Efficient video hashing based on low‐rank frames

IET Image ProcessingVolume 16, Issue 2 p. 344-355 ORIGINAL RESEARCH PAPEROpen Access Efficient video hashing based on low-rank frames Zhenhai Chen, Zhenhai Chen orcid.org/0000-0001-9414-6278 Guangxi Key Lab of Multi-Source Information Mining & Security, and School of Computer Science and Engineering, Guangxi Normal University, Guilin, ChinaSearch for more papers by this authorZhenjun Tang, Corresponding Author Zhenjun Tang tangzj230@163.com orcid.org/0000-0003-3664-1363 Guangxi Key Lab of Multi-Source Information Mining & Security, and School of Computer Science and Engineering, Guangxi Normal University, Guilin, China Correspondence Zhenjun Tang, Guangxi Key Lab of Multi-Source Information Mining, and Security, and School of Computer Science and Engineering, Guangxi Normal University, Guilin 541004, China. Email: tangzj230@163.comSearch for more papers by this authorXinpeng Zhang, Xinpeng Zhang School of Computer Science, Fudan University, Shanghai, ChinaSearch for more papers by this authorRonghai Sun, Ronghai Sun Guangxi Key Lab of Multi-Source Information Mining & Security, and School of Computer Science and Engineering, Guangxi Normal University, Guilin, ChinaSearch for more papers by this authorXianquan Zhang, Xianquan Zhang Guangxi Key Lab of Multi-Source Information Mining & Security, and School of Computer Science and Engineering, Guangxi Normal University, Guilin, ChinaSearch for more papers by this author Zhenhai Chen, Zhenhai Chen orcid.org/0000-0001-9414-6278 Guangxi Key Lab of Multi-Source Information Mining & Security, and School of Computer Science and Engineering, Guangxi Normal University, Guilin, ChinaSearch for more papers by this authorZhenjun Tang, Corresponding Author Zhenjun Tang tangzj230@163.com orcid.org/0000-0003-3664-1363 Guangxi Key Lab of Multi-Source Information Mining & Security, and School of Computer Science and Engineering, Guangxi Normal University, Guilin, China Correspondence Zhenjun Tang, Guangxi Key Lab of Multi-Source Information Mining, and Security, and School of Computer Science and Engineering, Guangxi Normal University, Guilin 541004, China. Email: tangzj230@163.comSearch for more papers by this authorXinpeng Zhang, Xinpeng Zhang School of Computer Science, Fudan University, Shanghai, ChinaSearch for more papers by this authorRonghai Sun, Ronghai Sun Guangxi Key Lab of Multi-Source Information Mining & Security, and School of Computer Science and Engineering, Guangxi Normal University, Guilin, ChinaSearch for more papers by this authorXianquan Zhang, Xianquan Zhang Guangxi Key Lab of Multi-Source Information Mining & Security, and School of Computer Science and Engineering, Guangxi Normal University, Guilin, ChinaSearch for more papers by this author First published: 04 October 2021 https://doi.org/10.1049/ipr2.12351AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Video hashing is a useful technology for diverse video applications, such as digital watermarking, copy detection and content authentication. This paper proposes a novel efficient video hashing based on low-rank frames. A key contribution is the low-rank frame calculation using the low-rank approximation of singular value decomposition (SVD). As the large singular values of SVD are stable to digital operations, video hash extraction using low-rank frames can provide good robustness. Since most energy is contained within the large singular values, low-rank frames also contribute to discrimination. Moreover, two-dimensional discrete wavelet transform (DWT) is applied to every low-rank frame and the mean of low-frequency DWT coefficients is selected as a hash element. Since these coefficients can represent input data approximately, hash discrimination is thus ensured. Experiments with 16,850 videos are carried out to test performances of the proposed algorithm. The results show that the proposed algorithm outperforms some well-known video hashing algorithms in computational time and classification about discrimination and robustness. 