coefficientsa0, a1, . . . , ar−1of the cor-responding polynomial
qj(x) = a0+ a1× x + · · · + ar−1× xr−1(mod p) are assigned as the r gray values of the r pixels in the sector. Thewth shadow image is the collection {qj(w)|j = 1, 2, . . . , original image size/r}. Since each sector j, which has r pixels, contributes only one pixelqj(w) to the wth shadow image, the size of thewth shadow image is only 1/r of the secret image. This property holds for every w ∈ {1, 2, 3, . . . , p − 1}.
3. Proposed method
The scheme is illustrated inFig. 1. First, a bit-plane scan-ning method rearranges the gray value information of the original image. Then, the rearranged data are shared. Finally, the shared results are hidden in some host images.
Step 1. Bit plane scanning to rearrange data: First, let the k threshold values r1, r2, . . . , rk be assigned so that
r1r2 · · · rk= r. The k thresholds denote the distinct number of shared results needed to recover the image with distinct quality levels. For example, a quite rough image can be recovered whenr1shared results are received. Then, the image quality gradually improves when more shared results are received. Finally, the image can be recovered completely whenrk shared results are received.
Bit-plane scanning Hiding Process Original image Rearranged data Sharing Shared result n Host images n Stego images
Fig. 1. The flow chart of our method (the encoding phase).
MSB LSB 128 6 4 32 16 8 4 2 1 Pixel A = 166 = ( 1 0 1 0 0 1 1 0 )2 Pixel B = 167 = ( 1 0 1 0 0 1 1 1 ) 2 Pixel C = 164 = ( 1 0 1 0 0 1 0 0 ) 2 Pixel D = 166 = ( 1 0 1 0 0 1 1 0 ) 2 Pixel E = 168 = ( 1 0 1 0 1 0 0 0 ) 2 Pixel F = 165 = ( 1 0 1 0 0 1 0 1 ) 2 Pixel G = 163 = ( 1 0 1 0 0 0 1 1 ) 2 Pixel H = 166 = ( 1 0 1 0 0 1 1 0 ) 2 Pixel I = 168 = ( 1 0 1 0 1 0 0 0 ) 2 Fig. 2. The sequence of scanning the 8 bit planes. Note that RSUM= r1+ r2+ · · · + rk pixels are scanned for each sector.
Below, we discuss how to scan the image and generate the rearranged data. To begin the process, the original image is divided into several non-overlapping sectors, and the sectors are processed one by one. Each sector always has RSUM pixels(RSUM = r1+ r2+ · · · + rk). Since each pixel has 8 bits in the gray-level image, each sector has 8(r1+ r2+ · · · + rk) bits. We then rearrange these bits to get another r1+ r2+ · · · + rk values; each of them is still an 8-bit number ranging from 0 to 255. For example, ifk = 3 and 2, 3, 4 as the three threshold values, then each sector contains 2+ 3 + 4 = 9 pixels. Without the loss of generality, let the 9 pixels of the sector being discussed have the gray values 166, 167, 164, 166, 168, 165, 163, 166, and 168, respectively. By the rearranging process, shown inFig. 2, the 9 transformed
values of the sector are 255, 128, 63, 224, 0, 143, 171, 76, and 140. The details are shown below. The 9 input pixels {A, B, C, D, E, F, G, H, I} are A = 166 = (1010 0110)2= (A1A2A3A4A5A6A7A8)2, B = 167 = (1010 0111)2= (B1B2B3B4B5B6B7B8)2, C = 164 = (1010 0100)2= (C1C2C3C4C5C6C7C8)2, D = 166 = (1010 0110)2= (D1D2D3D4D5D6D7D8)2, E = 168 = (1010 1000)2= (E1E2E3E4E5E6E7E8)2, F = 165 = (1010 0101)2= (F1F2F3F4F5F6F7F8)2, G = 163 = (1010 0011)2= (G1G2G3G4G5G6G7G8)2, H = 166 = (1010 0110)2= (H1H2H3H4H5H6H7H8)2, I = 168 = (1010 1000)2= (I1I2I3I4I5I6I7I8)2.
