Augmented Lagrangian without alternating directions
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Augmented Lagrangian without alternating directions
don't want to also tune the optimization algorithm parameters avoid interplay ... the parameter(s) image deconvolution [Matakos, Ramani & Fessler 2013].
Augmented Lagrangian without alternating directions: practical algorithms for inverse problems in imaging Rahul Mourya, Loïc Denis, Eric Thiébaut, Jean-Marie Becker Starting point
Typical optimization problems in imaging
Many image processing tasks are formulated as a large scale optimization problem:
The augmented Lagrangian reformulation the optimization problem is too hard
separate variables
convex
smooth
turn the constrained problem into an unconstrained form with the augmented Lagrangian: optimization parameter Lagrange multiplier
non-smooth e.g., L1 norm to promote sparsity or indicator function for constraints
large scale problems: typically image restoration
image segmentation
to
often requires to tune the regularization parameter need fast and reliable optimization methods don't want to also tune the optimization algorithm parameters avoid interplay between regularization & optimization parameter tuning
Lagrangian
unknowns
limited-memory quasi-Newton methods are very efficient cannot apply to non-smooth problems many algorithms for non-smooth optimization can be very slow
distance to the solution after 5s [Goldstein, Bresson & Osher 2010]
ADMM 1 splitting
can generally be computed in closed-form the partially optimized subproblem is smooth can be solved approximatively using a quasi-Newton method with and a line search
works best whith multiple splittings so that each sub-problem can be solved exactly [Matakos, Ramani & Fessler 2013] requires fine-tuning the parameter(s)
proposed proposed 1 splitting
starting point
starting point
almost insensitive to optimization parameter
reaches state-of-the art performance with
optimization parameter
no parameter tuning
[Matakos, Ramani & Fessler 2013] "Accelerated edge-preserving image restoration without boundary artifacts," IEEE Transactions on Image Processing [Houhou, Thiran & Bresson 2009] "Fast texture segmentation based on semi-local region descriptor and active contour," Numerical Mathematics: Theory, Methods and Applications [Goldstein, Bresson & Osher 2010] "Geometric applications of the split Bregman method: segmentation and surface reconstruction," Journal of Scientific Computing