1 INTRODUCTION Social networks (e.g. Facebook, TikTok, YouTube and Instagram) are very popular in the Internet era. Many people use their smartphones to capture images/videos and share them via the social networks. And some of them would like to download and edit their favourite images/videos shared by friends and redistribute these processed copies on the Internet. Consequently, there is a huge number of images/videos in cyberspace [1], and it is a challenge to detect similar copies from massive images/videos [2, 3]. To handle this problem, many researchers proposed to use hashing algorithms [4-9] to efficiently process multimedia data. Generally, there are two fundamental performances of video hashing [10-12]. The first fundamental performance is the robustness, which requires that video hashing algorithm should show robustness against digital video operations. This means that video hashing algorithm should correctly recognize those videos processed by normal operations. This performance can make the hashing suitable for video copy detection. The other fundamental performance is the discrimination which means that hashing algorithm should map different videos to different hashes. This performance ensures that video hashing algorithm can efficiently reduce videos falsely returned when searching similar videos from massive videos. It is clear that hashing algorithm is expected to reach good performances of discrimination and robustness. In the literature, researchers have proposed diverse video hashing algorithms for many applications [13-16], such as copy detection, digital watermarking, tampering detection, content authentication and video retrieval. Some typical algorithms related to our used techniques are introduced below. For example, De Roover et al. [10] built an effective hashing algorithm for still image by calculating discrete cosine transform (DCT) coefficients of the radial projection (RP) of image pixels and extended their image hashing to video hashing by extracting hashes of key frames. Weng and Preneel [17] calculated image hash based on higher-order cumulants and exploited the image hashing scheme to extract hash of every frame. This scheme has high computational cost and long hash length. Wang et al. [18] employed the visual attention model with cross-layer saliency optimization [19] to generate a visual hash and then combined the visual hash and spatio-temporal hash to make the final video hash. This scheme can resist median filtering and Gaussian noise. Li and Monga [11] proposed to calculate video hash by tensor-based subspace projections. Saikia [12] used 3D discrete wavelet transform (DWT) to extract features from group-of-frames and calculated video hash by compressing them via 1D DCT. Liu et al. [20] exploited the dynamic and static attention models to construct a temporally visual weighting method for calculating representative frame, which can improve robustness and discrimination of video hashing. Nie et al. [21] extended the ring partition of image [22, 23] to the spherical torus (ST) partition for video feature extraction, and learned hash code by non-negative matrix factorization (NMF). Sandeep et al. [24] randomly selected 3D sub-blocks from normalized video and derived hash from these sub-blocks by Tucker decomposition (TD). In another study, Sandeep et al. [25] extended the 2D RP in image [10] to 3D RP in video and established an improved hashing algorithm, which is better than the scheme [10] in resisting robust attacks. Recently, Rameshnath and Bora [26] exploited Achlioptas's random matrix (ARM) and temporal wavelet transform (TWT) to design video hashing. This algorithm is robust to the robust attacks of brightness modification, Gaussian blurring and watermark insertion. Khelifi and Bouridane [27] jointly used DCT and discrete sine transform (DST) to design a video hashing system. This hashing system can be applied to the applications of authentication and identification by using different similarity measures. Tang et al. [28] extracted video hash from feature matrices of dominant DCT coefficients by using NMF. This algorithm can resist MPEG-2 compression and random frame dropping. Chen et al. [29] decomposed every video frame by low-rank and sparse decomposition (LRSD) and exploited the means of low-frequency DWT coefficients to construct hash. The LRSD-DWT hashing reaches a high accuracy in classification. In another work, Tang et al. [30] proposed to construct a high-dimensional feature matrix using gradient features in the DWT domain, learned low-dimensional features from the high-dimensional matrix using multidimensional scaling, and quantified features via ordinal measures. This video hashing shows better classification performances than several existing algorithms [21, 26, 29]. Motivated by the success of deep learning in computer vision, some researchers [31, 32] exploited deep neural networks to design various video hashing algorithms. These algorithms can make good classification performance, but they have high costs in computation and storage of network parameters. Currently, most of the existing video hashing techniques do not make desirable classification about discrimination and robustness. Aiming at this issue, this paper presents a novel efficient video hashing algorithm based on low-rank frames, which