Simultaneous Encryption/Compression of Images Using Alpha Rooting Eric Wharton 1, Karen Panetta 1, and Sos Agaian 2 1 Tufts University, Dept. of Electrical and Computer Eng., Medford, MA 02155 2 The University of Texas at San Antonio, College of Eng., San Antonio, TX 78249 Abstract Significant work has been done in the areas of image encryption and compression as two independent problems, yet the two areas are strongly interleaved and should be considered simultaneously. Consider that current practice in image processing generally uses encryption on compressed image data or attempts to compress data that has already been encrypted. However, it is generally seen that traditional encryption techniques degrades compression. In this paper, we propose a novel method which prepares an image for compression while simultaneously encrypting the image data. We perform this using the well known alpha rooting algorithm. By increasing the relative magnitudes of the DCT transform coefficients, we are able to simplify the image in a way which makes it indiscernible, while also higher compression ratios. We demonstrate encryption and compression performance using the independent group JPEG algorithm, showing compression ratios and error metrics including MSE and SSIM for a collection of images. 1. Introduction Significant work has been performed on encrypting images and compressing images as two separate problems, but traditional encryption techniques geneally degrade the performance of multimedia compression systems [1]. To get around this difficulty, two methods have been used. The first employs known encryption algorithms on compressed image data [2][3][4][5]. The second attempts to develop compression algorithms which work well for typical encrypted data [6][7][8]. Algorithms which rely on encrypting compressed data generally rely on well known encryption methods, such as Digital Encryption Standard (DES) and Advanced Encryption Standard (AES), to encrypt portions of a compressed stream. In [2], DES is used to encrypt portions of an SPIHT compressed image. [3] and [4] detail two other ways of using well-known encryption methods to encrypt compressed data. Also, scrambling methods can be used as in [5], where scrambling techniques are used on JPEG image data to encrypt the image. However, such scrambling techniques have been cryptanalyzed and found to be lacking in security [24][25]. Excellent work has also been performed in the domain of compressing encrypted image data. In [6] and [7], compression of encrypted data is seen as a problem of source coding with side information. In order to achieve this, a distributed source coding scheme is used whose inter source correlations match the unencrypted source s statistics [6]. Another method is presented in [8], where an entropy coder is employed which combines run-length and Golomb-Rice encoders to compress Gaussian sources.
The work of Johnson et al. indicates that data should be able to be encrypted to the original source entropy regardless of encryption methods [6][7]. This is based on the assumption that, no matter what, the encrypted data is still related to the original image data with some amount of side information. In this paper, we develop a novel method to work towards this goal, using a method which both encrypts the image data and prepares it for compression. Alpha rooting has been used effectively for image enhancement [9][10][11][12]. It functions in the transform domain, reducing the magnitudes of the coefficients while leaving the phase unchanged. In [13] it was first shown that, by increasing instead of decreasing the magnitudes of the coefficients, an image could be encrypted instead of enhanced. By increasing the relative magnitudes of the coefficients, placing more emphasis on the less important information, a blurred and degraded image is produced. This affect can be undone by using the same alpha rooting equation and the inverse of the original encryption key. In this paper, we investigate the compression of these images, using alpha rooting as a pre-processing stage for compression as well as to perform the encryption of the image. For simplicity, we use the well known JPEG image compression algorithm, both in the lossless and lossy modes, to demonstrate the effectiveness of alpha rooting for simultaneous encryption and compression. We will present sample results and metrics to demonstrate the performance of this method, including compression ratio, mean squared error (MSE) of the encoded and decoded images, and the structural similarity (SSIM) [14] measure of the encoded and decoded images. This paper is organized as follows: section 2 presents necessary background information, including the alpha rooting algorithm and the metrics used. Section 3 presents the use of alpha rooting for simultaneous encryption and compression. Section 4 presents experimental results, giving compression ratios and error metrics to quantify performance. Section 5 is a discussion of results and some concluding comments are made. 2. Background In this section, we present necessary background information, including alpha rooting, the relational MSE measurement, and the relational SSIM measurement. 