An Extended Image Hashing Concept: Content-Based Fingerprinting Using FJLT
© X. Lv and Z. J.Wang. 2009
Received: 27 March 2009
Accepted: 23 September 2009
Published: 15 November 2009
Dimension reduction techniques, such as singular value decomposition (SVD) and nonnegative matrix factorization (NMF), have been successfully applied in image hashing by retaining the essential features of the original image matrix. However, a concern of great importance in image hashing is that no single solution is optimal and robust against all types of attacks. The contribution of this paper is threefold. First, we introduce a recently proposed dimension reduction technique, referred as Fast Johnson-Lindenstrauss Transform (FJLT), and propose the use of FJLT for image hashing. FJLT shares the low distortion characteristics of a random projection, but requires much lower computational complexity. Secondly, we incorporate Fourier-Mellin transform into FJLT hashing to improve its performance under rotation attacks. Thirdly, we propose a new concept, namely, content-based fingerprint, as an extension of image hashing by combining different hashes. Such a combined approach is capable of tackling all types of attacks and thus can yield a better overall performance in multimedia identification. To demonstrate the superior performance of the proposed schemes, receiver operating characteristics analysis over a large image database and a large class of distortions is performed and compared with the state-of-the-art image hashing using NMF.
Digital media has profoundly changed our daily life during the past decades. However, the massive proliferation and extensive use of media data arising from its easy-to-copy nature also pose new challenges to effectively manage such abundance of data (e.g., fast media searching, indexing) and protection of intellectual property of multimedia data. Among the various techniques proposed to address these challenges, image hashing has been proven to be an efficient tool because of its robustness and security.
An image hash is a compact and exclusive feature descriptor for a specific image. Robustness and security are its two desired properties [1, 2]. Different from traditional hash, image hash does not suffer from the sensitivity to minor degradations of original data because of its perceptual robustness. Such a property requires two images that are perceptually identical in human visual system (HVS) and are mapped to similar hash values. Obviously, the more robust a hash is, the less sensitive it is to large distortions upon the original images, which in turn inevitably incurs another problem that distinct images may be misclassified to the same group. Hence, tradeoff between robustness and anticollision of distinct images is of great concern. Additionally, by incorporating the pseudorandomization techniques, a hash is hardly obtained by unauthorized adversaries without the secret key. Therefore, the unpredictability encrypts the image hash and guarantees its security against illegal access.
Behaving as a secure tag for image data, image hashing facilitates significant developments in many areas such as image and video watermarking . It is worth mentioning that different applications may impose different requirements in a hashing design. For the purpose of image authentication, it is required that minor unmalicious modifications which do not alter the content of the data should preserve the authenticity of the data [4, 5]. The robustness of image hash assures its capability to authenticate the content by ignoring the effect of minor unmalicious modifications on the original data. For the management of large image databases , image hashing allows efficient media indexing, identification, and retrieval by avoiding exhaustively searching through all the entries, thus reducing computational complexity of similarity measurements. Moreover, specific hashing designed based on some specific features of image data, such as color, edges, and other information, obviously contributes to the content-based image retrieval (CBIR) system  at the semantic level. In this paper, we are particularly interested in image identification and explore the application of image hashing in this direction.
Although there exist various frameworks to design robust and secure hashes [8–10], a hashing scheme generally consists of two aspects: one is feature extraction and the other is pseudorandomization technique. Most hashing schemes combine both aspects to generate an intermediate hash as the first step and then incorporate a compression operation in postprocessing to generate the final hash [1, 10, 11]. Obviously, the robustness and security, two principal properties of hashing, lie in the first step. In order to resist routine unmalicious degradations (e.g., noising, compression) and other malicious attacks (e.g., cropping, rotation), the more invariant features are extracted, the more robust a hash scheme is. However, using features directly makes the scheme susceptible to forgery attacks. Therefore, pseudorandomization techniques should be employed in the hash schemes to assure the security.
Aiming at resisting both routine unmalicious degradations and malicious attacks, various approaches have been proposed in literatures for constructing image hashes, although there is no universallyoptimal hashing approach that is robust against all types of attacks. For example, Radon Soft Hash algorithm (RASH)  shows robustness against geometric transformation and some image processing attacks using Radon transform and principle component analysis (PCA). Swaminathan's hashing scheme  incorporates pseudorandomization into Fourier-Mellin transform to achieve better robustness to geometric operations. However, it suffers from some classical signal processing operations such as noising. It was also proposed in  to generate the hash by detecting invariant feature points, though the expensive searching and removal of feature points by malicious attacks such as cropping and blurring limit its performance in practice. Other content-preserving features based on statistics  and spectrum information [2, 13] have also contributed to the development of image hashing and enlightened some novel directions.
