Error correcting codes for robust color wavelet watermarking

The watermark construction

The initial matrix or the signature is constructed by a random number generator according to the user specified parameters. After we apply the encoding scheme to construct the watermark which is then embedded into the image with certain limitations depending on image size and user defined parameters. The signature size is chosen to be 8×8 as it corresponds to a compromise between a sufficient size for a copyright application and a minimum robustness.

In the literature, we have many options to construct the final watermark from the initial matrix. For the purpose of illustration let us consider the very basic encoding scheme—the repetition codes. This signature or the initial matrix is repeated four times into an intermediate matrix which is again repeated according to the image capacity.

The human visual system is sensitive to changes in the lower frequencies as they are associated to the more significant characteristics of the image. The higher frequencies give the details of the image but changes in the higher frequencies could be easily eliminated by a low pass filter. Therefore the proposed algorithm uses middle frequencies for the insertion of the mark as both invisibility and robustness against low pass filter attacks is required in such an algorithm.

Wavelet decomposition

The wavelet decomposition is applied to each color component R, G, and B. The wavelet decomposition gives us the decomposition of the signal into different frequency bands. This decomposition is done by a filter bank in such a way that we split the low frequency band into small segments in order to separate all the components of the signal and we split the higher frequency bands into large segments as they contain less information. We embed the mark into middle frequencies as the higher frequencies could be simply eliminated by a low pass filter and the lower frequencies carry the overall form of the image and changing these lower frequencies may make the watermark visible.

Vector definition

With a = {R, G, B}, [n, m] representing the coordinates, and (d_j,L)_{j = 1, 2, 3} the sub-bands of the wavelet decomposition at the L th level, as shown in Figure 1. The top right side of the Figure 1 (after wavelet decomposition) corresponds to the result of a low pass operation and the corresponding detail bands generated by the sub band filtering operation for the red color component. This process is repeated for each of the following wavelet decompositions until we reach the L th level where we are interested to insert the watermark. Then the vectors are defined for each of the color component. The bottom part of Figure 1 shows the vector definition for the red color component at scale L = 1 and position [n, m]. Here (d_1,L) (resp., (d_2,L) and (d_3,L)) corresponds to the horizontal (resp., vertical and diagonal) details of the image.

The maximum information (capacity) that can be embedded using the watermarking scheme is calculated using $\frac{D_x}{2^L}\ast \frac{D_y}{2^L}$, where D _x and D _y are the horizontal and vertical dimensions of the image and L is the level of the wavelet decomposition where we are interested to insert the watermark M.

Watermark insertion

In order to define the insertion process, we propose to adapt the QIM principle to vectorial case. For this, we will introduce a modification rule of color wavelet coefficients. As we have said the QIM uses a quantizer that is a function that maps a value to the nearest point belonging to a class of pre-defined discontinuous points. For non-adaptive QIM, the quantization step size is independent of the content. However, it is well known that the ability to perceive a change depends on the content. For example, the HVS is much less sensitive to changes in heavily textured regions and much more sensitive to changes in uniform regions. Moreover, the coefficient modifications and QIM process pose some challenges when applied generally to the color domain.

To account for this, we propose to use a method to automatically adapt the quantization step size at each sample. First, the step value is controlled by the wavelet coefficients that measure the spatial local activity. Second, the watermark insertion process is based upon moving one of the three color vectors (R, G, and B). A better candidate is defined in order to minimize the distortion at each insertion space.

With c ∈ {R, G, B}, c ≠ a and c ≠ b

Figure 2 shows that P_ref1 and P_ref are the most distant points from each pair of points and that is why P _R is chosen as P _M . Here P _x refers to the extreme point of the vector ${\overrightarrow{V}}_x$. P _M corresponds to ${\overrightarrow{V}}_M$ which is marked with the contents of the watermarking matrix M.

The watermarking convention is presented in Figure 3, where the watermarked vector ${\overrightarrow{V}}_{M,W}$, that corresponds to the original vector ${\overrightarrow{V}}_M$.

