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Keywords
(8)
Conjugate Gradient
Iterative Solution
Linear System
Matrix Equation
Numerical Solution
poisson equation
Surface Reconstruction
Alternating Direction Implicit
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An O(N) iterative solution to the Poisson equation in lowlevel vision problems
An O(N) iterative solution to the Poisson equation in lowlevel vision problems,10.1109/CVPR.1994.323804,S. H. Lai,B. C. Vemuri
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An O(N) iterative solution to the Poisson equation in lowlevel vision problems
(
Citations: 7
)
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S. H. Lai
,
B. C. Vemuri
In this paper, we present a novel iterative
numerical solution
to the
Poisson equation
whose solution is needed in a variety of lowlevel vision problems. Our algorithm is an O(N) (N being the number of discretization points) iterative technique and does not make any assumptions on the shape of the input domain unlike the polyhedral domain assumption in the proof of convergence of multigrid techniques. We present two major results namely, a generalized version of the capacitance matrix theorem and a theorem on O(N) convergence of the
alternating direction implicit
method (ADI) used in our algorithm. Using this generalized theorem, we express the
linear system
corresponding to the discretized
Poisson equation
as a Lyapunov and a capacitance matrix equation. The former is solved using the ADI method while the solution to the later is obtained using a modified biconjugate gradient algorithm. We demonstrate the algorithm performance on synthesized data for the
surface reconstruction
and the SFS problems
Conference:
Computer Vision and Pattern Recognition  CVPR
, pp. 914, 1994
DOI:
10.1109/CVPR.1994.323804
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Citation Context
(6)
...In [52], [36], [
32
], lowlevel vision problems such as optical flow, surface reconstruction, and shape from shading are formulated using variational principles whose EulerLagrange solutions take the form of a Poisson equation providing a necessary condition for a minimum...
Lena Gorelick
,
et al.
Shape Representation and Classification Using the Poisson Equation
...considerable attention. Accordingly, several techniques have been introduced to solve the equation by numerical, analytical, and iterative approaches [21,22,
25
]...
Sangmook Lee
,
et al.
A Shape Representation for Planar Curves by Shape Signature Harmonic E...
...In [52], [36], [
32
], lowlevel vision problems such as optical flow, surface reconstruction, and shape from shading are formulated using variational principles whose EulerLagrange solutions take the form of a Poisson equation providing a necessary condition for a minimum...
Lena Gorelick
,
et al.
Shape Representation and Classification Using the Poisson Equation
...The surface reconstruction problem is to apply some form of regularization to approximate the rough edge map by a smooth function . Variational approaches to this problem specify the solution as a stationary point which minimizes the energy functional [12], [
21
]:...
Jia Li
,
et al.
A Fast Spectral Method for Active 3D Shape Reconstruction
...We can prove that the ADI method takes only OÖNÜ time to converge [
29
] with soft data constraints where N is the number of nodes in the rectangular finite element discretization of the snake in our model...
Baba C. Vemuri
,
et al.
Snake Pedals: Compact and Versatile Geometric Models with PhysicsBase...
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Citations
(7)
Shape Representation and Classification Using the Poisson Equation
(
Citations: 40
)
Lena Gorelick
,
Meirav Galun
,
Eitan Sharon
,
Ronen Basri
,
Achi Brandt
Journal:
IEEE Transactions on Pattern Analysis and Machine Intelligence  PAMI
, vol. 28, no. 12, pp. 19912005, 2006
A Shape Representation for Planar Curves by Shape Signature Harmonic Embedding
(
Citations: 5
)
Sangmook Lee
,
A. Lynn Abbott
,
Neil A. Clark
,
Philip A. Araman
Conference:
Computer Vision and Pattern Recognition  CVPR
, vol. 2, pp. 19401947, 2006
Shape Representation and Classification Using the Poisson Equation
(
Citations: 67
)
Lena Gorelick
,
Meirav Galun
,
Eitan Sharon
,
Ronen Basri
,
Achi Brandt
Conference:
Computer Vision and Pattern Recognition  CVPR
, vol. 2, pp. 6167, 2004
A Fast Spectral Method for Active 3D Shape Reconstruction
(
Citations: 5
)
Jia Li
,
Alfred O. Hero
Journal:
Journal of Mathematical Imaging and Vision  JMIV
, vol. 20, no. 12, pp. 7387, 2004
Snake Pedals: Compact and Versatile Geometric Models with PhysicsBased Control
(
Citations: 12
)
Baba C. Vemuri
,
Yanlin Guo
Journal:
IEEE Transactions on Pattern Analysis and Machine Intelligence  PAMI
, vol. 22, no. 5, pp. 445459, 2000