On the detection of motion and the computation of optical flow

A method for the detection of motion in image sequences is presented. In this method, the intensity history at each pixel is convolved with the second derivative in time of a temporal Gaussian smoothing function. The zero crossings in a single frame of the resulting function indicate the positions of moving edges. Intensity changes in time due to illumination effects do not produce zero crossings; thus, they are not interpreted as motion by the present method. It is also shown that the spatial and temporal derivatives of this function can be used to compute the component of the optical flow that is normal to the zero-crossing contours. This computation is also insensitive to nonconvective temporal and spatial variations in the image intensity that are caused by illumination effects     

Data errors in the computation of free motion

Input data in the computation of free motion (the inertia and stiffness matrices) contain some errors. These errors generate errors of output data (the natural frequency vector, the natural mode vectors). In this paper the relationships between errors of input and output data in the computation of free motion are derived. This analysis is applied to the finite element method. The paper presents program ERROR, which computes the errors of the natural frequencies created by the inertia matrix errors. An example of the influence of the errors of the mass and of the mass moment of inertia on the natural frequencies of a ship hull is presented.     

Direct computation of qualitative 3-D shape and motion invariants

Structure from motion often refers to the computation of three-dimensional structure from a matched sequence of images. However, a depth map of a surface is difficult to compute and may not be a good representation for storage and recognition. Given matched images it is shown that the sign of the normal curvature in a given direction at a given point in the image can be computed from a simple difference of slopes of line segments in one image. Using this result, local surface patches can be classified as convex, concave, cylindrical, hyperbolic (saddle point), or planar. At the same time, the translational component of the optical flow, from which the focus of expansion can be computed, is obtained     

Region tracking via level set PDEs without motion computation

We propose an approach to region tracking that is derived from a Bayesian formulation. The novelty of the approach is twofold. First, no motion field or motion parameters need to be computed. This removes a major burden since accurate motion computation has been and remains a challenging problem and the quality of region tracking algorithms based on motion critically depends on the computed motion fields and parameters. The second novelty of this approach, is that very little a priori information about the region being tracked is used in the algorithm. In particular, unlike numerous tracking algorithms, no assumption is made on the strength of the intensity edges of the boundary of the region being tracked, nor is its shape assumed to be of a certain parametric form. The problem of region tracking is formulated as a Bayesian estimation problem and the resulting tracking algorithm is expressed as a level set partial differential equation. We present further extensions to this partial differential equation, allowing the possibility of including additional information in the tracking process, such as priors on the region's intensity boundaries and we present the details of the numerical implementation. Very promising experimental results are provided     

Optimal motion and structure estimation

The causes of existing linear algorithms exhibiting various high sensitivities to noise are analyzed. It is shown that even a small pixel-level perturbation may override the epipolar information that is essential for the linear algorithms to distinguish different motions. This analysis indicates the need for optimal estimation in the presence of noise. Methods are introduced for optimal motion and structure estimation under two situations of noise distribution: known and unknown. Computationally, the optimal estimation amounts to minimizing a nonlinear function. For the correct convergence of this nonlinear minimization, a two-step approach is used. The first step is using a linear algorithm to give a preliminary estimate for the parameters. The second step is minimizing the optimal objective function starting from that preliminary estimate as an initial guess. A remarkable accuracy improvement has been achieved by this two-step approach over using the linear algorithm alone     

Recovery of nonrigid motion and structure

The authors introduce a physically correct model of elastic nonrigid motion. This model is based on the finite element method, but decouples the degrees of freedom by breaking down object motion into rigid and nonrigid vibration or deformation modes. The result is an accurate representation for both rigid and nonrigid motion that has greatly reduced dimensionality, capturing the intuition that nonrigid motion is normally coherent and not chaotic. Because of the small number of parameters involved, this representation is used to obtain accurate overstrained estimates of both rigid and nonrigid global motion. It is also shown that these estimates can be integrated over time by use of an extended Kalman filter, resulting in stable and accurate estimates of both three-dimensional shape and three-dimensional velocity. The formulation is then extended to include constrained nonrigid motion. Examples of tracking single nonrigid objects and multiple constrained objects are presented     

On implementing a computation

To clarify the notion of computation and its role in cognitive science, we need an account of implementation, the nexus between abstract computations and physical systems. I provide such an account, based on the idea that a physical system implements a computation if the causal structure of the system mirrors the formal structure of the computation. The account is developed for the class of combinatorial-state automata, but is sufficiently general to cover all other discrete computational formalisms. The implementation relation is non-vacuous, so that criticisms by Searle and others fail. This account of computation can be extended to justify the foundational role of computation in artificial intelligence and cognitive science.     

On the computation of natural modes of an unsupported vibrating structure by simultaneous iteration

A method is suggested for applying the simultaneous vector iteration of Bauer/Rutishauser in the case where the mass and stiffness matrices involved are banded and non-negative definite. The numerical properties of the method are discussed.     

On two-tape real-time computation and queues

A Turing machine with two storage tapes cannot simulate a queue in both real-time and with at least one storage tape head always within o(n) squares from the start square. This fact may be useful for showing that a two-head tape unit is more powerful in real-time than two one-head tape units, as is commonly conjectured.     

