Curve Reconstruction via the Global Statistics of Natural Curves

Reconstructing the missing parts of a curve has been the subject of much computational research, with applications in image inpainting, object synthesis, etc. Different approaches for solving that problem are typically based on processes that seek visually pleasing or perceptually plausible completions. In this work we focus on reconstructing the underlying physically likely shape by utilizing the global statistics of natural curves. More specifically, we develop a reconstruction model that seeks the mean physical curve for a given inducer configuration. This simple model is both straightforward to compute and it is receptive to diverse additional information, but it requires enough samples for all curve configurations, a practical requirement that limits its effective utilization. To address this practical issue we explore and exploit statistical geometrical properties of natural curves, and in particular, we show that in many cases the mean curve is scale invariant and oftentimes it is extensible. This, in turn, allows to boost the number of examples and thus the robustness of the statistics and its applicability. The reconstruction results are not only more physically plausible but they also lead to important insights on the reconstruction problem, including an elegant explanation why certain inducer configurations are more likely to yield consistent perceptual completions than others.Comment: CVPR versio

contour interpolation by vector field combination

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Affine differential geometry analysis of human arm movements

Humans interact with their environment through sensory information and motor actions. These interactions may be understood via the underlying geometry of both perception and action. While the motor space is typically considered by default to be Euclidean, persistent behavioral observations point to a different underlying geometric structure. These observed regularities include the “two-thirds power law” which connects path curvature with velocity, and “local isochrony” which prescribes the relation between movement time and its extent. Starting with these empirical observations, we have developed a mathematical framework based on differential geometry, Lie group theory and Cartan’s moving frame method for the analysis of human hand trajectories. We also use this method to identify possible motion primitives, i.e., elementary building blocks from which more complicated movements are constructed. We show that a natural geometric description of continuous repetitive hand trajectories is not Euclidean but equi-affine. Specifically, equi-affine velocity is piecewise constant along movement segments, and movement execution time for a given segment is proportional to its equi-affine arc-length. Using this mathematical framework, we then analyze experimentally recorded drawing movements. To examine movement segmentation and classification, the two fundamental equi-affine differential invariants—equi-affine arc-length and curvature are calculated for the recorded movements. We also discuss the possible role of conic sections, i.e., curves with constant equi-affine curvature, as motor primitives and focus in more detail on parabolas, the equi-affine geodesics. Finally, we explore possible schemes for the internal neural coding of motor commands by showing that the equi-affine framework is compatible with the common model of population coding of the hand velocity vector when combined with a simple assumption on its dynamics. We then discuss several alternative explanations for the role that the equi-affine metric may play in internal representations of motion perception and production

A unified account of tilt illusions, association fields, and contour detection based on Elastica

As expressed in the Gestalt law of good continuation, human perception tends to associate stimuli that form smooth continuations. Contextual modulation in primary visual cortex, in the form of association fields, is believed to play an important role in this process. Yet a unified and principled account of the good continuation law on the neural level is lacking. In this study we introduce a population model of primary visual cortex. Its contextual interactions depend on the elastica curvature energy of the smoothest contour connecting oriented bars. As expected, this model leads to association fields consistent with data. However, in addition the model displays tilt-illusions for stimulus configurations with grating and single bars that closely match psychophysics. Furthermore, the model explains not only pop-out of contours amid a variety of backgrounds, but also pop-out of single targets amid a uniform background. We thus propose that elastica is a unifying principle of the visual cortical network

Computational models for image contour grouping

Contours are one dimensional curves which may correspond to meaningful entities such as object boundaries. Accurate contour detection will simplify many vision tasks such as object detection and image recognition. Due to the large variety of image content and contour topology, contours are often detected as edge fragments at first, followed by a second step known as {u0300}{u0300}contour grouping'' to connect them. Due to ambiguities in local image patches, contour grouping is essential for constructing globally coherent contour representation. This thesis aims to group contours so that they are consistent with human perception. We draw inspirations from Gestalt principles, which describe perceptual grouping ability of human vision system. In particular, our work is most relevant to the principles of closure, similarity, and past experiences. The first part of our contribution is a new computational model for contour closure. Most of existing contour grouping methods have focused on pixel-wise detection accuracy and ignored the psychological evidences for topological correctness. This chapter proposes a higher-order CRF model to achieve contour closure in the contour domain. We also propose an efficient inference method which is guaranteed to find integer solutions. Tested on the BSDS benchmark, our method achieves a superior contour grouping performance, comparable precision-recall curves, and more visually pleasant results. Our work makes progresses towards a better computational model of human perceptual grouping. The second part is an energy minimization framework for salient contour detection problem. Region cues such as color/texture homogeneity, and contour cues such as local contrast, are both useful for this task. In order to capture both kinds of cues in a joint energy function, topological consistency between both region and contour labels must be satisfied. Our technique makes use of the topological concept of winding numbers. By using a fast method for winding number computation, we find that a small number of linear constraints are sufficient for label consistency. Our method is instantiated by ratio-based energy functions. Due to cue integration, our method obtains improved results. User interaction can also be incorporated to further improve the results. The third part of our contribution is an efficient category-level image contour detector. The objective is to detect contours which most likely belong to a prescribed category. Our method, which is based on three levels of shape representation and non-parametric Bayesian learning, shows flexibility in learning from either human labeled edge images or unlabelled raw images. In both cases, our experiments obtain better contour detection results than competing methods. In addition, our training process is robust even with a considerable size of training samples. In contrast, state-of-the-art methods require more training samples, and often human interventions are required for new category training. Last but not least, in Chapter 7 we also show how to leverage contour information for symmetry detection. Our method is simple yet effective for detecting the symmetric axes of bilaterally symmetric objects in unsegmented natural scene images. Compared with methods based on feature points, our model can often produce better results for the images containing limited texture

Combining contour and region for closed boundary extraction of a shape

This study explored human ability to extract closed boundary of a target shape in the presence of noise using spatially global operations. Specifically, we investigated the contributions of contour-based processing using line edges and region-based processing using color, as well as their interaction. Performance of the subjects was reliable when the fixation was inside the shape, and it was much less reliable when the fixation was outside. With fixation inside the shape, performance was higher when both contour and color information were present compared to when only one of them was present. We propose a biologically-inspired model to emulate human boundary extraction. The model solves the shortest (least-cost) path in the log-polar representation, a representation which is a good approximation to the mapping from the retina to the visual cortex. Boundary extraction was framed as a global optimization problem with the costs of connections calculated using four features: distance of interpolation, turning angle, color similarity and color contrast. This model was tested on some of the conditions that were used in the psychophysical experiment and its performance was similar to the performance of subjects