After watermarking, if ${\overrightarrow{V}}_M$ denotes ${\overrightarrow{V}}_R$, then:

One of the most important possibilities lies on the ability of tuning the P_M,W shift in order to limit the visual degradations on the image. Figure 4 shows the possible shifts of P_M,W. Two cases are considered, Shift 1 and Shift 2. The limit of the two possible modifications is the median line between P_ref1 and P_ref. To be more robust, we define around this line a particular area (Figure 4) such that after the watermarking if P_M,W is in this area, it has to be moved out by increasing the strength of the insertion process. The border of this area can be equivalent to ±5% of the distance between P_ref1 and P_ref.

With this approach, there exist two cases of Shift. In the first case, P _M is already nearest to P_ref1 and the possible positions of P_M,W after watermarking belongs to the segment ${\overrightarrow{P}}_M{P}_{\text{ref1}}$ (if P _M is out of the median area). In the second case, P _M is not already nearest to P_ref1 and we create an intermediate point P_int, defined by:

where i = {1, 2}, a = {R, G, B}, 0 ≤ F _a [n, m] ≤ 1, S = {M;i nt} and F _a represents the weighted matrix for watermarking for each location [n, m].

In the case of maximum force of insertion F _a  = 1, a conflict problem is highlighted, as shown on Figure 3 with a circle. It is observed that the bit value between P _R and P _B can be different: M can receive 0 or 1. This means that, in the detection step, the vector identification (V _R , V _R , and V _R to ${\overrightarrow{V}}_M$, ${\overrightarrow{V}}_{\text{ref1}}$, and ${\overrightarrow{V}}_{\text{ref}}$) could be false. Thus, to avoid this configuration, F _a must be set inferior to 1.

The modification operation is applied on the whole host image in the wavelet domain. The last step in the watermark insertion process is the reconstruction of the image in to the spatial domain by inverse wavelet transform.

Watermark extraction

The first step of the extraction process consists also in a decomposition of the image with the same wavelet basis used in the insertion step. The watermark M _D is detected by measuring the largest distance between $\left|\right|{\overrightarrow{V}}_{v\text{ref1}}-{\overrightarrow{V}}_M\left|\right|$ and $\left|\right|{\overrightarrow{V}}_{v\text{ref}}-{\overrightarrow{V}}_M\left|\right|$. Following the convention used in insertion, the watermark is thus reconstructed, bit by bit. The signature S _D is obtained by making an average and a binarization that corresponds to the coding method used for the creation of the mark M. In order to decide if S _D corresponds to S, a threshold is fixed to accept an extracted signature. This threshold is based on the acceptable level of bit error rate for the watermarking system.

Based on this we can define an acceptance threshold which decides as to whether to accept the detected watermark or to reject it. The different modes and families of error correcting codes, discussed in Section 3 help to lower this acceptance threshold, thus increasing the robustness of the watermarking scheme.

Characteristics of the watermarking channel

Due to the requirement of watermark invisibility, the watermarks are weakly inserted in to the image. This makes the watermark signal prone to errors or attacks. The watermark channel is very noisy due to the different types of intentional or unintentional attacks. We consider the problem of watermark robustness against different errors or attacks analogous to the transmission of a signal over a noisy channel. To correctly transmit a signal over a noisy channel error correcting codes are used to protect the signal from the effects of the channel. The characteristics of the watermarking channel depend upon the type of attacks experienced by the watermarked image. Like in the transmission of a signal over a noisy channel, error correcting codes are used to protect the signature in the form of a watermark so that the effects of the channel are reduced or minimized. The underlying characteristics of an image, e.g., the texture and color information also determine the effect an attack has on the watermarked image. The watermarking algorithm and the type and mode of error correcting codes also play an important role in defining the combined performance of robustness and invisibility.

The characteristics of the watermarking channel are primarily determined by the different attacks. We consider JPEG compression, additive white Gaussian noise, low pass filter, hue, saturation, and brightness as the underlying characteristics of the watermarking channel. Each of the different error correcting codes presented in the following Section exhibit different properties against these attacks.

The watermarking channel is characterized by very high error rates. To correct these errors we use different error correction schemes. Four families of error correcting schemes are used in our study to enhance the robustness of the watermark—repetition codes, Hamming codes, BCH codes and Reed-Solomon codes. We explore the use and effectiveness of the concatenation of these different families of error correcting codes to enhance the robustness of the watermarking scheme.

