## Navier-Stokes, fluid dynamics, and image and video inpainting (2001)

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Venue: | Proc. IEEE Computer Vision and Pattern Recognition (CVPR |

Citations: | 116 - 16 self |

### BibTeX

@INPROCEEDINGS{Bertalmio01navier-stokes,fluid,

author = {M. Bertalmio and A. L. Bertozzi and G. Sapiro},

title = {Navier-Stokes, fluid dynamics, and image and video inpainting},

booktitle = {Proc. IEEE Computer Vision and Pattern Recognition (CVPR},

year = {2001},

pages = {355--362}

}

### Years of Citing Articles

### OpenURL

### Abstract

Image inpainting involves filling in part of an image or video using information from the surrounding area. Applications include the restoration of damaged photographs and movies and the removal of selected objects. In this paper, we introduce a class of automated methods for digital inpainting. The approach uses ideas from classical fluid dynamics to propagate isophote lines continuously from the exterior into the region to be inpainted. The main idea is to think of the image intensity as a ‘stream function ’ for a two-dimensional incompressible flow. The Laplacian of the image intensity plays the role of the vorticity of the fluid; it is transported into the region to be inpainted by a vector field defined by the stream function. The resulting algorithm is designed to continue isophotes while matching gradient vectors at the boundary of the inpainting region. The method is directly based on the Navier-Stokes equations for fluid dynamics, which has the immediate advantage of well-developed theoretical and numerical results. This is a new approach for introducing ideas from computational fluid dynamics into problems in computer vision and image analysis.

### Citations

1392 |
Nonlinear total variation based noise removal algorithms
- Rudin, Osher, et al.
- 1992
(Show Context)
Citation Context ...rformed independently of the one reported in [1]). The authors present a clear and intuitive axiomatic approach to the problem. The main algorithm they propose is to minimize the Total Variation (TV) =-=[33]-=-, of the image inside the hole (they also use, as proposed in [1, 2] and here, a band surrounding the region). As in the work of Masnou and Morel, their interpolation is limited to creating straight i... |

880 | Fronts propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations
- Osher, Sethian
- 1988
(Show Context)
Citation Context ...rior to determine the boundary vorticity. We evolve the vorticity stream form (7), using a simple forward Euler time stepping, with centered differences in space for the diffusion and a minmod method =-=[29]-=- for the convection term. The diffusion is anisotropic. After one time step of (7) we compute the image intensity ¡ by solving the Poisson equation (8) using the Jacobi iteration method. From this upd... |

769 | Texture synthesis by non-parametric sampling
- Efros, Leung
- 1999
(Show Context)
Citation Context ...n to be inpainted occupies a large number of frames. Another area related to our work is texture synthesis, in which a texture is selected and synthesized inside the region to be filled-in (the hole) =-=[9, 13, 15, 34]-=-. These algorithms often require the user to select the texture and are not often well-designed to fill in structure from boundary data. The closest methods to our approach are the fundamental works o... |

314 |
Image selective smoothing and edge detection by nonlinear diffusion
- Alvarez, Lions, et al.
- 1992
(Show Context)
Citation Context ...own to be ill-posed for a wide class of � functions , however the ill-posedness can be easily removed with a small amount of smoothness applied to the gradient inside the � function . 5 Catte et. al. =-=[6]-=- established an existence and uniqueness theory which could be extended to include nonlinear transport terms from the Navier-Stokes equations. In fact, nonlinear diffusion has a natural physical analo... |

258 |
The mathematical theory of viscous incompressible flow. Second English edition
- Ladyzhenskaya
- 1969
(Show Context)
Citation Context ..., with appropriate boundary conditions, are globally well-posed in two space dimensions. Solutions exist for any smooth initial condition and they depend continuously on the initial and boundary data =-=[17, 18, 23, 24, 36, 40]-=-. In terms of the stream function, equation (5) implies that steady state inviscid flows must satisfy ¦ ¨ � ��¦�� which says that the Laplacian of the stream function, and hence the vorticity, must ha... |

205 |
An Introduction to the Mathematical Theory of the Navier-Stokes equations, vol
- Galdi
- 1994
(Show Context)
Citation Context ...ions satisfying the same zero velocity boundary data. At the other extreme when viscosity is sufficiently large, for smooth boundaries there is a unique solution of the steady Navier-Stokes equations =-=[11]-=-. For viscous flows with moderate and small viscosity the problem is more complex; for example, there are classical experimental examples in which known steady state solutions have varying stability p... |

