Alexander B. Kyatkin’s research while affiliated with Johns Hopkins University and other places

... As a further example let us mention the field of materials science where bases consisting of (symmetrized) basis functions have recently been introduced for the representation of various functions of interest (Mason and Schuh, 2008;Patala, Mason and Schuh, 2012;Mason and Patala, 2019). In particular, the manifold SO(3) (the rotation group in dimension three) arises naturally in various applications from engineering and crystallography (Chirikjian and Kyatkin, 2021;Schaeben et al., 2007;Hielscher, 2013;Kovacs et al., 2003). ...

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Optimal Designs for Regression on Lie Groups

... This interval is the difference between the fixed length of each trial (2 s) and the length of the segments, like in the earlier work (Kipiński et al., 2011-see Fig. 2a). For consecutive trials, the spectral power was calculated as the discrete Fourier transform using the fast Fourier transform (FFT) algorithm (see, for example, Burrus, 2012). Furthermore, sequences of selected coefficients of these spectra, z t , t = 1, …, T, were derived for some fixed frequency f. ...

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Time-series analysis of trial-to-trial variability of MEG power spectrum during rest state, unattended listening, and frequency-modulated tones classification

... They employed closed-form convolution of realvalued functions on the Special Euclidean Group for generating workspaces. A method based on the Fourier transform of the discrete-motion group for computing the configuration and workspaces of robot manipulators was proposed in [13], [25], [26]. ...

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Obstacle Avoidance Path Planning of Planar Redundant Manipulators Using Workspace Density

... Angles are widely used to represent information in meteorology [37], ecology [53,108], medicine [2,19,20], astronomy [9,70] and engineering [11]. Moreover, many natural and social phenomena have circular-linear correlation on some time scale. ...

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Hyperdimensional computing: a framework for stochastic computation and symbolic AI

... Convolution is a mathematical operation with several applications in pure and applied mathematics such as numerical analysis, numerical linear algebra, and design of filters in signal processing [1][2][3][4]. The convolution operations associated with integral transforms have important theoretical value and many applications to numerous problems such as sampling [5], evaluation of integrals and convolution sums [6,7], summing of series, and solving equations of mathematical physics [8][9][10][11][12]. ...

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Fractional convolution associated with a class of integral equations

... Due to the left invariance of the kernel, it must depend solely on the group difference g −1 0 g P t|0 (∆g g|∆g g 0 ) = P t|0 (g|g 0 ) ∀ ∆g, g ⇒ P t|0 (g|g 0 ) = P t|0 (g −1 0 g|I) ∀ g Therefore, we can simply denote any left-invariant kernel as K t (g) = P t|0 (g|I). The right invariance requires this kernel to satisfy K t (∆g g ∆g −1 ) = K t (g), meaning that it is a class function, which does not exist for L 2 (SE(3)) [86,85]. ...

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Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulation

... Exhaustive search-based methods, on the other hand, could return more accurate results but have formidable costs if implemented naively. Methods exploiting convolution structures of 2 (22) can lead to great computational speed up but are still considered expensive for large volumes. Motivated by these issues, in this paper we shall propose an alignment algorithm based on solving (2) in the 1-Wasserstein distance, which is known to better reflect rigid transformations than Euclidean distances and hence creates a better loss landscape. ...

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Alignment of Density Maps in Wasserstein Distance

... Convolution incorporating rotational steering has been recognized as an effective tool for template matching since the early work of Freeman and Adelson [FA91,SF96,THO99], and has been extended to incorporate the actions of larger groups [KC99a,KC99b,CK16]. More recently, the advent of deep learning in imaging, vision and graphics has coincided with the development of group equivariant [CW16a, WHS18, BLV*18, WB18, WW19] and steerable [CW16b, WGTB17, WGW*18, EABMD18, WC19] convolutional neural networks (CNNs), which present a general notion of equivariant convolutions on homogeneous spaces. ...

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ECHO: Extended Convolution Histogram of Orientations for Local Surface Description

... Convolution incorporating rotational steering has been recognized as an effective tool for template matching since the early work of Freeman and Adelson [FA91,SF96,THO99], and has been extended to incorporate the actions of larger groups [KC99a,KC99b,CK16]. More recently, the advent of deep learning in imaging, vision and graphics has coincided with the development of group equivariant [CW16a, WHS18, BLV*18, WB18, WW19] and steerable [CW16b, WGTB17, WGW*18, EABMD18, WC19] convolutional neural networks (CNNs), which present a general notion of equivariant convolutions on homogeneous spaces. ...

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ECHO: Extended Convolution Histogram of Orientations for Local Surface Description