4.2.1. Vector quantization in the RGB space

Vector quantization allows us to insert information about the values of a color pixel (three dimensions). It can be considered as a quantization on a line segment generated by a direction vector (Figure 4.1).

Let P be a value of a color pixel. The result of quantization, denoted by P′ is defined by:

[4.1]

with u_P, a direction vector, and s =< P, u_P >, the canonical scalar product of P by u_P.

This equation modifies the color P according to s along the direction axis u_P. Depending on the color P on which the vector quantization is applied, the color distortion perceived by the HVS is different.

At the detection stage, the value to be estimated is s′. If the color is modified after the insertion of a watermark, we have:

[4.2]

with P″, a color modified after inserting information, and u_P″, its associated direction vector. We then have a decoding condition on the direction vectors: it is necessary that the direction vectors u_P and u_P″ are close enough to have a good estimate of s′.

Assuming we are in a case where detection is possible, that is, the color difference P″ − P ′ is small enough, we then have:

4.2.2. Choosing a color direction

The main differences between color specific watermarking methods are as follows:

• – the insertion method (as for grayscale approaches);
• – the choice of “color” information that will carry the message.

Whether the approaches are direct (use of a color channel), indirect (change of spaces) or even adapted (construction of a specific transform), in general, they can be modeled by the insertion of a message following a certain direction in the RGB cube. Therefore, the aim is to find a direction that minimizes the color quantization noise for the HVS.

To start with, we will study a basic approach that involves choosing a fixed insertion direction for the whole image. This approach illustrates the influence of this choice of direction when there is no local adaptation. If we randomly choose a direction vector constant for every color in the image, the HVS detects color distortions very easily (Figure 4.2(b)) since the choice of direction introduces colors that are not relevant to the content of the image.

We can reduce the detected quantization noise by using a direction that is judiciously fixed for the whole image; this is the case for a large number of algorithms in this book. A direction vector that limits color distortions is vector , which represents the gray level axis in the RGB space. An example is illustrated in Figure 4.2(c). In fact, compared to Figure 4.2(b), for the same level of modification, we no longer see false colors appearing.

However, these first results are not satisfactory in terms of watermarking invisibility. Depending on the color we modify, the detection of quantization noise is different, we have to introduce a so-called adaptive approach. For each color P, we must determine which direction vector choice minimizes the quantization noise, in the sense of the HVS. Figure 4.2(d) shows an example of a watermarked image with an adaptive approach that we will describe in section 4.4.5. We can see that the quantization noise is less visible. However, on closer inspection, we observe a slight colored noise in the homogeneous (or non-textured) areas of the image, such as the green and brown background; the local context of the image is also important.

As we suggested in the introduction, approaches using a more evolved representational space must address the same question: which is the best direction? To show this, we intend to recall the context of image watermarking according to quaternionic transforms.

4.3.1. Quaternions and color images

The quaternion algebra is algebra, , of dimension 4 over , made up of the elements of the form:

[4.3]

with the real number S(p), the scalar part, and the vector V (p), the vector part of the quaternion q.

All the usual geometric operations of ³ can be found when we limit the operations of quaternions to the set ₀ of pure quaternions (if q = (S(q), V (q)) then S(q) = 0). Let q₁ and q₂ be two pure quaternions representing the vectors V₁ and V₂ of ³, respectively, so:

• 1) q₁ + q₂ = V₁ + V₂;
• 2) q₁.q₂ = V₁.V₂;
• 3) q₁q₂ = −V₁.V₂ + V₁ ⌃ V₂.

Note that the quaternionic product of two pure quaternions contains, in its real part, the scalar product of the two vectors represented, and, in its imaginary part, the cross product. This result is present in some uses of quaternions for color images.

The algebra is provided with the product:

where V (p).V (q) (respectively, V(p) × V (q)) denotes the scalar product (respectively, mixed or vectorial) of two vectors V (p) and V (q) of ³.

The algebra is associative but not commutative.

