Discrete Angle Watermark Encoding and Recovery Imants Svalbe+, Andrew Tirkel* and Ron van Schyndel+ ‘Department of Physics, Monash University, Australia *Scient@ Technology, East Brighton, Australia
[email protected]
Abstract Angles derived from sets of image pixel values present an effective medium for embedding “invisible” watermarks. The derived angles can be dithered by the addition of small offset angles. The watermark is a pseudo-noise sequence of dither angles. The watermark embedding is followed by re-quantisation for image storage or transmission. Watermarkrecovery is achieved by performing a complex correlation between the watermarked image and the reference pseudo-noise sequence. This occurs without recourse to the original image. This paper analyses the form of the observed decorrelation as the watermark to image ratio is decreased. The dependence of sequence recovery on 1) the magnitude of the added dithered angle 2) the angle derivation scheme used and 3) on the level of guantisation in the original data is discussed. The novel spatially distributed embedding method used here also ofers potential robust data encryption advantages.
1. Introduction Traditional watermarking schemes modulate the amplitude of pixel values to embed a message, copyright or descriptors into images or other data [l]. This paper describes results for an angle based embedding scheme that exploits discrete Cartesian to discrete polar conversions to watermark arbitrary image data types. The method described in this paper can be applied to pixel values in any transform domain of the image, as well as to the original image values. This research follows an investigation of encoding watermarks in the hue angle of colour image data [ 2 ] . Chae et a1 [3] adopt a related pixel dithering strategy to that used here for the embedding of grey or colour images into the Y channel of W V coded colour images, [5] also adopts a similar “constellation” embedding approach. Watermarking in colour image data is also reported for
0-7695-0750-6/00 $10.00 0 2000 IEEE
246
example in [4], where the blue channel is encrypted to minimise perceptual changes in the watermarked image. Related angle encoding [SI aimed to detect the digital delay times for the fundamental and overtones in discrete phase-angle encrypted carrier wave signals. The objective there was to characterise the properties of non-linear transmission channels rather than watermark. The angles used here are not hue related, but are obtained, for example, as the inverse tangent derived from pairs of pixel values found at a pre-determined pattern of image locations. The set of angles thus derived fiom the image is dithered by adding perturbing angles. The set of derived angles is pre-quantised, to permit the addition of dither to the derived angle vectors without overlap between adjacent angle bins. The perturbing angles are typically downscaled values from some selected alphabet of angles, usually comprised of equal subdivisions over the range 0-2.x. The pattern of perturbing angles is chosen to be isomorphic to a pseudonoise sequence or array of angles with near optimal autocorrelation properties [6, 71. This helps maximise the watermark recovery process after estimation of the sequence of encoded dither angles. Further work is required to make this scheme more robust. The effects of JPEG image compression and small spatial distortions such as local warping and image cropping may be addressed as in [3], by encoding images after mapping to an appropriate transform domain using, for example, wavelets. Section 2 outlines several schemes to derive angles from quantised pixel data and comments on the distribution of angles produced. Quantisation of these derived angles and the subsequent dither encoding mechanism is presented in Section 3. Section 4 briefly describes the auto- and cross-correlation properties of’the pseudo-noise sequences used to select the dither angles. A typical “s-shaped” form for the decrease in correlative recovery results as the size of the embedded angles is decreased to make the watermark less perceptible. Section 5 analyses the contributing effects that cause the s-shaped loss of angle watermark recovery.
negatively offset, or interpreted as two’s complement values, to produce a (related) 360” set, with 8-fold symmetry in the derived angles.