Now we scan these 8× 9 = 72 bits according to the order specified inFig. 2; i.e. the 9 most significant bits (MSB) first (A1, B1, . . . , I1); then the 9 second-most significant bits(A2, B2, . . . , I2); then the 9 third-most significant bits
(A3, B3, . . . , I3), etc. Therefore we obtain a rearranged 72-bit data set (A1, B1, . . . , I1, A2, B2, . . . , I2, A3, B3, . . . ,
I3, . . . , A8, B8, . . . , I8). If we read these 72 bits (according to the above order), and explain them as 9 numbers (each one is an 8-bit number), then we can obtain the rearranged result for this sector, namely, the following 9 values:
(A1B1C1D1E1F1G1H1)2= (1111 1111)2= 255, (I1A2B2C2D2E2F2G2)2= (1000 0000)2= 128, (H2I2A3B3C3D4E4F3)2= (0011 1111)2= 63, (G3H3I3A4B4C4D4E4)2= (1110 0000)2= 224, (F4G4H4I4A5B5C5D5)2= (0000 0000)2= 0, (E5F5G5H5I5A6B6C6)2= (1000 1111)2= 143, (D6E6F6G6H6I6A7B7)2= (1010 1011)2= 171, (C7D7E7F7G7H7I7A8)2= (0100 1100)2= 76, (B8C8D8E8F8G8H8I8)2= (1000 1100)2= 140.
Step 2. Sharing: Recall that in our example above, we assumed that r1= 2, r2= 3, r3= 4, and therefore each sector has 9 pixels. In Step 1, we already transformed the 9 pixels {166, 167, 164, 166, 168, 165, 163, 166, 168} of the current sector (say, Sector j) into 9 new values {255, 128, 63, 224, 0, 143, 171, 76, 140}. Now, the first r1= 2 transformed values {255, 128} forms the first poly-nomial
fj(1)(x) = (255 + 128x) mod 257 (1) for the current sector. Then, the next r2= 3 transformed values{63, 224, 0} forms the second polynomial
fj(2)(x) = (63 + 224x + 0x2) mod 257 (2) for the current sector. Finally, the finalrk=r3=4 transformed values{143, 171, 76, 140} forms the final polynomial f3
j(x) = (143 + 171x + 76x2+ 140x3) mod 257 (3)
for the current sector. For each participantw ∈ {1, 2, 3, . . . , p − 1}, the collection
fj(i)(w)|j = 1, 2, . . . ,original image size
RSUM ,
and i = 1, 2, . . . , k
(4)
is called the wth shared result SR(w). Notably, since each sector has RSUM = r1 + r2 + · · · + rk pixels, the total number of sectors that we have is jMAX = original image size/RSUM. In addition, each shared result SR(w) receives only k numbers fj(1)(w) ∼ fj(k)(w) gener-ated from sector j, which is an RSUM-pixel region of the original image. Therefore, the size of each shared result is k/RSUM = k/(r1+ r2+ · · · + rk) of the original image. In the above example, this ratio is 2+3+43 =13.
(Optional) Step 3: Hiding the shared results in some host images: The contents of the shared results look noisy. There-fore, in the special case where the original image is an im-portant secret image, the shared results should be hidden in some host images to form stego images which look ordinary (non-noisy) to avoid attracting an attacker’s attention. The hiding algorithm that we use here is one we developed earlier (similar to the one used in Section 2.3 of Ref.[7]). Note that the size of each shared result is aboutk/(r1+ r2+ · · · + rk) of the original image; therefore, the size of each stego image is about 2k/(r1 + r2+ · · · + rk) of the original image.
4. Decoding phase
The recovery of the original image includes three steps: extracting the shared results from stego images; recovering the rearranged values from the shared results; and restoring the pixel values from the rearranged values.
Step 1: Extracting the shared results from stego images: This step needs only simple operations such as division, addition, and multiplication. The procedure is similar to the one used in Section 2.3 of Ref. [7], and hence omitted.
Step 2: Recovering the rearranged values from the shared results: To illustrate this step, let us inspect the example given in Eqs. (1)–(3) wherek=3 threshold values were used
(r1= 2, r2= 3, r3= 4). According to Eq. (4), the shared result held by participantw is
SR(w) = j
{fj(1)(w), fj(2)(w), fj(3)(w)}, (5) where j ranges through all possible sectors contained in the original image. Now, in the decoding phase, assume that we receive two shared results, say, SR(1) and SR(4).
