Exact Algorithms for $L^1$-TV Regularization of Real-Valued or Circle-Valued Signals

Storath, Martin; Weinmann, Andreas; Unser, Michaël

doi:10.1137/15m101796x

Cited by 19 publications

(20 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An 1 deviation function is a sum of (weighted) absolute values of the differences between the estimated values and the respective observation values. There are a number of advantages for the use of the 1 deviation function: This function gives the exact maximum likelihood estimate if the noises in the observation values follow the Laplacian distribution [29,10]; it is in general robust to heavy-tailed noises and to the presence of outliers [31,48,44]; it provides better preservation of the contrast and the invariance to global contrast changes [11,44]. The 1 deviation function, in weighted or unweighted form, was used in a number of models and applications.…”

Section: Models With Deviation Terms Onlymentioning

confidence: 99%

“…In signal processing, Storath, Weinmann, and Unser [44], considered a fused lasso model with 1 deviation functions:…”

Section: Models That Include Separation/regularization Termsmentioning

confidence: 99%

“…For the 1 -FL problem (1.9), Dümbgen and Kovac in [14] gave the best algorithm to-date with O(n log n) complexity. Recently Storath, Weinmann, and Unser in [44] proposed an algorithm for the 1 -FL problem of complexity O(n 2 ). GIMR-Algorithm solves all these problems in O(n log n) complexity.…”

Section: Best Algorithms For Total Order Estimationmentioning

confidence: 99%

See 2 more Smart Citations

A Faster Algorithm Solving a Generalization of Isotonic Median Regression and a Class of Fused Lasso Problems

Lu¹

2017

SIAM J. Optim.

View full text Add to dashboard Cite

Many applications in the areas of production, signal processing, economics, bioinformatics, and statistical learning involve a given partial order on certain parameters and a set of noisy observations of the parameters. The goal is to derive estimated values of the parameters that satisfy the partial order that minimize the loss of deviations from the given observed values. A prominent application of this setup is the well-known isotonic regression problem where the partial order is a total order. A commonly studied isotonic regression problem is the isotonic median regression (IMR) where the loss function is a sum of absolute value functions on the deviations of the estimated values from the observation values. We present here the most efficient algorithm to date for a generalization of IMR with any convex piecewise linear loss function as well as for variant models that include 1 norm penalty separation/regularization functions in the objective on the violation of the total order constraints. Such penalty minimization problems that feature convex piecewise linear deviation functions and 1 norm separation/regularization functions include classes of nearly isotonic regression and fused lasso problems as special cases. The algorithm devised here is therefore a unified approach for solving the generalized problem and all known special cases with a complexity that is the most efficient known to date. We present an empirical study showing that our algorithm outperforms the run times of a linear programming software, for simulated data sets.

show abstract

Section: Models With Deviation Terms Onlymentioning

confidence: 99%

“…In signal processing, Storath, Weinmann, and Unser [44], considered a fused lasso model with 1 deviation functions:…”

Section: Models That Include Separation/regularization Termsmentioning

confidence: 99%

See 1 more Smart Citation

A Faster Algorithm Solving a Generalization of Isotonic Median Regression and a Class of Fused Lasso Problems

Lu¹

2017

SIAM J. Optim.

View full text Add to dashboard Cite

show abstract

“…Besides real-valued data, there is an emerging interest in estimation of circle-valued data. Regression of circular data has been considered by Fisher & Lewis (1983) and by Downs & Mardia (2002), and LASSO/TV type problems by Giaquinta et al (1993), , Lellmann et al (2013), Weinmann et al (2014), Bergmann et al (2014), Storath et al (2016). This is motivated by their appearance as data spaces in various contexts including phase data, orientation data, as well as nonlinear color spaces.…”

Section: Prior and Related Workmentioning

confidence: 99%

Jump-penalized least absolute values estimation of scalar or circle-valued signals

Storath

Weinmann

Unser

2017

Information and Inference

Self Cite

View full text Add to dashboard Cite

We study jump-penalized estimators based on least absolute deviations which are often referred to as Potts estimators. They are estimators for a parsimonious piecewise constant representation of noisy data having a noise distribution which has heavier tails or which leads to many severe outliers. We consider real-valued data as well as circle-valued data which appear, for instance, as time series of angles or phase signals. We propose efficient algorithms that compute Potts estimators for real-valued scalar as well as for circlevalued data. The real-valued version improves upon the state-of-the-art solver w.r.t. to computational time. In particular for quantized data, the worst case complexity is improved. The circle-valued version is the first efficient algorithm of this kind. As an illustration, we apply our method to estimate the steps in the rotation of the bacterial flagella motor based on real biological data, and to the estimation of wind directions.

show abstract

“…Approaches for TV regularization for manifold-valued data are considered in [61] which proposes a reformulation as multi-label optimization problem and a convex relaxation, in [50] which proposes iteratively reweighted minimization, and in [93] which proposes cyclic and parallel proximal point algorithms. An exact solver for the TV problem for circle-valued signals has been proposed in [80]. Furthermore, [15] considers half-quadratic minimization approaches, which may be seen as an extension of [50], and [19] considers an extension of the Douglas-Rachford algorithm for manifold-valued data.…”

Section: Introductionmentioning

confidence: 99%

Total Generalized Variation for Manifold-Valued Data

Bredies¹,

Höller²,

Storath³

et al. 2018

SIAM J. Imaging Sci.

View full text Add to dashboard Cite

In this paper we introduce the notion of second-order total generalized variation (TGV) regularization for manifold-valued data in a discrete setting. We provide an axiomatic approach to formalize reasonable generalizations of TGV to the manifold setting and present two possible concrete instances that fulfill the proposed axioms. We provide well-posedness results and present algorithms for a numerical realization of these generalizations to the manifold setup. Further, we provide experimental results for synthetic and real data to further underpin the proposed generalization numerically and show its potential for applications with manifold-valued data.Mathematical subject classification: 94A08, 68U10, 90C90 53B99, 65K10.

show abstract

Exact Algorithms for $L^1$-TV Regularization of Real-Valued or Circle-Valued Signals

Cited by 19 publications

References 42 publications

A Faster Algorithm Solving a Generalization of Isotonic Median Regression and a Class of Fused Lasso Problems

A Faster Algorithm Solving a Generalization of Isotonic Median Regression and a Class of Fused Lasso Problems

Jump-penalized least absolute values estimation of scalar or circle-valued signals

Total Generalized Variation for Manifold-Valued Data

Contact Info

Product

Resources

About