that images are primarily defined on [0, 1]2 and extended to C = [−1, 1]2 via mirror symmetry. NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 7/30
that images are primarily defined on [0, 1]2 and extended to C = [−1, 1]2 via mirror symmetry. NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 7/30
that images are primarily defined on [0, 1]2 and extended to C = [−1, 1]2 via mirror symmetry. NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 7/30
that images are primarily defined on [0, 1]2 and extended to C = [−1, 1]2 via mirror symmetry. NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 7/30
that images are primarily defined on [0, 1]2 and extended to C = [−1, 1]2 via mirror symmetry. • L1# (C ) = u defined on C , such that Z u(x ) dx < +∞. C NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 7/30
we need a prior bound on the growth rate of u for |x |→ ∞ 2 |u(x , t)|≤ Ae a|x | , (x ∈ Rn , 0 ≤ t ≤ T ) • On bounded domain Ω: u0 continuous or L2 with Dirichlet or Neumann BC. NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 7/30
= T (u n ) ∂u n+1 = div (g(| (∇Gσ ∗ u n ) |)∇u n+1 ) Ω × ]0, +∞[ , ∂t u n+1 (x , 0) = u0 (x ) Ω, ∂un+1 = 0 ∂Ω × ]0, +∞[ ∂n (u n )n converges in C ([0, T ]; L2 (Ω)) to the unique solution. NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 12/30
account ∂u = div(D(∇u)∇u) ∂t where D represents a matrix-valued diffusion tensor that describes the smoothing directions and the corresponding diffusivities. • D(∇u) = I: constant diffusion coefficient in all directions • D(∇u) = g(|∇uσ |)I: Perona-Malik type nonlinear diffusion • D is a function of the structure tensor " ux2 ∇u∇u T = ux uy ux uy uy2 # • Eigenvectors: v1 ∥∇u (λ1 = |∇u|) and v2 ⊥∇u (λ2 = 0). NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 14/30
account ∂u = div(D(J(∇u))∇u) ∂t • How to choose the matrix D(J(∇u))? • Keep the eigenvectors of J(∇u). • Choose the eigenvalues of D, λ1 and λ2 , depending on the eigenvalues of J(∇u), µ1 and µ2 . NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 14/30
L∞ (Ω), ρ ≥ 0, and σ, T > 0. Let a := ess inf u0 , b := ess sup u0 , and consider the problem ∂u = div (D(Jρ (∇uσ |)∇u) Ω × ]0, T ] , ∂t u(x , 0) = u0 (x ) Ω, u = 0 ∂Ω × ]0, T ] n where D ∈ C ∞ (R2×2 , R2×2 ),d12 (J) = d21 (J) for all symmetric matrices J, and ∀w ∈ L∞ (Ω, R2 ), there exists a positive lower bound for the eigenvalues of D(Jρ (w )). Then, the problem has a unique solution which satisfies u ∈ C ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H 1 (Ω)), ∂t u ∈ L2 (0, T ; H 1 (Ω)). NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 14/30
curvature-driven motions, formulated either in a scalar or in a geometrical way ∂u = |∇u|F (curv (u)), ∂t ∂C F (κ)N, ∂t the AMSS special case is affine invariant: if u is a solution then so is (x , t) 7→ u(Ax , t). NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 20/30
also been applied to denoising. • The special invariances of AMSS are useful for shape recognition tasks, for corner detection, and for texture discrimination. • MCM and AMSS: ”Viscosity” solutions exist! • Min-Max flow (Malladi and Sethian): use either min(κ, 0) or max (κ, 0), depending on the average grey value within a certain neighbourhood. • Good denoising properties. But, well-posedness results are not known up to now, since the theory of viscosity solutions is no longer applicable. NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 20/30
∗ u • Let ĝ = 1 − ĥ so that we can write formally ! +∞ X uˆ0 û = = ĝ k uˆ0 . 1 − ĝ k=0 If we put ûn = Pn k=0 ĝ k uˆ0 , we get ûn+1 = uˆ0 + ĝ ûn = uˆ0 + (1 − ĥ)ûn , un+1 = u0 + un − h ∗ un NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 22/30
damaged domain • Penalizes length of edges: cannot connect contours across very large distances • Can result in corners across the inpainting domain NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 26/30
interpolated linearly • connected by a straight line across the missing domain • The direction of the level lines is not preserved • Solution: use of higher-order derivatives • Encode additional information (not only the length of level lines) such that the curvature NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 27/30
in missing parts in images by using local structural information only. • L = Information and D = Direction ∇L.D = 0 , ∂I = ∇L.D , ∂t NHSM - 5th year: Digital Image Processing Steady state equation Evolution equation - Introduction (Week 1) - M. Hachama ([email protected]) 28/30
Processing”; Applied Mathematical Sciences 147; Springer, New York; 2006. • T. Chan and J. Shen, ”Image Processing And Analysis: Variational, PDE, Wavelet, And Stochastic Methods”, Society for Industrial and Applied Mathematics, 2005. • Joachim Weickert, ”Anisotropic Diffusion in Image Processing, ECMI Series, Teubner-Verlag, Stuttgart, Germany, 1998. NHSM - 5th year: Digital Image Processing - Introduction (Week 1) - M. Hachama ([email protected]) 30/30
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