---
OA_type: closed access
_id: '18252'
abstract:
- lang: eng
  text: 'Vector fields arise in a variety of quantity measure and visualization techniques,
    such as fluid flow imaging, motion estimation, deformation measures, and color
    imaging, leading to a better understanding of physical phenomena. Recent progress
    in vector field imaging technologies has emphasized the need for efficient noise
    removal and reconstruction algorithms. A key ingredient in the successful extraction
    of signals from noisy measurements is prior information, which can often be represented
    as a parameterized model. In this work, we extend the overparameterization variational
    framework in order to perform model-based reconstruction of vector fields. The
    overparameterization methodology combines local modeling of the data with global
    model parameter regularization. By considering the vector field as a linear combination
    of basis vector fields and appropriate scale and rotation coefficients, we can
    reduce the denoising problem to a simpler form of coefficient recovery. We introduce
    two versions of the overparameterization framework: a total variation-based method
    and a sparsity-based method, which relies on the cosparse analysis model. We demonstrate
    the efficiency of the proposed frameworks for two- and three-dimensional vector
    fields with linear and quadratic overparameterization models.'
article_processing_charge: No
article_type: original
author:
- first_name: Keren
  full_name: Rotker, Keren
  last_name: Rotker
- first_name: Dafna Ben
  full_name: Bashat, Dafna Ben
  last_name: Bashat
- first_name: Alexander
  full_name: Bronstein, Alexander
  id: 58f3726e-7cba-11ef-ad8b-e6e8cb3904e6
  last_name: Bronstein
  orcid: 0000-0001-9699-8730
citation:
  ama: Rotker K, Bashat DB, Bronstein AM. Overparameterized models for vector fields.
    <i>SIAM Journal on Imaging Sciences</i>. 2020;13(3):1386-1414. doi:<a href="https://doi.org/10.1137/19m1280697">10.1137/19m1280697</a>
  apa: Rotker, K., Bashat, D. B., &#38; Bronstein, A. M. (2020). Overparameterized
    models for vector fields. <i>SIAM Journal on Imaging Sciences</i>. Society for
    Industrial &#38; Applied Mathematics. <a href="https://doi.org/10.1137/19m1280697">https://doi.org/10.1137/19m1280697</a>
  chicago: Rotker, Keren, Dafna Ben Bashat, and Alex M. Bronstein. “Overparameterized
    Models for Vector Fields.” <i>SIAM Journal on Imaging Sciences</i>. Society for
    Industrial &#38; Applied Mathematics, 2020. <a href="https://doi.org/10.1137/19m1280697">https://doi.org/10.1137/19m1280697</a>.
  ieee: K. Rotker, D. B. Bashat, and A. M. Bronstein, “Overparameterized models for
    vector fields,” <i>SIAM Journal on Imaging Sciences</i>, vol. 13, no. 3. Society
    for Industrial &#38; Applied Mathematics, pp. 1386–1414, 2020.
  ista: Rotker K, Bashat DB, Bronstein AM. 2020. Overparameterized models for vector
    fields. SIAM Journal on Imaging Sciences. 13(3), 1386–1414.
  mla: Rotker, Keren, et al. “Overparameterized Models for Vector Fields.” <i>SIAM
    Journal on Imaging Sciences</i>, vol. 13, no. 3, Society for Industrial &#38;
    Applied Mathematics, 2020, pp. 1386–414, doi:<a href="https://doi.org/10.1137/19m1280697">10.1137/19m1280697</a>.
  short: K. Rotker, D.B. Bashat, A.M. Bronstein, SIAM Journal on Imaging Sciences
    13 (2020) 1386–1414.
date_created: 2024-10-08T13:06:25Z
date_published: 2020-01-01T00:00:00Z
date_updated: 2024-10-15T10:43:38Z
day: '01'
doi: 10.1137/19m1280697
extern: '1'
intvolume: '        13'
issue: '3'
language:
- iso: eng
month: '01'
oa_version: None
page: 1386-1414
publication: SIAM Journal on Imaging Sciences
publication_identifier:
  eissn:
  - 1936-4954
publication_status: published
publisher: Society for Industrial & Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Overparameterized models for vector fields
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 13
year: '2020'
...
