Symmetry

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1732 KiB

Open AccessArticle

Redundant Symmetry Influences Perceptual Grouping (as Measured by Rotational Linkage)

by Barbara Gillam

Symmetry 2017, 9(5), 67; https://doi.org/10.3390/sym9050067 - 09 May 2017

Cited by 2 | Viewed by 4184

Abstract

Symmetry detection has long been a major focus of perception research. However, although symmetry is often cited as a “grouping principle”, the effect of symmetry on grouping, an important form of perceptual organization, has been little measured. In past research, we found little [...] Read more.

Symmetry detection has long been a major focus of perception research. However, although symmetry is often cited as a “grouping principle”, the effect of symmetry on grouping, an important form of perceptual organization, has been little measured. In past research, we found little spatio-temporal grouping for oblique lines symmetric around a horizontal axis during ambiguous rotary motion in depth. Grouping was measured by the degree to which the ambiguous motion direction was resolved for two elements in common (rotational linkage). We hypothesized that symmetry-based grouping would be stronger if symmetry was redundant i.e., carried by elements of greater complexity. Using the rotational linkage measure, we compared grouping for horizontally symmetric simple oblique lines and for lines composed of multiple conjoined orientations and found greater grouping for the more complex symmetric lines. A control experiment ruled out possible confounding factors and also showed a grouping effect of vertically aligned endpoints. We attribute the stronger grouping effect of redundant symmetry to the fact that it has a lower probability than does simple symmetry of arising from an accidental environmental arrangement. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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3837 KiB

Open AccessArticle

Binocular 3D Object Recovery Using a Symmetry Prior

by Aaron Michaux, Vikrant Kumar, Vijai Jayadevan, Edward Delp and Zygmunt Pizlo

Symmetry 2017, 9(5), 64; https://doi.org/10.3390/sym9050064 - 28 Apr 2017

Cited by 9 | Viewed by 5030

Abstract

We present a new algorithm for 3D shape reconstruction from stereo image pairs that uses mirror symmetry as a biologically inspired prior. 3D reconstruction requires some form of prior because it is an ill-posed inverse problem. Psychophysical research shows that mirror-symmetry is a [...] Read more.

We present a new algorithm for 3D shape reconstruction from stereo image pairs that uses mirror symmetry as a biologically inspired prior. 3D reconstruction requires some form of prior because it is an ill-posed inverse problem. Psychophysical research shows that mirror-symmetry is a key prior for 3D shape perception in humans, suggesting that a general purpose solution to this problem will have many applications. An approach is developed for finding objects that fit a given shape definition. The algorithm is developed for shapes with two orthogonal planes of symmetry, thus allowing for straightforward recovery of occluded portions of the objects. Two simulations were run to test: (1) the accuracy of 3D recovery, and (2) the ability of the algorithm to find the object in the presence of noise. We then tested the algorithm on the Children’s Furniture Corpus, a corpus of stereo image pairs of mirror symmetric furniture objects. Runtimes and 3D reconstruction errors are reported and failure modes described. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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1660 KiB

Open AccessArticle

Matching Visual and Acoustic Mirror Forms

by Ivana Bianchi, Roberto Burro, Roberta Pezzola and Ugo Savardi

Symmetry 2017, 9(3), 39; https://doi.org/10.3390/sym9030039 - 10 Mar 2017

Cited by 9 | Viewed by 5168

Abstract

This paper presents a comparative analysis of the ability to recognize three mirror forms in visual and acoustic tasks: inversion (reflection on a horizontal axis), retrograde (reflection on a vertical axis) and retrograde inversion (reflection on both horizontal and vertical axes). Dynamic patterns [...] Read more.

