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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 FabraGloria 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 HYDROSTATIC MODEL: SIMPLE MODEL:
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I. Shallow water equations Rarefaction waves Shock waves Contact discontinuities Vacuum formation
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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(h r ) Time discretization: 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
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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: Froude number:
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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-FriedrichsHarten
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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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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 chromaticity II.
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Use of the luminous efficiency functions tabulated by the CIE: Modification of luminance II.
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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: Modification of contrast II.
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Highest level of acuity achieved at photopic levels. Spatial summation principle [Cornsweet & Yellott]. 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 D55Using 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/m 2, 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/m 2 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/m 2 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.
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