---
_id: '18430'
abstract:
- lang: eng
  text: Sparse models in dictionary learning have been successfully applied in a wide
    variety of machine learning and computer vision problems, and as a result have
    recently attracted increased research interest. Another interesting related problem
    based on linear equality constraints, namely the sparse null space (SNS) problem,
    first appeared in 1986 and has since inspired results on sparse basis pursuit.
    In this paper, we investigate the relation between the SNS problem and the analysis
    dictionary learning (ADL) problem, and show that the SNS problem plays a central
    role, and may be utilized to solve dictionary learning problems. Moreover, we
    propose an efficient algorithm of sparse null space basis pursuit (SNS-BP) and
    extend it to a solution of ADL. Experimental results on numerical synthetic data
    and real-world data are further presented to validate the performance of our method.
article_processing_charge: No
author:
- first_name: Xiao
  full_name: Bian, Xiao
  last_name: Bian
- first_name: Hamid
  full_name: Krim, Hamid
  last_name: Krim
- first_name: Alexander
  full_name: Bronstein, Alexander
  id: 58f3726e-7cba-11ef-ad8b-e6e8cb3904e6
  last_name: Bronstein
  orcid: 0000-0001-9699-8730
- first_name: Liyi
  full_name: Dai, Liyi
  last_name: Dai
citation:
  ama: 'Bian X, Krim H, Bronstein AM, Dai L. Sparsity and nullity: Paradigms for analysis
    dictionary learning. <i>SIAM Journal on Imaging Sciences</i>. 2016;9(3):1107-1126.
    doi:<a href="https://doi.org/10.1137/15m1030376">10.1137/15m1030376</a>'
  apa: 'Bian, X., Krim, H., Bronstein, A. M., &#38; Dai, L. (2016). Sparsity and nullity:
    Paradigms for analysis dictionary learning. <i>SIAM Journal on Imaging Sciences</i>.
    Society for Industrial &#38; Applied Mathematics. <a href="https://doi.org/10.1137/15m1030376">https://doi.org/10.1137/15m1030376</a>'
  chicago: 'Bian, Xiao, Hamid Krim, Alex M. Bronstein, and Liyi Dai. “Sparsity and
    Nullity: Paradigms for Analysis Dictionary Learning.” <i>SIAM Journal on Imaging
    Sciences</i>. Society for Industrial &#38; Applied Mathematics, 2016. <a href="https://doi.org/10.1137/15m1030376">https://doi.org/10.1137/15m1030376</a>.'
  ieee: 'X. Bian, H. Krim, A. M. Bronstein, and L. Dai, “Sparsity and nullity: Paradigms
    for analysis dictionary learning,” <i>SIAM Journal on Imaging Sciences</i>, vol.
    9, no. 3. Society for Industrial &#38; Applied Mathematics, pp. 1107–1126, 2016.'
  ista: 'Bian X, Krim H, Bronstein AM, Dai L. 2016. Sparsity and nullity: Paradigms
    for analysis dictionary learning. SIAM Journal on Imaging Sciences. 9(3), 1107–1126.'
  mla: 'Bian, Xiao, et al. “Sparsity and Nullity: Paradigms for Analysis Dictionary
    Learning.” <i>SIAM Journal on Imaging Sciences</i>, vol. 9, no. 3, Society for
    Industrial &#38; Applied Mathematics, 2016, pp. 1107–26, doi:<a href="https://doi.org/10.1137/15m1030376">10.1137/15m1030376</a>.'
  short: X. Bian, H. Krim, A.M. Bronstein, L. Dai, SIAM Journal on Imaging Sciences
    9 (2016) 1107–1126.
date_created: 2024-10-15T11:20:55Z
date_published: 2016-08-09T00:00:00Z
date_updated: 2024-12-19T13:05:21Z
day: '09'
doi: 10.1137/15m1030376
extern: '1'
intvolume: '         9'
issue: '3'
language:
- iso: eng
month: '08'
oa_version: None
page: 1107-1126
publication: SIAM Journal on Imaging Sciences
publication_identifier:
  eissn:
  - 1936-4954
publication_status: published
publisher: Society for Industrial & Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Sparsity and nullity: Paradigms for analysis dictionary learning'
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 9
year: '2016'
...