Recently, several image hashing schemes based on dimension reduction have been developed and reported to outperform previous techniques. For instance, using low-rank matrix approximations obtained via singular value decomposition (SVD) for hashing was explored in . Its robustness against geometric attacks motivated other solutions in this direction. Monga introduced another dimension reduction technique, called nonnegative matrix factorization (NMF) , into their new hashing algorithm . The major benefit of NMF hashing is the structure of the basis resulting from its nonnegative constraints, which lead to a parts-based representation. In contrast to the global representation obtained by SVD, the non-negativity constraints result in a basis of interesting local features . Based on the results in , the NMF hashing possesses excellent robustness under a large class of perceptually insignificant attacks, while it significantly reduces misclassification for perceptually distinct images. Note that, for simplicity, we sometimes refer the NMF-NMF-SQ hashing scheme, which was shown to provide the best performance among NMF-based hashing schemes investigated in , simply as NMF hashing in this paper.
Inspired by the potential of dimension reduction techniques for image hashing, we introduced Fast Johnson-Lindenstrauss transform (FJLT), a dimension reduction technique recently proposed in , into our new robust and secure image hashing algorithm . FJLT shares the low-distortion characteristics of a random projection process but requires a lower computational complexity. It is also more suitable for practical implementation because of its high computational efficiency and security due to the random projection. Since we mainly focus on invariant feature extraction and are interested in image identification applications, the FJLT hashing seems promising because of its robustness to a large class of minor degradations and malicious attacks. Considering the fact that NMF hashing was reported to significantly outperform other existing hashing approaches , we use it as the comparison base for the proposed FJLT hashing. Our preliminary experimental results in  showed that FJLT hashing provides competitive or even better identification performance under various attacks such as additive noise, blurring, and JPEG compression. Moreover, its lower computational cost also makes it attractive.
However, geometric attacks such as rotation could essentially tamper the original images and thus prevent the accurate identification if we apply the hashing algorithms directly on the manipulated image. Even for the FJLT hashing, it still suffers from the rotation attacks with low identification accuracy. To address this concern, motivated by the work [8, 20], we plan to apply the Fourier-Mellin transform (FMT) on the original images first to make them invariant to geometric transform. Our later experimental results show that, under rotation attacks, the FJLT hashing combined with the proposed FMT preprocessing yields a better identification performance than that of the direct FJLT hashing.
Considering that a specific feature descriptor may be more robust against certain types of attacks, it is desirable to take advantage of different features together to enhance the overall robustness of hashing. Therefore we further propose an extended concept, namely, content-based fingerprinting, to represent a combined, superior hashing approach based on different robust feature descriptors. Similar to the idea of having the unique fingerprint for each human being, we aim at combining invariant characteristics of each feature to construct an exclusive (unique) identifier for each image. Under the framework of content-based fingerprinting, the inputs to the hashing algorithms are not restricted to the original images only, but can also be extendable to include various robust features extracted from the images, such as color, texture, and shape. An efficient joint decision scheme is important for such a combinational framework and significantly affects the identification accuracy. Our experimental results demonstrate that the content-based fingerprinting using a simple joint decision scheme can provide a better performance than the traditional onefold hashing approach. More sophisticated joint decision-making schemes are worth further being investigated in the future.
The rest of this paper is organized as follows. We first introduce the background and theoretic details about FJLT in Section 2. We then describe the proposed hashing algorithm based on random sampling and FJLT in Section 3. In Section 4, we propose the RI-FJLT hashing by combining the Fourier-Mellin transform and FJLT hashing to achieve better geometric robustness. To combine the advantages of both FJLT and RI-FJLT hashing algorithms, a general framework and experimental results of content-based fingerprinting using FJLT hashing for multimedia identification are presented in Section 5. The analytical and experimental results are exhibited in Section 6 to demonstrate the superior performance of the proposed schemes. The conclusion and suggestions for future work are given in Section 7.
2. Theoretical Background
Based on the literature review in Section 1, the current task of image hashing is to extract more robust features to guarantee the identification accuracy under manifold manipulations (e.g., noising, blurring, compression, etc.) and incorporate the pseudorandomization techniques into the feature extraction to enhance the security of the hash generation. According to the information theory , if we consider the original image as a source signal, similar to a transmission channel in communication, the feature extraction process will make the loss of information inevitable. Therefore, how to efficiently extract the robust features as lossless as possible is a key issue that the hashing algorithms such as SVD , NMF , and our FJLT hashing want to tackle.