2.1. Alpha Rooting Alpha rooting is a straightforward method originally proposed by Jain [9]. This algorithm was later modified by Aghagolzadeh and Ersoy [10] as well as by Agaian [11][12]. Alpha rooting can be used in combination with any orthogonal transform such as Fourier, Hartley, Haar wavelet and cosine transforms. Alpha rooting modifies these transform coefficients in the following manner: α 1 O( p, s) = X ( p, s) X ( p, s) (1) Where X( p, s ) is the 2-D orthogonal transform of the input image, O( p, s ) is the 2-D orthogonal transform of the output image, and α is a user-defined operating parameters,
Fig. 1. Sample results for alpha rooting enhancement, top row: Original images, bottom row: Alpha Rooting enhanced images with range 0 < α < 1 when used for enhancement. For this paper, we will use the Discrete Cosine Transform (DCT). Sample results for the alpha rooting enhancement method are shown in figure 1, showing enhancement of the fine details in the image. 2.2. Error Metrics A number of image similarity metrics have been proposed. Two common methods are the well known mean squared error (MSE), and Structural Similiarity (SSIM) measure [14]. MSE takes the sum of the squared error over the entire image and scales it by the number of pixels, returning smaller numbers for a more perfect match. The SSIM measure uses image statistics to compare luminance, contrast, and structure; outputting a number between 0 and 1 with 1 being a perfect match. In [15], a measure was proposed comparing the error of the decrypted signal to the error of the encrypted signal. Using the assumption that an encrypted image should have as large a difference from the original as possible while the decrypted image should have as small an error as possible, the measure divides the MSE of the encrypted image by the MSE of the decrypted image to measure encryption quality. This measure can be modified to use the original SSIM. Continuing to use the assumption that the error between the decoded and original images should be as small as possible while the error between the encrypted image and original should be as large as possible, we introduce the Relational SSIM (ReSSIM) measure as follows: SSIM decrypted Re SSIM = (2) SSIM encrypted 3. Alpha Rooting for Simultaneous Encryption and Compression Every day massive quantities of two-dimensional data are produced, stored, and transmitted [16]. As the amount of data generated continues to grow, it is clear that the role of data compression will be crucial in this development [17]. As well as the data compression techniques outlined in [6][7][8], there exist methods based on compression-
. boosting transforms [16], conditional averages [17], wavelet transforms and hierarchical trees [18], and adaptive linear predictors [19]. Both government and industry entities are interested in protecting information [20]. The government requires data protection to ensure national security. Industry requires data protection for information vital to it economic endeavors. As well as the methods in [2][3][4] which rely on DES or AES, there exist encryption algorithms which rely on scrambling [5][21] and methods which rely on chaotic systems [22][23]. It can be seen that there is a large amount of overlap between the need for compression and encryption. For example, government and corporations with proprietary documents demand effective encryption to protect their interests, while also requiring efficient compression to meet storage requirements. Despite this, there is little in the literature to address simultaneous encryption and compression. In [7] it is shown that, theoretically, it should be possible to compress encrypted data to the same limit as the unencrypted data that produced it. In [6], a method is proposed which can compress encrypted video frames by about 3:1, however it relies on temporal relations between different frames, making it unable to compress still images. In [8], a method is proposed which is specifically designed to compress data with Gaussian statistics, which could be useful as an ideally encrypted image would have a Gaussian distribution. In [13], the alpha rooting algorithm was first used for image encryption. It was shown that this could achieve scalable encryption, where a larger α-value would result in better encryption. In this paper, we will show that, not only does this simplification of the image destroy the visual image in a recoverable way, but it also simplifies the image in a way which makes it better for compression. Section 4 will demonstrate the encryption performance for a collection of test images, showing the previously presented error statistics to quantify performance. In