Small perturbations in the LSB channel are imperceptible to humans [188] but are statisti- cally detectable for image analysis. Here, we introduce the Progressive Randomization image meta-description approach for Image Categorization and Steganalysis. It is a new image meta- description approach that captures the differences between broad-image classes using the statis- tical artifacts inserted during the perturbation process. Our experiments in Sections 4.4 and 4.5 demonstrate PR captures the image class separability allowing us to successfully infer high-level information about images.
Algorithm 5 summarizes the four stages of PR applied to Image Categorization and Ste- ganalysis: (1) the randomization process (c.f., Sec. 4.3.2); (2) the selection of feature re- gions (c.f. Sec. 4.3.3); (3) the statistical descriptors analysis (c.f. Sec. 4.3.4); and (4) the in- variance transformation (c.f. Sec. 4.3.5).
In summary, if we use n = 6 controlled transformations, we have to analyze the perturbation artifacts in seven images (the input plus the perturbed ones). Furthermore, if we analyze r = 8 regions per image and use m = 2 statistical descriptors for each region, we have to assess r × m = 16 features for each image. In this context, the final PR description vector amounts to (n + 1) × r × m = 112 features. If we perform the last step of invariance (which depends on the application), we normalize each group of features of one perturbed image with respect to the feature values in the input image. Therefore, the final description vector after normalization amounts to n × r × m = 96 features.
4.3.1 Pixel perturbation
Let x be a Bernoulli distributed random variable with P rob{x = 0}) = P rob({x = 1}) = 12,
B be a sequence of bits composed by independent trials of x, p be a percentage, and S be a random set of pixels of an input image.
Given an input image I of |I| pixels, we define the LSB pixel perturbation T (I, p) the process of substitution of the LSBs of S of size p × |I| according to the bit sequence B. Consider a pixel pxi ∈ S and an associated bit bi∈ B
L(pxi) ← bi for all pxi∈ S. (4.2)
where L(pxi) is the LSB of the pixel pxi.
Algorithm 5 The PR image meta-description approach Require: Input image I; Percentages P = {P1, . . . Pn};
1: Randomization: perform n LSB pixel disturbances of the input image ⊲ Sec. 4.3.2
{Oi}i=0...n.= {I, T (I, P1), . . . , T (I, Pn)}.
2: Region selection: select r feature regions of each image i ∈ {Oi}i=0...n ⊲ Sec. 4.3.3 {Oij} i = 0 . . . n,
j = 1 . . . r. = {O01
, . . . , Onr}.
3: Statistical descriptors: calculate m descriptors for each region ⊲ Sec. 4.3.4 {dijk} = {dk(Oij)} i = 0 . . . n,
j = 1 . . . r, k = 1 . . . m.
4: Invariance: normalize the descriptors based on their behavior in the input image I ⊲ Sec. 4.3.5 F = {fe}e=1...n×r×m = dijk d0jk i = 0 . . . n, j = 1 . . . r, k = 1 . . . m. , (4.1)
5: Use {dijk} ∈ R(n+1)×r×m (non-normalized) or {dijk} ∈ Rn×r×m (normalized) features in your favorite machine learning black box.
4.3.2 The randomization process
Given an input image I, the randomization process consists in the progressive application I, T (I, P1), . . . , T (I, Pn) of LSB pixel disturbances. The process returns n images that only
differ in the LSB from the input image, and are identical to the naked eye.
The T (I, Pi) transformations are perturbations of different percentages of the available LSBs.
Here, we use n = 6 where P = {1%, 5%, 10%, 25%, 50%, 75%}, Pi ∈ P denotes the relative sizes
of the set of selected pixels S. The greater the LSB pixel disturbance, the greater the resulting LSB entropy of the transformation. Figure 4.2 shows that the T (I, Pi) transformations does not
introduce visual changes in the images. The perturbations are performed only over the LSBs of the input image (Figure 4.3). Hence, the differences between the input image and any other perturbed image are in the LSB channel only.
4.3.3 Feature region selection
Local image properties do not show up under a global analysis [188]. We use statistical descrip- tors on local regions to capture the changing dynamics of the statistical artifacts inserted during the randomization process (c.f., Sec. 4.3.2).
Given an image I, we use r regions with size l × l pixels to produce localized statistical descriptors. In Figure 4.4, we show the m = 8 overlapping regions we use in this paper.