We can find the so-called Hamilton form q = q₀ + iq₁ + jq₂ + kq₃, of quaternion q, by writing: i = (0, 1, 0, 0), j = (0, 0, 1, 0) and k = (0, 0, 0, 1). The calculation rules then become:

We find a vocabulary that is similar to that of complex numbers. The conjugate of a quaternion q is defined by: = q₀ − iq₁ − jq₂ − kq₃. Any quaternion q that is not zero accepts an inverse, called q⁻¹ = q/|q|², where . The positive real number is the magnitude of q, denoted |q|. As for complex numbers, it is possible to define combined quaternions (if q = (S(q), V (q)) then S(q)² + ‖V(q)‖² = 1).

Since a color contains only three components in the RGB space, it has been suggested that we describe the color information on the imaginary part of the quaternions (Sangwine 2000):

The color of a pixel of an image f at spatial coordinates (m, n) is then coded in the following way:

[4.4]

with f_r[m, n], f_v[m, n] and f_b[m, n] as the red, green and blue components of the pixel of coordinates (m, n), respectively. This is how, in the vast majority of quaternion based color watermarking algorithms, the information is modified.

Consider a pixel color f[m, n] = (f_r[m, n], f_v[m, n], f_b[m, n]) coded by a pure quaternion q:

[4.5]

We want to extract the colorimetric coordinates (intensity, hue and saturation). First we need to define a vector µ (pure unitary quaternion) representing the axis of intensities. The intensity I of q is the projection of this on µ:

This axis is generally defined as the grayscale axis, so , but in a watermarking strategy a choice for this direction µ can be more adaptive. Following this idea, we can generalize the operations for manipulating color information through the quaternion formalism (Carré et al. 2012):

with the result:

These few simple operations allow us to illustrate the first links between quaternions and color. We note that, again, the central element to all modifications is the definition of an axis µ, around which the different transforms will be configured. This is the case for quaternionic watermarking algorithms. However, most of them use a transformed domain, particularly a frequency domain.

4.3.2. Quaternionic Fourier transforms

Quaternionic Fourier transforms (QFT) were introduced in the context of color images by Sangwine and Ell (2001) in order to generalize the complex Fourier transform for color images. There are several versions of quaternionic Fourier transforms with different aims, particularly because the quaternionic product is not commutative. Sangwine offers a color version by adding a direction that is characterized by a pure unitary quaternion, to the expression of the exponential µ (Ell and Sangwine 2006). Most often, in order to avoid favoring any color for the spectral analysis, the neutral quaternion µ = µ_gris = (i + j + k)/ is chosen and corresponds to the achromatic axis of the RGB space. The definition is as follows:

[4.6]

It is possible to place the exponential to the right of the function f. We then obtain the definition of another Fourier transform since the quaternionic product is not commutative.

Different authors have looked to give an interpretation to the information described by the spectrum coefficients. We can identify three main approaches:

• – describe the quaternion spectrum with the exponential representation (Assefa et al. 2010);
• – break down each spectral coefficient into two parts, one simplex and one perplex, in order to separate the luminosity data from the chrominance data (Sangwine and Ell 2001);
• – carry out an analysis of elementary atoms (Denis et al. 2007).

This frequency information present in the quaternionic spectrum is especially used in digital image watermarking (Bas et al. 2003). To analyze watermarking in quaternion space, we introduce the Cayley–Dickson construction, which allows the separation of data into a luminosity part and a chromaticity part (Ell and Sangwine 2000). This rewriting has made it possible to analyze the transformations carried out with the quaternions through a “colormetric” view. We have seen that a quaternion can be considered as a vector of the base (e, i, j, k) with e = 1, the real unitary vector:

with a, b, c, d as any four real numbers of which a is the real part and (b, c, d) is the imaginary part.

We can choose another base, for example (e, µ, ν, µν) with µ and ν, two pure unitary orthogonal quaternions. Their product µν is also a pure and orthogonal quaternion to the other vectors of the base. We can write q in this base:

with, for example, the real number q_e = q.e defined by the scalar product and which therefore corresponds to the projection of q on e.

