consider an image i ( x , y ) and a rotated , scaled , and translated ( rst ) version of this image , i ′ ( x , y ). then we can write i ′ ( x , y )= i ( σ ( x cos α + y sin α )− x o , σ (− x sin α + y cos α )− y o ) ( 2 ) where the rst parameters are α , σ , and ( x o , y o ) respectively . the fourier transform of i ′ ( x , y ) is , i ′ ( f x , f y ) the magnitude of which is given by : | i ′ ( f x , f y |=| σ | − 2 | i ( σ − 1 ( f x cos α + f y sin α ), σ − 1 (− f x sin + f y cos α ))| ( 3 ) equation 3 is independent of the translational parameters , ( x o , y o ) . this is the well known translation property of the fourier transform . | i ′( f x , f y )|=| σ | − 2 | i ( σ − 1 e ρ cos ( θ − α ), σ − 1 e ρ sin ( θ − α ))| ( 6 ) =| σ | − 2 i ( e ( ρ - logσ ) cos ( θ − α ), e ρ - logσ ) sin ( θ − α )| ( 7 ) 1 . that the amplitude of the log - polar spectrum is scaled by | σ | − 2 , 2 . that image scaling results in a translational shift of log σ along the ρ axis , and 3 . that image rotation results in a cyclical shift of α along the θ axis . we need not be concerned with the amplitude scaling of the spectrum , since we intend to perform watermark detection using the correlation coefficient , which is invariant to this scaling . next , g ( θ ) is defined to be a one - dimensional projection of | i ( ρ , θ )| such that g  ( θ ) = ∑ j   log   (  i  ( ρ j ,  θ )  ) ( 9 ) the reason for summation of the log values rather than the magnitudes themselves is discussed below . due to the symmetry of the spectra of real images , we find it convenient to add the two halves of g ( θ ) together , obtaining with θ ′ ε [ 0 ° . . . 90 °). the reasons for this are also discussed below . clearly , g 1 ( θ ), is invariant to both translation and scaling . however , rotations result in a ( circular ) shift of the values of g 1 ( θ ). if θ is quantized to the nearest degree , then there are only 90 discrete shifts , and we handle this by an exhaustive search . in principle , detectors may be built that can handle watermarks encoding several bits . however , the detector of the present invention determines only whether or not a given watermark has been embedded in a given image . it takes as input , an image i , and a watermark w , and the output is a single bit indicating whether image i contains watermark w . the watermark w , is expressed as a vector of length n . to determine whether the watermark is present , we first compute an “ extracted signal ” v , from image i , by computing the descriptor g 1 ( θ ), for n values of θ evenly spaced between 0 ° and 90 °. we then compute the correlation coefficient between w and v . if the correlation coefficient is above a detection threshold t , then the image is judged to contain the watermark . the use of correlation coefficient as a detection measure is known in the art . one benefit of this metric is its independence to scaling of the signal amplitudes . 1 . compute a discrete log - polar fourier transform of the input image . this can be thought of as an array of m rows by n columns , in which each row corresponds to a value of ρ , and each column corresponds to a value of θ . 2 . sum the logs of all the values in each column , and add the result of summing column j to the result of summing column j + n / 2 ( j = 0 . . . ( n / 2 − 1 )) to obtain an invariant descriptor v , in which where θ j is the angle that corresponds to column j in the discrete log - polar fourier transform matrix . 3 . compute the correlation coefficient d , between v and the input watermark vector w , as d = w · v ( w · v )   ( v · v ) ( 13 ) 4 . if d is greater than a threshold t , then indicate that the watermark is present . otherwise , indicate that it is absent . once a method for detecting watermarks has been defined , a watermark embedding algorithm can be constructed according to methodology known in the art . in the methods known in the art , watermarking is cast as a case of communications with side information at the transmitter . the difference between this view of watermarking , and a more common view known in the art , is as follows . in most public watermarking methods known in the art , the original image is considered to be noise . the embedder adds a small - amplitude signal to this noise , and the detector must be sensitive enough to work with the small signal - to - noise ratio that results . however , this common approach ignores the fact that the embedder has complete knowledge of the “ noise ” caused by the original image . if we view the embedder as a transmitter and the cover image as a communications channel , then this knowledge amounts to side - information about the behavior of that channel . when the transmitter knows ahead of time what noise will be added to the signal , its optimal strategy is to subtract that noise from the signal before transmission . the noise then gets added back by the communications channel , and the receiver receives a perfect reconstruction of the intended signal . in the case of watermarking , it is unacceptable for the embedder to subtract the original image from the watermark before embedding the watermark , because it would result in unacceptable fidelity loss . in fact , if the watermark is expressed as a pattern that is the same size as the image , then this strategy simply replaces the image with the watermark pattern , which is clearly too drastic . however , when the watermark is expressed as a signal in a lower - dimensional space , as is the case with the present system , the results need not be so