Conference Paper

Wavelet-Based Fluid Motion Estimation

DOI: 10.1007/978-3-642-24785-9_62 Conference: Scale Space and Variational Methods in Computer Vision - Third International Conference, SSVM 2011, Ein-Gedi, Israel, May 29 - June 2, 2011, Revised Selected Papers
Source: DBLP


Based on a wavelet expansion of the velocity field, we present a novel optical flow algorithm dedicated to the estimation of continuous motion fields such as fluid flows. This scale-space representation, associated to a simple gradient-based optimization algorithm, naturally sets up a well-defined multi-resolution analysis framework for the optical flow estimation problem, thus avoiding the common drawbacks of standard multi-resolution schemes. Moreover, wavelet properties enable the design of simple yet efficient high-order regularizers or polynomial approximations associated to a low computational complexity. Accuracy of proposed methods is assessed on challenging sequences of turbulent fluids flows.

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    • "Variational approaches to optical flow estimation were pioneered by Horn and Schunck [34] followed by a vast number of refinements and extensions, including sophisticated data fidelity terms going beyond the brightness [7] [13] [31] and nonsmooth regularizers, e.g., TV-like ones [2] [33] including also higher order derivatives [62] [63] [64] and nonlocal regularizers [59], to mention only few of them. In general multiscale approaches have to be taken into account to correctly determine larger and smaller flow vectors [1] [12] [23]. A good overview is given in [7]. "
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    ABSTRACT: This paper addresses the problems of disparity and optical flow partitioning based on the brightness invariance assumption. We investigate new variational approaches to these problems with Potts priors and possibly box constraints. For the optical flow partitioning, our model includes vector-valued data and an adapted Potts regularizer. Using the notion of asymptotically level stable (als) functions, we prove the existence of global minimizers of our functionals. We propose a modified alternating direction method of multipliers. This iterative algorithm requires the computation of global minimizers of classical univariate Potts problems which can be done efficiently by dynamic programming. We prove that the algorithm converges both for the constrained and unconstrained problems. Numerical examples demonstrate the very good performance of our partitioning method.
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    • "For the sake of conciseness, we will limit the presentation to linear operators. Note however that extension to the non-linear case by successive linearization is possible [10]. Based on (1), one can for instance define an observation operator by using the function Π d : "

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    • ". Convergence of the Fourier integral is therefore guaranteed, and the concept of fBm may be extended to " integrated " forms. The third argument is related to the special context of optimization in Bayesian analysis: wavelet multiresolution analyses of unknowns proved experimentally to be relevant optimization strategy for non-convex inverse problems [9]. Moreover, optimization can be led with a low-complexity since divergence-free wavelet decomposition [11] or the one-dimensional fractional wavelet decomposition [1] have an advantageous fast implementation with recursive filter banks [22]. "
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    ABSTRACT: This paper presents a method for regularization of inverse problems. The vectorial bi-dimensional unknown is assumed to be the realization of an isotropic divergence-free fractional Brownian Motion (fBm). The method is based on fractional Laplacian and divergence-free wavelet bases. The main advantage of these bases is to enable an easy formalization in a Bayesian framework of fBm priors, by simply sampling wavelet coefficients according to Gaussian white noise. Fractional Laplacians and the divergence-free projector can naturally be implemented in the Fourier domain. An interesting alternative is to remain in the spatial domain. This is achieved by the analytical computation of the connection coefficients of divergence-free fractional Laplacian wavelets, which en-ables to easily rotate this simple prior in any sufficiently "regular" wavelet basis. Taking advantage of the tensorial structure of a separable fractional wavelet basis approximation, isotropic regulariza-tion is then computed in the spatial domain by low-dimensional matrix products. The method is successfully applied to fractal image restoration and turbulent optic-flow estimation.
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