This paper presents a comparative analysis of the ability to recognize three mirror forms in visual and acoustic tasks: inversion (reflection on a horizontal axis), retrograde (reflection on a vertical axis) and retrograde inversion (reflection on both horizontal and vertical axes). Dynamic patterns consisting of five tones in succession in the acoustic condition and five square dots in succession in the visual condition were presented to 180 non‐musically expert participants. In a yes/no task, they were asked to ascertain whether a comparison stimulus represented the “target” transformation (i.e., inversion, retrograde or retrograde inversion). Three main results emerged. Firstly, the fact that symmetry pertaining to a vertical axis is the most easily perceived does not only apply to static visual configurations (as found in previous literature) but also applies to dynamic visual configurations and acoustic stimuli where it is in fact even more marked. Secondly, however, differences emerged between the facility with which the three mirror forms were recognized in the acoustic and visual tasks. Thirdly, when the five elements in the stimulus were not of the same duration and therefore a rhythmic structure emerged, performance improved not only in the acoustic but also (even more significantly) in the visual task. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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879 KiB

Open AccessArticle

Using Convolutional Neural Network Filters to Measure Left-Right Mirror Symmetry in Images

by Anselm Brachmann and Christoph Redies

Symmetry 2016, 8(12), 144; https://doi.org/10.3390/sym8120144 - 01 Dec 2016

Cited by 24 | Viewed by 8020

Abstract

We propose a method for measuring symmetry in images by using filter responses from Convolutional Neural Networks (CNNs). The aim of the method is to model human perception of left/right symmetry as closely as possible. Using the Convolutional Neural Network (CNN) approach has [...] Read more.

We propose a method for measuring symmetry in images by using filter responses from Convolutional Neural Networks (CNNs). The aim of the method is to model human perception of left/right symmetry as closely as possible. Using the Convolutional Neural Network (CNN) approach has two main advantages: First, CNN filter responses closely match the responses of neurons in the human visual system; they take information on color, edges and texture into account simultaneously. Second, we can measure higher-order symmetry, which relies not only on color, edges and texture, but also on the shapes and objects that are depicted in images. We validated our algorithm on a dataset of 300 music album covers, which were rated according to their symmetry by 20 human observers, and compared results with those from a previously proposed method. With our method, human perception of symmetry can be predicted with high accuracy. Moreover, we demonstrate that the inclusion of features from higher CNN layers, which encode more abstract image content, increases the performance further. In conclusion, we introduce a model of left/right symmetry that closely models human perception of symmetry in CD album covers. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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1379 KiB

Open AccessArticle

Affine Geometry, Visual Sensation, and Preference for Symmetry of Things in a Thing

by Birgitta Dresp-Langley

Symmetry 2016, 8(11), 127; https://doi.org/10.3390/sym8110127 - 14 Nov 2016

Cited by 9 | Viewed by 6197

Abstract

Evolution and geometry generate complexity in similar ways. Evolution drives natural selection while geometry may capture the logic of this selection and express it visually, in terms of specific generic properties representing some kind of advantage. Geometry is ideally suited for expressing the [...] Read more.

Evolution and geometry generate complexity in similar ways. Evolution drives natural selection while geometry may capture the logic of this selection and express it visually, in terms of specific generic properties representing some kind of advantage. Geometry is ideally suited for expressing the logic of evolutionary selection for symmetry, which is found in the shape curves of vein systems and other natural objects such as leaves, cell membranes, or tunnel systems built by ants. The topology and geometry of symmetry is controlled by numerical parameters, which act in analogy with a biological organism’s DNA. The introductory part of this paper reviews findings from experiments illustrating the critical role of two-dimensional (2D) design parameters, affine geometry and shape symmetry for visual or tactile shape sensation and perception-based decision making in populations of experts and non-experts. It will be shown that 2D fractal symmetry, referred to herein as the “symmetry of things in a thing”, results from principles very similar to those of affine projection. Results from experiments on aesthetic and visual preference judgments in response to 2D fractal trees with varying degrees of asymmetry are presented. In a first experiment (psychophysical scaling procedure), non-expert observers had to rate (on a scale from 0 to 10) the perceived beauty of a random series of 2D fractal trees with varying degrees of fractal symmetry. In a second experiment (two-alternative forced choice procedure), they had to express their preference for one of two shapes from the series. The shape pairs were presented successively in random order. Results show that the smallest possible fractal deviation from “symmetry of things in a thing” significantly reduces the perceived attractiveness of such shapes. The potential of future studies where different levels of complexity of fractal patterns are weighed against different degrees of symmetry is pointed out in the conclusion. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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4819 KiB