drastic , since a wide variety of full - resolution images project into the same extracted signal and the embedder may choose the one that most resembles the original . but even in the case of lower - dimensional watermarks , it is not always possible to completely replace the extracted signal with the watermark signal while maintaining acceptable fidelity . to make maximal use of the side - information at the embedder , while maintaining acceptable fidelity , a method known in the art introduces the idea of a “ mixing function ”, f ( v , w ). this takes an extracted signal v , and a watermark vector w , as input , and the output is a signal s , which is perceptually similar to v , and has a high correlation with w . since s is something between v and w , it is referred to as the “ mixed signal ”. it is this mixed signal that the embedder transmits , by modifying the image so that the extraction process in the detector will produce s . the basic pattern for this type of embedding described in the art consists of three steps : 1 . apply the same signal - extraction process to the unwatermarked image as will be applied by the detector , thus obtaining an extracted vector , v . in our case , this means computing g 1 ( θ ). 2 . use the mixing function , s = f ( v , w ), to obtain a mixture between v and the desired watermark vector , w . at present , a mixing function which simply computes a weighted average of w and v is utilized . this is obviously not a preferred or optimal approach . however , it should be apparent to those skilled in the art that more sophisticated methods now known or later developed can also be used . 3 . modify the original image so that , when the signal - extraction process is applied to it , the result will be s instead of v . step 3 is the most difficult . a natural approach would be to modify all the values in column j of the log - polar fourier transform so that their logs sum to s j instead of v j . this can be done , for example , by adding ( s j − v j )/ m to each of the m values in column j . next , the log - polar resampling of the fourier magnitudes would be inverted , thus obtaining a modified , cartesian fourier magnitude . finally , the complex terms of the original fourier transform would be scaled to have the new magnitudes found in the modified fourier transform , and the inverse fourier transform would be applied to obtain the watermarked image . the main implementation issue in such an approach is the inherent instability in inverting the log - polar resampling . the methods of the present invention therefore approximate this step with an iterative method in which a local inversion of the interpolation function is used for the resampling . this method is described below . there are a number of problems that arise when implementing the previous methods discussed above . several of these are addressed below . the log - polar fourier transform of an image can be computed by resampling the image dft with a log - polar grid . some interpolation method must be used during the resampling , since the log - polar sample points don &# 39 ; t generally coincide with the cartesian sample points in the dft . the dft is conventionally assumed to represent a tiled version of an image , as illustrated in fig1 ( a ). those in the art have suggested that this tiling pattern represents an inherent problem for any algorithm that relies on the rotational properties of fourier transforms , since , when the content of an image is rotated , the rectilinear tiling grid is not rotated along with it . thus , the dft of a rotated image is not the rotated dft of that image . this problem is illustrated in fig1 ( b ) and ( c ). one possible solution is to compute the log - polar fourier transform directly , without using the cartesian dft as an intermediate step . in the continuous fourier domain , each point has a value determined by correlating the image with a complex , planar sinusoid . if we wish to obtain a value for a point between those that would be sampled in a dft , we can find the corresponding sinusoid and directly compute its correlation with the image . this amounts to assuming that all the pixel values outside the bounds of the image are black , rather than assuming they are tiled copies of the image . of course , the direct approach described above doesn &# 39 ; t take advantage of the efficient methods available for computing dft &# 39 ; s , and is thus likely to be prohibitively expensive . instead , one can approximate the log - polar fourier transform with the following steps : 1 . pad the image with black to obtain a larger image . 2 . take the dft of the padded image . this yields a more finely sampled version of the continuous fourier transform . 