jointly exploits SVD and DWT to calculate video hash. Different from the reported video hashing algorithms, the proposed algorithm has below contributions. A low-rank frame calculation is proposed based on the low-rank approximation of SVD using large singular values. Specifically, video is first divided into some video groups of frames. Each video group is used to construct a high dimensional matrix, SVD is then applied to the high dimensional matrix, and finally a low-rank frame for representing the video group is calculated by low-rank approximation of SVD using large singular values. As the large singular values of SVD are stable to digital operations, video hash extraction using low-rank frames can provide good robustness. In addition, the low-rank frames also contribute to discrimination since most energy is contained within the large singular values. Video hash is derived from the low-rank frames by using two-dimensional (2D) DWT. The 2D DWT is applied to every low-rank frame and the mean of low-frequency DWT coefficients is selected as a hash element. Since these coefficients can represent input data approximately, hash discrimination is thus ensured. Experiments with 16,850 videos are conducted to test performances of the proposed algorithm. Comparison results illustrate that the proposed algorithm is superior to some well-known video hashing algorithms in computational time and classification about discrimination and robustness. The remainder of this paper is structured as follows. The proposed algorithm is explained in Section 2. Sections 3 and 4 discuss the experimental results and performance comparison, respectively. Section 5 concludes this paper. 2 PROPOSED ALGORITHM The proposed algorithm consists of three steps. In the first step, an input video is processed to make a normalized video. In the second step, low-rank frames are then calculated based on low-rank approximation of SVD. Finally, the low-rank frames are compressed by 2D DWT. Figure 1 demonstrates the procedure of the proposed algorithm. Sections 2.1, 2.2, 2.1, 2.2 and 2.3 describe the details of the preprocessing, low-rank frame calculation and low-rank frame compression, respectively. Section 2.4 presents the pseudo-code of the proposed algorithm and Section 2.5 explains the used measure of video hashes. FIGURE 1Open in figure viewerPowerPoint Three steps of the proposed algorithm 2.1 Preprocessing The input RGB video is firstly mapped to the YCbCr space by Equation (1). Y C b C r = 65.481 128.553 24.966 − 37.797 − 74.203 112 112 − 93.786 − 18.214 R G B + 16 128 128 . (1)The luminance component Y is then selected as the video representation. Next, the resolution of each frame is resized to α × β by bilinear interpolation and the frame number is adjusted to γ by linear interpolation. The two resampling operations ensure that the proposed algorithm can resist the operations of frame manipulation, for example, frame addition, frame dropping and frame scaling. After the preprocessing, a normalized video sized α × β × γ is obtained. 2.2 Low-rank frame calculation Video partition is conducted by dividing the normalized video into some video groups of frames. Here the number of frames in each video group is selected as N. For simplicity, suppose that γ is an integral multiple of N. Thus, there are L = γ / N video groups in total. For each group, a low-rank frame is calculated by low-rank approximation of SVD using large singular values. The calculation involves the following steps. (1) High dimensional matrix construction. For each frame of a video group, its pixels are concatenated row by row to form a vector. Consequently, a high dimensional matrix is determined by Equation (2). G i = g i , 1 , g i , 2 , … , g i , N , (2)where g i , j is the vector of the j th frame of the i th video group, and G i is the constructed matrix of the i th video group ( i = 1 , 2 , … , L ) . (2) Singular value decomposition. SVD is applied to the high dimensional matrix of each video group. In general, the SVD definition [33, 34] of G i can be given by Equation (3). G i = U i S i V i T , (3)where S i is the real diagonal matrix with singular values arranged in descending order, U i and V i are the left and right singular vectors of G i , respectively. (3) Frame calculation with low-rank approximation. The first r largest singular values of S i are used to find the low-rank approximation of G i . This is done by Equation (4). G i ( r ) = U i ( r ) S i ( r ) V i ( r ) T , (4)where G i ( r ) is the low-rank approximation of G i , U i ( r ) is the matrix constructed by the first r columns of U i , V i ( r ) is the matrix constructed by the first r columns of V i , and