Likewise, we can calculate q_µ = q.µ, q_ν = q.ν and q_µν = q.µν of real numbers.

Notice that it is possible to regroup the quaternion q in two isomorphic subspaces in . In fact, the first block (q_e, q_µ) corresponds to the real part and the projection term on µ, the remainder being expressed in the plane (ν, µν), which is the plane perpendicular to µ. The quaternion can therefore be written as:

with q_‖ = q_ee + q_µµ and q_⊥ = q_νe + q_µνµ, q_‖ being the parallel part to µ and q_⊥ the perpendicular part.

In the case of pixel color manipulation, two pure quaternions are used for this decomposition: the first vector µ is, for example, the gray axis in the cube RGB:

The second is the axis perpendicular to µ in the direction of the red color (pointing out axis i):

since the red is used as a reference for a zero-value hue in the chromticity plane.

In this case, when a color image is divided into parts parallel and perpendicular to µ, it seems that this decomposition allows the separation of luminosity data on the parallel part, and chromatic data on the part perpendicular to µ. We find the classic representation associated with luminance–chrominance spaces.

When a watermarking algorithm suggests using the quaternion space to carry out the insertion of a message, especially in the transformed domain, in many cases the operation can be interpreted through the Cayley–Dickson representation. This results in a modification of the color pixels in one of the directions: the impact of the modification will therefore depend on the explicit or implicit choice of the µ axis. Although the approach uses a more complex model which seems to take the color into account, the algorithm remains configured by a wise choice of insertion color direction.

In section 4.4, we propose to discuss a psychovisual model that makes the choice of direction as adaptive as possible, that is, we show how to determine the optimal direction vectors for any RGB color.

4.4.1. Neurogeometry and perception

Understanding how humans perceive color differences can allow us to minimize insertion color distortions perceptually; in other words, to improve watermark invisibility. To achieve this objective, we need a color discrimination model. This model would be three dimensional, due to the human trichromy, and should mimic the representation of light by the cones of the human retina for their spectral sensitivities as well as their intensity encoding dynamics.

This type of model, known as neurogeometry, was suggested in the context of the perception of shapes based on achromatic data. A detailed description of the construction of forms by the primary visual cortex is given in Petitot (2003, 2008). Few other contributions have considered neurogeometry in the context of color vision (Koenderink et al. 1970; Zrenner et al. 1999; Alleysson and Méary 2012), and even less in the application context of color watermarking.

In this section, we limit our study to the modeling part and its application for color watermarking: more precisely, we use a chromatic discrimination ellipsoid model based on the Naka–Rushton law of photoreceptor dynamics.

A simple and effective way to model color discrimination data involves considering color discrimination to be a nonlinear operation in a three-dimensional Euclidean vector space φ ≅ ³. The space has the standard scalar product xy = x₁y₁ + x₂y₂ + x₃y₃, where x = [x₁, x₂, x₃] and y = [y₁, y₂, y₃] are two vectors in ³ linked to two points, M and N, as and with O, the origin of vectorial space associated with ϕ. The magnitude of a vector x is given by . The distance between x and y is given by d(x, y) = ‖x − y‖.

Assume that this space φ is a physical space in which the HVS maps another representation ψ, marked by a quadratic form q(x) = x^tGx where G is a symmetrical positive definite matrix representing the metric brought about by the nonlinearity of the HVS. This quadratic form defines a new form for vectors in the visual space, such as , and a scalar product balanced such that:

[4.7]

In this model, we assume that the HVS maps the physical space of light φ and the perceptual space ψ. An equal discrimination contour is represented by a circle on an isoluminant surface and the physical space is a space in which this constant discrimination circle in ψ corresponds to an ellipse in φ. The mapping function between these spaces is given by the neuronal function. This model is illustrated in Figure 4.3. In this figure, we assume that the HVS is a nonlinear function f between the flat physical space of light φ and the curved space of perception ψ. In this section, we use a modification of the Naka–Rushton transduction equation to account for visual nonlinearity.