2.1. Fast Johnson-Lindenstrauss Transform
The Johnson-Lindenstrauss (JL) theorem has found numerous applications, including searching for approximate nearest neighbors (ANNs)  and dimension reduction in database, and so forth, by the JL lemma , points in Euclidean space can be projected from the original dimensions down to lower dimensions while just incurring a distortion of at most in their pairwise distances, where . Based on the JL theorem, Alion and Chazelle  proposed a new low-distortion embedding of into , called Fast Johnson-Lindenstrauss transform (FJLT). FJLT is based on preconditioning of a sparse projection matrix with a randomized Fourier transform. Note that we will only consider the case because our hash is measured by the norm. For the case, interested readers please refer to .
Briefly speaking, FJLT is a random embedding, denoted as , that can be obtained as a product of three real-valued matrices:
where the matrices and are random and is deterministic .
(i) is a -by- matrix whose elements are drawn independently according to the following distribution, where means a Normal distribution with zero-mean and variance ,
for a large enough constant .
(ii) is a -by- normalized Hadamard matrix with the elements as
where is the dot-product of the -bit vectors of expressed in binary.
(iii) is a -by- diagonal matrix, where each diagonal element is drawn independently from with probability 0.5.
Therefore, is a -by- matrix, where is the original dimension number of the data and is the lower dimension number, which is set to be . Here, is the number of data points, is the distortion rate, and is a constant. Given any data point from a -dimension space, it is intuitively mapped to the data point at a lower -dimension space by the FJLT and the distortion of their pairwise distances could be illustrated by Johnson-Lindenstrauss lemma .
2.2. The Fast Johnson-Lindenstrauss Lemma
Fix any set of vectors in , , and let . With probability at least , the following two events occur.
() For all ,
() The mapping requires
Proofs of the previous theorems can be found in . Note that the probability of being successful (at least ) arises from the random projection and could be amplified to for any , if we repeat the construction times . Since the random projection is actually a pseudorandom process determined by a secret key in our case, most of the keys (at least ) are satisfied with the distortion bound described in FJLT lemma and could be used in our hashing algorithm. Hence, the FJLT will make our scheme widely applicable for most of the keys and suitable to be applied in practice.
3. Image Hashing via FJLT
Motivated by the hashing approaches based on SVD  and NMF , we believe that dimension reduction is a significantly important way to capture the essential features that are invariant under many image processing attacks. For FJLT, three benefits facilitate its application in hashing. First, FJLT is a random projection, enhancing the security of the hashing scheme. Second, FJLT's low distortion guarantees its robustness to most routine degradations and malicious attacks. The last one is its low computation cost when implemented in practice. Hence, we propose to use FJLT for our new hashing algorithm. Given an image, the proposed hashing scheme consists of three steps: random sampling, dimension reduction by FJLT, and ordered random weighting. Due to our purpose, we are only interested in feature extraction and randomization. The hash generated by FJLT is just an intermediate hash. For readers who are interested in generating the final hash by compression step, as in the frameworks[8, 9], they are suggested to refer [1, 11] for details.
3.1. Random Sampling
Given an original color image, we first convert it to a gray image and pseudorandomly select subimages depending on the secret key and get , for . Each is a vector with length by concatenating the columns of the corresponding subimage. Then we construct our original feature as.
The advantage of forming such a feature is that we can capture the global information in the matrix and local information in each component . Even if we lose some portions of the original image under geometric attacks such as cropping, it will only affect one or a few components in our matrix and have no significant influence on the global information. However, the matrix with the high dimension (e.g., , when ) is too large to store and match, which motivates us to employ dimension reduction techniques.
3.2. Dimension Reduction by FJLT
Based on the theorems in Section 2, FJLT is able to capture the essential features of the original data in a lower-dimensional space with minor distortion, if the factor is close to 0. Recall the construction , our work is to map the matrix from a high-dimensional space to a lower-dimensional space with minor distortion. We first get the three real-valued matrices , , and in our case, which is , where is deterministic but and are pseudorandomly dependent on the secret key. The lower dimension is set to be and is a constant. Then we can get our intermediate hash ( ) as
Here, the advantage of FJLT is that we can determine the lower dimension by adjusting the number of data points, which is the number of image blocks by random sampling in our case, and the distortion rate . This provides us with a good chance to get a better identification performance. However, the smaller is, the larger is. Hence we need to make a tradeoff between and in a real implementation.
