# Reflection symmetry detection of shapes based on shape signatures

TL;DR: Wang et al. as mentioned in this paper presented two shape signature-based reflection symmetry detection methods with their theoretical underpinning and empirical evaluation, which can effectively deal with compound shapes which are challenging for traditional contour-based methods.

Abstract: • A solid theoretical foundation of R-signature and LIP-signature about symmetric properties of a given shape is represented. • A verification process is theoretically justified to remove the false candidates based on an efficient symmetry measure. • Two novel datasets (UTLN-SRA & UTLN-MRA) with single & multiple reflections are designed for evaluating symmetry detectors. • A new evaluation protocol based on a lost measure is presented to evaluate reflectional symmetry detectors. • Comprehensive evaluations have verified that our proposed detectors perform well on binary images compared to state of the art. We present two novel shape signature-based reflection symmetry detection methods with their theoretical underpinning and empirical evaluation. LIP-signature and R-signature share similar beneficial properties allowing to detect reflection symmetry directions in a high-performing manner. For the shape signature of a given shape, its merit profile is constructed to detect candidates of symmetry direction. A verification process is utilized to eliminate the false candidates by addressing Radon projections. The proposed methods can effectively deal with compound shapes which are challenging for traditional contour-based methods. To quantify the symmetric efficiency, a new symmetry measure is proposed over the range [0, 1]. Furthermore, we introduce two symmetry shape datasets with a new evaluation protocol and a lost measure for evaluating symmetry detectors. Experimental results using standard and new datasets suggest that the proposed methods prominently perform compared to state of the art.

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TL;DR: In this article , a semi-shape signature is proposed to detect rotational symmetry of a binary shape by considering the correlation between the signature and its circular shift, and a new meaningful measure, ranging from 0 to 1, is also introduced to indicate how perfect the symmetry would be.

Abstract: Efficient detectors of rotationally symmetric shapes are proposed by introducing a novel concept of semi-shape signatures to overcome the main problem of projection-based approaches for studying the rotationally symmetric properties of an arbitrary binary shape. Indeed, the fact that the projection cues in these conventional approaches are periodical with a period of π has restricted an applicable exploitation of rotational symmetry detection. To this end, we propose a new concept of the profile of semi-shapes as a shape signature together with a simple yet efficient technique so that the rotational symmetry of the binary shape can be determined by considering the correlation between this signature and its circular shift. Moreover, a new meaningful measure, ranging from 0 to 1, is also introduced to indicate how perfect the rotational symmetry would be. Experimental results of detecting on single/compound shapes have clearly corroborated the competence of our proposal.

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21 Aug 2022TL;DR: In this paper , the authors proposed the profile of semi-shapes as a signature of the shape together with a simple yet efficient technique to determine the rotation symmetry of an arbitrary shape by considering the correlation of this signature and its circular shift.

Abstract: A novel method for detecting rotational symmetry is addressed in this paper by introducing a new concept of semi-shapes to overcome the main problem of projection-based approaches for studying rotational symmetric properties of an arbitrary shape. It is due to the fact that in the classical approaches, projection cues are periodical with a period of π preventing exploitation of rotational properties. We then propose the profile of semi-shapes as a signature of the shape together with a simple yet efficient technique to determine the rotation symmetry of an arbitrary shape by considering the correlation of this signature and its circular shift. A new measure is also introduced to determine how good the rotational symmetry would be. Experiments on single/compound-contour shapes have clearly corroborated the efficacy of our proposal.

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04 Sep 2022

TL;DR: In this paper , the Radon transform is used to estimate the upper bound of the Jaccard index for a given line to find the optimal reflection symmetry axis of an object in a binary image.

Abstract: This study determines the optimal reflection symmetry axis of an object in a binary image using the Jaccard index. We propose to find the global optimum on the grid by estimating the upper bounds of the Jaccard index with the Radon transform. Several approaches to improving the Jaccard index computation for a given line are considered. It is shown that for the shear symmetry the optimal skew for a candidate symmetry axis can be found analytically. Some strategies for enumerating possible symmetry axes and selecting the initial approximation are proposed. The experiments show that the Jaccard index for the optimal axis found by the proposed method is not inferior to the exhaustive brute-force of the axes passing through the points of the object contour considering the rasterization error. In terms of speed, the proposed method significantly exceeds the previously developed methods for limiting the exhaustive search.

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04 Sep 2022

TL;DR: The experiments show that the Jaccard index for the optimal axis found by the proposed method is not inferior to the exhaustive brute-force of the axes passing through the points of the object contour considering the rasterization error.

Abstract: This study determines the optimal reflection symmetry axis of an object in a binary image using the Jaccard index. We propose to find the global optimum on the grid by estimating the upper bounds of the Jaccard index with the Radon transform. Several approaches to improving the Jaccard index computation for a given line are considered. It is shown that for the shear symmetry the optimal skew for a candidate symmetry axis can be found analytically. Some strategies for enumerating possible symmetry axes and selecting the initial approximation are proposed. The experiments show that the Jaccard index for the optimal axis found by the proposed method is not inferior to the exhaustive brute-force of the axes passing through the points of the object contour considering the rasterization error. In terms of speed, the proposed method significantly exceeds the previously developed methods for limiting the exhaustive search.

