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
Gloria Haro Ortega
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
I. Objective
Find a model based on PDEs (Partial Differential Equations) of the erosion and sedimentation processes produced by the action of rivers.
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.
I. Proposed model
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HYDROSTATIC MODEL:
SIMPLE MODEL:
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I. Shallow water equations
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Rarefaction waves
Shock waves
Contact discontinuities
Vacuum formation RLLR ghghuu 22
I. Numerical implementation
Homogeneous system: Upwind flux difference ENO with Marquina’s Flux Splitting [Fedkiw et al.]
ENO TV(R(ŵ)) TV(w) + O(hr)
0)( xt UFUTime discretization:
)(UAU t Runge-Kutta
Spatial discretization:
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I. Numerical implementation
Source Term extension: Write source as a divergence [Gascón & Corberán]
Dry fronts and vacuum formation:
Special treatment
I. Evaluation and results
Dealing with vacuum:
Riemann invariantsWater elevation
I. Evaluation and results
Steady flow over a hump:
gh
uFr Froude number:
1rF
1rF
1rF
I. Evaluation and results
Drain on a non-flat bottom:
I. Evaluation and results
Vacuum occurrence over a step:
Lax-Friedrichs Harten
I. Evaluation and results
2D evolution test:
Dam break over three mounds.
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).
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.
II. Day for night
CONTENTS:
1. Objective
2. Algorithm
3. Some examples
4. Conclusion
5. Future work
OBJECTIVE: Transform a ‘day image’ into a ‘night’ version of it including the loss of acuity at low luminances.
+ desired luminance level =
II. Day for night
TRANSFORMATION IN 5 STEPS
1. 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.
II. Day for night algorithm
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Characteristic curve of the film
Estimation of reflectance values and modification of illuminant
Color-matching functions
II.
- 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.
Modification of chromaticityII.
Use of the luminous efficiency functions tabulated by the CIE:
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Modification of luminanceII.
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:
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Modification of contrastII.
Highest level of acuity achieved at photopic levels.
Spatial summation principle [Cornsweet & Yellott].
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Results of existence and uniqueness results, also monotonicity preserving and well-posed [Vázquez et al.].
Loss of acuity: Diffusion
Particular case:
Fast Diffusion Equations
Underlying family of PDE´s:
II.
Using night spectrum Palomar 1972Using night spectrum CA 1990
Using standard day illuminant D55 Using standard day illuminant D75
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.
II. OTHER EXAMPLES
Emulating human vision at night.
Emulating a photographic film (n=3, =1).
Simulated scene at 0.3 log cd/m2
II. OTHER EXAMPLES
Without changing the variance, a=1
Increasing the variance, a=0.1
Simulated scene at 0.1 log cd/m2
II. OTHER EXAMPLES
Video sequence
II. OTHER EXAMPLES
-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.
II. CONCLUSIONS
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.