We employ a concatenation model where two of these error correcting codes are concatenated so that the two error correcting codes can facilitate one another. The outer error correcting codes are a second version of the repetition codes: the watermark is built up from some repetitions of the signature. The outer error help in reducing the error rates so that the inner error correcting codes (repetition, Hamming, or BCH) could then further reduce the errors so that the decision that the received watermark is valid or not could be taken.

Error correcting codes are expressed in the following article in the form of (n,k,d), where n is the length of the code, k is the dimension and d is the minimum Hamming distance between any pair of different codewords. The Hamming distance, H _d is based on the Hamming weight of a codeword c given by H _w (c), the number of non zero elements in a vector. The Hamming distance H _d between two codewords is the number of elements in which they differ. The minimum Hamming distance d, between any two different codewords defines the error correcting capability of the particular error correcting code. An (n, k, d) error correcting code is capable of correcting t errors where $t<\frac{d}{2}$.

Concatenated error correcting codes

As we have said, the robustness of the watermarking scheme could be improved by concatenating these codes using signature repetition codes as outer codes and bit repetition, Hamming or BCH codes as inner codes. The outer coding is adaptive and is in accordance to the size of the image and user parameters, it is always repetition coding as shown in Figure 5.

At the receiver side an exact opposite procedure is applied to decode the signature from the watermark, i.e., we decode the watermark using repetition decoding first and then we decode the resulting information using repetition, Hamming or BCH decoding and we have the signature.

Such a concatenation mode has been selected because error correcting codes cannot display their potential unless the error rate induced by the channel is reduced below a critical value which brings about the possibility of first improving the channel error rate via repetition coding to an acceptable level, before any further decoding. The watermark channel may have to operate at very high bit error rates and codes such as BCH stop bringing in any advantage while the repetition codes continue with their modest protection. However concatenation of repetition and BCH codes is a way to improve the decoding performance when the error rates are high [3]. The BCH codes can correct up to t = ⌊(d − 1) / 2⌋ errors, all errors exceeding t may cause the decoder to decode erroneously. The repetition codes display better characteristics than BCH under high error rates. This could be seen in Section 4.2 (noise attack) where the repetition codes perform much better than the BCH (63,16,23) codes when the SNR<2.

As mentioned in the introduction Reed-Solomon codes are used in a standalone mode to correct burst errors. The decoding of Reed-Solomon codes is carried out using list decoding algorithms [1, 6]. The list decoding algorithms offer enhanced performance over bounded distance algorithms when the code rates are low.

Repetition codes

Repetition codes are used to construct the watermark from the signature and they are expressed in the form of (n, k, d). They are always used as (n, 1, n) where each codeword is repeated n number of times. The repetition codes are used as inner codes in the construction of the watermark. They are also used as outer codes in all cases. The decoding of the repetition codes is always done using a mean operation on the received codeword to distinguish between a 0 or a 1.

Hamming codes

Hamming codes are linear block codes. For an integer m > 1, we have the following representation for the binary Hamming codes in the form (n, k, d) = (2 ^m  − 1, 2 ^m  − 1 − m, m).

For m = 3, we have (7,4,3) Hamming error correcting codes. These Hamming codes encode 4 bits of data into 7 bit blocks (a Hamming code word). The extra 3 bits are parity bits. Each of the 3 parity bits is parity for 3 of the 4 data bits, and no 2 parity bits are for the same 3 data bits. All of the parity bits are even parity. The (7,4,3) Hamming error correcting code can correct 1 error in each of the Hamming codeword.

When we multiply the received codeword with the parity check matrix we get the corresponding parity ranging from 000 to 111. These three bits give us the error location. 000 indicating that there were no errors in transmission and the rest from 001 to 111 indicate the error location in our seven bit received codeword. Here we can correct one error according to t = ⌊(d − 1) / 2⌋ as the minimum Hamming distance between our code words is 7 − 4 = 3, we have 1 as the number of correctable errors. Now we have the error location, we could simply flip the bit corresponding to the error location and the error will be corrected. Then we discard the parity bits from position one, two and four we have our received data words. Hamming Codes are perfect 1 error correcting codes. That is, any received word with at most one error will be decoded correctly and the code has the smallest possible size of any code that does this. The Hamming codes that we used could correct 1 error in each codeword. There was a need to test other types of codes which can correct more errors. We selected the BCH codes which are explained in the following section.