For the sake of generality, we must consider that G depends on the physical stimulus x and on the adaptation state of the observer x₀, so we write G(x, x₀). The question to be solved for a color discrimination space is to find the relationship between the nonlinearity brought about by the HVS, and the shape of the metric G(x, x₀). One way to clarify this relationship is to fix the adaptive nonlinearity function. This cannot be done in general, but we assume that the nonlinearity of the HVS operates on each color channel separately and that it is parametric with an adaptation parameter, which modifies the curve of the nonlinearity.

4.4.2. Photoreceptor model and trichromatic vision

Based on this principle, we introduce a three-dimensional construction of an HVS trichomatic model (Alleysson and Méary 2012).

An HVS is a complex system that adapts according to environmental conditions, and it is no surprise that color vision is a nonlinear, adaptive phenomenon.

Assume for now that x is a physical one-dimensional variable corresponding to light entering the eye. y = f(x, x₀) is the transformation of this variable by the HVS according to an adaptation factor x₀.

We define f as a simplification of the Naka–Rushton function, which checks the transduction of light by the photoreceptor (Alleysson and Hérault 2001):

[4.8]

where y is the level of transduction. In other words, y represents the electric response of a cone as a function of x, the level of excitation of the cone produced by light and x₀, the state of adaptation.

x₀ is modified according to the average excitation level of the photoreceptor. Equation [4.8] represents the behavior of the photoreceptor according to two constants, α and x₀, set by the environmental conditions and the HVS state. In fact, the adaptation process of a photoreceptor is only partially understood and this model is probably a simplification.

We assume that this is the nonlinear, adaptive function of equation [4.8], which matches physical space to perceptual space. The function saturates α when x → ∞ and x₀ gives the half-maximum of the curve. The function’s curve changes according to x₀. A low x₀ gives a high curvature, while a high x₀ gives a low curvature. This nonlinear function is shown in Figure 4.4.

Therefore, considering the three-dimensional space of the physical colors φ, we establish the link between the physical and perceptual space by:

[4.9]

where x = [L, M, S] and x₀ = [L₀, M₀, S₀] are coordinates of the physical stimulus in the space φ, and the adaptation of the observer expressed as a parameter in the space φ, respectively.

Here, we implicitly assume, without losing generality, that the coordinates in the space φ correspond to the responses of cones L, M and S. L, M and S are then the encoding of the physical stimulus x by the HVS, suitable for x₀.

To summarize, the human retina possesses three types of photoreceptors, which are cones L, M and S. The responses of each of these cones are processed by a nonlinear parametric function that defines the perceptual variables. For each color channel, there are two parameters, α and x₀:

[4.10]

where the parameters α_L, α_M and α_S are gains on the respective components L, M and S. The constants L₀, M₀ and S₀ are the adaptation states of the respective cones.

In the trichromatic model of color vision, it is possible to find a three-dimensional construction of color discrimination with the photoreceptor model described above. In the space of transduction lms, pairs of points separated by the same distance have the same level of perception of the color difference. When these pairs are converted in the excitation space LMS, Euclidean distances are no longer equal even though the same level of perception is maintained.

This distortion phenomenon can be better observed by constructing a sphere centered at P_lms = (l_c, m_c, s_c) in the space lms defined as:

[4.11]

with r, the radius of the sphere, −π ≤ u ≤ π and −π/2 ≤ v ≤ π/2.

By converting the sphere in the space LMS, we get a volume defined by:

[4.12]

The distances between the center of the volume P_lms and the points on the surface of the volume are therefore constant:

[4.13]

with a constant radius of r.

Note that r is no longer constant in the case of the volume of center P_LMS = f⁻¹(P_lms).

