Minimizing Flows for the Monge--Kantorovich Problem

Angenent, Sigurd; Haker, Steven; Tannenbaum, Allen

doi:10.1137/s0036141002410927

Cited by 152 publications

(172 citation statements)

References 14 publications

Supporting

Mentioning

172

Contrasting

Order By: Relevance

“…A corresponding gradient descent technique is given in Section III and two examples based on real imagery are shown in Section IV. Finally, we note that our methods may be rigorously justified; see [2] for the mathematical details.…”

Section: A Image Registrationmentioning

confidence: 99%

“…Next, rather than working with directly, we solve the polar factorization problem via gradient descent. We should note that the algorithm described below converges to a global optimum [2]. Accordingly, we will assume that is a function of time, and then determine what …”

Section: Removing the Curlmentioning

confidence: 99%

“…We may also derive a second order local evolution equation for § ¦ For example, in the @ ¡ case (see [2] for generalizations), we use the divergence theorem with (25) …”

Section: Removing the Curlmentioning

confidence: 99%

“…It will mainly focus on the proof of the properties of MP mapping and the deduction of our gradient descent method. Full details of the mathematics underlying our methodology (including convergence to thee optimal solution) may be found in [2].…”

Section: Acknowledgementsmentioning

confidence: 99%

See 3 more Smart Citations

Optimal Mass Transport for Registration and Warping

Haker

Zhu

Tannenbaum

et al. 2004

International Journal of Computer Vision

Self Cite

352

356

View full text Add to dashboard Cite

Image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times. In this paper we present a method for computing elastic registration and warping maps based on the Monge-Kantorovich theory of optimal mass transport. This mass transport method has a number of important characteristics. First, it is parameter free. Moreover, it utilizes all of the grayscale data in both images, places the two images on equal footing and is symmetrical: the optimal mapping from image to image ¡ being the inverse of the optimal mapping from ¡ to £ ¢ The method does not require that landmarks be specified, and the minimizer of the distance functional involved is unique; there are no other local minimizers. Finally, optimal transport naturally takes into account changes in density that result from changes in area or volume. Although the optimal transport method is certainly not appropriate for all registration and warping problems, this mass preservation property makes the Monge-Kantorovich approach quite useful for an interesting class of warping problems, as we show in this paper. Our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the ¤ ¦ ¥ Kantorovich-Wasserstein or "Earth Mover's Distance" under a mass preservation constraint. We show how this approach leads to practical algorithms, and demonstrate our method with a number of examples, including those from the medical field. We also extended this method to take into account changes in intensity, and show that it is well suited for applications such as image morphing.

show abstract

Section: A Image Registrationmentioning

confidence: 99%

Section: Removing the Curlmentioning

confidence: 99%

“…We may also derive a second order local evolution equation for § ¦ For example, in the @ ¡ case (see [2] for generalizations), we use the divergence theorem with (25) …”

Section: Removing the Curlmentioning

confidence: 99%

Section: Acknowledgementsmentioning

confidence: 99%

See 2 more Smart Citations

Optimal Mass Transport for Registration and Warping

Haker

Zhu

Tannenbaum

et al. 2004

International Journal of Computer Vision

Self Cite

352

356

View full text Add to dashboard Cite

show abstract

“…Here M(R n ) is the Banach space of Radon measures, with M + (R n ) being the subset of nonnegative measures. In the relaxed variational formulation by Kantorovich, the MK problem consists in determining the optimal transport plan; i.e., the measure γ ∈ M + (R n × R n ) having projections f + and f − , such that U×R n γ = U f + and R n ×U γ = U f − for all Borel sets U ⊂ R n , and minimizing the cost of transportation Several numerical algorithms, see [4,9] and the references therein, have been derived for the quadratic case (p = 2), where the cost function is smooth and strictly convex. In this case the unique optimal transport plan is a map; and moreover, this map is the gradient of a convex function.…”

Section: Introductionmentioning

confidence: 99%

A Mixed Formulation of the Monge-Kantorovich Equations

Barrett¹,

Prigozhin

2007

ESAIM: M2AN

View full text Add to dashboard Cite

Abstract. We introduce and analyse a mixed formulation of the Monge-Kantorovich equations, which express optimality conditions for the mass transportation problem with cost proportional to distance. Furthermore, we introduce and analyse the finite element approximation of this formulation using the lowest order Raviart-Thomas element. Finally, we present some numerical experiments, where both the optimal transport density and the associated Kantorovich potential are computed for a coupling problem and problems involving obstacles and regions of cheap transportation.

show abstract

Adaptive Calibration of Soft Sensors Using Optimal Transportation Transfer Learning for Mass Production and Long‐Term Usage

Kim

Kwon

Jeon

et al. 2020

Advanced Intelligent Systems

View full text Add to dashboard Cite

Soft sensors suffer from high manufacturing tolerances and signal drift from long‐term usage, which degrades their practicality. Although deep learning has recently been proposed to address these issues, it is expensive in terms of data collection and processing. Therefore, an adaptive calibration method is proposed for soft sensors, suitable for mass production and long‐term usage. In addition to maintaining the original benefits of deep learning characterization, this method enables fast and accurate calibration by capturing the change in the characteristics of the sensor through domain adaptation, using optimal transportation. An offline calibration method is first described, which is for alleviating the difficulty in calibrating every single unit from mass produced soft sensors. The main advantage is that identically manufactured soft sensors in a large volume with variations can be calibrated with reduced time and effort for collecting and processing data. Online calibration is then discussed, which compensates for the parameter changes when a soft sensor is continuously used for an extended period of time. For a single sensor, even though the sensor shows signal drift from the long‐term usage, the calibrated network weights can be quickly adjusted online. Finally, the proposed adaptive calibration is experimentally evaluated using actual soft sensors.

show abstract

Minimizing Flows for the Monge--Kantorovich Problem

Cited by 152 publications

References 14 publications

Optimal Mass Transport for Registration and Warping

Optimal Mass Transport for Registration and Warping

A Mixed Formulation of the Monge-Kantorovich Equations

Adaptive Calibration of Soft Sensors Using Optimal Transportation Transfer Learning for Mass Production and Long‐Term Usage

Contact Info

Product

Resources

About