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Research Projects
Fast Solutions to Variational Optimization Problems in Image Processing
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Abstract
Energy minimization problems arise naturally in a lot of graphics and
vision applications, such as image editing, surface deforming or
variational motion registration, as they allow one to compute a
"natural" effect of a set of given actions on the image data. The
minimization problem can be expressed as the variational minimization of
the energy functional, or as a system of partial differential equations
known as the Euler-Lagrange (E-L) equations. Traditional methods approach
this problem almost exclusively through the E-L system, which is linear for
the case of the energy functional. However, this entails computing all
stationary functionals, only one of which is the desired minimizer
(analogously, one could talk about local and global minima). This can be an
inhibiting factor for many real-time applications, for which efficiency (in
both performance and accuracy) is instrumental.
We propose to explore the use of the Conjugate Gradient (CG) method in a hierarchical fashion, operating on both the the Variational and the E-L forms, in order to compute the optimal solution while avoiding divergence towards false candidate solutions. The CG method implicitly leverages the spectral properties of the system it is applied to and could be useful in filtering out unnecessary spectral components to obtain a coarse direction towards the true minimizer. Also, different discretization models that lead to a sparse gradient representation of the system will be considered.
Advisor(s): Xiaobai Sun
Carlo Tomasi, Nikos Pitsianis