Imaging with Kantorovich--Rubinstein Discrepancy

Lellmann, Jan; Lorenz, Dirk A.; Schönlieb, Carola-Bibiane; Valkonen, Tuomo

doi:10.1137/140975528

Cited by 92 publications

(96 citation statements)

References 57 publications

(65 reference statements)

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“…Based on this property, Brauer and Lorenz [52] proposed a three-part image decomposition with transport norms. Lellmann et al [53] proposed the use of transport norms for image denosing and two-part image decomposition. We believe that the commonalities between the Vese-Osher model [9], Meyer's G-norm, and the recently proposed models using transport norms deserve further research.…”

Section: Discussionmentioning

confidence: 99%

Directional global three-part image decomposition

Thai

Gottschlich

2016

J Image Video Proc.

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We consider the task of image decomposition, and we introduce a new model coined directional global three-part decomposition (DG3PD) for solving it. As key ingredients of the DG3PD model, we introduce a discrete multi-directional total variation norm and a discrete multi-directional G-norm. Using these novel norms, the proposed discrete DG3PD model can decompose an image into two or three parts. Existing models for image decomposition Advantages of the DG3PD model over existing ones lie in the properties enforced on the cartoon and texture images. The geometric objects in the cartoon image have a very smooth surface and sharp edges. The texture image yields oscillating patterns on a defined scale which are both smooth and sparse. Moreover, the DG3PD method achieves the goal of perfect reconstruction by summation of all components better than the other considered methods. Relevant applications of DG3PD are a novel way of image compression as well as feature extraction for applications such as latent fingerprint processing and optical character recognition.

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Section: Discussionmentioning

confidence: 99%

Directional global three-part image decomposition

Thai

Gottschlich

2016

J Image Video Proc.

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“…The optimal transport problem has also been used in several image denoising and restoration problems [30]. The goal in these applications is to restore or reconstruct an image from noisy or incomplete observation.…”

Section: Applicationsmentioning

confidence: 99%

“…The goal in these applications is to restore or reconstruct an image from noisy or incomplete observation. Lellmann et al [30] utilized the Kantorovich-Rubinsten discrepancy term together with a Total Variation term in the context of image denoising. They called their method Kantorovich-Rubinstein-TV (KR-TV) denoising.…”

Section: Applicationsmentioning

confidence: 99%

“…These techniques have been used to simplify and augment the accuracy of numerous pattern recognition-related problems. Some examples covered in this tutorial include image retrieval [32, 44], signal and image representation [25, 27, 40, 50], inverse problems [30], cancer detection [4, 39], texture and color modeling [18, 41], shape and image registration [22, 29], and machine learning [12, 17, 19, 28, 36, 42], to name a few. This tutorial is meant to serve as an introductory guide to those wishing to familiarize themselves with these emerging techniques.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Mass Transport: Signal processing and machine-learning applications

Kolouri

Park

Thorpe

et al. 2017

IEEE Signal Process. Mag.

299

221

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Transport-based techniques for signal and data analysis have received increased attention recently. Given their ability to provide accurate generative models for signal intensities and other data distributions, they have been used in a variety of applications including content-based retrieval, cancer detection, image super-resolution, and statistical machine learning, to name a few, and shown to produce state of the art results in several applications. Moreover, the geometric characteristics of transport-related metrics have inspired new kinds of algorithms for interpreting the meaning of data distributions. Here we provide a practical overview of the mathematical underpinnings of mass transport-related methods, including numerical implementation, as well as a review, with demonstrations, of several applications. Software accompanying this tutorial is available at [43].

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“…Recent applications include astronomy [9, 18, 19], biomedical sciences [3, 25–27, 77, 81, 82, 88, 89], colour transfer [14, 17, 49, 62, 63], computer vision and graphics [7, 44, 60, 65, 68, 74, 75], imaging [36, 40, 64], information theory [78], machine learning [1, 15, 20, 34, 37, 48, 76], operational research [69] and signal processing [54, 58]. …”

Section: Introductionmentioning

confidence: 99%

A Transportation $$L^p$$ Distance for Signal Analysis

et al. 2017

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Transport based distances, such as the Wasserstein distance and earth mover'sdistance, have been shown to be an effective tool in signal and image analysis. The success of transport based distances is in part due to their Lagrangian nature which allows it to capture the important variations in many signal classes. However these distances require the signal to be nonnegative and normalized. Furthermore, the signals are considered as measures and compared by redistributing (transporting) them, which does not directly take into account the signal intensity. Here we study a transport-based distance, called the TLp distance, that combines Lagrangian and intensity modelling and is directly applicable to general, non-positive and multi-channelled signals. The distance can be computed by existing numerical methods. We give an overview of the basic properties of this distance and applications to classification, with multi-channelled non-positive one-dimensional signals and two-dimensional images, and color transfer.

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Imaging with Kantorovich--Rubinstein Discrepancy

Cited by 92 publications

References 57 publications

Directional global three-part image decomposition

Directional global three-part image decomposition

Optimal Mass Transport: Signal processing and machine-learning applications

A Transportation $$L^p$$ Distance for Signal Analysis

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