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.
We propose a novel unsupervised method for discovering recurring patterns from a single view. A key contribution of our approach is the formulation and validation of a joint assignment optimization problem where multiple visual words and object instances of a potential recurring pattern are considered simultaneously. The optimization is achieved by a greedy randomized adaptive search procedure (GRASP) with moves specifically designed for fast convergence. We have quantified systematically the performance of our approach under stressed conditions of the input (missing features, geometric distortions). We demonstrate that our proposed algorithm outperforms state of the art methods for recurring pattern discovery on a diverse set of 400+ real world and synthesized test images. The chromatic polynomial @math gives the number of @math -colourings of a graph. If @math , then the graph @math is said to have a chromatic factorisation with chromatic factors @math and @math . It is known that the chromatic polynomial of any clique-separable graph has a chromatic factorisation. In this paper we construct an infinite family of graphs that have chromatic factorisations, but have chromatic polynomials that are not the chromatic polynomial of any clique-separable graph. A certificate of factorisation, that is, a sequence of rewritings based on identities for the chromatic polynomial, is given that explains the chromatic factorisations of graphs from this family. We show that the graphs in this infinite family are the only graphs that have a chromatic factorisation satisfying this certificate and having the odd cycle @math , @math , as a chromatic factor. Detection of repetitive patterns in images has been studied for a long time in computer vision. This paper discusses a method for representing a lattice or line pattern by shift-invariant descriptor of the repeating element. The descriptor overcomes shift ambiguity and can be matched between different a views. The pattern matching is then demonstrated in retrieval experiment, where different images of the same buildings are retrieved solely by repetitive patterns. Repeated structures such as building facades, fences or road markings often represent a significant challenge for place recognition. Repeated structures are notoriously hard for establishing correspondences using multi-view geometry. Even more importantly, they violate the feature independence assumed in the bag-of-visual-words representation which often leads to over-counting evidence and significant degradation of retrieval performance. In this work we show that repeated structures are not a nuisance but, when appropriately represented, they form an important distinguishing feature for many places. We describe a representation of repeated structures suitable for scalable retrieval. It is based on robust detection of repeated image structures and a simple modification of weights in the bag-of-visual-word model. Place recognition results are shown on datasets of street-level imagery from Pittsburgh and San Francisco demonstrating significant gains in recognition performance compared to the standard bag-of-visual-words baseline and more recently proposed burstiness weighting. We introduce a novel framework for automatic detection of repeated patterns in real images. The novelty of our work is to formulate the extraction of an underlying deformed lattice as a spatial, multi-target tracking problem using a new and efficient Mean-Shift Belief Propagation (MSBP) method. Compared to existing work, our approach has multiple advantages, including: 1) incorporating higher order constraints early-on to propose highly plausible lattice points; 2) growing a lattice in multiple directions simultaneously instead of one at a time sequentially; and 3) achieving more efficient and more accurate performance than state-of-the-art algorithms. These advantages are demonstrated by quantitative experimental results on a diverse set of real world photos. Structural semantics are fundamental to understanding both natural and man-made objects from languages to buildings. They are manifested as repeated structures or patterns and are often captured in images. Finding repeated patterns in images, therefore, has important applications in scene understanding, 3D reconstruction, and image retrieval as well as image compression. Previous approaches in visual-pattern mining limited themselves by looking for frequently co-occurring features within a small neighborhood in an image. However, semantics of a visual pattern are typically defined by specific spatial relationships between features regardless of the spatial proximity. In this paper, semantics are represented as visual elements and geometric relationships between them. A novel unsupervised learning algorithm finds pair-wise associations of visual elements that have consistent geometric relationships sufficiently often. The algorithms are efficient - maximal matchings are determined without combinatorial search. High-order structural semantics are extracted by mining patterns that are composed of pairwise spatially consistent associations of visual elements. We demonstrate the effectiveness of our approach for discovering repeated visual patterns on a variety of image collections. We propose a novel and robust computational framework for automatic detection of deformed 2D wallpaper patterns in real-world images. The theory of 2D crystallographic groups provides a sound and natural correspondence between the underlying lattice of a deformed wallpaper pattern and a degree-4 graphical model. We start the discovery process with unsupervised clustering of interest points and voting for consistent lattice unit proposals. The proposed lattice basis vectors and pattern element contribute to the pairwise compatibility and joint compatibility (observation model) functions in a Markov random field (MRF). Thus, we formulate the 2D lattice detection as a spatial, multitarget tracking problem, solved within an MRF framework using a novel and efficient mean-shift belief propagation (MSBP) method. Iterative detection and growth of the deformed lattice are interleaved with regularized thin-plate spline (TPS) warping, which rectifies the current deformed lattice into a regular one to ensure stability of the MRF model in the next round of lattice recovery. We provide quantitative comparisons of our proposed method with existing algorithms on a diverse set of 261 real-world photos to demonstrate significant advances in accuracy and speed over the state of the art in automatic discovery of regularity in real images.