993 research outputs found
Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods
The convex feasibility problem (CFP) is at the core of the modeling of many
problems in various areas of science. Subgradient projection methods are
important tools for solving the CFP because they enable the use of subgradient
calculations instead of orthogonal projections onto the individual sets of the
problem. Working in a real Hilbert space, we show that the sequential
subgradient projection method is perturbation resilient. By this we mean that
under appropriate conditions the sequence generated by the method converges
weakly, and sometimes also strongly, to a point in the intersection of the
given subsets of the feasibility problem, despite certain perturbations which
are allowed in each iterative step. Unlike previous works on solving the convex
feasibility problem, the involved functions, which induce the feasibility
problem's subsets, need not be convex. Instead, we allow them to belong to a
wider and richer class of functions satisfying a weaker condition, that we call
"zero-convexity". This class, which is introduced and discussed here, holds a
promise to solve optimization problems in various areas, especially in
non-smooth and non-convex optimization. The relevance of this study to
approximate minimization and to the recent superiorization methodology for
constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio
Convex Relaxations of SE(2) and SE(3) for Visual Pose Estimation
This paper proposes a new method for rigid body pose estimation based on
spectrahedral representations of the tautological orbitopes of and
. The approach can use dense point cloud data from stereo vision or an
RGB-D sensor (such as the Microsoft Kinect), as well as visual appearance data.
The method is a convex relaxation of the classical pose estimation problem, and
is based on explicit linear matrix inequality (LMI) representations for the
convex hulls of and . Given these representations, the relaxed
pose estimation problem can be framed as a robust least squares problem with
the optimization variable constrained to these convex sets. Although this
formulation is a relaxation of the original problem, numerical experiments
indicate that it is indeed exact - i.e. its solution is a member of or
- in many interesting settings. We additionally show that this method
is guaranteed to be exact for a large class of pose estimation problems.Comment: ICRA 2014 Preprin
Statistical Model of Shape Moments with Active Contour Evolution for Shape Detection and Segmentation
This paper describes a novel method for shape representation and robust image segmentation. The proposed method combines two well known methodologies, namely, statistical shape models and active contours implemented in level set framework. The shape detection is achieved by maximizing a posterior function that consists of a prior shape probability model and image likelihood function conditioned on shapes. The statistical shape model is built as a result of a learning process based on nonparametric probability estimation in a PCA reduced feature space formed by the Legendre moments of training silhouette images. A greedy strategy is applied to optimize the proposed cost function by iteratively evolving an implicit active contour in the image space and subsequent constrained optimization of the evolved shape in the reduced shape feature space. Experimental results presented in the paper demonstrate that the proposed method, contrary to many other active contour segmentation methods, is highly resilient to severe random and structural noise that could be present in the data
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Amortised MAP Inference for Image Super-Resolution
Image super-resolution (SR) is an underdetermined inverse problem, where a large number of plausible high resolution images can explain the same downsampled image. Most current single image SR methods use empirical risk minimisation, often with a pixel-wise mean squared error (MSE) loss. However, the outputs from such methods tend to be blurry, over-smoothed and generally appear implausible. A more desirable approach would employ Maximum a Posteriori (MAP) infer- ence, preferring solutions that always have a high probability under the image prior, and thus appear more plausible. Direct MAP estimation for SR is non- trivial, as it requires us to build a model for the image prior from samples. Here we introduce new methods for amortised MAP inference whereby we calculate the MAP estimate directly using a convolutional neural network. We first introduce a novel neural network architecture that performs a projection to the affine subspace of valid SR solutions ensuring that the high resolution output of the network is always consistent with the low resolution input. Using this architecture, the amor- tised MAP inference problem reduces to minimising the cross-entropy between two distributions, similar to training generative models. We propose three methods to solve this optimisation problem: (1) Generative Adversarial Networks (GAN) (2) denoiser-guided SR which backpropagates gradient-estimates from denoising to train the network, and (3) a baseline method using a maximum-likelihood- trained image prior. Our experiments show that the GAN based approach per- forms best on real image data. Lastly, we establish a connection between GANs and amortised variational inference as in e. g. variational autoencoders
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