Download - Numerical simulation of water erosion models and some physical models in image processing
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Numerical simulation of water erosion models and some physical models in
image processing
Gloria Haro Ortega
December 2003 Universitat Pompeu Fabra
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CONTENTS
I. Water, erosion and sedimentation
II. Day for night
December 2003 - Universitat Pompeu Fabra
Gloria Haro Ortega
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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
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I. Objective
Find a model based on PDEs (Partial Differential Equations) of the erosion and sedimentation processes produced by the action of rivers.
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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.
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I. Proposed model
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HYDROSTATIC MODEL:
SIMPLE MODEL:
2|)(|1
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hzC
hzhgv
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I. Shallow water equations
SUGUFU yxt )()(
2
1
q
q
h
U
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qh
q
ghqh
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ghqh
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qh
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UG
y
x
ghz
ghzS
0
Rarefaction waves
Shock waves
Contact discontinuities
Vacuum formation RLLR ghghuu 22
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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:
x
FFUF ii
x
2/12/1)(
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I. Numerical implementation
Source Term extension: Write source as a divergence [Gascón & Corberán]
Dry fronts and vacuum formation:
Special treatment
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I. Evaluation and results
Dealing with vacuum:
Riemann invariantsWater elevation
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I. Evaluation and results
Steady flow over a hump:
gh
uFr Froude number:
1rF
1rF
1rF
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I. Evaluation and results
Drain on a non-flat bottom:
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I. Evaluation and results
Vacuum occurrence over a step:
Lax-Friedrichs Harten
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I. Evaluation and results
2D evolution test:
Dam break over three mounds.
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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).
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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.
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II. Day for night
CONTENTS:
1. Objective
2. Algorithm
3. Some examples
4. Conclusion
5. Future work
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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
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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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dzSkZ
dySkY
dxSkX
)()()(
)()()(
)()()(
Characteristic curve of the film
Estimation of reflectance values and modification of illuminant
Color-matching functions
II.
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- 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.
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Use of the luminous efficiency functions tabulated by the CIE:
bb
aL
L
dVSL
0
0
''
)(')(')('
Modification of luminanceII.
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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:
rwat
atn L
LLLL
Lrw
)()'(
nLcLLn )('
1
Modification of contrastII.
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Highest level of acuity achieved at photopic levels.
Spatial summation principle [Cornsweet & Yellott].
))(( 2 III t
0));1(log(1
II
II t
I
II
)1log()(2
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.
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Using night spectrum Palomar 1972Using night spectrum CA 1990
Using standard day illuminant D55 Using standard day illuminant D75
II. OTHER EXAMPLES
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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
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Emulating human vision at night.
Emulating a photographic film (n=3, =1).
Simulated scene at 0.3 log cd/m2
II. OTHER EXAMPLES
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Without changing the variance, a=1
Increasing the variance, a=0.1
Simulated scene at 0.1 log cd/m2
II. OTHER EXAMPLES
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Video sequence
II. OTHER EXAMPLES
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-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
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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.