addition, further demonstration will show how this method prepares an image for compression using the common JPEG algorithm, both in the lossless and lossy modes. The experimental results will show that, as α increases, the compression ratio and encryption strength increases as well, with a tradeoff of less precise decryption. Figure 2 demonstrates the encryption method alone, giving the image histogram and error metrics to quantify the performance of alpha rooting encryption using the arctic hare image. It can be seen that, as α increases, the encryption strength increased and the image histogram moves toward a Gaussian distribution. This also results in a more readily compressed transform domain representation of the image, which lends itself to common compression techniques, such as JPEG but also compression methods intended for encrypted data such as those from [8]. It is noted that the error in the figure 2 comes from machine precision. If the decrypted images could be stored as floating point numbers instead of integers, the MSE would be significantly decreased. This is one issue which must be overcome for compression of these images. While lossless JPEG only rounds off the fractional portion resulting in
α = 1 (Original) α = 2 α = 3 α = 4 2500 2000 1500 1000 900 800 700 600 500 400 300 800 700 600 500 400 300 800 700 600 500 400 300 500 200 200 200 100 100 100 0 0 0 0 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 MSE = 0 (original) SSIM = 1 ReSSIM = 1 MSE = 5.2347x10-20 SSIM = 1 ReSSIM = 1.2769 MSE = 7.1042x10-9 SSIM = 1.0000 ReSSIM = 1.4667 MSE = 8.2445x10-4 SSIM = 0.9948 ReSSIM = 1.6982 Fig. 2. Sample results for alpha rooting; top row: encrypted images, middle row: histogram of encrypted images, bottom row: decrypted images with MSE values and SSIM values similar results as those shown here, lossy JPEG encryption slightly changes these values, which will result in a larger amount of error in the decrypted signal. However, it will be seen that this can be easily corrected by using a simple low pass filter or other methods. 4. Experimental Results In this section, we present experimental results for a collection of images using alpha rooting to simultaneously encrypt an image and prepare it for compression. We show results for a collection of test images. These consist of common test images such as Lena and the pentagon, camera phone images, and a collection of documents. Figure 3 shows experimental results using the pentagon image to demonstrate compression using different alpha values and lossless JPEG. The decryption becomes increasingly lossy as α increases, however this could be improved by a compression technique which can compress floating point data values instead of integers. This can also be improved using simple denoising techniques, for example when the image using α = 4 is processed using a simple low pass filter, the SSIM = 0.5318 and the ReSSIM = 1.7797, both significant gains. It is further noted that the alpha rooting processed images achieve a higher compression ratio than the original, unencrypted image. Figure 4 shows experimental results for lossy JPEG compression with Q=90. Again, the compression ratio increases as alpha increases, compressing the encrypted images better than the unencrypted images. It can be seen that, with acceptable amounts of error, compression ratios around 5:1 can be achieved with acceptable security for this image.
α = 1 (Original) CR = 1.39:1 α = 2 CR = 1.64:1 α = 3 CR = 1.95:1 α = 4 CR = 2.34:1 MSE = 0 (Original) SSIM = 1 ReSSIM = 1 MSE = 45.6677 SSIM = 0.9822 ReSSIM = 1.3585 MSE = 1.0500x10 3 SSIM = 0.7046 ReSSIM = 1.5799 MSE = 2.1070x10 3 SSIM = 0.3678 ReSSIM = 1.2437 Fig. 3. Sample results for simultaneous encryption and compression using Lossless JPEG; top row: encrypted images with compression ratio, bottom row: decrypted images with MSE values and SSIM values α = 1 (Original) CR = 1.84:1 α = 2 CR = 3.52:1 α = 3 CR = 5.45:1 α = 4 CR = 7.80:1 MSE = (Original) SSIM = 1 ReSSIM = 1 MSE = 632.9958 SSIM = 0.9128 ReSSIM = 1.2106 MSE = 1.3615x10 3 SSIM = 0.5924 ReSSIM = 1.2708 MSE = 1.0788x10 3 SSIM = 0.2930 ReSSIM = 0.9914 Fig. 4. Sample results for simultaneous encryption and compression using Lossy JPEG with Q=90; top row: encrypted images with compression ratio, bottom row: decrypted images with MSE values and SSIM values Tables I and II demonstrate the encryption and compression properties for a number of images. Table I shows the compression ratios and error metrics for a number of compressed images using lossless JPEG compression. Table II shows the compression ratios and error metrics for a number of compressed images using lossy JPEG compression. Of the images, Pentagon, Lena, and Cronkite are all well known test images. Cave and Faces are images captured using a Motorola RAZR V3 0.3 megapixel camera phone. Patent and Summons are examples of archived legal documents, and Resume is another document example.