Bose Chaudhuri Hocquenghem (BCH) codes

BCH codes are cyclic block codes such that for any positive integers m ≥ 3 and t with t ≤ 2^m − 1 − 1, there is a BCH codes of length n = 2 ^m  − 1 which is capable of correcting t error and has dimension k = n − m ∗ t.

Let C be a linear block code over a finite field F of block length n. C is called a cyclic code, if for every codeword c = (c₁,…,c _n ) from C, the word (c _n , c₁, …, c_n − 1) in F ⁿ obtained by a cyclic right shift of components is also a codeword from C.

We have selected the BCH (15,7,5) and the BCH (63,16,23) error correcting codes for the purpose of our experimentation. The BCH (63,16,23) is in line with our algorithm testing parameters since the size of our initial matrix is 8×8 bits.

Reed-Solomon codes

Reed-Solomon codes [49, 54] are q-ary [n, k, d] error correcting codes of length n, dimension k and Hamming minimum distance d equal to n − k + 1. These codes can decode in a unique way up to $\frac{n-k}{2}$ errors, and there exists the possibility to decode them beyond the classical bounded radius $\frac{n-k}{2}$. Usually these codes are considered over the Galois field G F(p ^m ) (for p a prime) and have parameters [p ^m  − 1, p ^m  − 1 − 2t, 2t + 1]. In particular the case p = 2 is often considered for applications since in that case any symbol of the code can be described with m bits. It is also possible either by considering less coordinates in their definition, either by shortening them, to construct Reed-Solomon codes over G F(p ^m ) with parameters [p ^m  − 1 − s, p ^m  − 1 − 2t − s, 2t + 1], which can be decoded in the same way that non shortened Reed-Solomon codes.

Reed-Solomon codes are particularly useful against burst noise. This is illustrated in the following example.

Consider an (n, k, d) = (40, 11, 30) Reed-Solomon code over G F(2⁶), where each symbol is made up of m = 6 bits as shown in Figure 6. As d = 30 indicates that this code can correct any t = 14 symbol error in a block of 40. Consider the presence of a burst of noise lasting 60 bits which disturbs 10 symbols as highlighted in Figure 6. The Reed-Solomon (40,11,30) error correcting codes can correct any 14 symbol errors using the bounded distance decoding algorithm without regard to the type of error induced by the attack. The code corrects by blocks of 6 bits and replaces the whole symbol by the correct one without regard to the number of bits corrupted in the symbol, i.e., it treats an error of 1 bit in the symbol in the same way as it treats an error of 6 bits of the symbol—replacing them with the correct 6 bit symbol. This gives the Reed-Solomon codes a tremendous burst noise advantage over binary codes. In this example, if the 60 bits noise disturbance can occur in a random fashion rather than as a contiguous burst, that could effect many more than 14 symbols which is beyond the capability of the code.

In the watermarking channel, the errors, characterized by the different attacks, occur in random or burst manner. Depending on the placement of the watermark in an image and the use of error correcting codes, the robustness of the signature can be increased against the attacks.

For Reed-Solomon codes the conventionally used, bounded distance decoding algorithms correct up to t = ⌊(n − k) / 2⌋ symbol errors as shown in the above example. Using list decoding, Sudan [1] and later Guruswami-Sudan [6] showed that the error correcting capability of Reed Solomon could be improved to $t_S=n-\sqrt{2\mathit{\text{kn}}}$ and $t_{\text{GS}}=n-\sqrt{\mathit{\text{nk}}}$ respectively.

List decoding of Reed-Solomon codes

It is well known that for a linear code [n,k,d] _q over the field G F(q), of length n, dimension k and distance d, it is possible to decode the code in a unique way up to a number of errors: t = [(d − 1) / 2]. Now what happens if the number of errors is greater than t? Clearly there will always be cases where a unique decoding will not occur. For instance if d is odd and a codeword c has weight d, any element x of weight (d + 1) / 2 (which support the set of non zero coordinates) is included in the support of c, will be at distance (d + 1) / 2 of two codewords: x and (0, 0, …, 0), which gives two possibilities for decoding. Meanwhile if one considers a random element of weight (d + 1) / 2 the probability that such a situation occurs is very unlikely. A closer look at probabilities leads to the fact that in fact even for larger t (but with t bounded by a certain bound, called the Johnson bound) the probability of a random element to be incorrectly decoded is in fact very small.

