Patent Application
20080175429
1. Field of the Invention
Embodiments of the invention relate to steganography. Steganography is the art and science of secret communication, whose purpose is to hide the very presence of information in content such as images, video, and audio. Steganalysis is the art and science of detecting information hidden in content (e.g., images, video and audio) through the use of steganography. In particular, embodiments of the invention relate to the detection of hidden information in texture images.
2. Description of Background Art
The goals of steganalysis are to: (1) identify suspect content; (2) detect whether or not the content has embedded information; and, (3) if possible, recover that information. Steganalysis is complicated primarily by four things: the suspect images may or may not have any information embedded into them in the first place; the information, if any, may have been encrypted before being embedded into the content; some of the suspect content may have had noise or irrelevant data embedded into them (which reduces stealth but can make steganalysis very difficult); and unless you can completely recover, decrypt, and inspect the information, you often can't be sure whether you really have content that is being used for transport of hidden information or not.
Images are one form of content in which information may be hidden. Since images are diverse and there are wide variations in data embedding approaches, effective methods for steganalysis can be particularly difficult to implement for images. However, since the cover medium has been modified by the step of embedding information, a cover image and an associated steganographic (stego) version of the cover image (i.e., the cover image with embedded information) generally will differ in some respect.
In particular, reliable steganalysis of texture images, which have characteristics that are different from smooth (i.e., non-texture) images, has proven difficult. Specifically, texture images have characteristics that are similar to noise. As a result, the embedded information is submerged in noise, and when methods for steganalysis are applied to texture images the embedded information is often very hard to detect with background art methods for steganalysis that were not developed with the characteristics of texture images in mind.
Our own background art methods for steganalysis, which are hereby incorporated by reference, are described in G. Xuan, Y. Q. Shi, J. Gao, D. Zou, C. Yang, Z. Zhang, P. Chai, C. Chen, W. Chen, “Steganalysis Based on Multiple Features Formed by Statistical Moments of Wavelet Characteristic Functions”, Information Hiding Workshop 2005, Barcelona, Spain, June 2005; and Y. Q. Shi, G. Xuan, D. Zou, J. Gao, C. Yang, Z. Zhang, P. Chai, W. Chen, C. Chen, “Steganalysis Based on Moments of Characteristic Functions Using Wavelet Decomposition, Prediction-Error Image, and Neural Network”, IEEE International Conference on Multimedia & Expo 2005, Amsterdam, Netherlands, July, 2005 (hereinafter Shi et al.). Our background art methods perform well on standard smooth image datasets, say, CorelDraw™ image dataset. However, the performance of our background art methods deteriorates dramatically for texture images and thus, these background art methods are not satisfactory for steganalysis for texture image applications.
Other examples of background art that do not provide the desired level of performance for detection of information in texture images include: J. Fridrich, “Feature-based steganalysis for JPEG images and its implications for future design of steganographic schemes,” 6th Information Hiding Workshop, Toronto, ON, Canada, 2004 (hereinafter Fidrich); H. Farid, “Detecting hidden messages using higher-order statistical models”, International Conference on Image Processing, Rochester, N.Y., USA, 2002, (hereinafter Farid); C. Chen, Y. Q. Shi, W Chen, “Statistical Moments Based Universal Steganalysis Using JPEG 2-D Array and 2-D Characteristic Function”, IEEE International Conference on Image Processing 2006, Atlanta, Ga., USA, Oct. 8-11, 2006 (hereinafter Chen et al.); D. Zou, Y. Q. Shi, W. Su, G. Xuan, “Steganalysis based on Markov model of thresholded prediction-error image”, IEEE International Conference on Multimedia and Expo, Toronto, ON, Canada, Jul. 9-12, 2006 (hereinafter Zou et al.); and Y. Q. Shi, C. Chen, W. Chen, “A Markov Process Based Approach to Effective Attacking JPEG Steganography”, Information Hiding Workshop 2006, Old Town Alexandria, Va., USA, Jul. 10-12, 2006 (hereinafter Shi et al. 11).
Additionally, recognition that background art methods for steganalysis have difficulty detecting information in texture images has also been reported in R. Bohme, “Assessment of Steganalytic Methods Using Multiple Regression Models”, Information Hiding Workshop 2005, Barcelona, Spain, June 2005 (hereinafter Bohme). In particular, Bohme found that images with noisy textures yield the least accurate detection results when two other exemplary background art methods for steganalysis: “Regular-Singular (RS)” and “Weighted Stego Image (WS)” are used. Therefore, there is clearly a need in the art for improved methods for steganalysis for texture images.
