Patent Application: US-5180802-A

Abstract:
the present invention relates to the methods of estimation and recovering of general affine geometrical transformations which were applied to data , extensible to any other defined class of geometrical transformations , according to the preamble of the dependent claims . the parameters of the undergone deformation are robustly estimated based on maxima given by a parametric transform such as hough transform or radon transform of some embedded information with periodical or any other known regular structure . the main applications of this invention are robust digital still image / video watermarking , document authentication , and detection of periodical or hidden patterns . in the case of periodical watermarks , the watermark can also be predistorted before embedding based on a key to defeat block - by - block removal attack .

Description:
we formulate the method as recovering of the general affine transformation , or other global geometrical distortion , applied to an image , using regular grids of points for the estimation of affine matrix coefficients , based on the hough transform ( ht ), the radon transform ( rt ), or any other parametric transform . the entire approach is based on a regular grid of points , extracted from an input and possibly distorted image , and compared with a known reference grid , as explained below . if it is a color image , in rgb or any other color representation , then points can be extracted , and the described method applied , either from the luminance component , or from each color - plane separately , or any combination of these color - planes . the synchronization information can be embedded either in the spatial domain , or in any transform domain such as discrete fourier transform ( dft ), discrete cosine transform ( dct ) or discrete wavelet transform ( dwt ), or in the succession of frames along the time axis in the case of video applications . the first important point to mention is that the approach described here is not suitable for completely random or pseudo - random templates , since our method relies on an a priory known regularity for the extraction of the wanted features . in the case of periodical synchronization data or template undergoing an affine transformation , points are expected to be placed at intersections between two classes of equidistant spaced parallel lines as shown in fig2 . theses two classes of lines should present distinct directions , distances between lines should be non - null and equal inside each class ( therefore points are equally spaced along lines ), but distances between points may differ from one class to the other . our method is typically applicable to local maxima , or peaks , resulting from the auto - correlation function ( acf ) or the magnitude spectrum ( ms ) generated by periodic patterns , and which keeps this regularity under any affine transformation ; for this purpose a periodic pattern w should first be embedded to the original image . the acf approach consists in computing the auto - correlation ŵ * ŵ of the periodical pattern ŵ which has been estimated from the possibly distorted image , where “*” is the convolution operator . the acf can be computed as ŵ * ŵ = f − 1 ( f ( ŵ ) 2 ), where f is the dft , ( . . . ) 2 the complex square , and f − 1 the inverse dft ( idft ). peaks can also be extracted from the ms without computing the acf ( thus avoiding the idft computation ), i . e . from the magnitude of the fourier transform , expressed as m =| f ( ŵ )| where | . . . | denotes the magnitude . the second important point concerns the robustness of the proposed approach with respect to one or more distortions , including affine transformation combined with lossy compression like jpeg or any signal degradation or alteration . as mentioned in the previous section , the estimated input points are generally noisy , with missing points , additional false points , and errors in positions of the remaining correct points such as rounding errors . therefore using directly these points would lead to a wrong estimation of the applied geometrical distortion . using the a priory knowledge of the expected regularity or any special shape of the underlying grid , we can increase the contribution of correct points with respect to that regularity , while wrong points will tend to cancel their contribution due to their randomness properties . in the case a periodical grid , random points are less likely to present significant alignments than correct ones , and therefore the ht or rt can be used to robustly detect such alignments . as already mentioned , the method is extensible to any kind of regular grid , including not only straight lines , but curves , by using generalized ht or rt ; for other kinds of geometrical transformations , any parametric transform which gives a robust estimate of the desired parameters could be used ; and the input points can be extracted from any suitable representation of a regular synchronization pattern like acf or ms peaks , or any regular template . in the case of affine transformations , we can represent the 2 main axes , and periods , of the periodic reference grid by two vectors { right arrow over ( u )} o and { right arrow over ( v )} o , and of which norms ∥{ right arrow over ( u )} o ∥, ∥{ right arrow over ( v )} o ∥ are the respective periods between points along each direction ( fig2 a ). within the acf or ms domain , an affine transform maps the reference grid to another one represented by vectors { right arrow over ( u )} and { right arrow over ( v )}. the four parameters a , b , c , d have to be estimated , corresponding to the affine transform matrix a of equation g1 ( fig1 , block 6 ). assuming that we know the reference grid , specified by { right arrow over ( u )} o ,{ right arrow over ( v )} o , and that we got the correct estimations of { right arrow over ( u )} and { right arrow over ( v )} from the extracted points , the matrix can be expressed as : where the ( . . . ) − 1 denotes the matrix inversion , with : t = ( x u x v y u y v ) , ⁢ t 0 = ( x uo x vo y uo y vo ) , u -& gt ; = ( x u y u ) , ⁢ v -& gt ; = ( x v y v ) , ⁢ u -& gt ; o = ( x uo y uo ) , ⁢ v -& gt ; o = ( x vo y vo ) . the directions and norms of the two vectors { right arrow over ( u )} o ,{ right arrow over ( v )} o ( represented by t o ) correspond to the periodicity along the two main axes of the embedded pattern w over the 2 - dimensional image . the above corresponds to the general case . in practice however we typically embed a square pattern , which is repeated horizontally and vertically along the x - and y - axes with the same period τ , and in this case t 0 becomes : depending on the transform used to extract points , further corrections in { right arrow over ( u )} and { right arrow over ( v )} may have to be applied in order to reflect the main axes orientations and period values in the spatial domain of the input data . let say that { right arrow over ( u )}′, { right arrow over ( v )}′ are the vectors coming from the estimated synchronization information : if the acf was used , then we just take { right arrow over ( u )}={ right arrow over ( u )}′ and { right arrow over ( v )}={ right arrow over ( v )}′; but if the ms domain is used , one should calculate : u → = n  u → ′  · e ^ u ⁢ ⁢ and ⁢ ⁢ v → = n  v → ′  · e ^ v ( g11 ) e ^ u = u → ′  u → ′  , ⁢ e ^ v = v → ′  v → ′  are the unitary vectors along each main axis , using a n × n ( square ) domain size for the the dft in order to preserve angles . a last point to mention is the possibility of ambiguities in the estimate of the affine transformation . the extracted grid of points contains no information on the orientation of the vectors . therefore the estimated vectors { right arrow over ( u )} and { right arrow over ( v )} may have wrong orientations , resulting in 8 ambiguities made of horizontal , vertical flips , and / or 90 °- rotations . then all of them may need to be checked . one mean to overcome this is to use patterns with central symmetry , presenting in this case only 2 ambiguities for 90 °- rotations . the computation of the acf or ms shows peaks , corresponding to local maxima , reflecting the repetitive nature of the synchronization data . in order to estimate the affine transform , one can either extract the peaks positions , using any method and resulting in a discrete set of points , or simply take the acf or ms domain as an input p for the following parametric transform : it will be the ht in case of extracted points , or the rt if the values from the transformed domain have been taken too , or any other parametric transform ( fig1 , block 1 ). fig3 a shows this set of peaks , which is noisy due to some signal degradation , and after the application of an affine transform ( here , a rotation ). after extracting main axes and periods , it will then be possible to fit parallel lines to each alignment of points , thus resulting in 2 main axes and 2 periods . fig3 b shows such fitted lines , along 2 main axes , and along diagonals too since any diagonal also corresponds to alignments . taking into account all points from p for the fitting ensures at the same time better precision and higher robustness . we can then compute the parametric transform h from p ( fig1 , block 1 ). ht or rt converts the x , y - representation a θ , ρ - representation , giving a projection distance from the origin in function of an angle of projection . after that a function of one of the parametric transform variable helps us in extracting the wanted features . fig4 a shows the ht of points from fig3 a , the projection ρ - axis being vertical and the angle θ - axis being horizontal ; strong vertically aligned peaks are clearly visible in h , corresponding to the main axes angles . by vertically adding the contributions of peaks in h , we can obtain a function h ( θ ) as shown in fig4 b , showing the largest peaks at the angles which correspond to the main axes ( fig1 , block 2 ). note also that the diagonals can yield an ambiguity , since they obviously present alignments too ; a solution to this problem can be offered by the h ( θ ) function , by taking into account the number of peaks added from the ht or the rt , thus exploiting the fact that diagonal orientations contain usually less parallel lines than correct ones : diagonal angles then result in much lower peaks in h ( θ ) as shown in fig4 b ( angles θ 3 and θ 4 ). the interest of a parametric representation is to allow the robust estimation of the correct parameters corresponding to the applied geometrical transform , which could be estimated from h directly . in our example , the angles of maximum contribution correspond to the orientations of the main axes . however this could lead to a lack in precision due to interpolation errors while computing the parametric transform h . it is therefore also possible to directly use points from p in order to refine the precision of the wanted features which have been estimated from the parametric transform h . in this case h gives us the information we need in order to select the correct points in p , and a precise estimate of parameters is directly computed from these points . for affine transforms , the ht or rt helps us in selecting points aligned along the correct main axes ( fig1 , block 3 ). once the correctly aligned points have been selected in p , any procedure for robust fitting of lines , curves , or any other feature can be used . in our example lines are fitted to the selected alignments corresponding to the 2 distinguished main axes ; least square , least median square , linear regression , or any robust line fitting algorithm can be used for this purpose ( fig1 , block 4 ). any feature can be robustly fitted to the correct points of p , based on the distinguishing information obtained from h . for affine transforms and ht or rt , since each main axis exhibits a group of parallel lines , the combination ( average , median , etc .) of all orientations in a class of parallel lines gives the precise orientation of the corresponding main axis . further , any manipulation of the set of points , which could increase the robustness and / or the precision of the estimation , can be used ; such manipulation are — but are not restricted to : additional point estimation in intersections between the identified lines , estimation of additional lines where points are missing , etc . at this stage , it is possible that not all the needed parameters have been estimated . in the affine transforms , in the ht or rt example , only the main axes orientation have been estimated ( giving the ū and { overscore ( v )} directions only ), thus intervals between points have to be estimated too ( fig1 , block 5 ). in general , any robust approach can be used for the estimation of the remaining parameters , including — but not limited to — another parametric transform , as well as a direct estimation using the selected points from p , or the lines or curves which have been fitted in the previous steps . in the case of affine transform , we can select all aligned points belonging the each class of parallel lines , corresponding to each main axis . fig5 illustrates the problem of estimating the best interval between points , even in the case of noise : points on correct lines have been selected , but wrong points may still be present on correct lines ; several or all parallel alignments ( thus belonging to the same class ) can be used ( fig5 a ). a correlation - like approach , based on a matching function m ( τ ) of a candidate period τ with respect the alignments , can be used ( fig5 b ); we can see that m ( τ ) gives the highest peak when τ is equal to the correct period τ o = 37 in this example . at this point the characteristics from the embedded and distorted grid of points have been estimated . the estimated parameters can be used in order to further estimate the applied global geometrical transform , by comparison with known parameter values corresponding to the originally embedded synchronization data ( fig1 , block 6 ). the idea in the previous steps is to estimate more than the strictly needed number of parameter values , in order to generate more than one geometrical transform candidate . therefore , additional reference information in the synchronization data , or any other mean , can help us in selecting the correct distortion — giving , if wanted , a trade - off between unique distortion estimation and fully exhaustive search , increasing the robustness of the method . for affine transformations , the matrix a can be estimated from the ( distorted ) vectors { right arrow over ( u )},{ right arrow over ( v )} and ( original ) vectors { right arrow over ( u )} 0 ,{ right arrow over ( v )} 0 as described in equation g9 , and possibly g10 or g11 . in the previous steps , we can compute n axes ≧ 2 possible main axes ( for a minimum required of 2 ), and n periods ≧ 1 possible periods ( at least 1 ) per axis ; 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