Improvement of DWT-SVD with Curve Fitting and Robust Regression : An Application to Astronomy Images

DWT-SVD is a frequency domain based eigenanalysis watermarking technique. In this work, we improve this method by exploring the relationship between the cover image’s DWT singular values and those of the watermark. We show that, via the usage of curve fitting and robust regression, it is possible to achieve accurate results. We also demonstrate that the improved scheme is suitable for the watermarking of astronomy images. In addition to encoding and decoding examples, statistical results on stealth and robustness are deduced from the experiments so that the clear advance can be observed. Quality of the watermark is measured by testing against various attack types. DOI: http://dx.doi.org/10.5755/j01.eie.22.3.15319


I. INTRODUCTION
Digital watermarking has a wide area of application. The most important one of these is copyright security; a secret message is embedded into the media so that the owner knows it stands as a signature. For the battle against piracy and for the management of media tracking, it is impossible to neglect the benefit of information hiding [1]- [5].
The domain of visual watermarking is classified into two types. The first one is the obvious one where the pixels reside; spatial domain. The second one is the frequency domain, where we embed the secret message into the frequency coefficients obtained via an analysis of the cover image. Discrete Wavelet Transform (DWT) and Discrete Cosine Transform (DCT) are examples of the state-of-theart frequency domain analysis. In basic DWT embedding, the visual watermark is hidden into the wavelet coefficients of the cover image [6]- [10]. That is, coefficients are modified by using the intensity values of the watermark. In DWT-SVD embedding, singular values of the frequency coefficients are modified by using the singular values of the visual watermark. In this work, we focus on the improvement of DWT-SVD technique [11].
Although the central technique DWT is somehow old, its modified and upgraded versions, that is, strengthened ways of frequency domain analysis via SVD or other transformations, are still in the field of active research [12], [13]. Moreover, even DWT itself is engineered in more articulated techniques [14], which is a proof of the solid state of this classical algorithm. Variants such as DCT are also up-to-date base selections for implementations of watermarking [15].
Influenced by [10], we have derived a technique which will be explained as follows: in II.A base theoretical tools, in II.B and II.C extensions such as curve fitting are noted. Overall scheme development and experiment details are given in II.D, II.E, respectively.

II. IMPROVEMENT OF DWT-SVD
Our aim at this work was to establish an improvement of DWT-SVD [10], [11], [14] via an additional analysis on singular values. For this, we utilized curve fitting and robust regression. Thus, the derived technique can be summarized as DWT-SVD-CF-RR (DSCR). The ideal advance of a watermarking scheme must be in both robustness and stealth. Robustness is the 'strength' measure of the encoding to a range of attacks. Stealth, on the other hand, is the transparency of the hidden message; this is essential for the overall commercial quality of the modified image.
Before carrying out the experiments, our main idea was this: what if one embeds an approximation of watermark singular values as a function of the cover DWT singular values rather than directly using the initial (original) ones? This corresponds to a more consistent modification since the same function is used for the band's all singular values; the resulting sequence is retrieved by a well-defined single variable function. On the other hand, since an approximation is used, the connection to the watermark is almost retained. Hence, our initial guess was that the final PSNR value of the encoded image should be higher than that of the standard DWT-SVD's output and the correlation values of the decoded singular values should have a tolerable distance to those of the standard strategy. After experiments, we have seen that, both PSNR values and correlation measurements were better.

A. DWT-SVD
Assume that we have a 2n × 2n cover and an n × n Improvement of DWT-SVD with Curve Fitting and Robust Regression: An Application to Astronomy Images x y x y x y  . What we do at (leastsquares) curve fitting is to find a d degree polynomial curve fitting for the LL band as follows: for a given degree Hence, we fit a polynomial of LL band singular values to approximate the singular values of the watermark.