4.4.3. Model approximation

We then need to explain how to calculate G(x, x₀), the metric from f. To do this, we consider that in the perceptual space ψ, an equal discrimination contour around a point (ℓ_c, m_c, s_c) is a sphere S with radius 1. So we write:

[4.14]

As we have seen, in the physical space, the resulting surface is given by the linear approximation of the nonlinearity around x_c. As we can consider dy to be very small, it is given by the Jacobian of f:

[4.15]

[4.16]

This approximation makes it possible to establish a direct correspondence between dy = [ℓ − ℓ_c, m − m_c, s − s_c] and dx = [L − L_c, M − M_c, S − S_c]:

[4.17]

As we have mentioned, a circle in perceptual space ψ corresponds to an ellipse in physical space φ expressed as:

[4.18]

G is equal to:

[4.19]

We see that, from a color defined by coordinates (L_c, M_c, S_c), we have two groups of parameters in this transformation, (L₀, M₀, S₀) and (α_L, α_M, α_S).

4.4.4. Parameters of the model

To use this model, the constants L₀, M₀ and S₀ and the gains α_L, α_M and α_S can be calculated using MacAdam ellipses (MacAdam 1942) (Figure 4.5).

If we want to calculate these parameters from the MacAdam ellipses, we have to first transform the MacAdam measurement made in the CIE-xyl space into space LMS. To do this, we use the following transformation:

[4.20]

With this transformation of (x, y, l) into (L, M, S) and with the parameter (α_L = 1665, α_M = 1665, α_S = 226), (L₀ = 66, M₀ = 33, S₀ = 0.16), it has been shown that this allows a good reconstruction of MacAdam ellipses (Figure 4.6) (Alleysson 1999).

4.4.5. Application to watermarking color images

In section 4.4.4, we introduced a color vision model to develop a less visible method of watermarking. In the color space LMS, we constructed ellipsoids that concretely illustrate how to control psychovisual distortion while ensuring satisfactory robustness.

In fact, the idea of psychovisual distorsion represents the perception of the color difference on insertion, as opposed to digital (Euclidean) distortions obtained with metrics such as the PSNR. As psychovisual distortions better describe the invisibility of the watermark for the HVS than digital distortion measurements, the theory is that it becomes possible to obtain a better compromise between invisibility and robustness: for the same level of psychovisual distortion, there are different levels of digital distortion, depending on the insert setting, and choosing maximum digital distortion improves robustness.

In this section, we intend to remove, for each color pixel P, the direction vector u_P which has the largest magnitude in its ellipsoid.

4.4.6. Conversions

Here we give the color space conversion matrices to work from XYZ and LMS psychovisual spaces to RGB space. The transformations are chosen according to the 1931 CIE standard¹. The conversion between color spaces is represented by the following matrices:

Let P_RGB, P_XYZ and P_LMS be the color pixels in their respective RGB spaces, XYZ and LMS. We have:

[4.21]

In section 4.2, we have seen that the choice of a direction vector has an enormous impact on the invisibility of a watermark. When we choose a fixed direction for all the colors, we note strong psychovisual distortions in certain areas of the image. However, invisibility is greatly improved when the best direction vector is chosen for each color pixel. Since we assume that the psychovisual distortions are the same for all the elements of ψ, we can choose the point furthest away from P, indicated by P_f, to allow the maximum numerical distortions authorized during integration in order to reinforce robustness. The direction vector u_P is defined such that .

Note that ℰ is the set of points belonging to the volume of perception associated with P. We have the farthest point P_f from P:

[4.22]

To estimate P_f or u_P, we use the Jacobian: to extract an optimal direction, we simply use the fact that the main axes of the ellipses can be calculated from the eigenvectors of the metric. Note that, from equation [4.19], it is possible to directly calculate the expression of the direction u_p for each color.

4.4.7. Psychovisual algorithm for color images

The algorithm that we present in this section makes it possible to insert a mark in a color image whose quality is improved from a psychovisual point of view. Compared to a classical insertion scenario, we suggest adding a step to adapt insertion to the color space (illustrated in Figure 4.7).

