Planar ornaments, a.k.a. wallpapers, are regular repetitive patterns which exhibit translational symmetry in two independent directions. There are exactly 17 distinct planar symmetry groups. We present a fully automatic method for complete analysis of planar ornaments in 13 of these groups, specifically, the groups called p6, p6m, p4g, p4m, p4, p31m, p3m, p3, cmm, pgg, pg, p2 and p1. Given the image of an ornament fragment, we present a method to simultaneously classify the input into one of the 13 groups and extract the so-called fundamental domain, the minimum region that is sufficient to reconstruct the entire ornament. A nice feature of our method is that even when the given ornament image is a small portion such that it does not contain multiple translational units, the symmetry group as well as the fundamental domain can still be defined. This is because, in contrast to common approach, we do not attempt to first identify a global translational repetition lattice. Though the presented constructions work for quite a wide range of ornament patterns, a key assumption we make is that the perceivable motifs (shapes that repeat) alone do not provide clues for the underlying symmetries of the ornament. In this sense, our main target is the planar arrangements of asymmetric interlocking shapes, as in the symmetry art of Escher.
In @cite_0 , for translational symmetry, model based lattice estimation is performed where the model comparison for hypotheses generated via peaks of the autocorrelation is implemented using approximate marginal likelihood.
@cite_0
1993992624 2060321377 1512015702 2407166119
The analysis of regular texture images is cast in a model comparison framework. Texel lattice hypotheses are used to define statistical models which are compared in terms of their ability to explain the images. This approach is used to estimate lattice geometry from patterns that exhibit translational symmetry (regular textures). It is also used to determine whether images consist of such regular textures. A method based on this approach is described in which lattice hypotheses are generated using analysis of peaks in the image autocorrelation function, statistical models are based on Gaussian or Gaussian mixture clusters, and model comparison is performed using the marginal likelihood as approximated by the Bayes Information Criterion (BIC). Experiments on public domain images and a commercial textile image archive demonstrate substantially improved accuracy compared to several alternative methods. A method for efficient detection of the dominant local reflectional symmetry in grey level images is described. The general approach is to define a local measure of reflectional symmetry that transforms the symmetry detection problem to an optimization problem, and obtain the symmetric regions by an efficient global optimization algorithm. The symmetry of a 1D function can be measured in the frequency domain as the fraction of its energy that resides in symmetric Fourier basis functions. This approach is extended to two dimensions. Locality can be formally treated in terms of the Gabor decomposition and implemented via soft windowing. The resulting measure is a complicated multimodal function of the location of the center of the supporting region, its size, and the orientation of the symmetry axis. A new probabilistic generic algorithm is applied to the determination of the global maximum of the reflectional symmetry function. Less than one thousand evaluations of the local symmetry measure are typically needed in order to locate the dominant symmetry in natural, wildlife test images. We address an open question, regarding whether a lattice code with lattice decoding (as opposed to maximum-likelihood (ML) decoding) can achieve the additive white Gaussian noise (AWGN) channel capacity. We first demonstrate how minimum mean-square error (MMSE) scaling along with dithering (lattice randomization) techniques can transform the power-constrained AWGN channel into a modulo-lattice additive noise channel, whose effective noise is reduced by a factor of spl radic (1+SNR SNR). For the resulting channel, a uniform input maximizes mutual information, which in the limit of large lattice dimension becomes 1 2 log (1+SNR), i.e., the full capacity of the original power constrained AWGN channel. We then show that capacity may also be achieved using nested lattice codes, the coarse lattice serving for shaping via the modulo-lattice transformation, the fine lattice for channel coding. We show that such pairs exist for any desired nesting ratio, i.e., for any signal-to-noise ratio (SNR). Furthermore, for the modulo-lattice additive noise channel lattice decoding is optimal. Finally, we show that the error exponent of the proposed scheme is lower bounded by the Poltyrev exponent. We investigate the use of hierarchical phrase-based SMT lattices in end-to-end neural machine translation (NMT). Weight pushing transforms the Hiero scores for complete translation hypotheses, with the full translation grammar score and full n-gram language model score, into posteriors compatible with NMT predictive probabilities. With a slightly modified NMT beam-search decoder we find gains over both Hiero and NMT decoding alone, with practical advantages in extending NMT to very large input and output vocabularies.