3.3. Ordered Random Weighting
Although the original feature set has been mapped to a lower-dimensional space with a small distortion, the size of intermediate hash can still be large. For instance, if we set , and , the size of will be -by- . To address this issue, similar to the NMF-NMF-SQ hashing in , we can introduce the pseudorandom weight vectors with drawn from the uniform distribution by the secret key, and we can calculate the final secure hash as
where is the th column in , and is the inner product of the vectors and . Hence, the final hash is obtained as a vector with length for each image, which is compact and secure. However, the weight vector drawn from could diminish the distance between the hash components and from two images and degrade the identification accuracy later. Here we describe a simple example to explain this effect. Suppose we have two vectors and , the Euclidean distance is 9. In the first case, if we assign the weight vector to and , after the inner product (9), the hash values of and will be 1.9 and 1, respectively. Obviously, the distance between and is significantly shortened. However, if we assign the weight to and in the second case, after the inner product (9), the hash values of and will be 9.1 and 1, respectively. The distance between and is still 8.1. We would like to maintain the distinction of two vectors and avoid the effect of an inappropriate weight vector as the first case.
3.4. Identification and Evaluation
3.4.1. Identification Process
Let be the set of original images in the tested database and define a space as the set of corresponding hash vectors. We use Euclidean distance as the performance metric to measure the discriminating capability between two hash vectors, defined as
where means the corresponding hash vector with length of the image . Given a tested image , we first calculate its hash and then obtain its distances to each original image in the hash space . Intuitively, the query image is identified as the th original images which yields the minimum corresponding distance, expressed as
The simple identification process described above can be considered as a special case of the -nearest-neighbor classification approach with . Here is set as 1 since we only have one copy of each original image in the current database. For a more general case, if we have multiple copies of each original image with no distortion or with only slight distortions, we could adopt the -nearest neighbor (KNN) algorithm for image identification in our problem.
3.4.2. Receiver Operating Characteristics Analysis
Except investigating identification accuracy, we also study the receiver operating characteristics (ROC) curve  to visualize the performance of different hashing approaches, including NMF-NMF-SQ hashing, FJLT hashing, and Content-based fingerprinting proposed later. The ROC curve depicts the relative tradeoffs between benefits and cost of the identification and is an effective way to compare the performances of different hashing approaches.
To obtain ROC curves to analyze the hashing algorithms, we may define the probability of true identification and probability of false alarm as
where is the identification threshold. The images and are two distinct original images and the images and are manipulated versions of the image and , respectively. Ideally, we hope that the hashes of the original image and its manipulated version should be similar and thus be identified accurately, while the distinct images and should have different hashes. In other words, given a certain threshold , an efficient hashing should provide a higher with a lower simultaneously. Consequently, when we obtain all the distances between manipulated images and original images, we could generate a ROC curve by sweeping the threshold from the minimum value to the maximum value, and further compare the performances of different hashing approaches.
4. Rotation Invariant FJLT Hashing
Although the Fast Johnson-Lindenstrauss transform has been shown to be successful in the hashing in our previous preliminary work , the FJLT hashing can still be vulnerable to rotation attacks. Based on the hashing scheme described in Section 3, random sampling can be an effective approach to reduce the distortion introduced by cropping, and scaling attack can be efficiently tackled by upsampling and downsampling in the preprocessing. However, to successfully handle the rotation attacks, we need to introduce other geometrically invariant transform to improve the performance of the original FJLT hashing.
4.1. Fourier-Mellin Transform
The Fourier-Mellin transform (FMT) is a useful mathematical tool for image recognition and registration, because its resulting spectrum is invariant to rotation, translation, and scaling [8, 20]. Let denote a gray-level image defined over a compact set of , the standard FMT of in polar coordinates (log-polar coordinates) is given by
If we make , (13) is clearly a Fourier transform like
Therefore, the FMT could be divided into three steps, which result in the invariance to geometric attacks.
(i)Fourier Transform. It converts the translation of original image in spatial domain into the offset of angle in spectrum domain. The magnitude is translation invariant.
(ii)Cartesian to Log-Polar Coordinates. It converts the scaling and rotation in Cartesian coordinates into the vertical and horizontal offsets in Log-Polar Coordinates.