Before transforming the host image in the chosen quantization space, we calculate a direction vector for each color pixel of the image χ. This direction vector will be defined from the optimal axis presented in section 4.4.6. Then we introduce the scalar product, s (the “Proj” step in Figure 4.7).

[4.23]

We then obtain a scalar value image . Considering the knowledge of the set of computed direction vectors, we have a one-to-one correspondence between χ and . In Figure 4.8, we can see that image is a dark grayscale version of the color image χ.

The next step then involves changing the representation of . A good first choice of representation is the spatial domain, but a transformed space can be used (transformed into discrete cosine [DCT] or transformed into wavelet coefficients [DWT]). These different representations (denoted Tr and Tr⁻¹ for the inverse operation) of the image make it possible to obtain useful properties according to requirements, such as breaking the image down into independent frequency bands.

An extraction function Extr must then be chosen in order to select the coefficients to be changed. In this chapter, we have chosen a function which randomly determines insertion sites of the image . Another method for selecting coefficients was presented in Chapter 3, which involves breaking an image down into blocks and then randomly selecting coefficients in each block. The result of this extraction takes place in the watermarking space, which is a secret space (known only to the creator of the mark and its recipient). Access to this space can be secured with a secret key k.

Once the coefficients are selected, they are modified by the insertion method (the LQIM method, but it is also possible to use other watermarking methods) according to the message m, then integrated into the representation of the image used for extraction and we get the marked image ′. So, for any modified color P′ (QVC stage), we have:

[4.24]

where s′ is the scalar modified by the chosen insertion method.

The set of s′ forms the grayscale image . Of course, note that the use of different error correcting codes, such as those proposed in Chapter 3 (e.g. Hamming, BCH, Reed–Solomon and Gabidulin codes), is completely possible and straightforward. In this chapter, we focus only on the “invisibility control” aspect and its optimization with respect to robustness.

At the detection stage (Figure 4.9), we identify the grayscale image by recalculating the direction vectors associated with each color of the image received (equation [4.23]), and then accessing the watermark channel using the Extr extraction function and the key k.

We then apply the detector associated with the chosen insertion method to the coefficients . If the power of the attack is reasonable and the mark robustness parameters are well chosen, the estimate m′ should be the same as m.

As we said for the detection, the calculation of the direction vectors is an important step. Assuming that the colors are reasonably modified, the ability to correctly detect the message lies in the variation of the direction vector. In our experiments we will see that the variations of the direction vectors are acceptable and make it possible to ensure good detection.

We have detailed a classical watermarking algorithm for color images using a color vector quantization method based on a psychovisual model of the HVS. The LQIM method was chosen to be adapted to watermarking color images (introduced in section 4.2). To be able to use it, we adapt the classical insertion diagram with our psychovisual quantization model 4.7, then we integrate the quantifier Q_m into the insertion function of the LQIM method. The insertion sites are chosen randomly from the coefficients available for insertion. One of the advantages of this insertion strategy is improving the invisibility of the mark, especially in textured areas of the image. At the detection stage, we also combine the extraction stage of the received coefficients (Figure 4.9) with the classical decoding scheme using the LQIM detector.

Section 4.4.8 gives an evaluation of the robustness performance of the CLQIM method against various image modifications.

4.4.8. Experimental validation of the psychovisual approach for color watermarking

The images used for the experiments belong to the Corel database (1,000 random images selected from the 10,000 available). The insertion sites are determined randomly. To evaluate the invisibility, parameters such as the size of the message or the dimension of the Euclidean lattice L of the Lattice QIM method are calculated in order to obtain an adequate image quality and an adequate insertion rate.

4.4.8.1. Validation of invisibility

In this section, we propose different groups of marked images in order to appreciate the improvement in invisibility, compared to marked images following a vector quantization of constant direction vector.

We then denote the following two approaches by GA and AA:

• – the color version with a constant direction axis u = (1, 1, 1) (GA). We have chosen this direction vector because choosing the luminance axis is, in our opinion, a satisfactory compromise in order to guarantee a good level of invisibility of a mark;
• – the color version with an adaptive direction axis u_P (AA).