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01 Jan 2022

TL;DR: In this article , the optimal reflection symmetry axis of an object in a binary image using the Jaccard index was determined by estimating the upper bounds of the JACCard index with the Radon transform.

Abstract: This study determines the optimal reflection symmetry axis of an object in a binary image using the Jaccard index. We propose to find the global optimum on the grid by estimating the upper bounds of the Jaccard index with the Radon transform. Several approaches to improving the Jaccard index computation for a given line are considered. Some strategies for enumerating possible symmetry axes and selecting the initial approximation are proposed. The experiments show that the Jaccard index for the optimal axis found by the proposed method is not inferior to the exhaustive bruteforce of the axes passing through the points of the object contour considering the rasterization error. In terms of speed, the proposed method significantly exceeds the previously developed methods for limiting the exhaustive search.

##### References

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TL;DR: The results obtained with six natural images suggest that the orientation and the spatial-frequency tuning of mammalian simple cells are well suited for coding the information in such images if the goal of the code is to convert higher-order redundancy into first- order redundancy.

Abstract: The relative efficiency of any particular image-coding scheme should be defined only in relation to the class of images that the code is likely to encounter. To understand the representation of images by the mammalian visual system, it might therefore be useful to consider the statistics of images from the natural environment (i.e., images with trees, rocks, bushes, etc). In this study, various coding schemes are compared in relation to how they represent the information in such natural images. The coefficients of such codes are represented by arrays of mechanisms that respond to local regions of space, spatial frequency, and orientation (Gabor-like transforms). For many classes of image, such codes will not be an efficient means of representing information. However, the results obtained with six natural images suggest that the orientation and the spatial-frequency tuning of mammalian simple cells are well suited for coding the information in such images if the goal of the code is to convert higher-order redundancy (e.g., correlation between the intensities of neighboring pixels) into first-order redundancy (i.e., the response distribution of the coefficients). Such coding produces a relatively high signal-to-noise ratio and permits information to be transmitted with only a subset of the total number of cells. These results support Barlow's theory that the goal of natural vision is to represent the information in the natural environment with minimal redundancy.

3,077 citations

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TL;DR: The mathematical analysis of wavelet scattering networks explains important properties of deep convolution networks for classification.

Abstract: A wavelet scattering network computes a translation invariant image representation which is stable to deformations and preserves high-frequency information for classification. It cascades wavelet transform convolutions with nonlinear modulus and averaging operators. The first network layer outputs SIFT-type descriptors, whereas the next layers provide complementary invariant information that improves classification. The mathematical analysis of wavelet scattering networks explains important properties of deep convolution networks for classification. A scattering representation of stationary processes incorporates higher order moments and can thus discriminate textures having the same Fourier power spectrum. State-of-the-art classification results are obtained for handwritten digits and texture discrimination, with a Gaussian kernel SVM and a generative PCA classifier.

1,337 citations

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15 Jun 2000TL;DR: This paper reports on the MPEG-7 Core Experiment CE-Shape, which gave a unique opportunity to compare various shape descriptors for non-rigid shapes with a single closed contour and found that a more theoretical comparison of these descriptors seems to be extremely hard.

Abstract: The Core Experiment CE-Shape-1 for shape descriptors performed for the MPEG-7 standard gave a unique opportunity to compare various shape descriptors for non-rigid shapes with a single closed contour. There are two main differences with respect to other comparison results reported in the literature: (1) For each shape descriptor the experiments were carried out by an institute that is in favor of this descriptor. This implies that the parameters for each system were optimally determined and the implementations were thoroughly rested. (2) It was possible to compare the performance of shape descriptors based on totally different mathematical approaches. A more theoretical comparison of these descriptors seems to be extremely hard. In this paper we report on the MPEG-7 Core Experiment CE-Shape.

890 citations

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TL;DR: A simple yet powerful deep network architecture, U2-Net, for salient object detection (SOD), a two-level nested U-structure that enables us to train a deep network from scratch without using backbones from image classification tasks.

Abstract: In this paper, we design a simple yet powerful deep network architecture, U2-Net, for salient object detection (SOD). The architecture of our U2-Net is a two-level nested U-structure. The design has the following advantages: (1) it is able to capture more contextual information from different scales thanks to the mixture of receptive fields of different sizes in our proposed ReSidual U-blocks (RSU), (2) it increases the depth of the whole architecture without significantly increasing the computational cost because of the pooling operations used in these RSU blocks. This architecture enables us to train a deep network from scratch without using backbones from image classification tasks. We instantiate two models of the proposed architecture, U2-Net (176.3 MB, 30 FPS on GTX 1080Ti GPU) and U2-Net† (4.7 MB, 40 FPS), to facilitate the usage in different environments. Both models achieve competitive performance on six SOD datasets. The code is available: https://github.com/NathanUA/U-2-Net .

753 citations

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TL;DR: Galton's own 1890 account of the moment of discovery is discussed and contrasted with Karl Pearson's widely known association of correlation with a retreat into a recess at Naworth Castle as discussed by the authors, and the circumstances that led Galton to write the account are reviewed.

Abstract: Francis Galton's invention of correlation dates from late in the year 1888, and it arose when he recognized a common thread in three different scientific problems he was studying. Galton's own 1890 account of the moment of discovery is discussed and contrasted with Karl Pearson's widely known association of correlation with a retreat into a recess at Naworth Castle. The circumstances that led Galton to write the account are reviewed.

519 citations