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

SUGUFU yxt )()(

2

1

q

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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:

x

FFUF ii

x

2/12/1)(

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

dzSkZ

dySkY

dxSkX

)()()(

)()()(

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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:

bb

aL

L

dVSL

0

0

''

)(')(')('

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:

rwat

atn L

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Lrw

)()'(

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1

Modification of contrastII.

Highest level of acuity achieved at photopic levels.

Spatial summation principle [Cornsweet & Yellott].

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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.

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.