Embodiments of the invention are directed at overcoming the foregoing and other difficulties encountered by the background arts. In particular, embodiments of the invention provide improved methods for steganalysis that combine features extracted from the spatial representation; and the computation of multiple block discrete cosine transform (BDCT) with different block sizes.
In addition, embodiments of the invention provide a steganalysis framework that combines the features from the spatial representation, from the BDCT coefficient representation, and from the JPEG coefficient representation for JPEG images. Embodiments of the invention also combine features from the 1-D characteristic function, the 2-D characteristic function, moments and the discrete wavelet transform.
Further, images for steganalysis used by embodiments of the invention can be provided by, but are not limited to: raw images and JPEG images. If the given image is in JPEG format, the image is decompressed to the spatial representation and feature extraction is performed by BDCTs with block sizes of at least 2×2, 4×4, and 16×16. For JPEG images, if the block size is 8×8, the JPEG coefficients (i.e., the JPEG quantized 8×8 BDCT coefficients) are derived directly from the given JPEG image and are used for feature extraction.
Furthermore, embodiments of the invention can be used as an effective universal steganalyzer for both texture and non-texture images. That is, embodiments of the invention can provide an effective universal steganalysis framework that combines features extracted from the spatial representation and from the BDCT representation with different block sizes. Moreover, various embodiments of the invention provide improved steganalysis capability that can be applied to improve detection of information in texture images.
One exemplary embodiment of the invention is a method for steganalysis of an image comprising: inputting a two-dimensional (2-D) spatial representation of the image; generating non-overlapping, N×N block decompositions of the spatial representation of the image; applying a block discrete cosine transform (BDCT) to each of the non-overlapping, N×N block decompositions; determining BDCT coefficient arrays derived from the coefficients of all non-overlapping, N×N block decompositions; extracting moments of a characteristic function from the spatial representation of the image and each of the BDCT coefficient 2-D arrays; generating features based at least on the extracted moments of the characteristic function; and classifying the image based at least on the generated features. Preferably, in an exemplary embodiment of the invention: the block size N×N, with N being at least one of 2, 4, 8 and 16.
Another exemplary embodiment is a method for steganalysis of an image comprising: inputting a JPEG representation of the image; performing a JPEG decompression of the JPEG representation; reconstructing a spatial representation of the image from the JPEG decompression; performing entropy decoding of the JPEG representation; re-ordering the entropy decoded JPEG representation; generating non-overlapping, N×N block decompositions of the spatial representation of the image; applying block discrete cosine transform (BDCT) to each of the non-overlapping, N×N block decompositions; determining BDCT coefficient 2-D arrays; forming 8×8 JPEG coefficients from the re-ordering of the entropy decoding of the JPEG representation; extracting moments of a characteristic function from the BDCT coefficient 2-D arrays and 8×8 JPEG coefficients; generating features based at least on the extracted moments of the characteristic function; and classifying the image based at least on the generated features.
Yet another exemplary embodiment of the invention is a processor-readable medium containing software code that, when executed by a processor, causes the processor to implement a method for steganalysis of an image comprising: inputting a two-dimensional (2-D) spatial representation of the image; generating non-overlapping, N×N block decompositions of the spatial representation of the image; applying block discrete cosine transform (BDCT) to each of the non-overlapping, N×N block decompositions; determining BDCT coefficient 2-D arrays for each N×N block; extracting moments of a characteristic function from the spatial representation of the image and each of the BDCT coefficient 2-D arrays; generating features based at least on the extracted moments of the characteristic function; and classifying the image based at least on the generated features.
Another exemplary embodiment is a processor-readable medium containing software code that, when executed by a processor, causes the processor to implement a method for steganalysis of an image comprising: inputting a JPEG representation of the image; performing a JPEG decompression of the JPEG representation; reconstructing a spatial representation from the JPEG decompression; performing entropy decoding of the JPEG representation; generating non-overlapping, N×N block decompositions of the spatial representation of the image; applying block discrete cosine transform (BDCT) to each of the non-overlapping, N×N block decompositions; determining BDCT coefficient 2-D arrays; forming 8×8 JPEG coefficient from the re-ordering of the entropy decoding of the JPEG representation; extracting moments of a characteristic function from the BDCT coefficient 2-D arrays and 8×8 JPEG coefficients; generating features based at least on the extracted moments of the characteristic function; and classifying the image based at least on the generated features.