C. Robust Linear Regression
Ordinary Least Squares (OLS) is formulized as follows: given X , y matrices where the dimensions are n × D and n × 1 respectively, find D × 1 dimensional matrix β =β such that  Xβ y, (2) is minimized. That is, we model the relationship between an input vector x and a target value y in the form of a linear function characterized by β, where the predictionŷ is evaluated as so that the total distance of the estimated values to the actual ones are minimized.
While modeling via OLS, we do not take the weights of the samples into account. If we do it, the result is a new formulation which contains the analysis of outliers and leverage points; Weighted Least Squares (WLS). Outliers are points that are not fitted very well by the linear model (i.e. points having large residuals), whereas leverage points are those outlying in the x-space [3]. Since the weight assigning procedure is dependent to residuals and residuals are calculated at each iteration, weighting is done iteratively. Robust Regression can be summarized as solving at each j-th iteration. Weighting is done according to the residuals of the last fit r y y y Xβ (5) and the leverage matrix

D. Scheme
Assume that we have a 2n×2n cover and an n×n watermark image.

3) Summary
Given the details of the embedding and extraction routines, the overall algorithm can be summarized as follows: 1. Given the input and watermark images, calculate DWT transformations.

For input and watermark images:
a. Calculate SVD of each subband.     , we found optimal d , t and s . Optimality is calculated in terms of PSNR (stealth) and mean Pearson correlation value (robustness) under a range of attacks. We started with "1111" s  , tried several values for t and d , only if an improvement -a greater PSNR of the encoded image and a higher mean correlation value -couldn't be found, we modified s gradually; we first tried "1110" s  , later "1101" s  . and so on. Here, s is of length 4 for the sake of convenience; we never omitted curve fittingapproximation embedded in LL band -modification. Choosing the high-PSNR values of the original DWT-SVD algorithm, we set 1 0.05

Express the input singular values in terms of
used bisquare weighting and the same t for all robust regression steps. Reported mean correlation and PSNR values are obtained when DSCR got a better result compared to DWT-SVD, i.e. these are not the best values, albeit the superior outcome.

2) Astronomy Images
This work started with an application-driven idea of implementing a visual watermark algorithm to the domain of astrophotography. We used Hubble Site [4] for the dataset of images.
Examples of deep sky images are shown in Fig. 1.  Messier 101: a spiral galaxy ( Fig. 1(a)). Estimated number of stars it consists is about one trillion.  NGC 290: a star cluster in the Small Magellanic Cloud ( Fig. 1(b)).  Messier 74: also known as NGC 628, a spiral galaxy slightly smaller than Milky Way (Fig. 1(c)).  LH 95: a star forming region of glowing hydrogen in the Large Magellanic Cloud [3] (Fig. 1(d)). On the encoded image, we measured PSNR and mean correlation. Each correlation value is recorded as in [9], i.e. by taking the highest of all bands (including the fusion result 2.4.1). Differing from [1], we used mean correlation as a final robustness metric (7) where i λ and w λ are the i-th reconstructed singular value vector (from the i-th attacked image) and the visual watermark's singular value vector, respectively. corr is Pearson correlation function.
For NGC 290, the original, encoded and decoded images can be seen in Fig. 2. Detailed results for M101, NGC 290, M74 and LH 95 can be seen in Table I.  Hence, we have an improvement of DWT-SVD via curvefitting and robust regression for each astronomy image. The most striking result is obtained on NGC 290, where we have an obvious advance especially in terms of robustness: a boosted mean correlation value of 0.8511.

3) Classical Images
From the following results, one can easily see that, the proposed technique is not a specialized-for-deep-sky strategy. In this section we demonstrate DSCR embedding on well-known image processing data (https://homepages.cae.wisc.edu/~ece533/images/). For Goldhill, the original, encoded and decoded images can be seen in Fig. 4. Detailed results for Airplane, Baboon, Boat, Goldhill, Barbara, Lenna and Watch are shown in the Table II.

III. CONCLUSIONS
In this work, an idea based on expressing the set of watermark singular values by functions of the cover image's DWT domain singular values is implemented. Since such an approximation enforces a more consistent transform on the cover image's DWT singular values, a higher PSNR value is obtained. Moreover, together with the utilization of a twolevel robust regression, a boosting of the mean correlation value is achieved. Obviously, one can find more sophisticated ways of fusing the components, e.g. through Support Vector Regression, albeit the thematic concentration on iterative reweighted least-squares of this