The idea of list decoding is that for t > (d − 1) / 2 a list decoding algorithm will output a list of codewords rather than a unique codeword. List decoding was introduced by Elias [55], but the first usable algorithm for a family of codes, the Reed Solomon codes, was proposed by Sudan in [1], later the method was improved by Guruswami and Sudan [6].

The list decoding method is a very powerful method but it is slower than classical algorithms which decode less errors. For usual context in coding theory the decoding speed is a very important factor since one wants to optimize communications speed, but there exist contexts in which the use of such a decoding is not as important since the use of the algorithm is only causal in the overall process. This is for instance the case in cryptography and in traitor tracing schemes [56] where list decoding algorithms are used when one wants to search a corrupted mark (which does not occur all the time).

The principle of the algorithm is a generalization of the classical Welch-Berlekamp algorithm, the algorithm works in two steps: first construct a particular bivariate polynomial Q(x, y) over G F(q) and then factorize it for finding special factors. These factors lead to a list of decoded codewords.

The first algorithm by Sudan permits (for k / n < 1 / 3) to decode up to $n-\sqrt{2\mathit{\text{kn}}}$ errors rather than n / 2 for classical algorithms. This method is based on Lagrange interpolation.

The list decoding algorithm of Sudan [1, 57] is detailed in the following steps

The second method of Guruswami and Sudan [6, 57] permits to decode up to $n-\sqrt{\mathit{\text{kn}}}$ errors, but is trickier to use since it is based on Hermite bivariate interpolation and on the notion of Hasse derivative.

In practice the hard step of decoding is finding the polynomial Q(x, y). It can be done in cubic complexity in an elementary (but slow) way by the inversion of a matrix, or also in quadratic complexity but with a more hard to implement method [58].

where h + u < s, i = 1, 2, …, n, and s is a natural number.

The performance of Guruswami-Sudan algorithm is better than the algorithm proposed by Sudan when the code rate R = k / n is high. When the code rate is very low they have similar performance. The performance of both the list decoding algorithms shows clear improvement over the bounded distance (BD) algorithms when the code rate is low. We exploit this property of the list decoding algorithms to encode the signature in to the watermark. The improvement of performance is shown in Figure 7.

We select the Sudan’s algorithm for the purpose of decoding as the code rate R < 1 / 3 for the watermarking scheme presented in Section 2 [5] for a signature size of 64 bits and the actual performance gain by the Guruswami-Sudan over the Sudan algorithm is not significant. The parameters used to demonstrate the performance of the Reed-Solomon codes are RS (40,11,30), RS (127,9,119), and RS (448,8,441) and the code rates for these three cases are 0.275, 0.071, and 0.018 respectively. Therefore it is useless to use the high complexity Guruswami-Sudan algorithm as for the code rates are very low for the given cases and Sudan algorithm has similar performance, specially for RS (127,9,119) and RS (448,8,441), as seen in Figure 7.

According to the properties of the different codes, we are now going to study the integration of these tools in our color watermarking process.

JPEG compression

Due to the lossy JPEG compression we lose the higher frequency components which results in blurring of the image. As the watermarking algorithm embeds the watermark in middle frequencies, the attack does not completely remove the watermark at low levels. Blurring and blocking effects can be noticed in attacked images with a higher compression level causing random errors in the watermark.

We tested our method against JPEG attack by starting off the compression from a quality level of 1% to a quality level of 96% with a step size of 5% as one could see in Figure 9a.

In general, we can say that our watermarking method is robust against JPEG compression where the compression level is up to 50%. If the compression level of the image is reduced beyond 50% then the image is quite degraded. This is so because the JPEG compression starts to compress middle frequencies where the watermark resides.

If the quality level is reduced beyond 50% then we observe loss of high frequency information, artifacts on subimage boundaries and localized stronger artifacts. The loss of high frequency information has no effect, but the localized artifacts have influence to some wavelet coefficient values. Due to the relationship between the contents of an image and the position of the artifacts, the effects of the errors induced by the JPEG compression on the wavelet coefficients are random in nature.

