On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction

Snyder, Martin Avery

doi:10.1109/wvm.1989.47100

Cited by 17 publications

(14 citation statements)

References 10 publications

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“…This projection has been introduced by Nagel and Enkelmann [31,32] in the context of optic flow estimation. We use it here because of its simplicity (the underlying second order differential operator is linear) and because this method has demonstrated its performance numerous times in the context of optical flow estimations [3,4,8,14,31,32,45,47].…”

Section: The Energy Functionalmentioning

confidence: 99%

Dense Disparity Map Estimation Respecting Image Discontinuities: A PDE and Scale-Space Based Approach

Álvarez¹,

Deriche²,

Sánchez³

et al. 2002

Journal of Visual Communication and Image Representation

158

155

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We present an energy based approach to estimate a dense disparity map from a set of two weakly calibrated stereoscopic images while preserving its discontinuities resulting from image boundaries. We first derive a simplified expression for the disparity that allows us to estimate it from a stereo pair of images using an energy minimization approach. We assume that the epipolar geometry is known, and we include this information in the energy model. Discontinuities are preserved by means of a regularization term based on the Nagel-Enkelmann operator. We investigate the associated Euler-Lagrange equation of the energy functional, and we approach the solution of the underlying partial differential equation (PDE) using a gradient descent method. The resulting parabolic problem has a unique solution. In order to reduce the risk to be trapped within some irrelevant local minima during the iterations, we use a focusing strategy based on a linear scalespace. Experimental results on both synthetic and real images are presented to illustrate the capabilities of this PDE and scale-space based method.

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Section: The Energy Functionalmentioning

confidence: 99%

Dense Disparity Map Estimation Respecting Image Discontinuities: A PDE and Scale-Space Based Approach

Álvarez¹,

Deriche²,

Sánchez³

et al. 2002

Journal of Visual Communication and Image Representation

158

155

View full text Add to dashboard Cite

show abstract

“…Some [5] will do it blindly, while others will do it only at certain locations ( [10]) or along certain directions ( [9]). Still others will use a robust norm to minimize the influence of the outliers ( [2,8,7]). …”

Section: Methodsmentioning

confidence: 99%

“…inhomogeneous and anisotropic, image-driven h(tr(∇u∇u T + ∇v∇v T )) [2,8] inhomogeneous and isotropic, flow-driven tr h(∇u∇u T + ∇v∇v T ) [7] inhomogeneous and anisotropic, flow-driven Table 2. The six flow components that are induced by the six DOFs of a 3D rigid-body motion.…”

Section: Expressionmentioning

confidence: 99%

Regularizing optical-flow computation using tensor theory and complex analysis

Koppel

Tsai

Wang

2008

2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops

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“…Usually, these functionals consist of two terms measuring different properties of the admissible functions: closeness to the data and smoothness. Snyder (1989) has recently classified all smoothness terms involving first-order derivatives of the components of the vector field and of the image function g(x, t), which are positive definite quadratic forms and do not couple the two components vl and v2 of the vector field. It turned out that the only physically plausible smoothness terms of this class are those of Horn & Schunck (1981) …”

Section: Overviewmentioning

confidence: 99%

“…

In this paper we discuss the possibilities of approximating the solutions to the minimization problems of Horn & Schunk (1981) andNagel (1987).

…”

mentioning

confidence: 99%

Determining optical flow for irregular domains by minimizing quadratic functionals of a certain class

Schnörr

1991

Int J Comput Vision

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Snyder (1989) has recently classified all smoothness terms which involve first-order derivatives of the flowfield u (x,t) and of the image grey-value function g (x,t). The physically plausible smoothness terms belonging to this class are known from the work of Horn and Schunck (1981) and Nagel (1987). In this paper we discuss the possibilities of approximating the solutions to the minimization problems of Horn and Schunck (1981) and Nagel (1987). In particular, it is shown that these solutions exist, are unique, and depend continuously on the input data. These properties make it possible, while taking into consideration arbitrary models of the grey-value function, to approximate efficiently the (weak) solutions of the associated boundary-value problems in irregularly shaped domains (with a "sufficiently smooth" boundary) using finite elements. Experiments with image sequences from synthetical as well as outdoor scenes show how the orientation dependency of the smoothness term in Nagel' s approach influences the results

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On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction

Cited by 17 publications

References 10 publications

Dense Disparity Map Estimation Respecting Image Discontinuities: A PDE and Scale-Space Based Approach

Dense Disparity Map Estimation Respecting Image Discontinuities: A PDE and Scale-Space Based Approach

Regularizing optical-flow computation using tensor theory and complex analysis

Determining optical flow for irregular domains by minimizing quadratic functionals of a certain class

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