184 |
Navier-Stokes Equations
- Temam
- 1979
(Show Context)
Citation Context ..., with appropriate boundary conditions, are globally well-posed in two space dimensions. Solutions exist for any smooth initial condition and they depend continuously on the initial and boundary data =-=[17, 18, 23, 24, 36, 40]-=-. In terms of the stream function, equation (5) implies that steady state inviscid flows must satisfy ¦ ¨ � ��¦�� which says that the Laplacian of the stream function, and hence the vorticity, must ha... |

170 | Efficient and reliable schemes for nonlinear diffusion filtering
- Weickert, Romeny, et al.
- 1998
(Show Context)
Citation Context ...�� , ��� ¤ ��� ¤�� , ¤�� ¤ � , � � � Jacobi, steps of anisotropic diffusion of ¡ every �£� cycles. The evolution (7) might be made more efficient by using an ADI (alternate direction implicit) method =-=[39]-=-. When working with a color image, we perform inpainting on its three components separately (one luminance image and two chroma images), and join the results at the end (in the same was as in [2]). Th... |

137 |
Computational Methods for Fluid Flow
- Peyret, Taylor
- 1983
(Show Context)
Citation Context ...¡ � ��� ¤�¡�� � (8) ��¡�¤ � ¤�� For , the direct numerical solution of of (7-8) is a classical way to solve both the dynamic fluid equations and to evolve the dynamics towards a steady state solution =-=[30]-=-. For fluid problems with small � viscosity , the above dynamics can take a long time to converge to steady state, making the method less practical. Instead there are pseudosteady methods that involve... |

122 | Filling-in by joint interpolation of vector fields and gray levels
- Ballester, Bertalmio, et al.
(Show Context)
Citation Context ...he importance of propagating both the gradient direction (geometry) and gray-values (photometry) of the image in a band surrounding the hole to be filled-in. Some of the ideas of [2] where adopted in =-=[1]-=-, while deviating from the particular model in order to be able to define a formal variational approach to the filling-in/inpainting problem. The work in [2] inspired a very elegant approach to the - ... |

97 |
Vorticity and Incompressible Flow
- Majda, Bertozzi
- 2002
(Show Context)
Citation Context ..., with appropriate boundary conditions, are globally well-posed in two space dimensions. Solutions exist for any smooth initial condition and they depend continuously on the initial and boundary data =-=[17, 18, 23, 24, 36, 40]-=-. In terms of the stream function, equation (5) implies that steady state inviscid flows must satisfy ¦ ¨ � ��¦�� which says that the Laplacian of the stream function, and hence the vorticity, must ha... |

94 | Texture characterization via joint statistics of wavelet coefficient magnitudes
- Simoncelli, Portilla
- 1998
(Show Context)
Citation Context ...n to be inpainted occupies a large number of frames. Another area related to our work is texture synthesis, in which a texture is selected and synthesized inside the region to be filled-in (the hole) =-=[9, 13, 15, 34]-=-. These algorithms often require the user to select the texture and are not often well-designed to fill in structure from boundary data. The closest methods to our approach are the fundamental works o... |

85 | An axiomatic approach to image interpolation
- Caselles, Morel, et al.
- 1998
(Show Context)
Citation Context ...ally). pler filter is used for the chroma components. Better results may be achieved via harmonic map smoothing of vectors (see [35]). For an axiomatic/PDE based technique for image iterpolation, see =-=[5]-=-. 4. Summary and Conclusions In this paper, we show the importance of computational fluid dynamics (CFD) in general, and Navier-Stokes equations in particular, to vision problems. Although CFD ideas h... |

77 |
Detection of missing data in image sequences
- Kokaram, Morris, et al.
- 1995
(Show Context)
Citation Context ...s (e.g., removal of stamped date and redeye from photographs, the infamous “airbrushing” of political enemies [20]). A current active area of research is to automate digital techniques for inpainting =-=[2, 3, 16, 21, 22]-=-. In this paper, we introduce a novel algorithm for digital inpainting of still images that attempts to replicate the 1 basic techniques used by professional restorators. Our algorithm, motivated by a... |