Open AccessArticle

Two-Dimensional Hermite Filters Simplify the Description of High-Order Statistics of Natural Images

by Qin Hu and Jonathan D. Victor

Symmetry 2016, 8(9), 98; https://doi.org/10.3390/sym8090098 - 21 Sep 2016

Cited by 5 | Viewed by 6555

Abstract

Natural image statistics play a crucial role in shaping biological visual systems, understanding their function and design principles, and designing effective computer-vision algorithms. High-order statistics are critical for conveying local features but they are challenging to study, largely because their number and variety [...] Read more.

Natural image statistics play a crucial role in shaping biological visual systems, understanding their function and design principles, and designing effective computer-vision algorithms. High-order statistics are critical for conveying local features but they are challenging to study, largely because their number and variety is large. Here, via the use of two-dimensional Hermite (TDH) functions, we identify a covert symmetry in high-order statistics of natural images that simplifies this task. This emerges from the structure of TDH functions, which are an orthogonal set of functions that are organized into a hierarchy of ranks. Specifically, we find that the shape (skewness and kurtosis) of the distribution of filter coefficients depends only on the projection of the function onto a one-dimensional subspace specific to each rank. The characterization of natural image statistics provided by TDH filter coefficients reflects both their phase and amplitude structure, and we suggest an intuitive interpretation for the special subspace within each rank. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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3722 KiB

Open AccessArticle

The Conformal Camera in Modeling Active Binocular Vision

by Jacek Turski

Symmetry 2016, 8(9), 88; https://doi.org/10.3390/sym8090088 - 31 Aug 2016

Cited by 3 | Viewed by 5431

Abstract

Primate vision is an active process that constructs a stable internal representation of the 3D world based on 2D sensory inputs that are inherently unstable due to incessant eye movements. We present here a mathematical framework for processing visual information for a biologically-mediated [...] Read more.

Primate vision is an active process that constructs a stable internal representation of the 3D world based on 2D sensory inputs that are inherently unstable due to incessant eye movements. We present here a mathematical framework for processing visual information for a biologically-mediated active vision stereo system with asymmetric conformal cameras. This model utilizes the geometric analysis on the Riemann sphere developed in the group-theoretic framework of the conformal camera, thus far only applicable in modeling monocular vision. The asymmetric conformal camera model constructed here includes the fovea’s asymmetric displacement on the retina and the eye’s natural crystalline lens tilt and decentration, as observed in ophthalmological diagnostics. We extend the group-theoretic framework underlying the conformal camera to the stereo system with asymmetric conformal cameras. Our numerical simulation shows that the theoretical horopter curves in this stereo system are conics that well approximate the empirical longitudinal horopters of the primate vision system. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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1019 KiB

Open AccessArticle

Modeling Bottom-Up Visual Attention Using Dihedral Group D₄

by Puneet Sharma

Symmetry 2016, 8(8), 79; https://doi.org/10.3390/sym8080079 - 15 Aug 2016

Cited by 3 | Viewed by 6553

Abstract

In this paper, first, we briefly describe the dihedral group

D_{4}

that serves as the basis for calculating saliency in our proposed model. Second, our saliency model makes two major changes in a latest state-of-the-art model known as group-based asymmetry. First, based [...] Read more.

In this paper, first, we briefly describe the dihedral group

D_{4}

that serves as the basis for calculating saliency in our proposed model. Second, our saliency model makes two major changes in a latest state-of-the-art model known as group-based asymmetry. First, based on the properties of the dihedral group

D_{4}

, we simplify the asymmetry calculations associated with the measurement of saliency. This results is an algorithm that reduces the number of calculations by at least half that makes it the fastest among the six best algorithms used in this research article. Second, in order to maximize the information across different chromatic and multi-resolution features, the color image space is de-correlated. We evaluate our algorithm against 10 state-of-the-art saliency models. Our results show that by using optimal parameters for a given dataset, our proposed model can outperform the best saliency algorithm in the literature. However, as the differences among the (few) best saliency models are small, we would like to suggest that our proposed model is among the best and the fastest among the best. Finally, as a part of future work, we suggest that our proposed approach on saliency can be extended to include three-dimensional image data. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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6493 KiB

