Learning the Lie Groups of Visual Invariance

Abstract

A fundamental problem in biological and machine vision is visual invariance: How are objects perceived to be the same despite transformations such as translations, rotations, and scaling? In this letter, we describe a new, unsupervised approach to learning invariances based on Lie group theory. Unlike traditional approaches that sacrifice information about transformations to achieve invariance, the Lie group approach explicitly models the effects of transformations in images. As a result, estimates of transformations are available for other purposes, such as pose estimation and visuomotor control. Previous approaches based on first-order Taylor series expansions of images can be regarded as special cases of the Lie group approach, which utilizes a matrix-exponential-based generative model of images and can handle arbitrarily large transformations. We present an unsupervised expectation-maximization algorithm for learning Lie transformation operators directly from image data containing examples of transformations. Our experimental results show that the Lie operators learned by the algorithm from an artificial data set containing six types of affine transformations closely match the analytically predicted affine operators. We then demonstrate that the algorithm can also recover novel transformation operators from natural image sequences. We conclude by showing that the learned operators can be used to both generate and estimate transformations in images, thereby providing a basis for achieving visual invariance.

A Theory of Slow Feature Analysis for Transformation-Based Input Signals with an Application to Complex Cells

“…The tuning properties of the simulated cells can be understood as a way of generating harmonic oscillations as output signals if the transformations are applied with constant velocity, in line with previous analytical results (Wiskott, 2003). In addition, the analysis raises a link to previous group-theoretical approaches that learn the dynamical structure of image sequences in terms of the generators of the underlying transformations (Rao & Ruderman, 1998;Miao & Rao, 2007).…”

Invariance of visual operations at the level of receptive fields

“…In relation to visual invariances, Miao and Rao [109] have presented an expectation/maximization approach for learning Lie transformation operators and applied this approach for learning affine transformations in the spatial domain. There is a close relationship between derivatives of Lie groups and local linearizations of non-linear transformations as used in our work.…”

LIE operators for compressive sensing

“…, zL); thus, T = T (z) for each T ∈ T . Then, the group structure endowed on T enables the following convenient formula: x = k e z k G k x0, where G k represents the matrix logarithm of the k th generator T k [8]. Note that the transformation operators will not be commutative in general.…”

“…Manifolds defined by Lie operators are widely encountered in practice. For instance, such models can be used for families of 1D signals formed by shifting, scaling and dilating a fixed base signal x0 (applicable to radar, sonar and antenna arrays); families of 2D images subjected to transformations such as translation, rotation, scaling, and illumination change (applicable to multi-view camera networks); and a temporally varying sequence of images (applicable to video acquisition) [8,9]. Specifically, we study and solve two problems pertaining to CS sampling and recovery in this context:…”