Three-dimensional motion computation and object segmentation in a long sequence of stereo frames

We address the problem of computing the three-dimensional motions of objects in a long sequence of stereo frames. Our approach is bottom-up and consists of two levels. The first level deals with the tracking of 3D tokens from frame to frame and the estimation of their kinematics. The processing is completely parallel for each token. The second level groups tokens into objects based on their kinematic parameters, controls the processing at the low level to cope with problems such as occlusion, disappearance, and appearance of tokens, and provides information to other components of the system. We have implemented this approach using 3D line segments obtained from stereo as the tokens. We use classical kinematics and derive closed-form solutions for some special, but useful, cases of motions. The motion computation problem is then formulated as a tracking problem in order to apply the extended Kalman filter. The tracking is performed in a prediction-matching-update loop in which multiple matches can be handled. Tokens are labeled by a number called its support of existence which measures their adequation to the measurements. If this number goes beyond a threshold, the token disappears. The individual line segments can be grouped into rigid objects according to the similarity of their kinematic parameters. Experiments using synthetic and real data have been carried out and the results found to be quite good.     

On private Hamming distance computation

Finding similarities between two datasets is an important task in many research areas, particularly those of data mining, information retrieval, cloud computing, and biometrics. However, maintaining data protection and privacy while enabling similarity measurements has become a priority for data owners in recent years. In this paper, we study the design of an efficient and secure protocol to facilitate the Hamming distance computation between two semi-honest parties (a client and a server). In our protocol design, both parties are constrained to ensure that no extra information will be revealed other than the computed result (privacy is protected) and further, the output of the protocol is according to the prescribed functionality (correctness is guaranteed). In order to achieve these requirements, we utilize a multiplicative homomorphic cryptosystem and include chaff data into the computation. Two experimental results in this paper demonstrate the performance of both the client and the server.     

The accuracy of the computation of optical flow and of the recoveryof motion parameters

The accuracy and the dependence on parameters of a general scheme for the analysis of time-varying image sequences are discussed. The approach is able to produce vector fields from which it is possible to recover 3-D motion parameters such as time-to-collision and angular velocity. The numerical stability of the computed optical flow and the dependence of the recovery of 3-D motion parameters on spatial and temporal filtering is investigated. By considering optical flows computed on subsampled images or along single scanlines, it is also possible to recover 3-D motion parameters from reduced optical flows. An adequate estimate of time-to-collision can be obtained from sequences of images with spatial resolution reduced to 128×128 pixels or from sequences of single scanlines passing near the focus of expansion. The use of Kalman filtering increases the accuracy and the robustness of the estimation of motion parameters. The proposed approach seems to be able to provide not only a theoretical background but also practical tools that are adequate for the analysis of time-varying image sequences     

Automatic evaluation of sports motion: A generic computation of spatial and temporal errors

In this paper, we propose an innovative automatic evaluation process for any sport motions. Based on a 2-level Dynamic Time Warping, the process allows the evaluation of both spatial and temporal errors of a novice motion based on an experts' motion database and without any prior knowledge on the sport. This new methodology is evaluated with regards to coaches' assessment on two different kinds of motions: tennis serves and karate tsuki.     

On the computation of controllability subspaces

A procedure to determine parametric expressions for the basis matrices of all possible controllability subspaces (c.s.)  of a pair (A, B) is presented. Based on this procedure the paramotrization of the basis matrices corresponding to the family of c.s. of (A, B) that are contained in the kernel of the output map c becomes possible. This result leads to an alternative method of computing the maximal c.s.  of )A, B) contained in the kernel of c, which avoids the intermediate computation of the maximal (A, B)-invarianfc subspaco contained in ker C.     

On computation of Boolean involutive bases

Gröbner bases in Boolean rings can be calculated by an involutive algorithm based on the Janet or Pommaret division. The Pommaret division allows calculations immediately in the Boolean ring, whereas the Janet division implies use of a polynomial ring over field \(\mathbb{F}_2 \). In this paper, both divisions are considered, and distributive and recursive representations of Boolean polynomials are compared from the point of view of calculation effectiveness. Results of computer experiments with both representations for an algorithm based on the Pommaret division and for lexicographical monomial order are presented.     

On the computation of Kalman gains

Modern filtering methods often require high order matrix differential equations to be solved. Standard numerical methods are traditionally slow and prone to be unstable. A numerical approach to the problem of computing the Kalman gain matrix is developed which is both numerically efficient and stable. If a piecewise approximation of the Kalman gain matrix is desired, efficiencies of many orders of magnitude can be realized.     

Exact two-image structure from motion

For two-image structure from motion, we present a simple, exact expression for a least-squares image-reprojection error for finite motion that depends only on the motion. Optimal estimates of the structure and motion can be computed by minimizing this expression just over the motion parameters. Also, we present a solution to the triangulation problem: an exact, explicit expression for the optimal structure estimate given the motion. We identify a new ambiguity in recovering the structure for known motion. We study the exact error's properties experimentally and demonstrate that it often has several local minima for forward or backward motion estimates. Our experiments also show that the "reflected" local minimum of Oliensis (2001) and Soatto et al. (1998) occurs for large translational motions. Our exact results assume that the camera is calibrated and use a least-squares image-reprojection error that applies most naturally to a spherical imaging surface. We approximately extend our approach to the standard least-squares error in the image plane and uncalibrated cameras. We present an improved version of the Sampson error which gives better results experimentally.