Given the random comportment of the errors induced by the JPEG compression, the BCH codes show the best performance in all cases of image size, with and without the extra protection of repetition codes, and the Reed-Solomon codes give the worst performance due to the random nature of the attack.

As we have said, there is also a relationship between the contents of an image and the attack applied. The dependability of this relationship and the type of error correcting codes used is also significant. The test images have different distribution of frequency content. The watermarking scheme uses these characteristics to insert the watermark. The robustness of these images using Reed-Solomon codes against JPEG compression is shown in Figure 9b.

The effect of the use of the different types of error correcting codes and the relationship between the type of image used for the JPEG compression attack could be observed while considering repetition codes and Reed-Solomon codes. Let us consider repetition codes for the same images. In the results shown in Figure 9b,c we can notice the difference in performance of the repetition codes and Reed-Solomon codes for the same images. It is observed that the repetition codes protect the same images at higher compression levels than the Reed-Solomon codes but with the same sensibility to the characteristics of the image.

Notice the difference in robustness of a compression factor of 40 for the image Bear and the image Plant shown in Figure 8. This difference in performance for the two images is due to their characteristics, the manner in which watermarking scheme exploits these characteristics to insert the watermark and how these locations are effected by the attack. The frequency contents of the image Bear are generally high, very sensitive to the compression scheme. Whereas the figure Plant has relatively lower frequency content and has important discontinuities. The watermarking scheme uses middle frequencies to insert the watermark and the JPEG compression attack will first remove any high frequencies in an image, this explains the observed results.

To conclude, the analysis of robustness against JPEG compression using the different families of error correcting codes shows that the image play a very important role for the overall robustness of the watermarking algorithm, and also shows that different error correcting codes have different performance against the different attacks for every type of image. We consider that Color watermarking scheme with BCH codes is usually robust to JPEG compression.

Additive white Gaussian noise

The type of noise that we introduce into the image is additive white Gaussian noise (AWGN). The insertion of the noise or the attack is completely random in nature and there are no bursts of noise. The x-axis in the Figure 10a shows the signal to noise ratio (SNR) and the y-axis shows the bit error rate. The AWGN is distributed uniformly through the image and due to the randomness of the attack the wavelet coefficients are effected uniformly. The difference in the performance of the Reed-Solomon codes and the others is evident as all other codes except the Reed-Solomon codes are more capable to correct the random errors.

Low pass filtering

The low pass filter gives a blurring effect to an image as it filters out high frequency components from the image. The watermark is robust against low pass filtering as it is not embedded into high frequencies. The error correcting codes perform equally well for the low pass filter attack, the bit error rate is 0 (Figure 10b) except for some cases of large filter dimensions (9×9) and no code protection. This is because the low pass filter starts filtering frequencies where the watermark is embedded but the error correcting codes provide enough robustness so that this error remains negligible. This robustness against low pass filter is a generic feature of transform domain watermarking schemes as usually transform domain watermarking schemes do not insert the mark in high frequencies.

One of the main objectives of this article is to highlight the effects of modification of the color components for watermarking schemes and provide effective robustness against color attacks. We will now discuss some of these color attacks and effective counter measures using error correcting codes.

Hue

The term hue describes the distinct characteristics of color that distinguishes red from yellow and yellow from blue. These hues are largely dependent on the dominant wavelength of light that is emitted or reflected from an object. Hue is the angle between the color vector associated with the pixel and a color vector taken anywhere on the plan orthogonal to grey axis and which sets the reference zero Hue angle. This reference Hue value is often taken to represent the red color vector, so we decided arbitrarily to associate the red color vector and gave it a zero Hue value (Figure 11). Hue is the angle between this reference color vector and the color vector.

The scheme is not very resistant against changes in hue because they effect directly the color vectors that we use in the watermarking process.

In general, all the coding schemes except the Reed-Solomon codes give almost the same results as the error rates are beyond their error correcting capacity. To characterize the noise introduced by hue modification, we can say that changes in hue affect all coefficients representing a particular color similarly and this would have hue effect in blocks.

This is in accordance with the block design of Reed-Solomon codes which help protect the watermark to a certain limit. The progressive improvement of the performance of Reed-Solomon codes could be seen in Figure 12, where reasonable changes modulo 180° are correctable when using RS(448,8,441). The little resistance to the changes in hue is provided by list decoding of Reed-Solomon codes as changes in hue effect all the color components at the same time, consequently effecting the wavelet coefficients, where the watermark has been inserted.