(iii)Mellin Transform. It is another Fourier transform in Log-Polar coordinates and converts the vertical and horizontal offsets into the offsets of angles in spectrum domain. The final magnitude is invariant to translation, rotation, and scaling.
However, the inherent drawback of the Fourier transform makes FMT only robust to geometric transform, but vulnerable to many other classical signal processing distortions such as cropping and noising. As we know, when converting an image into the spectrum domain by 2D Fourier transform, each coefficient is contributed by all the pixels of the image. It means that the Fourier coefficients are dependent on the global information of the image in the spatial domain. Therefore, the features extracted by Fourier-Mellin transform are sensitive to certain attacks such as noising and cropping, because the global information is no longer maintained. To overcome this problem, we have modified the FMT implementation in our proposed rotation-invariant FJLT (RI-FJLT) hashing.
4.2. RI-FJLT Hashing
The invariance of FMT to geometric attacks such as rotation and scaling has been widely applied in image hashing [3, 8] and watermarking [20, 24]. It also motivates us to address the deficiency of FJLT hashing by incorporating FMT. Here, we propose the rotation-invariant FJLT hashing by introducing FMT into the FJLT hashing. Specially, the proposed rotation-invariant FJLT hashing (RI-FJLT) consists of three steps.
Applying Mellin transform (Fourier transform under Log-Polar coordinates) to the converted image and return the magnitude feature image.
Applying FJLT hashing in Section 3 to the magnitude feature image derived in Step 2.
For the conversion in Step 1, since the pixels in Cartesian coordinates are not able to be one-to-one mapped to pixels in the Log-Polar coordinates space, some value interpolation approaches are needed. We have investigated three different interpolation approaches for the proposed RI-FJLT hashing, including nearest neighbor, bilinear and bicubic interpolations, and found that the bilinear is superior to others. Therefore we only report the results under bilinear interpolation here. Note that we abandon the first step of FMT in RI-FJLT hashing, because we only focus on rotation attacks (other translations are considered as cropping) and it is helpful to reduce the influence of noising attacks by removing the Fourier transform step. The performance will be illustrated in Section 6. However, since Step 2 can inevitably be affected by attacks such as noising, some preprocessing such as median filtering can help improve the final identification performance.
5. Content-Based Fingerprinting
5.1. Concept and Framework
Specially, each vertical arrow in Figure 5 represents an independent hashing generation procedure, which consists of robust feature extraction and intermediate hash generation proposed by [8, 10]. Because it is the combination of various hash descriptors, the content-based fingerprinting can be considered as an extension and evolution of image hashing and thus offers much more freedom to accommodate different robust features (color, shape, texture, salient points, etc., ) and design efficient hashing algorithms to successfully against different types of attacks and distortions. Similar to the idea of finding one-to-one relationships between the fingerprints and an individual human being, the goal of content-based fingerprinting is to generate an exclusive digital signature, which is able to uniquely identify the corresponding media data no matter which content-preserving manipulation or attack is taken on.
Compared with the traditional image hashing concept, the superiority of content-based fingerprint concept lies in its potential high discriminating capability, better robustness, and multilayer security arising from the combination of various robust feature descriptors and a joint decision-making process. Same as in any information fusion processes, theoretically the discrimination capability of the content-based fingerprinting with effective joint decision-making scheme should outperform a single image hashing. Since the content-based fingerprint consists of several hash vectors, which are generated based on various robust features and different secret keys, it is argued that the framework of content-based fingerprinting results in a better robustness and multilayer security when an efficient joint decision-making is available. However, combining multiple image hashes approaches requires additional computation cost for the generation of content-based fingerprinting. The tradeoff between computation cost and performance is a concern with great importance in practice.
5.2. A Simple Content-Based Fingerprinting Approach
From the experimental results in Section 6, we note that FJLT hashing is robust to most types of the tested distortions and attacks except for rotation attacks and that RI-FJLT hashing provides a significantly better performance for rotation attacks at the cost of the degraded performances under other types of attacks. Recall an important fact that it is relatively easy to find a robust feature to resist one specific type of distortion; however it is very difficult, if not impossible, to find a feature which is uniformly robust to against all types of distortions and attacks. Any desire to generate an exclusive signature for the image by a single image hashing approach is infeasible. Here we plan to demonstrate the advantages of the concept of content-based fingerprinting by combining the proposed FJLT hashing and RI-FJLT hashing. The major components of the content-based fingerprinting framework include hash generations and the joint decision-making process which should take advantage of the combinations of the hashes to achieve a superior identification decision-making. Regarding the joint decision-making, there are many approaches in machine learning  that can be useful. Here we only present a simple decision-making process in rank level  to demonstrate the superiority of content-based fingerprinting.