2. Obtaining angles from pixel values Angles can be assigned to directly correspond to a range of pixel grey levels, but such mappings mirror the more traditional 1D amplitude encoding. By selecting, for example, pairs i and j of pixels, with values v, and v,, an angle can be obtained from the ratio of their values as artan[v,/ v]].Adding a dither angle to the derived angle and then re-quantising the resulting total angle components, as v,’and v,: perturbs each pair of original pixel values in a distributed, less perceptible, nonlinear fashion. This process effectively associates pairs of pixel values as projections onto an orthonormal basis. The stored value of either or both selected pixels may change (at the least significant bit level) with respect to the original values, for each encoded pixel entry. This reduces the dependence on and interaction with image content and greatly extends the available encoding space. It maintains, however, the same level of perceptible differences between original and watermarked images. A correlative approach is used to recover the set of dither angles, as outlined in section 5. This guards against sensitivity to individual erasures of the watermark dither angles caused by quantisation. The relative position of the selected pixel pair can be chosen arbitrarily. We chose to use a raster scan pattern that maintains a large distance between selected pixels, such as pairing pixels ti-om opposite ends of the image. This randomizes the derived angle distribution more effectively. Each image pixel can only be dithered once when embedding a single watermark This strategy has proven to be adequate for a range of natural images and for random test images with Gaussian or uniform distributions. Correlation effects between the chosen pixels appear to be negligible, but this still warrants further investigation. Taking the artan of integer ratios does place restrictions on the possible angles and all angles are not equally probable [9]. Some angles, such as 45”, occur ftequently and are also difficult to dither because of the large gaps to the next available angles with integer ratios. The addition of a small dither to a vector needs to survive the re-quantisation of the pixel derived angle components fiom v, to v,’, so length of the vectors associated with each pixel value pair also affects the dither encoding ability. The selected pixel pairs can also be filtered to ensure the vector length exceeds some minimum value. For images with a small number of quantisation levels, 0 I v IV, the shorter maximum length of these vectors also restricts the available angle space for encoding. A ratio of positive integers produces angles over the range 0 to 90 degrees, with an angle distribution symmetric around 45 degrees. Pixel values can be
3. Angle quantisation and dither encoding The set of derived angles is quantised to provide a fixed, minimum interval, of size de = 2 d s , between all derived angle members. The gaps between the allowed quantised angle values are used to store the added dither angles, ensuring separability of the image values and the embedded watermark. The scale parameter s can range from 1 to 400 before dithered angles fail to be encoded and recovered sufficiently for 8-bit pixel data. The quantisation method provides superior isolation of watermark from image content. Other schemes use the mean or median of the local angle distribution to estimate the angle onto which the dither is added [2]. Here the local angle associated with any pixel is largely unrelated to the angle associated with the adjoining pixels, because of the use of a spatially disjoint pixel pair selection mechanism. This angle-encoded method is immune to local angle estimation techniques. The dither angle to be embedded for each pixel pair corresponds to that of a unit vector with angles 2 d n , where n is an integer. For binary sequences, the dither angle is effectively f d 2 . When adding a binary dither angle to the derived angle, the unit dither angle vector is added in a direction perpendicular to the derived angle, to maximise the separability of the watermark signal ftom the image derived angle.
4. Pseudo-random dither angle sequences The sequence of dither angles to be embedded is chosen to be a 1D “complexified” Legendre sequence (or a related 2D Legendre array) [6,7, 101. Binary Legendre sequences, of (prime) length p , have optimal auto-correlation values of p@-I) or -1 and a cross-correlation magnitude of order p [6]. The strong Legendre correlation properties enable extremely robust extraction of the watermark signal. Recovery is possible even when the encoding losses are significant due to the smallness of the added dither signal or because of perturbation of the image due to noise added in the transmission channel. Legendre sequences have a second favorable property, in being invariant to Fourier transformation [ 1 11. Without requiring knowledge other than the length of the encoding sequence, a strong cross correlation will be obtained between the encoded sequence and its Fourier transform. The message can thus be determined as being watermarked, without needing to disclose the watermark content, although the encoded data capacity will be low.
247
Figure 3 shows a histogram of the recovered dither angles, again for the case of uniformly distributed %bit image data and a binary Legendre sequence. As the value of s increases, the two peaks in the recovered angles, centered on f nI2, spread and eventually merge so that the dither vectors are miss-classified. In Fig. 1, the curve marked “With binary Snap” is obtained by rounding the estimated dither angles to either 7d2 or 4 2 before correlation with the reference Legendre sequence. Using the “snapped” recovered angles increases the resulting complex correlation, as the estimated vectors all lie parallel or anti-parallel to the reference set. The same snapping approach can be used in the recovery of an arbitrary number of dither angles.
5. Mechanisms affecting angle recovery This section presents some of the results obtained for the dithered angle-watermarking scheme. Figure 1 shows the characteristic “s-shaped” loss of correlative recovery that occurs as the size of the scaling parameter s increases, making the added dither angles smaller. For 8-bit uniformly distributed data, the watermark can still be recovered for s = 400, where the added dither angle is about 0.2”, for the case of binary sequence encoding. The image signal to watermark ratio is about 50 dE3 in this case for a 25 1*251 watermark.
\-
08-
I
‘”4
0.04 c
f 0.035
,
n
A
g 0
s
0.03-
:;
/’
0.025 -
,:I
I
SFES
0
50
100
150 200 250 Scale Factor
300
350
400
Angle
Figure 1. Relation between encoding scale-factor and degree of correlative recovery for a uniformly distributed 8-bit noise image with and without snapping the recovered angles (see text).
Figure 3. Histogram of recovered angles for a binary watermark. Curves for scale factors, SF, of 2,5,10,20 are shown.
Figure 2 shows that the maximum value of s required to secure a fixed ii-action (here 90%) of the possible peak correlation value scales linearly with the number of quantisation bits, V, used to represent the original image data. The Legendre sequence correlation value also drops linearly with the number of sequence entries that are incorrectly encoded or recovered.
the estimated dither vector angles accounts for the loss of correlation. The cosine of the estimated angle for each unit vector gives the useful component that adds constructively to the correlation. The sum of the cosine weighted unit vectors reproduces the s-shaped curve of Figure 1, with excellent precision. The sum of the sines of the recovered vector angles converges to zero.
Figure 4 confms that the degree of misalignment of
:7]
350 300
+“Q,.