Yet another exemplary embodiment further comprises: determining a prediction error for at least one of the spatial representation 2-D array, the BDCT coefficient 2-D arrays and the JPEG coefficient 2-D arrays; computing a wavelet transform for at least one of the spatial representation 2-D array of the image, the BDCT coefficient 2-D arrays, the JPEG coefficient 2-D array and the prediction error; determining histograms of the wavelet subbands; determining at least one of horizontal, vertical, diagonal and minor diagonal 2-D histograms from at least one of the spatial representation 2-D array of the image, the BDCT coefficient 2-D arrays and the JPEG coefficient 2-D array; applying a discrete Fourier transform (DFT) to the histograms; applying a 2-D DFT to the 2-D histograms; determining at least one of the first order, second order and third order moments from the characteristic function (i.e., with the discrete Fourier transform of the histogram); and determining at least one of the first order, second order and third order marginal moments from the characteristic function; and generating features based at least on the moments.
Preferably, the wavelet transform of the above-discussed exemplary embodiments uses a one-level discrete wavelet transform. Alternatively, two-level and three-level discrete wavelet transforms can be used. In addition, in these exemplary embodiments, the wavelet transform is at least one of a Haar wavelet, Daubechies 9/7 wavelet and integer 5/3 wavelet.
Another exemplary embodiment of the invention is a method further comprising: computing sign and magnitude for at least one of the spatial representation 2-D array of the image, the BDCT coefficient 2-D arrays and the JPEG coefficient 2-D arrays; computing an expression |a|+|b|−|c|; and determining the product of the expression and the sign.
Yet another embodiment of the invention is an apparatus comprising: means for generating features based at least in part on moments of a characteristic function of said image; and means for classifying said image based at least in part on said generated features. Preferably said means for generating features comprises means for generating features based at least in part on moments of a set of decomposition of images. Preferably said set of decomposition of images is based at least in part on at least one of the discrete wavelet transform or the Haar wavelet transform. Preferably, the means for classifying comprises a means for classifying an image as either a stego-image or a non-stego image. Preferably said means for generating features includes means for generating a prediction error based at least in part on said image.
Another embodiment of the invention is an apparatus comprising: means for applying a trained classifier to an image; and means for classifying said image based at least in part on applying a trained classifier to a host of features generated from said image. Preferably, the means for classifying comprises means for classifying based at least in part on applying a trained classifier comprising at least one of a trained Support Vector Machine (SVM) classifier, a trained neural network classifier and a trained Bayes classifier. Preferably, the means for classifying includes means for classifying based at least in part on a host of features generated from a prediction error of said image.
As discussed above, texture images are generally noisy. As a result, when background art methods for steganalysis are applied to texture images, the embedded signal is submerged in noise and is very difficult to detect. A detailed description/definition of “texture” suggests: “[t]he notion of texture appears to depend upon three ingredients: (1) some local ‘order’ is repeated over a region which is large in comparison to the order's size; (2) the order consists in the nonrandom arrangement of elementary parts; and (3) the parts are roughly uniform entities having approximately the same dimensions everywhere within the textured region,” and was given in J. K. Hawkins, “Textural Properties for Pattern Recognition”, Picture Processing and Psychopictoris, B. C. Lipkin and A. Rosenfeld (editors), Academic Press, New York, 1970, pp 347-370. In addition, the term “texture is used to describe two dimensional arrays of variations” in R. M. Pickett, “Visual Analysis of Texture in the Detection and Recognition of Objects”, Picture Processing and Psychopictories, B. C. Lipkin and A. Rosenfeld (editors), Academic Press, New York, 1970, pp 198-216.
As mentioned in the Background Art section above, our background art method for steganalysis, as reported in Shi et al., works quite well on smooth CorelDraw™ images but almost fails on texture images. The reason for why this is the case is analyzed in the following paragraphs. This reasoning is further supported by the experimental results given in Table 1 to Table 4, which are also discussed below.