52 | Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case
- Tang, Sapiro, et al.
(Show Context)
Citation Context ...ng has been removed using Navier-Stokes inpainting (available electronically). pler filter is used for the chroma components. Better results may be achieved via harmonic map smoothing of vectors (see =-=[35]-=-). For an axiomatic/PDE based technique for image iterpolation, see [5]. 4. Summary and Conclusions In this paper, we show the importance of computational fluid dynamics (CFD) in general, and Navier-S... |

49 |
Level-lines based disocclusion
- Masnou, Morel
- 1998
(Show Context)
Citation Context ...e gray level. (Other researchers, e.g., D. Jacobs, R. Basri, and S. Zucker, have followed this interesting research area, mainly developing techniques for smooth curve continuation.) Masnou and Morel =-=[25, 26]-=- recently extended these ideas, presenting a formal variational formulation for disocclusion and a particular practical implementation. The algorithm uses geodesic curves to join the isophotes arrivin... |

46 |
Pyramid based texture analysis/synthesis
- Heeger, Bergen
- 1995
(Show Context)
Citation Context ...n to be inpainted occupies a large number of frames. Another area related to our work is texture synthesis, in which a texture is selected and synthesized inside the region to be filled-in (the hole) =-=[9, 13, 15, 34]-=-. These algorithms often require the user to select the texture and are not often well-designed to fill in structure from boundary data. The closest methods to our approach are the fundamental works o... |

44 | Combining frequency and spatial domain information for fast interactive image noise removal
- Hirani, Totsuka
- 1996
(Show Context)
Citation Context |

38 |
Spatial and temporal properties of illusory contours and amodal boundary completion. Vision Res 36: 3037-3050. 13 von der Heydt: Image Parsing Mechanisms of the Visual Cortex
- DL, Shapley
- 1996
(Show Context)
Citation Context ...ions (PDE’s) or systems of PDE’s, as done in [1, 2] and here (the authors of [7] have also recently introduced higher order models). Research in perception, from the Gestalt to more recent work (e.g. =-=[31]-=-) supports the idea of performing a smooth continuation of the angle of arrival of the level lines at the gap. The paper [2] proposes an algorithm designed to project the gradient of the smoothness of... |

31 |
Mathematical models for local deterministic inpaintings
- Chan, Shen
- 2000
(Show Context)
Citation Context ...l in order to be able to define a formal variational approach to the filling-in/inpainting problem. The work in [2] inspired a very elegant approach to the - 2 filling-in problem recently reported in =-=[7]-=- (this work was performed independently of the one reported in [1]). The authors present a clear and intuitive axiomatic approach to the problem. The main algorithm they propose is to minimize the Tot... |

26 | Detection and removal of line scratches in motion picture films
- Joyeux, Buisson, et al.
- 1999
(Show Context)
Citation Context ...s (e.g., removal of stamped date and redeye from photographs, the infamous “airbrushing” of political enemies [20]). A current active area of research is to automate digital techniques for inpainting =-=[2, 3, 16, 21, 22]-=-. In this paper, we introduce a novel algorithm for digital inpainting of still images that attempts to replicate the 1 basic techniques used by professional restorators. Our algorithm, motivated by a... |

23 |
On classical solutions of the two-dimensional non-stationary Euler equation
- Kato
- 1967
(Show Context)
Citation Context |

22 |
An Album of Fluid Motion, The Parabolic
- Dyke
- 1982
(Show Context)
Citation Context ...satisfies the Navier-Stokes equation for all viscosities. However this solution is unstable for sufficiently high viscosity, resulting in the creation of multiple eddies in such experiments (see e.g. =-=[37]-=- p. 80-81). We expect that Navier-Stokes based inpainting may inherit some of the stability and uniqueness issues known for incompressible fluids, although the effect of anisotropic diffusion is not c... |

17 |
The Restorer’s Handbook of Easel Painting
- Emile-Male
- 1976
(Show Context)
Citation Context ...theoretical and numerical results. This is a new approach for introducing ideas from computational fluid dynamics into problems in computer vision and image analysis. 1. Introduction Image inpainting =-=[2, 10, 20, 38]-=- is the process of filling in missing data in a designated region of a still or video image. Applications range from removing objects from a scene to re-touching damaged paintings and photographs. The... |

16 |
Filtrage et désocclusion d’images par méthodes d’ensembles de niveau
- Masnou
- 1998
(Show Context)
Citation Context ...e gray level. (Other researchers, e.g., D. Jacobs, R. Basri, and S. Zucker, have followed this interesting research area, mainly developing techniques for smooth curve continuation.) Masnou and Morel =-=[25, 26]-=- recently extended these ideas, presenting a formal variational formulation for disocclusion and a particular practical implementation. The algorithm uses geodesic curves to join the isophotes arrivin... |