135 = 1000 0111 138 = 1000 1010 114 = 0111 0010 46 = 0010 1110 135 138 114 46 135 139 115 46
Figure 4.1: An example of LSB perturbation using the bits B = 1110.
The curse of dimensionality keeps us from adding too many regions — we have found out, experimentally, that eight regions are a good tradeoff.
4.3.4 Statistical descriptors
The LSB perturbation procedure changes the contents of a selected number of pixels and induces local changes of pixel statistics. An L-bit pixel spans 2L possible values, and has 2L−1 classes of invariance under pixel perturbations (c.f., Sec. 4.3.1). Let’s call these invariant classes pair of values (PoV).
Figure 4.2: The input image I (leftmost) and its perturbed version T (I, 75%) (middle): differ- ences are not visible by the naked eye. However, they are present in the LSB channel (rightmost).
Input I LSBs(I) T (I, 1%) T (I, 5%) T(I ,10% ) T (I ,2 5% ) T(I ,7 5% ) T (I , 50% )
Figure 4.3: The Progressive Randomization behavior over the LSBs. T (I, Pi) represents a PR
perturbation with percentage Pi over the LSBs of the input image I.
When we disturb all the available LSBs in S with a sequence B, the distribution of 0/1 values of a PoV will be the same as in B. The statistical analysis compares the theoretically expected frequency distribution of the PoVs with the observed ones after the perturbation process.
We apply the χ2(chi-squared test) [191] and U
T (Ueli Maurer Universal Test) [102] to analyze
the images.
χ2 test
The χ2 test [48] compares two histograms fobs and fexp. Histogram fobs represents the obser-
vations and fexp represents the expected histogram. The procedure computes the sum of the square differences of fobs and fexp divided by fexp,
χ2=X
i
(fiobs− fiexp)2
fiexp . (4.3)
Ueli test
The Ueli test (UT) [102] is an effective way to evaluate the randomness of a given sequence of
Figure 4.4: The eight overlapping regions used in the experiments.
remaining blocks, looks for the most recent occurrence of bi, and takes the log of the summed
temporal occurrences. Let B(S) = (b1, b2, . . . , bN) be a set of n blocks such that ∪∀bi = S.
Let |bi| = L be the block size for each i and |B(S)| = N be the number of blocks. We define
UT : B(S) → R+ as a function UT(B(S)) = 1 K Q+K X i=Q ln A(bi), (4.4)
where K is the number of analyzed bits (e.g., K = N ), Q is a shift in B(S) (e.g., Q = K10 [102]), and
A(bi) =
(
i 6 ∃i′∈ N, i′ < i → bi′ = bi,
min{i′: bi′ = bi} otherwise.
In practice, if UT is close to 7.1836, we have a high randomness condition. On the other
hand, the lower UT, the more predictable is the condition in S.
4.3.5 Invariance
In some situations, it is necessary to use an image-invariant feature vector. For that, we use the variation rate of our statistical descriptors with regard to the PR, rather than their values. We normalize all descriptors from the transformations with regard to their values in the input image F = {fe}e=1...n×r×m = dijk d0jk i = 0 . . . n, j = 1 . . . r, k = 1 . . . m. , (4.5)
where d denotes a descriptor 1 ≤ k ≤ m of a region 1 ≤ j ≤ r of an image 0 ≤ i ≤ n and F is the final generated descriptor vector of the image I. Figures 4.5(a-b) show the behavior of our statistical descriptors along the progressive randomization of one selected image I.
Base T1 T2 T3 T4 T5 T6 T1 T2 T3 T4 T5 T6 PR transformation χ 2d es cr ip to r va lu e 0 0.2 0.4 0.6 0.8 1 1.2 (a) χ2 descriptor. Base T1 T2 T3 T4 T5 T6 T1 T2 T3 T4 T5 T6 PR transformation UT d es cr ip to r va lu e 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 (b) UT descriptor.
Figure 4.5: Normalized descriptor’s behavior along the progressive randomization. Ti represents
the PR operation T (I, Pi).
The need for invariance depends on the application. For instance, it is necessary for Ste- ganalysis but harmful for Image Categorization. In Steganalysis, we want to differentiate images that do not contain hidden messages from those that contain hidden messages, and the image class is not important. On the other hand, in Image Categorization, the descriptor values are important to improve the class differentiation. Different classes do have distinct behavior under Progressive Randomization approach (c.f., Sec. 4.4, 4.5, and 4.6).