Given an image with certain distortion, we, respectively, generate the hash vectors and by FJLT and RI-FJLT hashing. Suppose that the hash values of original images are and generated by FJLT and RI-FJLT hashing, respectively. We denote as the confidence measure that we identify image as image when applying the FJLT hashing. Similarly, is denoted for that of the RI-FJLT hashing. Here, we simply define
where and are preselected weights in the case of FJLT and RI-FJLT hashing, respectively, and means the Euclidean norm. Considering the poor performances of RI-FJLT hashing under many other types of attacks except for rotation ones, we intuitively introduce a weight , where , to the original confidence measures of FJLT and RI-FJLT hashing to decrease the possible negative influence of RI-FJLT hashing and maintain the advantages of both FJLT and RI-FJLT hashing in the proposed content-based fingerprinting under different attacks.
Regarding the identification decision making, given a tested image , we calculate all the confidence measures and over the image database of by using FJLT and RI-FJLT hashing, and make the identification decision correspondingly by selecting the highest one among and . Note that if a confidence measure is negative, it means that the image is outside the confidence interval of the image and the confidence measure is assigned to be zero.
6. Analytical and Experimental Results
6.1. Database and Content-Preserving Manipulations
Content-preserving manipulations and parameter settings.
Salt and Pepper noise
Filter size: , Sigma
5%, 10%, 20%, 25%, 30%, 35%
25%, 50%, 75%, 150%, 200%
6.2. Identification Results and ROC Analysis
Our preliminary study  on a small database showed that FJLT hashing provides nearly perfect identification accuracy for the standard test images such as Baboon, Lena, and Peppers. Here we will measure the FJLT hashing and the new proposed RI-FJLT hashing on the new database, which consists of 1000 nature images from ten categories. Ideally, to be robust to all routine degradations and malicious attacks, no matter what content-preserving manipulation is done, the image with any distortion should still be correctly classified into the corresponding original image.
It is worth mentioning that all the pseudorandomizations of NMF-NMF-SQ hashing, FJLT hashing, and content-based fingerprinting are dependent on the same secret key in our experiment. As discussed in , the secret keys, more precisely the key-based randomizations, play important roles on both increasing the security (i.e., making the hash unpredictable) and enhancing scalability (i.e., keeping the collision ability from distinct images low and thus yielding a better identification performance) of the hashing algorithm. Therefore, the identification accuracy of a hashing algorithm is determined simultaneously by both the dimension reduction techniques (e.g., FJLT and NMF) and the secret keys. As shown in NMF hashing in , if we generate hashes of different images with varied secret keys, the identification performance can be further improved significantly because the secret key boosts up the cardinality of the probability space and brings down the probability of false alarm. In this paper, because we mainly focus on examining the identification capacity of hashing schemes themselves rather than the effects of secret keys, to minimize the effects of the factor of the secret keys, we use the same key in generating hash vectors for different images.
6.2.1. Results of FJLT Hashing
Identification accuracy for manipulated images by NMF-NMF-SQ (NMF) hashing, FJLT hashing, and content-Based fingerprinting (CBF) based on FJLT and RI-FJLT hashing.
Salt and Pepper noise
Parameter setting in the FJLT hashing algorithm.
Size of the subimage
Length of the hash vector
Parameters of FJLT
Here we try to give some intuitive explanations regarding the observed performances of the two hashing algorithms. In NMF hashing, the dimension reduction technique is based on the approximative nonnegative matrix factorization, which factorizes the image matrix into two lower rank matrices. However, the problem of choosing a low rank (e.g., , in the NMF hashing) is of great importance, though it is observed to be sensitive to the data. While for FJLT hashing, the mapping is obtained by a coefficients matrix and a subimage is treated as a point in a high-dimensional space (in our case, the dimension is ). One advantage of FJLT hashing is that minor modifications in the content will not affect the integrity of the global information, which results in a better performance. However, as illustrated in Table 2 and the ROC curve in Figure 7(d), both FJLT hashing and NMF hashing provide poor performances under rotation attacks, and we shall investigate this problem further.
6.2.2. Results of RI-FJLT Hashing
In Table 2, we note that one drawback of FJLT hashing is its vulnerability to rotation attacks. Especially, as shown by an example in Figure 4, for a large rotation degree of 45, FJLT hashing failed to identify the image content. Here we apply the RI-FJLT hashing approach presented in Section 6 to overcome this drawback.