Open AccessArticle

Relationship between Fractal Dimension and Spectral Scaling Decay Rate in Computer-Generated Fractals

by Alexander J. Bies, Cooper R. Boydston, Richard P. Taylor and Margaret E. Sereno

Symmetry 2016, 8(7), 66; https://doi.org/10.3390/sym8070066 - 19 Jul 2016

Cited by 19 | Viewed by 8466

Abstract

Two measures are commonly used to describe scale-invariant complexity in images: fractal dimension (D) and power spectrum decay rate (β). Although a relationship between these measures has been derived mathematically, empirical validation across measurements is lacking. Here, we determine the relationship [...] Read more.

Two measures are commonly used to describe scale-invariant complexity in images: fractal dimension (D) and power spectrum decay rate (β). Although a relationship between these measures has been derived mathematically, empirical validation across measurements is lacking. Here, we determine the relationship between D and β for 1- and 2-dimensional fractals. We find that for 1-dimensional fractals, measurements of D and β obey the derived relationship. Similarly, in 2-dimensional fractals, measurements along any straight-line path across the fractal’s surface obey the mathematically derived relationship. However, the standard approach of vision researchers is to measure β of the surface after 2-dimensional Fourier decomposition rather than along a straight-line path. This surface technique provides measurements of β that do not obey the mathematically derived relationship with D. Instead, this method produces values of β that imply that the fractal’s surface is much smoother than the measurements along the straight lines indicate. To facilitate communication across disciplines, we provide empirically derived equations for relating each measure of β to D. Finally, we discuss implications for future research on topics including stress reduction and the perception of motion in the context of a generalized equation relating β to D. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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14765 KiB

Open AccessArticle

Anomalous Mirror Symmetry Generated by Optical Illusion

by Kokichi Sugihara

Symmetry 2016, 8(4), 21; https://doi.org/10.3390/sym8040021 - 08 Apr 2016

Cited by 10 | Viewed by 17347

Abstract

This paper introduces a new concept of mirror symmetry, called “anomalous mirror symmetry”, which is physically impossible but can be perceived by human vision systems because of optical illusion. This symmetry is characterized geometrically and a method for creating cylindrical surfaces that create [...] Read more.

This paper introduces a new concept of mirror symmetry, called “anomalous mirror symmetry”, which is physically impossible but can be perceived by human vision systems because of optical illusion. This symmetry is characterized geometrically and a method for creating cylindrical surfaces that create this symmetry is constructed. Examples of solid objects constructed by a 3D printer are also shown. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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Review

Jump to: Research

693 KiB

Open AccessReview

On the Legibility of Mirror-Reflected and Rotated Text

by Gennady Erlikhman, Lars Strother, Iskra Barzakov and Gideon Paul Caplovitz

Symmetry 2017, 9(3), 28; https://doi.org/10.3390/sym9030028 - 23 Feb 2017

Cited by 7 | Viewed by 11977

Abstract

We happened to observe that text that was reflected about either the horizontal or vertical axis was more difficult to read than text that was reflected about first one and then the other, which amounts to a 180-degree rotation. In this article, we [...] Read more.

We happened to observe that text that was reflected about either the horizontal or vertical axis was more difficult to read than text that was reflected about first one and then the other, which amounts to a 180-degree rotation. In this article, we review a number of studies that examine the nature of recognizing reflected and inverted letters, and the frequency of mirror reversal errors (e.g., confusing 'b' for 'd') in children and adults. We explore recent ideas linking the acquisition of literacy with the loss of mirror-invariance, not just for text, but for objects in general. We try to connect these various literatures to examine why certain transformations of text are more difficult to read than others for adults. Full article

(This article belongs to the Special Issue Symmetry in Vision)

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