Real-Time Computation of Optical Flow Along Contours of Significant Intensity Change

We propose a system for the real-time computation of optical flow along contours of significant intensity change. Hildreth 1] formulated the energy functional for this problem and presented a conjugate gradient method to find the global minimum of the quadratic energy functional. For a contour withNpoints, the conjugate gradient method requiresO(N) iterations (i.e.,O(N2) operations) to converge to a solution. The direct analytical methods we present here require onlyO(N) operations. Using current desktop computing power (a Sun SPARCstation 10), the direct methods make it possible to compute the optical flow in real time.In the finite difference formulation of the problem, the structure of the coefficient matrix for open contours is block tridiagonal, and that for closed contours is cyclic block tridiagonal 2,3]. Therefore, it is natural to consider block extensions of the tridiagonal matrix solvers abundant in mathematics literature. This approach is graceful in that the properties of the tridiagonal matrix solvers carry over to the corresponding block tridiagonal solvers. Some of these properties are low computational complexityO(N) operations), high numerical stability, and parallelism. Based on these guiding principles, we propose block tridiagonal Gaussian elimination for open contours and the generalized Ahlberg?Nilson?Walsh method for closed contours. Assuming that the computation of Laplacian of Gaussian of the images in a sequence, and the detection of the image contours, can be done in real time using parallel hardware, the computation of optical flow using the two methods can be done in real time with common desktop hardware (we report results using a 70 MHz Sun SPARCstation 10). Both of these methods can be further speeded up by implementation on parallel hardware using a block generalization of Wang's partition method.

[1]  Tomaso A. Poggio,et al.  Motion Field and Optical Flow: Qualitative Properties , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[2]  H. H. Wang,et al.  A Parallel Method for Tridiagonal Equations , 1981, TOMS.

[3]  J. L. Walsh,et al.  The theory of splines and their applications , 1969 .

[4]  Robert G. Voigt,et al.  The Solution of Tridiagonal Linear Systems on the CDC STAR 100 Computer , 1975, TOMS.

[5]  K. Nakayama,et al.  The aperture problem—II. Spatial integration of velocity information along contours , 1988, Vision Research.

[6]  K. Nakayama,et al.  The aperture problem—I. Perception of nonrigidity and motion direction in translating sinusoidal lines , 1988, Vision Research.

[7]  Atul Kumar Chhabra Algorithms and architectures for variational problems in early vision , 1991 .

[8]  Ellen C. Hildreth,et al.  Computations Underlying the Measurement of Visual Motion , 1984, Artif. Intell..

[9]  Stephan Olariu,et al.  A Novel Deterministic Sampling Scheme with Applications to Broadcast-Efficient Sorting on the Reconfigurable Mesh , 1986, J. Parallel Distributed Comput..

[10]  Harold S. Stone,et al.  Parallel Tridiagonal Equation Solvers , 1975, TOMS.

[11]  D. Heller A Survey of Parallel Algorithms in Numerical Linear Algebra. , 1978 .

[12]  Atul K. Chhabra Fast direct methods for computing optical flow along contours , 1994, Proceedings of 1st International Conference on Image Processing.

[13]  G. Wasilkowski,et al.  Computing optical flow , 1989, [1989] Proceedings. Workshop on Visual Motion.

[14]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[15]  Roger W. Hockney,et al.  A Fast Direct Solution of Poisson's Equation Using Fourier Analysis , 1965, JACM.

[16]  Clive Temperton,et al.  Algorithms for the Solution of Cyclic Tridiagonal Systems , 1975 .

[17]  Ellen C. Hildreth,et al.  Measurement of Visual Motion , 1984 .

[18]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[19]  D Marr,et al.  Directional selectivity and its use in early visual processing , 1981, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[20]  Timothy A. Grogan,et al.  Uniqueness, the minimum norm constraint, and analog networks for optical flow along contours , 1990, [1990] Proceedings Third International Conference on Computer Vision.

[21]  D Marr,et al.  Theory of edge detection , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[22]  Harold S. Stone,et al.  An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations , 1973, JACM.

[23]  F. Dorr The Direct Solution of the Discrete Poisson Equation on a Rectangle , 1970 .