Then, since SR(1) = j fj(1)(1), fj(2)(1), fj(3)(1) and SR(4) = j fj(1)(4), fj(2)(4), fj(3)(4),
we can reveal the valuesfj(1)(1) and fj(1)(4) for each sector j. Therefore, the coefficients 255 and 128 of the polynomial fj(1)(x) defined in Eq. (1) can be determined (through the two points(1, fj(1)(1)) and (4, fj(1)(4)), a unique line is determined; and the equation of this unique interpolation polynomial of degree 1 can be found by using Lagrange’s interpolation [see Section 3.2 of Ref.[4]]).
In the process of progressive transmission, assume that we receive one more shared result, say, besides SR(1) and SR(4), we also receive SR(5). Then, since
SR(5) = j
{fj(1)(5), fj(2)(5), fj(3)(5)},
we can extract the values fj(2)(1), fj(2)(4), and fj2(5). Again, by using Lagrange’s interpolation, we can find the unique interpolation polynomial through the three points (1, fj(2)(1)), (4, fj(2)(4)), and (5, fj(2)(5)). Therefore, the 3 coefficients{63, 224, 0} of fj(2)(x) defined in (2) can be obtained. Therefore, {255, 128; 63, 224, 0} are all known when we receive three shared results SR(1), SR(4), and SR(5).
An analogous argument shows that we can know all 9 coefficients in Eqs. (1)–(3) if we receive 4 shared results, say, SR(1), SR(4), SR(5), and SR(8).
Step 3: Restoring the pixel values from the rearranged values: After obtaining the rearranged values in the pvious step, the values can be transformed back to re-store the pixels of the original image. For example, if {255, 128} and {63, 224, 0} are recovered in the previ-ous step (assuming that three shared results are received), then 255 = (1111 1111)2, 128 = (1000 0000)2, 63 = (0011 1111)2, 224 = (1110 0000)2, and 0= (0000 0000)2 together form a sequence of 40 bits, i.e. (1111 1111 1000 0000 0011 1111 1110 0000 0000 0000). If we restore these 40 bits according to the scan order listed in Fig. 2, we can restore at least 40/9 = 4 of the most significant bits of the 9 pixels A.I. In fact, 40 − 9 × 4 = 4 implies that the first four pixels(A.D) can each recover one more bit. Therefore the40/9 = 5 of the most significant bits of the pixels A, B, C, and D are revealed as (10100), (10100), (10100), and (10100), respectively; while the 40/9 = 4 of the most significant bits of the pixels E, F , G, H ,
andI are revealed as (1010), (1010), (1010), (1010), (1010), respectively (seeFig. 2).
5. Experiment
The experimental result is shown in Figs. 3–5 and Table 1. The input is the image Lena shown in Fig. 3, which is shared by our progressive scheme. In the experiment, we use k = 4 thresholds, which are
(r1= 2) < (r2= 3) < (r3= 4) < (r4= rk= 5). We generate, say,n=6 shares, and then the 6 shared results are hidden in six host images to generate 6 stego images.Fig. 4shows the stego images. Their PSNRs range from 34.20 to 34.49. No-tably, the size of each host image and each stego image in this experiment is 388× 388 = 2 × [512 × 512 ×2+3+4+54 ], for 512× 512 ×2+3+4+54 (the original image size multi-plied byk/(r1+ r2+ · · · + rk)) is the size of each shared result. The factor 2 is due to the fact that the stego image size is two times greater than the data (the shared result) hidden inside.
Fig. 5shows the images recovered from various numbers of the stego images.Fig. 5(a) shows the recovered image when “any” two of the stego images inFig. 4are available. The recovered image has a poor quality because the PSNR is only 14.57 db.Fig. 5(b) shows the recovered image when any of the three stego images are available; the recovered image has a better quality (29.28 db).Fig. 5(c) shows the recovered image when any four of these stego images are available; the recovered image has an even better quality (48.46 db).Fig. 4shows the recovered image when any five of the six stego images are available; the recovered image is lossless.
Fig. 4. The six 388× 388 stego images (the PSNRs range from 34.20 to 34.49).
Fig. 5. The recovered images revealed from various numbers of stego images: (a) from any 2 stego images (PSNR = 14.57); (b) from any 3 stego images(PSNR = 29.28); (c) from any 4 stego images(PSNR = 48.46); (d) from any 5 stego images (lossless).