The method in Shi et al. is based on the moments of the characteristic function (i.e., the Fourier transform of the histogram) of an image, which are given by:
where H(xi) is the characteristic function (CF) component at frequency xi and N is the total number of different value level of image pixels (e.g., N=256 for 8-bit images). In S. Qian, D. Chen, Joint Time-Frequency Analysis-Methods and Applications, Prentice Hall, 1996, the first order moment (i.e., n=1 in Equation (1)) of the CF of the histogram is also referred to as the mean instantaneous frequency (MIF) of the histogram. It has proven, for example in: G. Xuan, Y. Q. Shi, J. Gao, D. Zou, C. Yang, Z. Zhang, P. Chai, C. Chen, W. Chen, “Steganalysis Based on Multiple Features Formed by Statistical Moments of Wavelet Characteristic Functions”, Information Hiding Workshop 2005, Barcelona, Spain, June 2005], that after information has been embedded into an image, the above defined moments will decrease or remain the same under the assumption that: (1) the embedded information (or hidden data) is additive to and independent of the cover image; and (2) the CF has a distribution such that its magnitude is non-increasing from 1 to N/2.
In addition, under the above assumptions, the additive information embedding of steganography acts as a low pass filtering process on the image histogram. Further, with respect to the use of the same embedded information and the same embedding process, it is observed from our experiments on two image databases (i.e., CorelDraw™ image database and the 798 texture images) that after information is embedded, the MIF of the histogram of images with a higher MIF will decrease more than that of the histogram of images with a lower MIF.
As discussed above, the smooth CorelDraw™ images have higher MIFs than the texture images before information embedding and the relative decrease in the value of the MIFs of the smooth CorelDraw™ images is larger than that of the texture images. This difference in the decrease of the MIFs after data embedding indicates the separation of the cover images from the stego version of the cover images is larger and hence the steganalysis is easier for the smooth CorelDraw™ images. Further, this suggests that the larger separation between the cover image and stego version of the cover image with smooth images is the reason for background art methods for steganalysis to perform better on the smooth CorelDraw™ images than on the texture images.
In order to improve the performance of methods for steganalysis for texture images, one embodiment of the invention computes moments generated from BDCT representations with different block size to enhance the capability of steganalysis. In particular, embodiments of the invention exploit the concept that the moments of the CF of the histogram of the BDCT coefficient 2-D arrays decreases or remains the same after embedding information under the assumption that the embedded information is additive to and independent of the cover image, and with such a distribution, the magnitude of the CF of the transformed embedded signal is non-increasing from 1 to N/2 (where N is the total number of different value levels of BDCT coefficients). In addition, the moments of 2-D characteristic functions may also be included. Currently, the advanced steganographic schemes aim at keeping the 1-D histogram of the image unchanged as much as possible during data embedding. Consequently, the 1-D histogram and hence the moments of 1-D characteristic function may vary very little after data embedding By exploiting moments derived from 2-D characteristic function (the DFT of 2-D histogram), these steganographic schemes may be detected.
The N×N block decompositions of the image data are input to BDCT functions 213, 215, 219. The outputs of the BDCTs 213, 215, 219 provide inputs for BDCT coefficient 2-D arrays 223, 225, 229. The JPEG Quantized coefficient 2-D array 226 and BDCT coefficient 2-D arrays 223, 225, 229 are each provide as inputs for Moment Extraction functions 237, 233, 235, 239. The outputs from the Moment Extraction function 231, 233, 235, 237, 239 are each ultimately 54-D feature vectors 241, 243, 245, 247, 249 which form a part of a final 270-D feature vector that is input to a Support Vector Machine (SVM) trainer and classifier.
A support vector machine (SVM) is used in the trainer and classifier function 250, as shown in
With regards to the decompression function 204 of
The JPEG decompression is a reverse process of the JPEG compression function discussed above. Hence, to form a 2-D array consisting of all of the JPEG quantized BDCT coefficients from all of the 8×8 blocks, as shown in
The moments of the CF of the histograms are given by Equation (1), as discussed above. For the spatial representation and each BDCT/JPEG coefficient representation, four second-order histograms (also referred to as 2-D histograms) are generated with the following four separations:
with the parameters ρ representing the distance between the pair of elements for which the two-dimensional histogram is considered, and θ representing the angle of the line linking these two elements with respect to the horizontal direction; which are called horizontal 2-D histogram, vertical 2-D histogram, diagonal 2-D histogram, and minor diagonal 2-D histogram, respectively. After applying the 2-D DFT 319 to the 2-D histograms 307, the marginal moments of 2-D CF's are calculated by:
where H(ui, vj) is the 2-D CF component at frequency (ui, vj) and N is the total number of different values of coefficients in a wavelet subband under consideration.
In implementing the flow diagram of
With block size N×N, where N is equal to 2, 4, 8, and 16, are chosen accordingly.
