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Numerical simulation of water erosion models and some physical models in image processing

Numerical simulation of water erosion models and some physical models in image processing. Gloria Haro Ortega. December 2003. Universitat Pompeu Fabra. CONTENTS. I. Water, erosion and sedimentation II. Day for night. December 2003 - Universitat Pompeu Fabra.

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Numerical simulation of water erosion models and some physical models in image processing

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  1. Numerical simulation of water erosion models and some physical models in image processing Gloria Haro Ortega December 2003 Universitat Pompeu Fabra

  2. CONTENTS I. Water, erosion and sedimentation II. Day for night December 2003 - Universitat Pompeu Fabra Gloria Haro Ortega

  3. I. Water, erosion and sedimentation CONTENTS: 1. Objective 2. State of the art 3. Proposed model 4. Shallow water equations 5. Numerical implementation 6. Evaluation and results 7. Conclusions 8. Future work

  4. I. Objective  Find a model based on PDEs (Partial Differential Equations) of the erosion and sedimentation processes produced by the action of rivers.

  5. I. State of the art • Models including only erosion • Models including both erosion and sedimentation but do not model water movement. • Models that include water thickness evolution and make a simplification of the velocity.

  6. I. Proposed model HYDROSTATIC MODEL: SIMPLE MODEL:

  7. I. Shallow water equations Rarefaction waves Shock waves Contact discontinuities Vacuum formation

  8. I. Numerical implementation Homogeneous system:Upwindflux difference ENO with Marquina’s Flux Splitting [Fedkiw et al.] Time discretization: Runge-Kutta Spatial discretization: ENO  TV(R(ŵ))  TV(w) + O(hr)

  9. I. Numerical implementation Source Term extension:Write source as a divergence [Gascón & Corberán] Dry fronts and vacuum formation: Special treatment

  10. I. Evaluation and results Dealing with vacuum: Water elevation Riemann invariants

  11. I. Evaluation and results Steady flow over a hump: Froude number:

  12. I. Evaluation and results Drain on a non-flat bottom:

  13. I. Evaluation and results Vacuum occurrence over a step: Lax-Friedrichs Harten

  14. I. Evaluation and results 2D evolution test: Dam break over three mounds.

  15. I. Conclusions - Physical model for the erosion and sedimentation processes. - Extension of a numerical scheme for homogeneous systems so as to include the source term. - Special treatment of wet/dry boundaries and vacuum formation. - Experimental evaluation in 1D (2D).

  16. I. Future work • - Experimental evaluation in 2D. • - Numerical study of the complete erosion-sedimentation model. • Simulations on real and synthetic topographies. • - Analyse the suitability to generate river networks. • - Study the possible use to interpolate Digital Elevation Maps.

  17. II. Day for night CONTENTS: 1. Objective 2. Algorithm 3. Some examples 4. Conclusion 5. Future work

  18. II. Day for night OBJECTIVE:Transform a ‘day image’ into a ‘night’ version of it including the loss of acuity at low luminances. + desiredluminancelevel =

  19. II.Day for night algorithm TRANSFORMATION IN 5 STEPS • Estimation of reflectance values and modification of illuminant. • 2. Modification of chromaticity. • 3. Modification of luminance. • 4. Modification of contrast. • 5. Loss of acuity: Diffusion.

  20. Estimation of reflectance values and modification of illuminant II. 1 Color-matching functions Characteristic curve of the film

  21. II. Modification of chromaticity 2 - The preceived chromaticity depends on the illumination level. - Difficult to emulate directly on film. - We use experimental data in [Stabell & Stabell] to modify the color matching functions.

  22. II. Modification of luminance 3 • Use of the luminous efficiency functions tabulated by the CIE:

  23. II. Modification of contrast 4 Human sensitivity to contrast depends on the adaptation luminance. Contrast in night image must be different than in the original daylight scene. Two ways: - Approximating the eye‘s performance: tone reproduction operator [Ward et al.]. - Emulating a photograpic film with a characteristic curve:

  24. II. Loss of acuity: Diffusion 5 Highest level of acuity achieved at photopic levels. Spatial summation principle [Cornsweet & Yellott]. Underlying family of PDE´s: Fast Diffusion Equations Particular case: Results of existence and uniqueness results, also monotonicity preserving and well-posed [Vázquez et al.].

  25. II. OTHER EXAMPLES Using standard day illuminant D55 Using standard day illuminant D75 Using night spectrum CA 1990 Using night spectrum Palomar 1972

  26. II. OTHER EXAMPLES Ambient luminance: 1, 0.6, 0.3, 0.1 and -0.1 log cd/m2, 5, 8, 10, 11 and 15 iterations of diffusion respectively from left to right and from top to bottom.

  27. II. OTHER EXAMPLES Emulating human vision at night. Simulated scene at 0.3 log cd/m2 Emulating a photographic film (n=3, =1).

  28. II. OTHER EXAMPLES Without changing the variance, a=1 Simulated scene at 0.1 log cd/m2 Increasing the variance, a=0.1

  29. II. OTHER EXAMPLES Video sequence

  30. II. CONCLUSIONS -Transformations based on real physical and visual perception experimental data. - Modification night illuminant spectrum. - Novel diffusion equation to simulate the loss of resolution(well-posed, existence and uniqueness results, no ringing suitable for video sequences). Limitation: assumption that all light in the scene is natural, i.e. one illuminant for the whole image.

  31. II. FUTURE WORK • Solve the constraint of one illuminant and simulate artificial lights. • Include emulations of the developing process, and reformulate the algorithm in terms and units that cinematographers use.

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