6. Quality control design
It is possible to control the quality of the image recov-ered from a small number of shared results. In some
com-Table 1
PSNRs for the three kinds of thresholds-setting that are all com-posed of 3, 4, and 5 shared results
Number of Thresholds-setting stego images
3, 4, 5 3, 4, 5, 5, 5 3, 3, 3, 4, 5 (12 pixels/sector) (22 pixels/sector) (18 pixel/sector)
3 20.01 13.15 31.11
4 42.70 20.00 46.38
5 Lossless Lossless Lossless
mercial applications (for example, a cable system whose image quality depends on the amount of money paid by the viewer), if a viewer has, say, only 3 shared results, the copyright owner may require that the unveiled quality should be poor. In some other applications (for example, the hot line between two police stations), the requirement might be that only 3 shared results can still provide good-quality recovery. To control good-quality, we may repeat some threshold values in the design. For example, consider a de-sign where the threshold values are{3, 4, 5}. If we use only three threshold values{r1= 3, r2= 4, r3= 5}, we get the results shown in the middle column ofTable 1; i.e. 20.01 and 42.7 db when we receive 3 and 4 shared results, respec-tively. (There are 3+ 4 + 5 = 12 pixels per sector in this case; for we construct 3 polynomials for each sector, and these 3 polynomials have 3, 4, and 5 coefficients, respec-tively.) In some business applications, their systems may want to reveal only poor quality images when customers purchase less than 5 shared results. The thresholds-setting
{r1= 3, r2= 4, r3= 5, r4= 5, r5= 5} is then a solution for this, because only about 3+4+5+5+53+4 × 8 =227 × 8 ≈ 2.5 of the 8 bits are revealed for each pixel when 4 shared results are received. (There are 3+ 4 + 5 + 5 + 5 = 22 pixels per sector in this thresholds-setting case; for we build 5 polynomials whose numbers of coefficients are 3, 4, 5, 5, and 5, respectively. Receiving 4 shared results can recover only the first two polynomials [because 34 and 44], and therefore recover only 3+ 4 = 7 of the 22 coefficients of the polynomials.) As shown inTable 1, the image quality is quite poor (13.15 or 20.00 db) when the image is recovered using 3 or 4 shared results. Finally, when fast revealing of the good-quality image is required, e.g. between police stations, the thresholds-setting{r1= 3, r2= 3, r3= 3, r4= 4, r5= 5} can achieve the goal, because 3+3+3+4+53+3+3 × 8 =12× 8 = 4 of the 8 bits are revealed for each pixel when only 3 shared results are received. (There are 3+ 3 + 3 + 4 + 5 = 18 pixels per sector in this thresholds-setting case; for we build 5 polynomials whose numbers of coefficients are 3, 3, 3, 4, and 5, respectively. Receiving 3 shared results can recover only the first three polynomials, and therefore recover 3+3+3=9 of the 18 coefficients). As shown in Table 1, the image quality is acceptable (31.11 db) when the image is recovered by 3 shared results. Moreover, the image quality is good (46.38 db) when the image is recovered by 4 shared results.
7. Conclusion
There are several characteristics in the proposed progres-sive image sharing scheme, including (1) the scheme is fault tolerant (allowingn−r stego images to be lost or damaged); (2) the shared results are equally important, so there is no need to worry about which part is lost or transmitted first; (3) the scheme is secure (less thanr1shared results cannot reveal any information about the image); and (4) quality-control design is possible (as explained in Section 6).
Acknowledgements
The authors would like to thank the reviewer for valuable comments. This work was supported by the National Science Council, Republic of China, under grant NSC92-2213-E-009-032.
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About the Author—SHANG-KUAN CHEN was born in 1972 in Taiwan, Republic of China. He received his B.S. degree in applied mathematics in 1994 from Fu Jen Catholic University, Taiwan. In 1998, he received his M.S. degree in applied mathematics from National Chiao Tung University, Taiwan. Since 1999, he has been studying for a Ph.D. degree in the Computer and Information Science Department of National Chiao Tung University. His research interests include visual cryptography, data hiding, and interconnecting network. About the Author—JA-CHEN LIN was born in 1955 in Taiwan, Republic of China. He received his B.S. degree in computer science in 1977 and M.S. degree in Applied Mathematics in 1979, both from National Chiao Tung University, Taiwan. In 1988, he received his Ph.D. degree in mathematics from Purdue University, USA. During 1981–1982, he was an instructor at National Chiao Tung University. From 1984 to 1988, he was a graduate instructor at Purdue University. He joined the Department of Computer and Information Science at National Chiao Tung University in August 1988, and became a professor there. His research interests include pattern recognition and image processing. Dr. Lin is a member of the Phi-Tau-Phi Scholastic Honor Society.
