Differential Invariants for SE(2)-Equivariant Networks

Sangalli, Mateus; Blusseau, Samy; Velasco-Forero, Santiago; Angulo, Jesús

doi:10.1109/icip46576.2022.9897301

Cited by 3 publications

(6 citation statements)

References 17 publications

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“…Particularly, in comparison with L-STN that used linear interpolation, ESTN shows significant improvements with very similar network settings. In addition, ESTN has higher accuracy compared to the recent developed methods such as ScDCFNet [34] and SE3Mov [23].…”

Section: Svhn and Celebamentioning

confidence: 99%

“…In [34], an equivariant CNN to space and scaling is proposed, which is equivariance for the regular representation of scaling and translation. Moreover, [23] proposed an efficient equivariant network by reducing the computation of moving frames at the input stage rather than repeating computations at all layers. In fact, these approaches support limited transformations because the computational cost increases linearly with the cardinality ratio of the transformation.…”

Section: Related Workmentioning

confidence: 99%

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Entropy Transformer Networks: A Learning Approach via Tangent Bundle Data Manifold

Shamsolmoali

Zareapoor

2023

2023 International Joint Conference on Neural Networks (IJCNN)

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This paper focuses on an accurate and fast interpolation approach for image transformation employed in the design of CNN architectures. Standard Spatial Transformer Networks (STNs) use bilinear or linear interpolation as their interpolation, with unrealistic assumptions about the underlying data distributions, which leads to poor performance under scale variations. Moreover, STNs do not preserve the norm of gradients in propagation due to their dependency on sparse neighboring pixels. To address this problem, a novel Entropy STN (ESTN) is proposed that interpolates on the data manifold distributions. In particular, random samples are generated for each pixel in association with the tangent space of the data manifold and construct a linear approximation of their intensity values with an entropy regularizer to compute the transformer parameters. A simple yet effective technique is also proposed to normalize the nonzero values of the convolution operation, to fine-tune the layers for gradients' norm-regularization during training. Experiments on challenging benchmarks show that the proposed ESTN can improve predictive accuracy over a range of computer vision tasks, including image reconstruction, and classification, while reducing the computational cost.

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Section: Svhn and Celebamentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Entropy Transformer Networks: A Learning Approach via Tangent Bundle Data Manifold

Shamsolmoali

Zareapoor

2023

2023 International Joint Conference on Neural Networks (IJCNN)

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“…In summary, this theory states that convolutions with Gaussian kernels and Gaussian derivatives constitutes a canonical class of image operations as a first layer of visual processing. Such spatial receptive fields, or approximations thereof, can, in turn, be used as basis for expressing a large variety of image operations, both in classical computer vision [8,12,15,42,48,49,51,53,62,63,79,89] and more recently in deep learning [26,32,57,58,69,72,78].…”

Section: Introductionmentioning

confidence: 99%

“…One such domain, and which motivates the present deeper study of discretization effects for Gaussian smoothing operations and Gaussian derivative computations at fine scales, is when applying Gaussian derivative operations in deep networks, as done in a recently developed subdomain of deep learning [26,32,57,58,69,72,78].…”

Section: Introductionmentioning

confidence: 99%

Discrete Approximations of Gaussian Smoothing and Gaussian Derivatives

Lindeberg

2024

J Math Imaging Vis

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This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and the Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways of discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region, to aim at suppressing some of the severe artefacts of sampled Gaussian kernels and sampled Gaussian derivatives at very fine scales, or (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data.We study the properties of these three main discretization methods both theoretically and experimentally and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and the sampled Gaussian derivatives as well as the integrated Gaussian kernels and the integrated Gaussian derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in most of the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing. Below a standard deviation of about 0.75, the derivative estimates obtained from convolutions with the sampled Gaussian derivative kernels are, however, not numerically accurate or consistent, while the results obtained from the discrete analogue of the Gaussian kernel, with its associated central difference operators applied to the spatially smoothed image data, are then a much better choice.

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“…Then this feature function is used to detect corners and edges depending where the function takes its local extrema. In DL, the use of intrinsic properties of invariance and equivariance from the data is an active area of research [6,7,8,9]. Two of the main motivations to study equi/in-variance in neural networks are quite intuitive: i) Objects in natural images are not always oriented, scaled or localized in the same way unless the environment of acquisition is highly constrained.…”

Section: Introductionmentioning

confidence: 99%

GenHarris-ResNet: A Rotation Invariant Neural Network Based on Elementary Symmetric Polynomials

Penaud--Polge

Velasco-Forero

Angulo

2023

Lecture Notes in Computer Science

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In this paper, we propose a rotation invariant neural network based on Gaussian derivatives. The proposed network covers the main steps of the Harris corner detector in a generalized manner. More precisely, the Harris corner response function is a combination of the elementary symmetric polynomials of the integrated dyadic (outer) product of the gradient with itself. In the same way, we define matrices to be the self dyadic product of vectors composed with higher order partial derivatives and combine the elementary symmetric polynomials. A specific global pooling layer is used to mimic the local pooling used by Harris in his method. The proposed network is evaluated through three experiments. It first shows a quasi perfect invariance to rotations on Fashion-MNIST, it obtains competitive results compared to other rotation invariant networks on MNIST-Rot, and it obtains better performances classifying galaxies (EFIGI Dataset) than networks using up to a thousand times more trainable parameters.

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Differential Invariants for SE(2)-Equivariant Networks

Cited by 3 publications

References 17 publications

Entropy Transformer Networks: A Learning Approach via Tangent Bundle Data Manifold

Entropy Transformer Networks: A Learning Approach via Tangent Bundle Data Manifold

Discrete Approximations of Gaussian Smoothing and Gaussian Derivatives

GenHarris-ResNet: A Rotation Invariant Neural Network Based on Elementary Symmetric Polynomials

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