14 |
The Ravished Image
- Walden
- 1985
(Show Context)
Citation Context ...theoretical and numerical results. This is a new approach for introducing ideas from computational fluid dynamics into problems in computer vision and image analysis. 1. Introduction Image inpainting =-=[2, 10, 20, 38]-=- is the process of filling in missing data in a designated region of a still or video image. Applications range from removing objects from a scene to re-touching damaged paintings and photographs. The... |

13 |
Mathematical analysis of viscoelastic flows
- Renardy
(Show Context)
Citation Context ...e Navier-Stokes equations. In fact, nonlinear diffusion has a natural physical analogy; Non-Newtonian fluids have a viscosity that depends locally and nonlinearly on the (� ¦ � � shear ) of the fluid =-=[32]-=-. Another form of viscosity for Navier-Stokes include fourth order hyperviscosity which is commonly used in numerical simulations of turbulence [4]. 3. Computational Examples We now show via some exam... |

12 |
The Commissar Vanishes
- King
- 1997
(Show Context)
Citation Context ...theoretical and numerical results. This is a new approach for introducing ideas from computational fluid dynamics into problems in computer vision and image analysis. 1. Introduction Image inpainting =-=[2, 10, 20, 38]-=- is the process of filling in missing data in a designated region of a still or video image. Applications range from removing objects from a scene to re-touching damaged paintings and photographs. The... |

10 | Photoshop retouching handbook - Braverman - 1998 |

10 |
New interpretation and improvement of the nonlinear anisotropic diffusion for image enhancement
- Monteil, Beghdadi
- 1999
(Show Context)
Citation Context ... problems. Although CFD ideas have been used in computer graphics for such problems as modeling natural phenomena, in shape analysis following [29], and as an interpretation for anisotropic diffusion =-=[27]-=-, the connection and application here is to the best of our knowledge novel. Having such a direct interdisciplinary connection has several benefits for the computer vision community. First it brings w... |

10 | Pyramid based texture analysis /synthesis - Heeger, Bergen - 1995 |

5 |
On DiPerna-Majda concentration sets for two-dimensional incompressible flow
- Greengard, Thomann
- 1988
(Show Context)
Citation Context ...nique steady-state solution. This can be seen in the following simple example, motivated by problems involving the mathematical concentration of vorticity in solution sequences of the Euler equations =-=[8, 12]-=-. Consider the problem ¤�� � (6) ��� with on . � Consider a , contained � ��� disk of ��� radius , centered at the � point � inside . ��������� Let � � ����� and be two different compactly supported f... |

3 |
Comparison of numerical methods for the calculation of two-dimensional turbulence
- Browning, Kreiss
- 1989
(Show Context)
Citation Context ...nd nonlinearly on the (� ¦ � � shear ) of the fluid [32]. Another form of viscosity for Navier-Stokes include fourth order hyperviscosity which is commonly used in numerical simulations of turbulence =-=[4]-=-. 3. Computational Examples We now show via some examples how the Navier-Stokes inpainting method performs. In each example described below, the Navier-Stokes equations are solved in the inpainting re... |

3 |
Majda Concentrations in Regularizations for 2-D
- DiPerna, A
- 1987
(Show Context)
Citation Context ...nique steady-state solution. This can be seen in the following simple example, motivated by problems involving the mathematical concentration of vorticity in solution sequences of the Euler equations =-=[8, 12]-=-. Consider the problem ¤�� � (6) ��� with on . � Consider a , contained � ��� disk of ��� radius , centered at the � point � inside . ��������� Let � � ����� and be two different compactly supported f... |

2 |
Über discontinuierliche Flüssigkeitsbewegungen, Monatsberichte der königl. Akad. Wissenschaften zu
- Helmholtz
- 1868
(Show Context)
Citation Context ...he dynamical evolution of the fluid equations and stability of solutions. For example it is possible that classical two-dimensional fluid dynamics instabilities, like the Kelvin-Helmholtz instability =-=[19, 14]-=- for shear layers, might allow us to generate interesting pattern formation in large inpainting regions. The control of such instabilities should be linked to the magnitude of diffusion present in the... |