S i ( r ) the diagonal matrix constructed by the first r singular values. Let G i ( r ) = [ g i , 1 ( r ) , g i , 2 ( r ) , … , g i , N ( r ) ] . Thus, the j th vector g i , j ( r ) is reshaped to the matrix F i , j ( r ) of size α × β by filling its element into the matrix row by row. Consequently, the low-rank frame R i of the i th video group can be calculated by Equation (5). R i ( k , l ) = 1 N ∑ j = 1 N F i , j ( r ) ( k , l ) , (5)where R i ( k , l ) and F i , j ( r ) ( k , l ) are the elements of R i and F i , j ( r ) in the k th row and l th column ( k = 1 , 2 , … , α ; l = 1 , 2 , … , β ) , respectively. Our considerations of selecting SVD for low-rank frame calculation are as follows. The large singular values of SVD are stable to digital operations [34]. Therefore, hash extraction using low-rank frames can provide good robustness of the proposed algorithm. Moreover, as most energy of the matrix is contained within the large singular values [34, 35], the low-rank frames are also discriminative. 2.3 Low-rank frame compression To conduct frame compression, 2D DWT is used here. Specifically, 2D DWT with one level is applied to the low-rank frame R i . Figure 2 is a diagram of one-level 2D DWT. Clearly, the results of one-level 2D DWT include four sub-bands, where the LL sub-band contains the approximation coefficients and other three sub-bands contain the detailed coefficients. In this work, the mean of the coefficients of LL sub-band is used to generate a hash element. Suppose that D i of size d α × d β represents those coefficients of the LL sub-band after one-level decomposition. Thus, d α = ⌈ α / 2 ⌉ and d β = ⌈ β / 2 ⌉ , where ⌈ . ⌉ is the operation of upward rounding. Consequently, the mean of D i can be computed by Equation (6). μ i = ∑ j = 1 d α ∑ k = 1 d β D i ( j , k ) d α × d β , (6)where D i ( j , k ) is the coefficient of D i in the j th row and k th column. The use of DWT can realize data compression and our selection of the LL sub-band is because its coefficients can approximately represent input data. FIGURE 2Open in figure viewerPowerPoint Block diagram of one-level 2D DWT Next, the floating-point number μ i is quantized to reduce storage cost by Equation (7). h ( i ) = μ i + 0.5 , (7)in which [ . ] represents the rounding operation. And finally, a hash sequence h with L integers is obtained by Equation (8). h = h ( 1 ) , h ( 2 ) , … , h ( L ) . (8)Clearly, the length of the video hash is equal to the number of video groups. 2.4 Pseudo-code of the proposed algorithm To make a better understanding of our detailed steps, the pseudo-code of the proposed algorithm is presented in Algorithm 1. ALGORITHM 1. The proposed algorithm Input: An input video, parameters: α, β, γ, N and r 1: If the input video is a color video, its Y component in the YCbCr space is taken for the representation. 2: Every frame is resized to α × β by bilinear interpolation. 3: Frame number is adjusted to γ by linear interpolation. 4: The normalized video sized α × β × γ is divided into L groups and a group is represented as G i ( i = 1 , 2 , … , L ) by Equation (2). 5: for i = 1 to i = L do 6: Apply SVD to G i and obtain its low-rank approximation G i ( r ) by Equation (4). 7: Calculate the low-rank frame R i by Equation (5). 8: 2D DWT is applied to R i and the mean of coefficients in the LL sub-band μ i is calculated by Equation (6). 9: Compute h ( i ) = [ μ i + 0.5 ] . 10: end for 11: Video hash is determined by h = [ h ( 1 ) , h ( 2 ) , … , h ( L ) ] . Output: A video hash h. 2.5 Similarity measure The Pearson correlation coefficient [22] is taken as the measure for evaluating similarity of our video hashes. Let h = [ h 1 ( 1 ) , h 1 ( 2 ) , … , h 1 ( L ) ] and h ′ = [ h ′ ( 1 ) , h ′ ( 2 ) , … , h ′ ( L ) ] represent two video hashes. Thus, the correlation coefficient is determined by Equation (9). S = ∑ i = 1 L h i − m 1 h ′ i − m 2 ∑ i = 1 L h i − m 1 2 ∑ i = 1 L h ′ i − m 2 2 + ε , (9)where ε is a very small positive constant close to zero which is used to avoid the division error when ∑ i = 1 L [ h ( i ) − m 1 ] 2 = 0 or ∑ i = 1 L [ h ′ ( i ) − m 2 ] 2 = 0 , and m 1 and m 2 are the means of h and h ′ , respectively. The interval of S is [ − 1 , 1 ] . A bigger S implies more similar videos of the input video hashes. In practice, a threshold T is used to classify videos. If S > T , the videos of input video hashes are considered as a similar pair. Otherwise, they are a pair of different videos. 