09-
-
19,
Sum of Cos-WeighledAngles NormalisedConelabon
i,
508-
v)
; 07-
b
0
e
z06e! $05-
/’
z“ 200
ff 150-
/
/’
0
-
E
k
xa 0 4 -
s
03-
\
202-
-0
200
400 600 800 1000 Number of QuantisationLevels
1200
0
%* a , .>
01 -
=* % = 50
100
150 200 250 Scale Factor
300
350
400
Figure 4. The sum of the recovered angles, weighted by the cosine of the distance to the template angle. This yields a distribution identical to the correlation curve of Fig. 1.
Figure 2. Dependence of degree of correlative recovery with image bit depth (expressed as number of quantisation levels).
248
7. Acknowledgments
Figure 5 shows the number of true and false classifications for the recovered angles of the unit vectors as a function of the scaling parameter s. The true and false curves converge when the dither angle estimation becomes random.
RvS is supported an Australian Postgraduate Award, provided through the Australian Research Council. Thanks are also due to Professors Charles Osborne and Tom Hall from Monash for useful discussions.
8. References [ 13 N. Nikolaidis, I. Pitas, “Digital Image Watermarking: an Overview”, ICMCS 99, Volume 1, Florence, Italy, pp 1 - 6.
0.35 0
e
0.3-
[2] R. van Schyndel, A. Z. Tirkel, I. D. Svalbe, “A MultiplicativeColor Watermark”, IEEE-EURASIP Workshop on Non-Linear Signal and Imaging Processing,Antalya, Turkey, 1999, pp. 336-340.
y,o.25 c 5 0.2 -
-
0.15
”
---
0
50
100
150 ZOO 250 Scale Factor
300
350
[3] J. J. Chae and B. S. Majunath, “A Technique for Image Data Hiding and Reconstructionwithout Host Image”, SPIE Conference on Securityand Watermarking of Multimedia Contents, San Jose, Califomia, USA, January, 1999, pp. 386 396.
400
Figure 5. The fraction of recovered angles classified as true and false matches to the watermark template as a function of scale factor
[4] M. Kutter, “Digital Signature of Color Images using Amplitude Modulation”,SPIE, vol. 3022, San Jose, Feb., 1997, pp. 518-525.
6. Conclusions
[5] B. Tao and M. Orchard, “Coding and Modulation in Watermarkingand Data Hiding”, SPIE Conference on Security and Watermarkingof Multimedia Contents, San Jose, California, USA, January, 1999, pp. 503 - 510.
An angle-encoding scheme has been described to embed watermarks in arbitrary image data. The angle-
encoding scheme uses pairs of pixel values to derive an angle, onto which a dither angle is added as the watermark content. The sequence or array of added dither values is derived fkom complexified Legendre sequences [6] to maximise the correlative recovery of the watermark. We have measured the decrease in correlation as the image to watermark signal ratio increases and have outlined the relative effect of contributing factors determining the decline in watermark recovery. The inclusion of a transform based embedding step is required to provide some resistance to compression losses. Future work would also include provision of an adaptive mechanism to adjust the scaling parameter s to better suit the individual pixel pair values for each discrete derived angle. This will increase the coding density and the recoverability of the watermark. A watermark can also be embedded in arbitrary images in higher dimensions by associating more than 2 pixels to form a vector . For example, using a triplet of pixel values (x, y, z) allows two dither angles to be added (as de and d@)to a 3D (r, e,@)vector. This extension should offer further enhancements to the coding advantages already described in this paper. We aim to investigate the generation of pseudo-random sequences or arrays for double angles.
[6] R. M. Schroeder, Number Theory in Science and Communications,2”dEdn. 1997, Springer-Verlag. [7] A. Z. Tirkel, R. van Schyndel, T. E. Hall, C. F. Osborne, “SecureArrays for Digital Watermarking”, IEEE Intemational Conference on Pattern Recognition,Brisbane, Australia, August 1998, pp. 1643-1645. [8] R. van Schyndel, A. Z. Tirkel, I. D. Svalbe, “DelayRecovery from a Non-Linear Polynomial Response System”,IEEE International Workshop on Intelligent Signal Processing and Communication Systems, Melbourne, November 1998, Vol. 1, pp. 294-298. [9] I. D. Svalbe, “Natural Representationsfor the Hough Transform”, IEEE Transactionson Pattern Analysis and Machine Intelligence”, vol. 12, no. 2, 1991, pp. 336-342. [lo] R. G. van Schyndel, A. Z . Tirkel, I. D. Svalbe, T. E. Hall and C. F. Osbome, “Algebraic Constructionof a new class of Quasi-orthogonal Arrays in Steganography”, SPIE Electronic Imaging 1999, San Jose, USA, January, 1999, pp. 354 - 364. [ 111R. van Schyndel, A. Z. Tirkel, I. D. Svalbe, “Key Independent WatermarkDetection”,IEEE International Conference on Multimedia Computing and Systems, Florence, Italy, Vol. 1, June 1999,pp. 580-585.
249