3 . resample in a log - polar grid , using an inexpensive interpolation technique . the preferable technique used is linear interpolation of the magnitudes of the coefficients . by padding with black to obtain a denser sampling of the fourier transform , we reduce the distances between the dft &# 39 ; s sample points and the log - polar sample points , thus reducing the error introduced by inexpensive interpolation . each element of the log - polar fourier magnitude array is a weighted average of up to four elements of the cartesian fourier magnitude array . thus , we can write where f is a column vector containing all the elements of the log - polar array , c is a column vector containing the elements of the cartesian array , and m contains the weights used to perform interpolation . if we wish to modify the log - polar array so that it contains the watermark , and then find the corresponding cartesian array , we have to find the inverse of m . unfortunately , m is ill - conditioned and it is not practical to perform this inversion precisely . instead , one can use an iterative process to perform an approximate inversion . let f ′ be the modified version of f . we begin by observing that the four non - zero values in each row of m sum to 1 . thus , if we add f ′ i - f i to each of the elements c j1 . . . c j4 , where m i , j1 . . . m i , j4 are non - zero , then the resulting cartesian array will yield f ′ i in its log - polar mapping . unfortunately , if we try to apply this method to change all the elements of f , we &# 39 ; ll have conflicting changes in the various elements of c . for example , both m i , j and m k , j might be non - zero , so that we &# 39 ; d want to change c j both when changing f i to f ′ i and when changing f k to f ′ k . the desired changes are unlikely to be the same . we resolve this problem by using a weighted average of all the desired changes to each element of c . so , in the above example , we would change the value of c j by m i , j  ( f i ′ - f i ) + m k ′  j  ( f k ′ - f k ) m i , j + m k , j ( 15 ) ( assuming that m i , j and m k , j are the only nonzero elements of column j ). the above method results in a rough approximation to the desired inversion . we can obtain successively better approximations by applying the operation iteratively . in practice , we find it most effective to iterate the entire watermark embedding process described above , using the above approximate log - polar inversion in each iteration . in practice , it has been found that three or four iterations usually suffice to produce an approximation that can be detected by the detector . it is well known that the rectangular boundary of an image usually causes a “ cross ” artifact in the image &# 39 ; s energy spectrum ( see fig2 a and 2 b ). this happens because there is usually a large discontinuity at each edge of the image due to the implicit tiling . the dft magnitude of such vertical and horizontal discontinuities has large energy in all the vertically and horizontally oriented frequencies , which results in the cross artifact . if the image is rotated , but padded with black so that no image content is cropped , then the cross in the dft magnitude will also rotate ( fig3 a and 3 b ). if , on the other hand , the rotated image is cropped , so that no black is added , then the new image boundaries cause a horizontal and vertical cross similar to that found in the original image , even though the rest of the dft magnitude is rotated ( fig4 a and 4 b ). since the cross has so much energy , it tends to cause two large bumps in the extracted watermark vector , which substantially reduce the correlation coefficient with the embedded watermark . a solution to this problem is to simply ignore the bumps in the extracted signal by ignoring a neighborhood around each of the two highest - valued elements . alternative solutions that appear in the literature include multiplication of the image by a circularly - symmetric window and blurring of the image edges . these solutions are probably more general than the one employed here , but would require modification to the watermark embedder . the magnitude of low frequencies can be very much larger than the magnitude of mid and high frequencies . in these circumstances , the low frequencies can become overwhelming . to reduce this problem , we sum the logs of the magnitudes of the frequencies along the columns of the log - polar fourier transform , rather than summing the magnitudes themselves . a beneficial side - effect of this is that a desired change in a given frequency is expressed as a fraction of the frequency &# 39 ; s current magnitude rather than as an absolute value . this is better from a fidelity perspective . it is well known that the lowest and highest frequencies in an image are usually unreliable for watermarking . the low frequencies are unreliable because they are difficult to modify without making visible changes in the image . the high frequencies are unreliable because they can be easily modified by common processes such as compression , printing , and analog transmission . our solution is to neglect these unreliable frequencies when extracting the watermark . a better solution would be to use a perceptual model to estimate the maximum amount of change that can be applied to each frequency and a model of specific attacks to estimate the degree of robustness . the amount of watermark energy embedded into each frequency would then be proportional to this perceptual significance and robustness . such an approach is known in the art . application of this idea can also be applied to the present watermarking method . the energy in an image is seldom evenly distributed in angular frequency . images frequently have a large amount of energy in one group of directions , while having much lower energy in an orthogonal group of directions . for example , images containing buildings and trees have significant vertical structure yielding more energy in the horizontal frequencies than in the vertical ( fig5 a and 5 b ), while seascapes or sunsets are strongly oriented in the horizontal direction yielding higher vertical frequencies ( fig6 a and 6 b ). spectra such as those of fig5 b and 6 b suggest an uneven masking ability in orthogonal directions . as a consequence , it may be much easier , from