3 EXPERIMENTAL RESULTS In the following sections, extensive experiments are conducted to test performances of the proposed algorithm. Specifically, the performances of robustness, discrimination, and hash length are discussed in the Sections 3.1, 3.2, 3.1, 3.2 and 3.3, respectively. Parameter selections about the rank of SVD and the frame number of a video group are discussed in the Sections 3.4 and 3.5, respectively. In these experiments, our parameter settings are as follows: α = 256 , β = 256 , γ = 256 , N = 16 and r = 2 . Thus, our video hash length is L = 16 integers. The used computer server is equipped with a 2.10 GHz Intel Xeon E5-2620 CPU, 256 GB RAM and a 24 GB graphics card, and the coding software is the MATLAB R2019b. 3.1 Robustness test The videos numbered from 1 to 150 in the open video database [36] are selected to generate similar videos, where some typical samples in the database are shown in Figure 3. These videos are in AVI format, their resolutions are all 384 × 288 , and their frame numbers range from 66 to 2751. In this experiment, the number of the used digital operations is 11 and different parameters are selected for each operation. Table 1 presents the detailed settings of the used operations. Clearly, the number of the total manipulations reaches 110. This means that there are 110 similar versions of every original video. And thus the total number of similar videos is 150 × 110 = 16500 . Therefore, the number of the used videos is 150 + 16500 = 16650 . To analyze the visual qualities between the original videos and their similar videos, the well-known metric of image quality assessment called SSIM [37] is taken. SSIM value ranges from − 1 to 1. A bigger SSIM means a better visual quality. For video quality assessment, SSIM is calculated frame by frame, and then the mean of all SSIM values is used for evaluation. As the frame scaling changes frame resolution and the frame addition/dropping alters frame number, similar videos are converted to the frame resolution/number equaling to those of their original videos by interpolation. Table 2 lists the average SSIM values under different operations. It is found that most SSIM values are high enough. The SSIM of additive white gaussian noise (AWGN) is much smaller than those of other operations. This is because it introduces more distortions to videos than other operations. Nevertheless, the attacked videos of AWGN are still similar with their original videos. TABLE 1. Detailed settings of the used video operations Operation Parameter Parameter value Number Brightness adjustment Magnitude − 30 , − 20 , − 10 , − 5 , 5, 10, 20, 30 8 Contrast adjustment Magnitude − 30 , − 20 , − 10 , − 5 , 5, 10, 20, 30 8 MPEG-2 compression Bit rate 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 10 MPEG-4 compression Compression quality 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 10 Frame rotation Angle − 90 ∘ , − 2 ∘ , − 1 . 5 ∘ , − 1 ∘ , − 0 . 5 ∘ , 0.5 ∘ , 1 ∘ , 1.5 ∘ , 2 ∘ , 90 ∘ 10 Frame scaling Ratio 0.5, 0.75, 0.9, 1.1, 1.5, 2 6 Frame translation Column number 2, 4, 6, 8, 10 5 Random frame swapping Pair of frames 1, 2, 3, 4, 5 5 Random frame addition Frame number 1, 2, 5, 10, 15, 20 6 Random frame dropping Frame number 1, 2, 5, 10, 15, 20 6 Additive white gaussian noise Signal noise ratio 1, 2, 3, 4, 5, 6 6 Salt and pepper noise Density 0.001, 0.002, 0.005, 0.007, 0.01, 0.02, 0.05, 0.07, 0.1, 0.2 10 Speckle noise Variance 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1 10 3 ×3 Gaussian low-pass filtering Standard deviation 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1 10 Total 110 TABLE 2. SSIM values under different operations Operation SSIM Brightness adjustment 0.95941 Contrast adjustment 0.98588 MPEG-2 compression 0.86706 MPEG-4 compression 0.89196 Frame rotation 0.57158 Frame scaling 0.97953 Frame translation 0.61027 Random frame swapping 0.94499 Random frame addition 0.72815 Random frame dropping 0.72460 Additive white gaussian noise 0.19020 Salt and pepper noise 0.66096 Speckle noise 0.57737 3 ×3 Gaussian low-pass filtering 0.93717 FIGURE 3Open in figure viewerPowerPoint Some videos in the database [36] Figure 4 presents the means of the correlation coefficients of 150 pairs of original videos and their similar videos under different operations with specific parameter, in which the x-axis represents the parameter of digital operation and the y-axis denotes the mean value of 150 correlation coefficients. From Figure 4, it can be seen that all means are bigger than 0.9, except several cases of random frame addition and random frame dropping. In addition, Table 3 lists the statistical results of correlation coefficients under different operations. It is found that 96.66% similar videos can be correctly recognized if the threshold is selected as 0.9. A high correct rate verifies good robustness of the proposed algorithm. TABLE 3. Statistical results under