Recovery of Tampered Pixels

In this section, we present how the altered pixels in the tampered regions can be recovered. As illustrated in Figure 4-10, if the extracted watermark wˆ(j) does not match the original watermark bit w(j) , there are four different cases how the watermark might be changed. Table 4-2 lists the four possible mismatch cases between the extracted and the original watermark bits.

The watermark mismatch is incurred by the change of the feature value, either by increasing or decreasing. Due to the introduction of the dummy entries “2”, in order to get the watermark change in each case listed in Table 4-2, the amount of increasing or decreasing the feature value is different. For example, Case 2 where the watermark is changed from “0” to “2” might be caused by a decrease of the feature value of [Q/2, 3Q/2] or an increase of [3/2Q, 5/2Q]. If we assume that the modification of the

……

verify

0 2

0 ……

… …

(3k-1)Q 3kQ (3k+1)Q

1 2 0

(3k+2)Q (3k+3)Q

feature value of a block is smaller than 3/2Q, the feature value can only be moved to its neighborhood. In practice, this assumption is reasonable because when the size of manipulated regions is far smaller than the image size, the total amount of pixel manipulation will be evenly distributed into quite a few blocks in the permuted image.

Hence, the amount of pixel modification in every single block will be quite limited. If the size of manipulated regions is relatively very large, the tampered area can not be localized any more due to the overwhelming noise-like potential unverified pixels in the authentication process. The limit of the manipulated region size that can be localized will be discussed in the next section.

Under this assumption, we can determine whether the feature value is shifted to the left or the right entry by the pixel manipulation, namely, whether it is increased or decreased. When Q=1, this means that only one pixel in a block is altered. Based on the shift direction of the feature value, we can determine what kind of pixel is changed in the block. In case the feature value is the number of black pixels or the pixels belonging to category c1 in color image, the original pixel can be recovered as listed in Table 4-3.

For example, in Case 2 where the extracted watermark wˆ(j) is “2” and the original one )

(j

w is “0”, it means the number of the black value is increased by one, namely, one of the white pixels in the jth block is modified to black. Therefore, the original pixel

) , (x y

I_o should be white. At this step, we can not exactly determine which white pixel in the block is changed, so we consider all the white pixels in the block as potential unverified pixels. Note that we can also mark all the pixels in the unverified blocks as potential unverified pixels instead of only white pixels. It will compensate the errors caused by the blocks in which not only one white pixel is changed and will therefore increase the density of the unverified pixels in the tampered region after the inverse permutation. In the other 3 cases, the original pixel color can be similarly deduced.

Based on the mismatch of the extracted watermark and the original one, we classify the unverified pixels into two categories. If the watermark change falls into Case 1 and 3, all the unverified pixels in the jth block are classified into category u1. If the watermark

into category u2. Thus, we obtain two different kinds of unverified pixels. After the inverse permutation in the second step of authentication process, the two categories of unverified pixels will be clustered in separate tampered regions according to the way of the pixel manipulation and form the shape of the removed original content or the added parts. When all pixels in unverified blocks are considered as unverified, there will be some mixture of unverified pixels of two categories in some tampered regions. In each tampered region, however, the unverified pixels of one category will be overwhelming majority. Therefore, the original pixel values of each tampered region can be easily estimated and recovered. After filtering the isolated noise-like unverified pixels in the third step of the authentication process, together with the reconstruction of the shape of the original content and the added part, the manipulated image content can be recovered accordingly. Note that for color synthetic images only the original category of the pixel can be recovered instead of the actual pixel color in binary images.

Since there are two categories of unverified pixels, the noise filter can be applied to every category separately. A properly designed noise filter can not only filter out the noise-like pixels, but also can compensate for detection errors. If the number of altered pixels in one block is larger than one, the filter can be used to smooth out the wrong points. For example, if only white or black pixels are modified in a block and the number of the modified pixels is a multiple of 3, the output of the quantization function will change from kQ to (k+3), which have the same mapping value and therefore the pixel modification will not be detected. Another possible case occurs if a certain number of black pixels are modified to white in a block, while the same amount of white pixels are modified to black in the same block: the feature value, i.e. the number of the black pixels, will keep the same and such modification can not be detected. Such tamper detection errors will result in some holes in the area of converged unverified pixels, the noise filter will fill these holes by checking their neighbor pixels.

Furthermore, the size of the applied noise filter can be scaled differently to detect manipulations at various scales according to the requirement of particular applications.

Table 4-3 Original pixel recovery based on the watermark comparison Case w(j) wˆ(j) I_o(x,y)

1 0 1 white/c2 2 0 2 black/c1 3 1 2 white/c2 4 1 0 black/c1