Recursive Motion Estimation on the Essential Manifold

Visual motion estimation can be regarded as estimation of the state of a system of difference equations with unknown inputs defined on a manifold. Such a system happens to be "linear", but it is defined on a space (the so called "Essential manifold") which is not a linear (vector) space. In this paper we will introduce a novel perspective for viewing the motion estimation problem which results in three original schemes for solving it. The first consists in "flattening the space" and solving a nonlinear estimation problem on the flat (euclidean) space. The second approach consists in viewing the system as embedded in a larger euclidean space (the smallest of the embedding spaces), and solving at each step a linear estimation problem on a linear space, followed by a "projection" on the manifold (see fig. 5). A third "algebraic" formulation of motion estimation is inspired by the structure of the problem in local coordinates (flattened space), and consists in a double iteration for solving an "adaptive fixed-point" problem (see fig. 6). Each one of these three schemes outputs motion estimates together with the joint second order statistics of the estimation error, which can be used by any structure from motion module which incorporates motion error [20, 23] in order to estimate 3D scene structure. The original contribution of this paper involves both the problem formulation, which gives new insight into the differential geometric structure of visual motion estimation, and the ideas generating the three schemes. These are viewed within a unified framework. All the schemes have a strong theoretical motivation and exhibit accuracy, speed of convergence, real time operation and flexibility which are superior to other existing schemes [1, 20, 23]. Simulations are presented for real and synthetic image sequences to compare the three schemes against each other and highlight the peculiarities of each one

Recursive Estimation of Camera Motion from Uncalibrated Image Sequences

In This memo we present an extension of the motion estimation scheme presented in a previous CDS technical report [14, 16], in order to deal with image sequences coming from an uncalibrated camera. The scheme is based on some results in epipolar geometry and invariant theory which can be found in [6]. Experiments are performed on noisy synthetic images

Robust and Efficient Recovery of Rigid Motion from Subspace Constraints Solved using Recursive Identification of Nonlinear Implicit Systems

The problem of estimating rigid motion from projections may be characterized using a nonlinear dynamical system, composed of the rigid motion transformation and the perspective map. The time derivative of the output of such a system, which is also called the "motion field", is bilinear in the motion parameters, and may be used to specify a subspace constraint on either the direction of translation or the inverse depth of the observed points. Estimating motion may then be formulated as an optimization task constrained on such a subspace. Heeger and Jepson [5], who first introduced this constraint, solve the optimization task using an extensive search over the possible directions of translation. We reformulate the optimization problem in a systems theoretic framework as the the identification of a dynamic system in exterior differential form with parameters on a differentiable manifold, and use techniques which pertain to nonlinear estimation and identification theory to perform the optimization task in a principled manner. The general technique for addressing such identification problems [14] has been used successfully in addressing other problems in computational vision [13, 12]. The application of the general method [14] results in a recursive and pseudo-optimal solution of the motion problem, which has robustness properties far superior to other existing techniques we have implemented. By releasing the constraint that the visible points lie in front of the observer, we may explain some psychophysical effects on the nonrigid percept of rigidly moving shapes. Experiments on real and synthetic image sequences show very promising results in terms of robustness, accuracy and computational efficiency

Dynamic Rigid Motion Estimation From Weak Perspective

“Weak perspective” represents a simplified projection model that approximates the imaging process when the scene is viewed under a small viewing angle and its depth relief is small relative to its distance from the viewer. We study how to generate dynamic models for estimating rigid 3D motion from weak perspective. A crucial feature in dynamic visual motion estimation is to decouple structure from motion in the estimation model. The reasons are both geometric-to achieve global observability of the model-and practical, for a structure independent motion estimator allows us to deal with occlusions and appearance of new features in a principled way. It is also possible to push the decoupling even further, and isolate the motion parameters that are affected by the so called “bas relief ambiguity” from the ones that are not. We present a novel method for reducing the order of the estimator by decoupling portions of the state space from the time evolution of the measurement constraint. We use this method to construct an estimator of full rigid motion (modulo a scaling factor) on a six dimensional state space, an approximate estimator for a four dimensional subset of the motion space, and a reduced filter with only two states. The latter two are immune to the bas relief ambiguity. We compare strengths and weaknesses of each of the schemes on real and synthetic image sequences

Three dimensional transparent structure segmentation and multiple 3D motion estimation from monocular perspective image sequences

A three dimensional scene can be segmented using different cues, such as boundaries, texture, motion, discontinuities of the optical flow, stereo, models for structure, etc. We investigate segmentation based upon one of these cues, namely three dimensional motion. If the scene contain transparent objects, the two dimensional (local) cues are inconsistent, since neighboring points with similar optical flow can correspond to different objects. We present a method for performing three dimensional motion-based segmentation of (possibly) transparent scenes together with recursive estimation of the motion of each independent rigid object from monocular perspective images. Our algorithm is based on a recently proposed method for rigid motion reconstruction and a validation test which allows us to initialize the scheme and detect outliers during the motion estimation procedure. The scheme is tested on challenging real and synthetic image sequences. Segmentation is performed for the Ullmann's experiment of two transparent cylinders rotating about the same axis in opposite directions

Observability/Identifiability of Rigid Motion under Perspective Projection

The "visual motion" problem consists of estimating the motion of an object viewed under projection. In this paper we address the feasibility of such a problem. We will show that the model which defines the visual motion problem for feature points in the euclidean 3D space lacks of both linear and local (weak) observability. The locally observable manifold is covered with three levels of lie differentiations. Indeed, by imposing metric constraints on the state-space, it is possible to reduce the set of indistinguishable states. We will then analyze a model for visual motion estimation in terms of identification of an Exterior Differential System, with the parameters living on a topological manifold, called the "essential manifold", which includes explicitly in its definition the forementioned metric constraints. We will show that rigid motion is globally observable/identifiable under perspective projection with zero level of lie differentiation under some general position conditions. Such conditions hold when the viewer does not move on a quadric surface containing all the visible points

Dynamic Estimation of Rigid Motion from Perspective Views via Recursive Identification of Exterior Differential Systems with Parameters on a Topological Manifold

We formulate the problem of estimating the motion of a rigid object viewed under perspective projection as the identification of a dynamic model in Exterior Differential form with parameters on a topological manifold. We first describe a general method for recursive identification of nonlinear implicit systems using prediction error criteria. The parameters are allowed to move slowly on some topological (not necessarily smooth) manifold. The basic recursion is solved in two different ways: one is based on a simple extension of the traditional Kalman Filter to nonlinear and implicit measurement constraints, the other may be regarded as a generalized "Gauss-Newton" iteration, akin to traditional Recursive Prediction Error Method techniques in linear identification. A derivation of the "Implicit Extended Kalman Filter" (IEKF) is reported in the appendix. The ID framework is then applied to solving the visual motion problem: it indeed is possible to characterize it in terms of identification of an Exterior Differential System with parameters living on a C0 topological manifold, called the "essential manifold". We consider two alternative estimation paradigms. The first is in the local coordinates of the essential manifold: we estimate the state of a nonlinear implicit model on a linear space. The second is obtained by a linear update on the (linear) embedding space followed by a projection onto the essential manifold. These schemes proved successful in performing the motion estimation task, as we show in experiments on real and noisy synthetic image sequences