different operations Operation Max Min Mean Standard deviation Brightness adjustment 1 0.80665 0.99806 0.00971 Contrast adjustment 1 0.06259 0.97705 0.07771 MPEG-2 compression 0.99999 0.69472 0.99129 0.02843 MPEG-4 compression 1 0.59736 0.98664 0.04053 Frame rotation 1 0.99999 0.99999 0.00001 Frame scaling 1 0.99892 0.99999 0.00001 Frame translation 0.99999 0.68629 0.99267 0.02997 Random frame swapping 0.99999 0.68630 0.99267 0.02997 Random frame addition 1 0.08190 0.94800 0.09964 Random frame dropping 1 0.00707 0.94236 0.11066 Additive white gaussian noise 0.99997 0.87626 0.99539 0.01244 Salt and pepper noise 1 0.99717 0.99998 0.00014 Speckle noise 1 0.69317 0.99179 0.02789 3 ×3 Gaussian low-pass filtering 1 0.04517 0.98540 0.05969 FIGURE 4Open in figure viewerPowerPoint Means of correlation coefficients under different operations FIGURE 4 (continued)Open in figure viewerPowerPoint Means of correlation coefficients under different operations 3.2 Discrimination validation The videos numbered from 151 to 350 in the video database [36] are exploited to test discrimination of the proposed algorithm. Hash similarity between each pair of different videos is calculated and thus there are C 200 2 = 200 × ( 200 − 1 ) / 2 = 19900 correlation coefficients in total. Figure 5 demonstrates the distribution of correlation coefficients, where the y-axis denotes the frequency a correlation coefficient value, and the x-axis is the value. It is found the distribution approximately follows a normal distribution. If the threshold is selected as 0.9, 0.23% different videos are wrongly identified as similar ones. The low false detection rate verifies good discrimination of the proposed algorithm. FIGURE 5Open in figure viewerPowerPoint Correlation coefficient distribution of different hashes 3.3 Hash length discussion To analyze our storage cost in binary form, all videos in Sections 3.1 and 3.2 are used here. In the above two sections, 16 , 650 + 200 = 16 , 850 hashes are extracted. Consequently, there are 16 × 16 , 850 = 269 , 600 hash elements since every video hash has 16 elements. Figure 6 presents the distribution of these hash elements, where the y-axis is the frequency of an element value and the x-axis is the value. It is observed that the minimum and maximum values are 17 and 331, respectively. So a hash element only requires 9 bits for storage. Therefore, the length of our video hash is 16 × 9 = 144 bits. FIGURE 6Open in figure viewerPowerPoint Element distribution 3.4 Rank selection During low-rank frame calculation, the rank of SVD, that is, the r value, is used to control the low-rank approximation. To find optimal rank selection, our classifications about discrimination and robustness under different rank settings are evaluated. The used rank settings are r = 1 , r = 2 and r = 3 . The rank r = 1 means that only the first singular value is used to construct low-rank approximation, the rank r = 2 means that only the first two singular values are used to construct low-rank approximation, and the rank r = 3 means that only the first three singular values are used to construct low-rank approximation. Here the receiver operating characteristic (ROC) graph [38] is taken to compare classification performances. In the ROC graph, a curve is generally drawn by some points whose x-coordinate and y-coordinate are the false positive rate (FPR) and the true positive rate (TPR), respectively. Their calculations can be determined by Equation (10) and Equation (11). FPR = # different videos detected as similar videos # different videos , (10) TPR = # similar videos identified as similar videos # similar videos . (11)Clearly, FPR and TPR are the measures of discrimination and robustness, respectively. In general, area under the curve (AUC) is used as the quantitative metric to evaluate classification performance. The AUC score varies between 0 and 1. A bigger value of AUC implies a better classification performance. Figure 7 presents the curves of different ranks. It is clear that the curves of different ranks are almost the same. The quantitative analysis shows that the AUCs of r = 1 , r = 2 and r = 3 are 0.99848, 0.99863 and 0.99863, respectively. Clearly, r = 2 and r = 3 are a little better than r = 1 . As to the computational time, the results of r = 1 , r = 2 and r = 3 are 0.3696, 0.3724 and 0.3768 s, respectively. It can be seen that the increase of r cannot significantly improve classification. But more time is required because the low-rank frame calculation uses more data. Therefore, the proposed algorithm can reach better whole performances when r = 2 or r = 3 from the viewpoint of classification and computational time. Table 4 shows the