Bounded variation | EPFL Graph Search

In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed x-axis and to the y-axis. Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltjes i

On the optimal sampling design for the Graph Total Variation decoder: recovering piecewise-constant graph signals from a few vertex measurements

Rodrigo Cerqueira Gonzalez Pena

Compressed Sensing teaches us that measurements can be traded for offline computation if the signal being sensed has a simple enough representation. Proper decoders can exactly recover the high-dimensional signal of interest from a lower-dimensional vector of that signal's observations. In graph domains -- like social, similarity, or interaction networks -- the relevant signals often have to do with the network's cluster structure. Partitioning a graph into different communities induces a piecewise-constant signal, an object that can be decoded via Graph Total Variation (G-TV) minimization even if it is not fully observed. In fact, assume that such a signal can only be accessed by querying vertices at random. Then, we could sensibly ask: what are the sampling probabilities that minimize the number of queries required for a successful G-TV recovery? This thesis is an attempt to answer this question through the study of the success conditions in G-TV minimization programs. I show that the recovery error in these programs undergoes a phase transition in terms of the number of measurements, with a threshold that explicitly depends on the vertex-sampling probabilities. It suffices to minimize this threshold to obtain an optimal sampling design. Yet, sampling optimally in practice has problems of its own. While numerical experiments reveal that it is important to focus on the places of the graph where the signal varies, implementing the optimal design without actually knowing the signal-to-be-sampled remains an open issue.

EPFL2020

Filtration Shrinkage, Strict Local Martingales And The Follmer Measure

Martin Larsson

When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated Follmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.

Inst Mathematical Statistics2014

Fast high-resolution brain metabolite mapping on a clinical 3T MRI by accelerated H-1-FID-MRSI and low-rank constrained reconstruction

Sébastien Courvoisier, Matthieu Guerquin-Kern, Jeffrey Adam Kasten, Michel Kocher, François Lazeyras, Dimitri Nestor Alice Van De Ville

Purpose: Epitomizing the advantages of ultra short echo time and no chemical shift displacement error, high-resolution-free induction decay magnetic resonance spectroscopic imaging (FID-MRSI) sequences have proven to be highly effective in providing unbiased characterizations of metabolite distributions. However, its merits are often overshadowed in high-resolution settings by reduced signal-to-noise ratios resulting from the smaller voxel volumes procured by extensive phase encoding and the related acquisition times. Methods: To address these limitations, we here propose an acquisition and reconstruction scheme that offers both implicit dataset denoising and acquisition acceleration. Specifically, a slice selective high-resolution FID-MRSI sequence was implemented. Spectroscopic datasets were processed to remove fat contamination, and then reconstructed using a total generalized variation (TGV) regularized low-rank model. We further measured reconstruction performance for random under-sampled data to assess feasibility of a compressed-sensing SENSE acceleration scheme. Performance of the lipid suppression was assessed using an ad hoc phantom, while that of the low-rank TGV reconstruction model was benchmarked using simulated MRSI data. To assess real-world performance, 2D FID-MRSI acquisitions of the brain in healthy volunteers were reconstructed using the proposed framework. Results: Results from the phantom and simulated data demonstrate that skull lipid contamination is effectively removed and that data reconstruction quality is improved with the low-rank TGV model. Also, we demonstrated that the presented acquisition and reconstruction methods are compatible with a compressed-sensing SENSE acceleration scheme. Conclusions: An original reconstruction pipeline for 2D H-1-FID-MRSI datasets was presented that places high-resolution metabolite mapping on 3T MR scanners within clinically feasible limits.

WILEY2019

Fast high-resolution brain metabolite mapping on a clinical 3T MRI by accelerated H-1-FID-MRSI and low-rank constrained reconstruction

Sébastien Courvoisier, Matthieu Guerquin-Kern, Jeffrey Adam Kasten, Michel Kocher, François Lazeyras, Dimitri Nestor Alice Van De Ville

WILEY2019