## 3 Related Works
### Watermarking algorithms based on Singular Values  
In the literature, watermarking methods from the first category insert the mark in different ways. The simplest embedding scheme consists in adding the watermark by modifying the SVs of the whole host image (e.g., a gray-scale image), where the embedded watermark is a logo binary image or a pseudo-random generated sequence [22; 23]. Another variant is to apply the SVD transform also to the watermark, followed by insertion of the watermark singular values into the SVs of the image [24]. These methods are non-blind or blind [22], and are characterized by a simple implementation. Instead of applying the SVD to the whole image, other schemes split the original image (and the watermark) into non-overlapping blocks, which are then transformed using the SVD. In the embedding process, SVs of the mark are added to SVs of the blocks of the image [25; 26]. The detection process can be blind [27; 28], semi-blind [26], or non-blind [29]. In [24] the authors proposed an algorithm which embeds the watermark (a gray-scale image) in the host image as follows: firstly, a block-based SVD transform (layer 1) is applied to the host image, while a whole SVD transform is performed on the watermark. Each SV of the watermark is then added to the largest SV of each block. Afterward, a global SVD (layer 2) is performed and the SVs of the watermark are added to those of the host image. In several schemes, the embedding space is chosen basing the selection on feature extraction tools, like the Canny edge detector [29] and an entropy masking function [30]. Other techniques embed the watermark, which can be a pseudo-random sequence [21, 30], or a gray-scale image [20] or a binary image sequence (technique 2 in [31]), directly into the largest SV or by quantizing the largest SV of each block of the host image. The detection process is non-blind for [30] and [31], whereas it does not use the original image for schemes that quantize the largest SV (in [15, 21]). The main drawbacks of the presented schemes can be summarized as follows:  
* _False positive._ As stated in [16, 31-35], these schemes are subject to the _false positive problem_, which is an erroneous detection of a watermark in a content which does not actually contain one [36]. However, a false positive may occur also when a specific watermark is detected from a content in which a different watermark was embedded, causing an _ambiguous situation_[32]. This fact does not allow one to solve the _rightful ownership problem_[37]. In general, SVD-based watermarking methods which embed the Singular Values of the watermark into the Singular Values of the host image are considered invertible schemes [31]. Thus, they are vulnerable to the false positive problem, which appears in the detection phase, during the watermark reconstruction. Recently, [PERSON] _et al_. [31] proposed a scheme which modifies the [PERSON]'s algorithm [23] in order to solve this problem. However, the algorithm cannot resist against common and geometric attacks even if the original image is used in the detection process. * _Quality of the watermarked image._ In these schemes, modifying the largest SV may degrade the watermarked image, thus it is necessary to use an adequate strength factor in the embedding process, which attenuates the energy of the watermark and thus lowers the resistance against attacks. * _Payload._ To increase the payload of these schemes the host image is split into small blocks; this fact lowers the robustness against common attacks (e.g., JPEG compression and noise addition) since the stability of the SVs decreases when reducing the size of the blocks. * _Robustness._ Some of the presented algorithms aren't robust to common image manipulation and geometric attacks [20, 26, 27, 30, technique 1 in 31]. * _BER._ Exact extraction of the watermark cannot be achieved, i.e., the bit error rate (BER) between the original watermark and the recovered one is not zero [23, 35, 38].