a fidelity perspective , to embed some portions of the watermark than other portions of the watermark . for example , when watermarking the image of tall buildings , we can more easily hide noise with a strong vertical component than noise with a strong horizontal component . this can be a problem if the difficult - to - modify portions of the watermark are critical in differentiating it from other watermarks . to reduce this problem , we divide the extracted signal into two halves , and add the two halves together . thus , rather than using g ( θ ) of equation 9 , we use g 1 ( θ ), of equation 11 . if we want to modify an element of g 1 ( θ ), we can do so by hiding noise that &# 39 ; s oriented along either angle θ or angle θ + 90 °. this increases the likelihood that each element of the watermark can be embedded within the fidelity constraints . for natural images , g 1 ( θ ) is likely to vary smoothly as a function of θ . in other words , the extracted signal will have more low - frequency content than high - frequency content . this reduces the effectiveness of the correlation coefficient as a detection measure . we improve the detection measure by applying a whitening filter to both the extracted signal and the watermark being tested for before computing the correlation coefficient . note that the whitening filter is employed only in the watermark detector ; the embedder is unchanged . the whitening filter was designed to decorrelate the elements of signals extracted from natural images , and was derived from signals extracted from 10 , 000 images from a publically available photo image library ( these images were not used in any of the subsequent experiments reported below ). the idea of using a whitening filter to improve watermark detection in this way is also known in the art . during watermark embedding , it is difficult to change the value of one element of the extracted watermark , without changing the values of its neighbors . this results primarily from the fact that any one frequency in the dft can effect several values of g 1 ( θ ), so changing that frequency can effect several elements of the watermark . because of this , it is difficult to embed a watermark that varies wildly from one element to the next . we reduce this problem by replicating elements of the desired watermark to obtain a lower - frequency watermark . for example , if the watermarks are extracted by computing 74 samples of g 1 ( θ ) ( after removing the samples that contain the “ bumps ” discussed above ), then we would define our desired watermark as a vector of 37 values , and duplicate each of its 37 values to obtain a length 74 vector . the following results were obtained by extracting a length 90 vector form the image and neglecting the 16 samples surrounding the peak ( assumed to correspond to the dft cross artifact ). this leaves a descriptor that is 74 samples in length . the detection process involves a comparison of the watermark with all cyclic rotations of the extracted descriptor . note that there are 90 such rotations . we begin our evaluation of the watermarking method of the present invention by finding the relationship between the threshold , t , and the probability of false positives , p fp . a false positive occurs when the detector incorrectly concludes that an unwatermarked image contains a given watermark . so where d max is a detection value obtained by running the detector on a randomly selected , unwatermarked image . to estimate p fp empirically , we ran the detector on 10 , 000 unwatermarked images from a commercially available image database , the 10 , 000 images used in this test were all different from the 10 , 000 images from the same database that were used to generate the whitening filter described above , testing for 10 different binary watermarks in each . fig7 ( a ) shows the 10 resulting histograms , superimposed on one another . fig7 ( a ) indicates that most detection values from unwatermarked images fall between 0 . 2 and 0 . 4 . this might seem surprising , since we might expect unwatermarked images to yield detection values closer to zero . the reason the values are so high is that each one is the maximum of 90 different correlation coefficients , computed during the cyclical search discussed above . this means that s ( t )= p { d max & gt ; t }= p { d 0 & gt ; t ) or ( d 1 & gt ; t ) or . . . ( d 89 & gt ; t } ( 17 ) where d 0 are the 90 correlation coefficients computed during the search . each of d 0 . . . d 89 is drawn from a distribution that is centered around zero , as shown in fig7 ( b ), which shows 10 superimposed histograms of the 90 , 000 correlation coefficients computer for each of the 10 watermarks during the experiment . the maximum of 90 values drawn from a distribution like that in fig7 ( b ) is likely to be higher than zero . during the experiment with unwatermarked images , the highest detection value obtained was 0 . 55 . thus , we have no data for estimating p fp for t & gt ; 0 . 55 . to estimate this , we must employ a theoretical model . one such theoretical model says that , if d is the correlation coefficient between a preselected d - dimensional watermark vector and a random vector drawn from a radially - symmetric distribution , then p  { d & gt ; t } = r  ( t , d ) = ∫ 0  cos - 1  ( t )  sin d - 2  ( u )    u 2  ∫ 0 n / 2  sin d - 2  ( u )   u ( 18 ) the whitening filter employed in our detector makes the distribution roughly spherical , so we can expect this model to apply to the present system , with d = 74 . however , the model only gives the probability that one correlation coefficient is greater than the threshold , not the probability that the maximum of several coefficients is greater . thus , it should give us p { d i & gt ; t }, iε [ 0 . . . 