Identification accuracy under rotation attacks by FJLT and RI-FJLT.
Although the invariance of Fourier-Mellin transform benefits the FJLT hashing with the robustness to rotation attacks, such robustness to rotation comes at the cost of degraded identification accuracy for other types of manipulations and attacks. We have intuitively discussed the reasons for this observation in Section 4. We argue that it may not be feasible to be robustly against various attacks by only depending on single feature descriptor. This observation motivates us to look for an alternative solution that is the content-based fingerprinting we proposed in Section 5 to tackle this problem.
6.2.3. Results of Content-Based Fingerprinting
Since FJLT hashing is demonstrated to be robust against a large class of distortions except for rotation attacks and RI-FJLT hashing achieves superior performance under rotation attacks at the cost of sensitivity to other manipulations, it accounts for the fact that it is very difficult to design a globally optimal hashing approach that could handle all of the distortions and manipulations. Hence, we combine FJLT hashing and RI-FJLT hashing following the framework of content-based fingerprinting proposed in Section 5 and test its performance on the database described in Section 6.1. Considering the poor performance of RI-FJLT hashing on other manipulations, we need to introduce an elaborate weight shown in Section 5.2 to the confidence measure of RI-FJLT hashing to get rid of its negative influence and try to maintain the advantages of both FJLT and RI-FJLT hashing in the proposed content-based fingerprinting. Based on our preliminary study, we set to keep the advantages of FJLT hashing and find that a good weight could be drawn from the interval range . We set in our implementation and exhibit the results in Table 2.
To have a fair comparison between different approaches, though we combine the FJLT hashing and the RI-FJLT hashing in the content-based fingerprinting, the length of the overall fingerprint vector is still chosen as 40 (with 20 components from the FJLT hashing and the left 20 from the RI-FJLT hashing), which is the same as that of the FJLT hashing and the NMF hashing. It is clear that the simple joint decisionmaking complements the drawback of FJLT hashing under rotation attacks by incorporating the RI-FJLT hashing into the proposed content-based fingerprinting. The ROC curves for FJLT hashing, NMF hashing, and the proposed content-based fingerprinting under rotation attacks are shown in Figure 7(d). Obviously, among the three approaches, the content-based fingerprinting yields the highest true positive rates when the same false positive rates are considered. The ROC curves of the content-based fingerprinting approach under other types of attacks are also illustrated in Figure 7. We note that the robustness of content-based fingerprinting to additive noise, cropping, and Gamma correction slightly degrades, as shown in Figure 7. One possible explanation could be that the current simple decision-making process is not the theoretically optimal one that could eliminate the negative effect of RI-FJLT hashing under these attacks. However, the overall performance of content-based fingerprinting as illustrated by the ROC curve in Figure 6 demonstrates that it is superior and more flexible than a single hashing approach, because the selection of features and secure hashes can be adapted to address different practical application concerns. Therefore, the proposed content-based fingerprinting can be a promising extension and evolution of traditional image hashing.
6.3. Unpredictability Analysis
Except for the robustness against different types of attacks, the security in terms of unpredictability that arises from the key-dependent randomization is another important property of hashing and the proposed content-based fingerprinting. Here we mainly focus on the unpredictability analysis of FJLT hashing, because the unpredictability of the RI-FJLT hashing and the content-based fingerprinting proposed arise from the FJLT hashing. Higher amount of the randomness in the hash values makes it harder for the adversary to estimate and forge the hash without knowing the secret keys. Since it is believed that a high differential entropy is a necessary property of secure image hashes, we evaluate the security in terms of unpredictability of FJLT hashing by quantifying the differential entropy of the FJLT hash vector, as proposed in . The differential entropy of a continuous random variable is given by
where means the determinant of the covariance matrix of the hash vector, and means the length of the FJLT hash vector.
From Figure 8(b) where an example of the covariance matrix of the FJLT hash vector is shown, we can see that the covariance matrix is approximately a diagonal matrix, meaning that the components are approximately statistically independent. Therefore, can be approximately estimated as
where means the variance of the component in the FJLT hash vector. Since from information theory, the differential entropy of a random vector is maximized when follows a multivariate normal distribution , we argue that the proposed FJLT hashing is highly secure (unpredictable) as it approximately follows . We note that NMF-NMF-SQ hashing also was shown to approximately follow a joint Gaussian distribution and a similar statement in terms of differential entropy was given in . Hence, we state that the proposed FJLT hash is comparably as secure as NMF hashing, which was shown to be presumably more secure than previously proposed schemes that are based on random rectangles alone .