Even Reed-Solomon codes are not able to resist the hue attack if the force of the attack is increased (the changes in hue are increased) as the blocks of Reed-Solomon codes and the blocking effect of changes in hue do not cater for exactly the same wavelet color vectors.

Saturation

Saturation is the measure of color intensity in an image, the Saturation is the distance between the color vector and the grey axis (Figure 11). The less saturated the image, the more washed-out it appears until finally, when saturation is at −100, the image becomes a monochrome or grayscale image and as the saturation increases the colors become more vivid until they no longer look real.

The negative saturation poses no problem to any of our error correction schemes except for a value of −100 but then the image is just a grayscale image. Now the image is desaturated and all the color planes have the same values and there is no difference between the different color vectors. Therefore the watermarking algorithm is not able to decode the watermark. When the image is not completely desaturated then the difference between the wavelet color coefficients is changed in relative proportion to the changes in saturation and the watermarking algorithm is able to decode the signature without the help of error correcting.

On the positive side, except the list decoding of Reed-Solomon codes, the coding schemes start giving a significant number of errors when the change in saturation exceeds a value of about 30.

As for the previous hue attack, the saturation attacks has effect in blocks. In fact, the saturation on the positive side might not effect the watermark if the change in saturation does not effect the color vectors strongly. But if the change in saturation effects the color components beyond a certain limit, then the error correcting codes are not able to provide robustness for these high levels. This is the case where the pixels values go out of bounds after processing. To take care of the R, G, and B values exceeding the bound, this problem is tackled by clipping the out of boundary values to the bounds. Clipping the values to the bounds create undesirable shift of hue and as in the previous section affect all coefficients representing a particular color similarly and this would have effect in blocks. This is in accordance with the block design of Reed-Solomon codes. The performance improvement using the Reed-Solomon codes is evident in Figure 13a, where the BER<0.1. Furthermore if the image is saturated beyond a value of 60, it ceases to have a commercial value as natural images are not saturated to such values on the positive side.

The saturation attack, like the other attacks, has dependency on the contents of the image and the robustness measure applied to protect the watermark. The BER increase in Figure 13a for high saturation values is due to the image Leaf and image Monkey (Figure 8) as shown in Figure 13b and it is not attributed entirely to the incapability of list decoding of Reed-Solomon codes to decode the watermark for highly saturated images. For example, the original image Leaf is highly saturated and the modification of the saturation is quickly associated with R, G, and B values exceeding the bound.

Brightness

The brightness of a color measures the intensity of light per unit area of its source. We consider that brightness is the norm of the color’s orthogonal projection vector on the grey axis (Figure 11). It is enough to say that brightness runs from very dim (dark) to very bright (dazzling).

Our scheme resists the modification in brightness in the negative side to the extent of −90, where the image is hardly invisible. On the positive side at a value after 40 none of the schemes could correct the errors but then the image is degraded to such an extent that it is no longer useful.

The modification of the Brightness distorts the saturation as a side effect and the effect of saturation change is more significant when the change in luminance is important. Contrary to the others attacks, it may be more difficult to fully characterize this transform. The attack impact will depend on a significant number of factors (the parameter of the modification, the saturation value, the intensity value). Furthermore, the modification of the brightness may be considered as a random degradation dependent on the characteristics of the image.

In general, all the coding schemes except the Reed-Solomon codes give almost the same results as the error rates are beyond their error correcting capacity and give better performance than RS code for large size images (Figure 14a). Moreover, the response of individual watermarked images is also dependent on the characteristics of the image as shown in Figure 14b. For example, using Reed-Solomon codes, the figure “Bear” has the same robustness as for other error correcting codes, this image is unsaturated. The rest of the images have different robustness.

The results and the special cases discussed in this section show that the characteristics of the image, the watermarking scheme, the use of error correcting codes and the effects of the attack all play a very important role determining the robustness for watermarking schemes. Each of these characteristics has to be considered separately and also the relationship between each of these issues is to be considered while determining the robustness performance of watermarking schemes.

On the whole, we can say that the strategy we propose based on wavelet domain and error correcting codes (with judicious choices) perform well for the different attacks.