89 ], rather than p { d max & gt ; t }. fig8 ( a ) indicates how well the model predicted p { di & gt ; t } in our experiment . we obtain an estimated upper bound on p { d max & gt ; t } by observing that p  { q 0   or   q 1   or    …   q n - 1 } ≤ min   ( 1 , ∑ i   p  { q i } ) ( 19 ) when qi corresponds to the event ( d i & gt ; t ), and n = 90 , we obtain fig8 ( b ) shows how well this function predicts p { d max & gt ; t } in our experiment . the tradeoff between fidelity and robustness is controlled by adjusting the weights used in the mixing function ( discussed above ). if the weight of the watermark is increased , then a stronger watermark will be embedded at the expense of lower fidelity . in our experiments , the same weights are used for every image . the weights were chosen so that we usually obtained a signal - to - noise ratio of about 40 db . here the “ signal ” is the image , and the “ noise ” is the watermark pattern . fig9 shows a histogram of the ratios obtained . fig1 shows an example of a watermarked image with little impact on fidelity . it must be noted , however , that signal - to - noise ratio is not a very effective predictor of perceptual quality . the fidelity of the image depends to a large degree on the perceptual relationship between the image and the noise . in general , noise that matches the underlying textures in an image is less perceptible than noise that is very different from the image , even at the same signal - to - noise ratios . the present system generates watermark patterns by making small percentage adjustments to the powers of frequencies in the image &# 39 ; s spectrum , so the resulting noise pattern is usually similar to the textures in the image . thus , when we watermark an image that contains a homogeneous texture , the watermark is well - hidden . but when we mark an image that contains widely varying textures , the mark can become visible . fig1 illustrates the problem . the watermark strength in fig1 was increased so that the problem should be visible after printing in a journal . solving the fidelity problem in non - homogeneous images would require a modification to the algorithm that attenuates or shapes the watermark according to local texture characteristics . two experiments were performed to test the robustness of a watermark inserted by the methods of the present invention against rotation . the first experiment was designed to isolate the effects of rotation from all other types of attack . the second was a more realistic test of the effects of rotation with cropping . 1 . pad an image with neutral gray , increasing its size . the amount of padding was chosen to allow rotation without any part of the original image going outside of the image boundaries ( fig1 ( a )). 3 . replace the padding with neutral gray again . this removes any watermark information from the neutral gray area . 4 . run the watermark detector on the image to obtain a detection value before rotation . 5 . rotate the image by a predetermined angle , and crop to the original size . fig1 ( b ) shows what an image looks like after this step . note that only the padding is cropped , so we do not crop off any of the watermark pattern . 6 . run the watermark detector on the image to obtain a detection value after rotation . since the padding that &# 39 ; s cropped off during rotation contains no watermark pattern , any difference between the “ before ” value obtained in step 4 and the “ after ” value obtained in step 6 can only result from the effects of rotation . this experiment was performed on 2 , 000 images with rotations of 4 °, 8 °, 30 °, and 45 °. we limited this test to a maximum rotation of 45 ° because rotations beyond 45 ° are equivalent to smaller rotations after a rotation of 90 °. an image that has been rotated 90 ° yields exactly the same extracted vector as an unrotated image , so a rotation of greater than 45 ° should behave the same as a smaller rotation . as indicated in fig1 ( a ), the different rotations yielded essentially the same results . fig1 ( b ) shows receiver - operating - characteristic ( roc ) curves before and after rotation . for each of the roc curves , the false - positive probabilities were estimated using the method described above . in the second experiment , we watermarked the original image without padding , and allowed part of the watermark pattern to be cropped off after rotation . fig1 ( f ) shows an example of what an image looked like after the rotation . this experiment was performed on 2 , 000 images with rotations of 4 °, 8 °, 30 °, and 45 °. fig1 a and 14 b shows the results . the results of these experiments indicate that rotation alone has only a small effect on the probability of detection . the primary effects on detection probability are how well the watermark is embedded initially , and how much of the watermark is cropped off . in the first experiment , all the cropping happens before we rotate when we replace the padding with unwatermarked gray . the second experiment shows a larger effect of rotation , and larger variation between the different rotation angles , because different rotations result in different amounts of cropping . to test robustness to scaling , three experiments were performed . the first and second test the effect of scaling