These two watermarking methods are the color adaptations of the LQIM method previously presented in this chapter. For each approach, the direction vectors are the same magnitude (arbitrarily set at 0.5). For the same level of digital distortion, it is a question of validating, through experimentation, that the AA approach inserts a less visible mark than the GA approach.

In these experiments, we evaluate the digital distortion thanks to the signal-to-noise ratio, characterizing the quantization noise of the LQIM method, denoted as DWR.

In Figure 4.10, we show examples of color images marked with the GA approach of the LQIM method. For each image in this figure, we can easily see that the colors saturate toward gray. It therefore becomes easier to perceive color differences by comparing them to neighboring pixels. The visual appearance of quantization noise is salt and pepper noise and it adds texture to the image.

With the AA approach (Figure 4.11), the quantization noise has the effect of saturating the colors toward blue and green. This adapts according to the modified color. It is therefore more difficult to perceive the noise. Compared to neighboring pixels in areas of the same color, we find that detecting color difference is difficult.

At equal digital distortion, images marked with the GA and AA approaches are not affected with the same quantization noise from a psychovisual point of view. The perception of quantization noise with the AA approach is much lower for the HVS compared to that of the AA approach. We can see this in the images of Figures 4.10 and 4.11, but these results were also confirmed in other tests.

To able to better observe the insertion noise, we have chosen images of size 60 × 60 and have chosen a strong insertion rate (ER = 0.5). In practice, the images are much larger in size (e.g. 1,080 × 1,920 for high definition) and the insertion rate is weaker, which makes the insertion noise less visible for the HVS.

We repeated the comparison experiment of the GA and AA approaches described above with different observers (15 in number) and with the Kodak image base, composed of 24 elements of size 768 × 512. For each image of this database, we offered a pair of images marked with the two color approaches GA and AA (still with the LQIM method). For the 24 pairs of images, each observer voted for the “least noisy” image.

The results of this experiment are presented in Table 4.1. Of these subjects, only 4% of images on average were described as “least noisy” with the GA approach. Of these results, we conclude that the use of a psychovisual model allows us to obtain a much better invisibility.

Approaches	GA approaches	AA approaches
Average votes	4 ± 3%	96 ± 3%

COMMENT ON TABLE 4.1.– Each person had to decide which of the images was more degraded (a watermarked image with the constant approach, and the other with the adaptive approach). The parameters are DWR = 20 dB, ER = 0.5 for each image. This table shows the percentage of images noted as less degraded for each approach.

We now intend to compare the two approaches in terms of their robustness performance against several image modifications. We offer a simple experiment for this. We propose to insert a message of size n = 128 bits. In order to ensure satisfactory image quality and mark invisibility, we determined the average maximum quantization step before a mark was visible for each method. We measured average bit error rates (100 repetitions for the same quantization step) depending on the strength of the applied image modification. We now propose to analyze the robustness against classical attacks.

4.4.8.2. Impact on robustness

4.4.8.2.1. Modification of luminance

We model the modification of luminance by equation [4.25]:

[4.25]

where y is the modified version of x, the value of a pixel.

The robustness results are shown in Figure 4.12. The error curves oscillate like a step function between 0 and 1 periodically. This attack is special because it applies modifications that are not dependent on the image. This behavior is justified, both by the use of the LQIM method, and by the structure of the error produced by the modification of luminance (studied in detail in Chapter 3).

For the LQIM AA curve, the oscillation period is greater than that of the LQIM GA curve, which gives a larger error-free interval around 0.

4.4.8.2.2. Modification in the HSV space

We now consider examples of color image modifications. By representing an image in the HSV color space, we modify each color component h, s and v.

In Figure 4.13(b), we observe that the LQIM AA SP curves are closer to 0 than the LQIM GA SP curves on each component, therefore showing that taking an HVS model into account allows us to improve the robustness of a watermarking algorithm, not just its invisibility.