In implementing the flow diagram of
{circumflex over (x)}=sign(x)·{|a|+|b|−|c|}. (6)
Considering the high dimensionality of the features leads to high computational complexity, we compute at least a one-level (1-level) discrete wavelet transform (DWT) in an exemplary embodiment of the invention. If we consider the spatial representation of the image, the BDCT coefficient 2-D array, or the JPEG coefficient 2-D array as LL0, we have five subbands for a 1-level DWT decomposition. As compared to a 3-level DWT decomposition, the feature dimensionality of a 1-level DWT is reduced to 38%. The wavelet transform for this embodiment can be, but is not limited to the Haar wavelet transform, which is the simplest wavelet.
The experimental investigation results are reported in Table 1 to Table 4 below. The leftmost column in these tables lists the data embedding methods used to generate stego images. These data embedding methods are: (1) Marvel et al's Spread Spectrum Image Steganography (SSIS) as given in L. M. Marvel, C. G. Boncelet, Jr, C. T. Retter, “Spread Spectrum Image Steganography,” IEEE Trans. on Image Processing, 8, pp. 1075-1083, August 1999 (0.1 bpp (bit per pixel); peak signal to noise ratio (PSNR) equal to 40 dB); (2) Cox et al.'s non-blind Spread Spectrum (SS) (a=0.1), as found in I. J. Cox, J. Kilian, T. Leighton, T. Shamoon, “Secure Spread Spectrum Watermarking for Multimedia,” IEEE Trans. on Image Processing, 6, 12, 1673-1687, 1997; (3) Piva et. al's blind SS, as found in A. Piva, M. Barni, E Bartolini, V. Cappellini, “DCT-based Watermark Recovering without Resorting to the Uncorrupted Original Image”, Proc. ICIP 97, vol. 1, pp. 520; (4) Huang and Shi's 8×8 block SS, as found in J. Huang, Y. Q. Shi, “An adaptive image watermarking scheme based on visual masking,” IEE Electronic Letters, vol. 34, no. 8, pp. 748-750, April 1998; (5a, 5b) generic LSB replacement (0.1 and 0.3 bpp; both the pixel position used for embedding data and the to-be-embedded bits are randomly selected), (6a, 6b) generic LSB modulation (0.1 and 0.3 bpp; both the pixel position used for embedding data and the to-be-embedded bits are randomly selected), (7) a generic Quantization Index Modulation (QIM) (0.1 bpp; step size=5), as found in B. Chen, G. W. Wornell, “Digital watermarking and information embedding using dither modulation,” Proceedings of IEEE MMSP 1998, pp 273-278; (8a, 8b) MB1 (0.1 and 0.2 bpc (bits per non-zero BDCT AC coefficient), step size=2), as found in P. Sallee, “Model-based steganography,” International Work-shop on Digital Watermarking, Seoul, Korea, 2003; (9a, 9b) F5 (0.1 and 0.2 bpc), as found in A. Westfeld, “F5 a steganographic algorithm: High capacity despite better steganalysis,” 4th International Workshop on Information Hiding, Pittsburgh, Pa., USA, 2001; and (10a, 10b) OutGuess (0.1 and 0.2 bpc), as found in N. Provos, “Defending against statistical steganalysis,” 10th USENIXSecurity Symposium, Washington D.C., USA, 2001. As can be seen from the experimental results of Table 1, our exemplary embodiment provides improved performance over the background art methods for steganalysis for texture images in general.
To evaluate the performance of an exemplary embodiment of the method for steganalysis of the invention on general smooth images, we also use all the 1096 sample images included in the CorelDraw™ Version 10.0 software for experiments. CorelDraw™ images are well known as natural images with good quality. The results shown in Table 2 are the arithmetic average of 20 random experiments. As can be seen from the experimental results of Table 2, our exemplary embodiment provides improved performance over the background art methods for steganalysis for smooth images in general.
Tests were also conducted to examine the effect on performance due to various block sizes. Specifically, the performance of an embodiment of the invention was tested without using features generated from 16×16 BDCT coefficient 2-D array. The results of the test are shown in Table 3 and are the arithmetic average of 20 random experimental results. From Table 3, it is observed that features from 16×16 BDCT have enhanced the detection rates by 5% to 10% (except when the detection rate has already been above 95%) in detecting steganography in a JPEG image. We randomly select 5/6 of the cover/stego image pairs to train the SVM classifier and the remaining 1/6 pairs to test the trained classifier. We apply our exemplary embodiment of a steganalyzer and a steganalyzer without features generated from 16×16 BDCT coefficient 2-D array in the same way to detect the above mentioned data hiding schemes. The results shown in Table 3 are the arithmetic average of 20 random experiments.