up , with and without cropping . the third tests the effect of scaling down , with padding . in the first scaling test , the steps performed for each trial were the same as those for the first rotation step , with the exception that instead of rotating the image we scaled the image up . fig1 ( c ) shows an example of an image that has been scaled up after padding and watermarking . the test was performed on 2 , 000 images at scales 5 %, 10 %, 15 %, and 20 % larger than the original . the results are shown in fig1 a and 15 b . the second test was the same as the first except without padding the images before scaling , so part of the image was cropped off after scaling . fig1 ( g ) illustrates the attack . the test was performed on 947 images at scales of 5 %, 10 %, 15 %, and 20 % larger than the original . the results are shown in fig1 a and 16 b . for the test of reduced scaling , we do not have to be concerned with cropping . rather , after watermarking and scaling , the image is padded back to its original size . since cropping is not an issue here , we only performed one version of this experiment , in which the image was not padded before watermarking as shown in fig1 ( h ). the test was performed on 2 , 000 images at scales 5 %, 10 %, 15 %, and 20 % smaller than the original . the results are shown in fig1 a and 17 b . as with rotation , scaling by up to 20 % appears to have very little impact on the probability of detection . in fact , decreasing size by up to 20 % has essentially no impact on detection , except at very high thresholds . the results show that the cropping which often accompanies an increase in scale can degrade the watermark detection . it is expected that translation alone would have no effect on the watermark , since the watermark is computed from the magnitudes of the fourier coefficients . to test this , two experiments were performed . the first experiment was similar to the first rotation and scaling experiments , in that the image was padded before watermarking and the padding was replaced after watermarking . we then translated the image by cropping gray off the top and right , and padding gray onto the bottom and left . fig1 ( d ) shows an example of such a translated image . the experiment was performed on 2 , 000 images at translations of 5 %, 10 %, and 15 % of the image size . the results are shown in fig1 a and 18 b . the second translation test was performed without padding the image before translation , so that part of the watermark pattern is cropped during translation . fig1 ( i ) shows an example of this attack . again , the experiment was performed on 2 , 000 images at translations of 5 %, 10 %, and 15 % of the image size . the results are shown in fig1 a and 19 b . the results of the first experiment show that translation has negligible effect on probability of detection . this means that the second test is more a test of robustness to cropping than to translation , and we see the same sort of pattern that was observed in the second rotation and scaling experiments . while the purpose of the present watermark design is to survive rst transformations , it is , of course , important that the watermarks also survive other common types of image processing . tests were also conducted on robustness to jpeg compression . after watermarking , images were jpeg compressed at quality factors of 100 %, 90 %, 80 %, and 70 %. the test was performed on 1 , 909 images . fig2 a and 20 b show the results . thus , the methods of the present invention also show some resistance to mild jpeg compression . geometric distortions continue to be a major weakness for many watermarking methods . the methods of the present invention provide a solution to the common problems of rotation , scale , and translation . this solution is related to earlier proposals in the pattern recognition literature regarding invariants of the fourier - mellon transform . however , unlike those proposals , an invariance relationship is not explicitly derived . instead of creating a truly rst invariant signal , a signal is created that can be searched for the effects of rst in a trivial manner . the calculation of this projection is performed by taking the fourier transform of the image , performing a log - polar resampling and then integrating along the radial dimension . it is important to note that an alternative implementation can be performed using other transforms such as the radon transform . the one - dimensional watermark has a many - to - one mapping to the two - dimensional image space . this is advantageous , especially when the embedder is based on the principle of communications with side information . the implementation disclosed herein is a very simple example of this principle can lead to significant improvements . experimental results on a database of over 2 , 000 images clearly demonstrate that the method of the present invention is robust to either rotations , scale changes or translations . it is also robust to some jpeg compression , and has some resistance to cropping . while there has been shown and described what is considered to be preferred embodiments of the invention , it will , of course , be understood that various modifications and changes in form or detail could readily be made without departing from the spirit of the invention . it is therefore intended that the invention be not limited to the exact forms described and illustrated , but should be constructed to cover all modifications that may fall within the scope of the appended claims .