It can be seen that the Lossy JPEG achieves a compression ratio of approximately 3:1, with the Summons document achieving a ratio so high as 5.59:1. Also, Lossy JPEG achieves appropriately larger compression ratios, with results so large as 10.40:1. Finally, an example is given in the appendix for the Resume document. As Table II shows, this image achieved 6.77:1 compression ratio with higher MSE than the other images and lower SSIM than the others. Despite this, the decrypted image is fully readable and contains all of the original information. Table I Encryption/Compression using Alpha Rooting and Lossless JPEG Image Compression MSE decrypted SSIM decrypted ReSSIM Ratio Pentagon 1.9524:1 1.0500x10 3 0.7046 1.5799 Lena 2.1699:1 926.3269 0.7812 1.116 Cronkite 3.1071:1 617.7411 0.8423 1.2111 Cave 2.6523:1 421.1964 0.9344 1.3326 Faces 2.8573:1 1.2814x10 3 0.8385 1.1562 Patent 3.0146:1 2.7028x10 3 0.8126 1.8014 Summons 5.5858:1 1.8623x10 3 0.8515 1.2287 Resume 3.6229:1 2.6787x10 3 0.8051 1.8650 Table II Encryption/Compression using Alpha Rooting and Lossy JPEG Image Compression MSE decrypted SSIM decrypted ReSSIM Ratio Pentagon 5.4500:1 1.3615x10 3 0.5924 1.2708 Lena 6.0845:1 631.5569 0.8687 1.0795 Cronkite 10.4009:1 1.0924x10 3 0.8525 1.1317 Cave 9.5957:1 1.5180x10 3 0.7128 1.0230 Faces 10.3252:1 1.0146x0 3 0.8581 1.0627 Patent 6.0147 5.3064x10 3 0.6809 1.2142 Summons 10.2971 5.3593x10 3 0.7916 1.0478 Resume 6.7671:1 6.5777x10 3 0.5164 1.1914 5. Conclusion In conclusion, we have presented a novel use of alpha rooting to simultaneously encrypt an image and prepare it for compression. We have used the well known JPEG standard to compress the image, quantifying performance using compression ratios and established error metrics. We have shown that using lossless JPEG on encrypted images, compression ratios of 3:1 are possible. Also, we have shown that, using lossy JPEG, compression ratios of 10:1 are possible. Future work includes developing methods to overcome the lossy nature of the decoding, either developing some denoising filter to improve the decoded image or developing a more application specific compression method. For example, the compression method in [8] is developed specifically for compression of sources with Gaussian distributions, further development of similar methods could increase both the compression and decoding of alpha rooting processed images. References [1]C. P. Wu and C. C. Jay Kuo, Design of Integrated Multimedia Compression and Encryption Systems, IEEE Tran. Multimedia, 7(5), pp. 828 839, October 2005. [2] H. Cheng and X. Li, Partial Encryption of Compressed Image and Videos, IEEE Tran. Signal Processing, 48(8), pp. 2439 2451, August 2000.
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APPENDIX B: