1,083 research outputs found

    On weak Mellin transforms, second degree characters and the Riemann hypothesis

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    We say that a function f defined on R or Qp has a well defined weak Mellin transform (or weak zeta integral) if there exists some function M_f(s)M\_f(s) so that we have Mell(ϕf,s)=Mell(ϕ,s)M_f(s)Mell(\phi \star f,s) = Mell(\phi,s)M\_f(s) for all test functions ϕ\phi in C_c(R)C\_c^\infty(R^*) or C_c(Q_p)C\_c^\infty(Q\_p^*). We show that if ff is a non degenerate second degree character on R or Qp, as defined by Weil, then the weak Mellin transform of ff satisfies a functional equation and cancels only for (s)=1/2\Re(s) = 1/2. We then show that if ff is a non degenerate second degree character defined on the adele ring A_QA\_Q, the same statement is equivalent to the Riemann hypothesis. Various generalizations are provided

    Robust and efficient Fourier-Mellin transform approximations for invariant grey-level image description and reconstruction

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    International audienceThis paper addresses the gray-level image representation ability of the Fourier-Mellin Transform (FMT) for pattern recognition, reconstruction and image database retrieval. The main practical di±culty of the FMT lies in the accuracy and e±ciency of its numerical approximation and we propose three estimations of its analytical extension. Comparison of these approximations is performed from discrete and ¯nite-extent sets of Fourier- Mellin harmonics by means of experiments in: (i) image reconstruction via both visual inspection and the computation of a reconstruction error; and (ii) pattern recognition and discrimination by using a complete and convergent set of features invariant under planar similarities. Experimental results on real gray-level images show that it is possible to recover an image to within a speci¯ed degree of accuracy and to classify objects reliably even when a large set of descriptors is used. Finally, an example will be given, illustrating both theoretical and numerical results in the context of content-based image retrieval

    A Kind of Affine Weighted Moment Invariants

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    A new kind of geometric invariants is proposed in this paper, which is called affine weighted moment invariant (AWMI). By combination of local affine differential invariants and a framework of global integral, they can more effectively extract features of images and help to increase the number of low-order invariants and to decrease the calculating cost. The experimental results show that AWMIs have good stability and distinguishability and achieve better results in image retrieval than traditional moment invariants. An extension to 3D is straightforward

    Multiscale Point Correspondence Using Feature Distribution and Frequency Domain Alignment

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    In this paper, a hybrid scheme is proposed to find the reliable point-correspondences between two images, which combines the distribution of invariant spatial feature description and frequency domain alignment based on two-stage coarse to fine refinement strategy. Firstly, the source and the target images are both down-sampled by the image pyramid algorithm in a hierarchical multi-scale way. The Fourier-Mellin transform is applied to obtain the transformation parameters at the coarse level between the image pairs; then, the parameters can serve as the initial coarse guess, to guide the following feature matching step at the original scale, where the correspondences are restricted in a search window determined by the deformation between the reference image and the current image; Finally, a novel matching strategy is developed to reject the false matches by validating geometrical relationships between candidate matching points. By doing so, the alignment parameters are refined, which is more accurate and more flexible than a robust fitting technique. This in return can provide a more accurate result for feature correspondence. Experiments on real and synthetic image-pairs show that our approach provides satisfactory feature matching performance

    Registration of holographic images based on the integral transformation

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    The paper describes the possibilities of using Fourier-Mellin transform for registering images of holographic interferograms. Registered holographic images will then allow automating their evaluation. Registration based on changes in image intensities using the discrete integral transforms was selected of the methods of registration. Whereas it was necessary to register the images, which are not only translated, but also rotated and with the changed of scale, the Fourier-Mellin transform was used. Use of the image discrete transforms is original in this field, proposed processing algorithm contains also simplified mean of calculating the angle of rotation of the test image instead of common Fourier-Mellin transformation method sequence
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