34 research outputs found
On Graduated Optimization for Stochastic Non-Convex Problems
The graduated optimization approach, also known as the continuation method,
is a popular heuristic to solving non-convex problems that has received renewed
interest over the last decade. Despite its popularity, very little is known in
terms of theoretical convergence analysis. In this paper we describe a new
first-order algorithm based on graduated optimiza- tion and analyze its
performance. We characterize a parameterized family of non- convex functions
for which this algorithm provably converges to a global optimum. In particular,
we prove that the algorithm converges to an {\epsilon}-approximate solution
within O(1/\epsilon^2) gradient-based steps. We extend our algorithm and
analysis to the setting of stochastic non-convex optimization with noisy
gradient feedback, attaining the same convergence rate. Additionally, we
discuss the setting of zero-order optimization, and devise a a variant of our
algorithm which converges at rate of O(d^2/\epsilon^4).Comment: 17 page
Speckle Noise Reduction via Nonconvex High Total Variation Approach
We address the problem of speckle noise removal. The classical total variation is extensively used in this field to solve such problem, but this method suffers from the staircase-like artifacts and the loss of image details. In order to resolve these problems, a nonconvex total generalized variation (TGV) regularization is used to preserve both edges and details of the images. The TGV regularization which is able to remove the staircase effect has strong theoretical guarantee by means of its high order smooth feature. Our method combines the merits of both the TGV method and the nonconvex variational method and avoids their main drawbacks. Furthermore, we develop an efficient algorithm for solving the nonconvex TGV-based optimization problem. We experimentally demonstrate the excellent performance of the technique, both visually and quantitatively
Total Variation Image Restoration Method Based on Subspace Optimization
The alternating direction method is widely applied in total variation image restoration. However, the search directions of the method are not accurate enough. In this paper, one method based on the subspace optimization is proposed to improve its optimization performance. This method corrects the search directions of primal alternating direction method by using the energy function and a linear combination of the previous search directions. In addition, the convergence of the primal alternating direction method is proven under some weaker conditions. Thus the convergence of the corrected method could be easily obtained since it has same convergence with the primal alternating direction method. Numerical examples are given to show the performance of proposed method finally
Image Reconstruction from Undersampled Confocal Microscopy Data using Multiresolution Based Maximum Entropy Regularization
We consider the problem of reconstructing 2D images from randomly
under-sampled confocal microscopy samples. The well known and widely celebrated
total variation regularization, which is the L1 norm of derivatives, turns out
to be unsuitable for this problem; it is unable to handle both noise and
under-sampling together. This issue is linked with the notion of phase
transition phenomenon observed in compressive sensing research, which is
essentially the break-down of total variation methods, when sampling density
gets lower than certain threshold. The severity of this breakdown is determined
by the so-called mutual incoherence between the derivative operators and
measurement operator. In our problem, the mutual incoherence is low, and hence
the total variation regularization gives serious artifacts in the presence of
noise even when the sampling density is not very low. There has been very few
attempts in developing regularization methods that perform better than total
variation regularization for this problem. We develop a multi-resolution based
regularization method that is adaptive to image structure. In our approach, the
desired reconstruction is formulated as a series of coarse-to-fine
multi-resolution reconstructions; for reconstruction at each level, the
regularization is constructed to be adaptive to the image structure, where the
information for adaption is obtained from the reconstruction obtained at
coarser resolution level. This adaptation is achieved by using maximum entropy
principle, where the required adaptive regularization is determined as the
maximizer of entropy subject to the information extracted from the coarse
reconstruction as constraints. We demonstrate the superiority of the proposed
regularization method over existing ones using several reconstruction examples