# INTERPOLATION BETWEEN IMAGES BY CONSTRAINED OPTIMAL TRANSPORT

### Said Kerrache, Yasushi Nakauchi

#### Abstract

In this paper, the recently proposed technique of constrained optimal transport is used to interpolate between images under specified constraints. The intensity values in both images are considered as mass distributions, and a flow of minimum kinetic energy is computed to transport the initial distribution to the final one, while satisfying specified constraints on the intermediate mass as well as the the velocity or the momentum field. As an application, the proposed method is used for interpolating between images under constraint on the volume expansion and contraction. This is achieved by imposing bounds on the divergence of the velocity field of the flow. This constraint is discretized then integrated into the problem Lagrangian using the augmented Lagrangian method. A variation of the solution is also presented, where the constraint is decoupled into two constraints coordinated by an additional Lagrange multiplier. This allows a considerable speedup, though numerical robustness decreases in certain cases. Constrained image interpolation by optimal transport has potential applications in image registration. In particular, the proposed method for controlling the volume change has potential application in registration of images under volume change constraints as it is the case for medical images depicting muscle movements or those with contrast enhancing structures.

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#### Paper Citation

#### in Harvard Style

Kerrache S. and Nakauchi Y. (2011). **INTERPOLATION BETWEEN IMAGES BY CONSTRAINED OPTIMAL TRANSPORT** . In *Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2011)* ISBN 978-989-8425-47-8, pages 75-84. DOI: 10.5220/0003375200750084

#### in Bibtex Style

@conference{visapp11,

author={Said Kerrache and Yasushi Nakauchi},

title={INTERPOLATION BETWEEN IMAGES BY CONSTRAINED OPTIMAL TRANSPORT},

booktitle={Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2011)},

year={2011},

pages={75-84},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003375200750084},

isbn={978-989-8425-47-8},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2011)

TI - INTERPOLATION BETWEEN IMAGES BY CONSTRAINED OPTIMAL TRANSPORT

SN - 978-989-8425-47-8

AU - Kerrache S.

AU - Nakauchi Y.

PY - 2011

SP - 75

EP - 84

DO - 10.5220/0003375200750084