An exemplary embodiment of the invention uses 270-D features. The first 54 feature vector components are derived from the spatial representation. The second, third, fourth, and fifth are derived from BDCT representation with block size 2×2, 4×4, 8×8 (alternatively, for a JPEG coefficient 2-D array for JPEG images at 8×8 BDCT), and 16×16, respectively. Experimental results using part of and all of these features are given in Table 4. In this table, “54-D” means only the first 54 feature components are used, “108-D” means the first and the second 54 feature components are used, and so on.
It is observed from the above results that each of the BDCT's makes contribution to our proposed steganalyzer. Moreover, the more the different BDCT sizes (up to 16×16) involved, the better the steganalysis performance. In other words, each feature subset associated with the specific-block-size BDCT contributes to the steganalyzer, just like each reflected signal contributes to the RAKE receiver to improve the received SNR (signal to noise ratio), which is widely used in wireless communication applications. We refer to this series of BDCT's as a RAKE transform. For steganographic methods applied to image spatial respresentation, it seems that the steganalyzer gives similar performance with 16×16 BDCT to that without 16×16 BDCT. However, 16×16 BDCT enhances the steganalyzer's performance on JPEG images significantly.
Our experimental results show that features generated from block DCT coefficient 2-D array with block size 2, 4, or 8 also make a contribution that improves the effectiveness of steganalysis significantly. However, block DCT coefficient 2-D arrays with block size larger than 16 contribute much less to improving performance. This is expected because the correlation between image pixels reduces dramatically when the distance between pixels is large, as is the case for block sizes larger than 16.
From the experimental results we conclude the following: (1) exemplary embodiments of the invention significantly improve the capability for steganalysis in detecting steganography that has been applied to the texture images in the spatial representation; (2) the exemplary embodiment also works quite well on general smooth images (e.g., the CorelDraw™ images); (3) a large number of experimental results indicate the exemplary embodiment outperforms background art universal steganalyzers in detecting steganography applied to both texture images and general smooth images; (4) the exemplary embodiment outperforms the background art universal steganalyzers (e.g., Farid, Shi et al) in detecting data hidden in JPEG images by a significant margin. Though the exemplary embodiment does not perform better in detecting modern JPEG steganographic schemes such as MB1, F5, and OutGuess than some steganalyzers (e.g., Fridrich and Shi et al II), these steganalyzers were specifically designed for detecting these modern JPEG steganographic schemes and perform poorly in steganalyzing steganographic schemes applied to images in spatial representation. In addition, these steganalyzers (e.g., Fridrich) sometimes cannot be carried out for all of 798 texture images in our experiments.
It will, of course, be understood that, although particular embodiments have just been described, the claimed subject matter is not limited in scope to a particular embodiment or implementation. For example, one embodiment may be in hardware, such as implemented to operate on a device or combination of devices, for example, whereas another embodiment may be in software. Likewise, an embodiment may be implemented in firmware, or as any combination of hardware, software, and/or firmware, for example. Likewise, although claimed subject matter is not limited in scope in this respect, one embodiment may comprise one or more articles, such as a storage medium or storage media. This storage media, such as, one or more CD-ROMs and/or disks, for example, may have stored thereon instructions, that when executed by a system, such as a computer system, computing platform, or other system, for example, may result in an embodiment of a method in accordance with claimed subject matter being executed, such as one of the embodiments previously described, for example. As one potential example, a computing platform may include one or more processing units or processors, one or more input/output devices, such as a display, a keyboard and/or a mouse, and/or one or more memories, such as static random access memory, dynamic random access memory, flash memory, and/or a hard drive. For example, a display may be employed to display one or more queries, such as those that may be interrelated, and or one or more tree expressions, although, again, claimed subject matter is not limited in scope to this example.
In the preceding description, various aspects of claimed subject matter have been described. For purposes of explanation, specific numbers, systems and/or configurations were set forth to provide a thorough understanding of claimed subject matter. However, it should be apparent to one skilled in the art having the benefit of this disclosure that claimed subject matter may be practiced without the specific details. In other instances, well known features were omitted and/or simplified so as not to obscure the claimed subject matter. While certain features have been illustrated and/or described herein, many modifications, substitutions, changes and/or equivalents will now occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and/or changes as fall within the true spirit of claimed subject matter.