However, the security of image hashing does not only lie on a higher differential entropy, which is only one aspect of a secure image hashing [8, 16], but also includes other factors such as key diversity and prior knowledge possessed by adversaries. Therefore, how to comprehensively evaluate the security of image hashing is still an open question. Interested readers could refer to the literatures [8, 27] regarding the security analysis issues.
6.4. Computational Complexity
NMF. In , the computational complexity of NMF-NMF-SQ hashing has been given as follows. It does a rank NMF on matrices and then a rank approximation from the resulting matrix in . At last, pseudorandom numbers are incorporated in the NMF-NMF vector of length , and the total computation cost is
FJLT. Based on the analysis in , given a , the computation cost of FJLT on is calculated as follows. Computing requires time and requires . For computing , it takes , where the is the number of nonzeros in , we know that the satisfies the Binomial distribution , therefore we take the mean value of as that equals , where is . Then, take the random weight incorporation into account, we have the total computation cost of the FJLT hashing as ( in our case)
RI-FJLT. Except for the cost of FJLT hashing, we need to take the bilinear interpolation that requires and Fourier transform that takes by FFT into account. Consequently, the cost of RI-FJLT is
Computational time costs for Lena with by FJLT, RI-FJLT and NMF-NMF-SQ hashing algorithms.
7. Discussions and Conclusion
In this paper, we have introduced a new dimension reduction technique—FJLT, and applied it to develop new image hashing algorithms. Based on our experimental results, it is noted that the FJLT-based hashing is robust to a large class of routine distortions and malicious manipulations. Compared with the NMF-based approach, the proposed FJLT hashing can achieve comparable, sometimes better, performances than that of NMF, while requiring less computational cost. The random projection and low distortion properties of FJLT make it more suitable for hashing in practice than the NMF approach. Further, we have incorporated Fourier-Mellin transform to complement the deficiency of FJLT hashing under rotation attacks. The experimental results confirm the fact that generating a hash descriptor based on a ceratin type of features to resist all types of attacks is highly unlikely in practice. However, for a particular type of distortion, it is feasible to find a specific feature to tackle it and obtain good performance. These observations motivate us to propose the concept of content-based fingerprinting as an extension of image hashing and demonstrate the superiority of combining different features and hashing algorithms.
We note that the content-based fingerprinting approach by using FJLT and RI-FJLT still suffers from some distortions, such as Gaussian noise and Gamma correction. One solution is to further find other features that are robust to these attacks/manipulations and incorporate them into the proposed scheme to enhance the performance. Future work will include how to incorporate other robust features (such as the popular SIFT-based features) and secure hashing algorithms to optimize the content-based fingerprinting framework and at the same time explore efficient hierarchical decision-making schemes for identification.
Furthermore, we plan to explore the variations of the current FJLT hashing. Similar to the NMF-based hashing approach (referred as NMF-NMF-SQ hashing in ) where the hash is based on a two-stage application of NMF, we can modify the proposed FJLT hashing into a two-stage FJLT-based hashing approach by introducing a second stage of FJLT as follows. Treat the intermediate hash as a vector with length , and then reapply FJLT to obtain a representation of the vector with further dimension reduction. Compared with our current one-stage FJLT-based hashing, the length of intermediate hash could be further shortened by the second FJLT and the security would be enhanced in the two-stage FJLT hashing. However, the robustness of a two-stage FJLT-based hashing under attacks such as cropping may degrade, since now each component in the modified hash vector is contributed by all the subimages by random sampling. Therefore, the distortion of local information in one subimage could affect the whole hash vector rather than a couple of hash components. The computation cost can also be a concern. We will investigate these issues in the future work.
Another concern that is of great importance in practice but is rarely discussed in the context of image hashing is automation. Automatic estimation/choice of design parameters removes the subjectivity from the design procedure and can yield better performances. For instance, algorithms for automating the design process of image watermarking have already been implemented in the literature [28–30]. However, to our knowledge, this automated solution has not yet been explored in the context of image hashing. Our preliminary study in  demonstrated that using a genetic algorithm (GA) for automatic estimation of parameters of the FJLT hashing using could improve the identification performance. However, choosing the appropriate fitness function is challenging in automated image hash. We plan to investigate different fitness functions and how the GA algorithm can incorporate other factors (such as keys) and other constraints (such as the hash length).
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