1 / 75

Lecture 3

Lecture 3. PDE Methods for Image Restoration. Overview. Generic 2 nd order nonlinear evolution PDE Classification: Forward parabolic (smoothing): heat equation Backward parabolic (smoothing-enhancing): Perona-Malik Hyperbolic (enhancing): shock-filtering. Artificial time (scales).

shawnj
Télécharger la présentation

Lecture 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 3 PDE Methods for Image Restoration

  2. Overview • Generic 2nd order nonlinear evolution PDE • Classification: • Forward parabolic (smoothing): heat equation • Backward parabolic (smoothing-enhancing): Perona-Malik • Hyperbolic (enhancing): shock-filtering Artificial time (scales) Initial degraded image

  3. Smoothing PDEs

  4. Heat Equation • PDE • Extend initial value from to • Define space be the extended functions that are integrable on C

  5. Heat Equation • Solution

  6. Heat Equation • Fourier transform: • Convolution theorem

  7. Heat Equation • Convolution in Fourier (frequency) domain Fourier Transform Attenuating high frequency

  8. Heat Equation • Convolution in Fourier (frequency) domain Fourier Transform Attenuating high frequency

  9. Heat Equation • Convolution in Fourier (frequency) domain Fourier Transform Attenuating high frequency

  10. Heat Equation • Isotropy. For any two orthogonal directions, we have • The isotropy means that the diffusion is equivalent in the two directions. • In particular

  11. Heat Equation • Derivation: • Let • Then • Finally, use the fact that D is unitary

  12. Heat Equation

  13. Heat Equation • Properties: let , then

  14. Heat Equation • Properties – continued • All desirable properties for image analysis. • However, edges are smeared out.

  15. Nonlinear Diffusion • Introducing nonlinearity hoping for better balance between smoothness and sharpness. • Consider • How to choose the function c(x)? We want: • Smoothing where the norm of gradient is small. • No/Minor smoothing where the norm of gradient is large.

  16. Nonlinear Diffusion • Impose c(0)=1, smooth and c(x) decreases with x. • Decomposition in normal and tangent direction • Let , then • We impose , which is equivalent to • For example: when s is large

  17. Nonlinear Diffusion • How to choose c(x) such that the PDE is well-posed? • Consider • The PDE is parabolic if • Then there exists unique weak solution (under suitable assumptions)

  18. Nonlinear Diffusion • How to choose c(x) such that the PDE is well-posed? • Consider • Then, the above PDE is parabolic if we require b(x)>0. (Just need to check det(A)>0 and tr(A)>0.) where

  19. Nonlinear Diffusion • Good nonlinear diffusion of the form if • Example: or

  20. The Alvarez–Guichard–Lions–Morel Scale Space Theory • Define a multiscale analysis as a family of operators with • The operator generated by heat equation satisfies a list of axioms that are required for image analysis. • Question: is the converse also true, i.e. if a list of axioms are satisfied, the operator will generate solutions of (nonlinear) PDEs. • More interestingly, can we obtain new PDEs?

  21. The Alvarez–Guichard–Lions–Morel Scale Space Theory • Assume the following list of axioms are satisfied

  22. The Alvarez–Guichard–Lions–Morel Scale Space Theory

  23. The Alvarez–Guichard–Lions–Morel Scale Space Theory

  24. The Alvarez–Guichard–Lions–Morel Scale Space Theory curvature

  25. Weickert’s Approach • Motivation: take into account local variations of the gradient orientation. • Observation: is maximal when d is in the same direction as gradient and minimal when its orthogonal to gradient. • Equivalently consider matrix • It has eigenvalues . Eigenvectors are in the direction of normal and tangent direction. • It is tempting to define at x an orientation descriptor as a function of .

  26. Weickert’s Approach • Define positive semidefinite matrix where and is a Gaussian kernel. • Eigenvalues • Classification of structures • Isotropic structures: • Line-like structures: • Corner structures:

  27. Weickert’s Approach • Nonlinear PDE • Choosing the diffusion tensor D(J): let D(J) have the same eigenvectors as J. Then, • Edge-enhancing anisotropic diffusion • Coherence-enhancing anisotropic diffusion

  28. Weickert’s Approach • Edge-enhancing Original Processed

  29. Weickert’s Approach • Coherence-enhancing Original Processed

  30. Smoothing-Enhancing PDEs Perona-Malik Equation

  31. The Perona and Malik PDE • Back to general 2nd order nonlinear diffusion • Objective: sharpen edge in the normal direction • Question: how?

  32. The Perona and Malik PDE • Idea: backward heat equation. • Recall heat equation and solution • Warning: backward heat equation is ill-posed! Backward Forward

  33. The Perona and Malik PDE • 1D example showing ill-posedness • No classical nor weak solution unless is infinitely differentiable.

  34. The Perona and Malik PDE • PM equation • Backward diffusion at edge • Isotropic diffusion at homogeneous regions • Example of such function c(s) • Warning: theoretically solution may not exist.

  35. The Perona and Malik PDE

  36. The Perona and Malik PDE • Catt′e et al.’s modification

  37. Enhancing PDEs Nonlinear Hyperbolic PDEs (Shock Filters)

  38. The Osher and Rudin Shock Filters • A perfect edge • Challenge: go from smooth to discontinuous • Objective: find with edge-sharpening effects

  39. The Osher and Rudin Shock Filters • Design of the sharpening PDE (1D): start from

  40. The Osher and Rudin Shock Filters • Transport equation (1D constant coefficients) • Variable coefficient transport equation • Example: Solution: Solution:

  41. The Osher and Rudin Shock Filters • 1D design (Osher and Rudin, 1990) Can we be more precise?

  42. Method of Characteristics • Consider a general 1st order PDE • Idea: given an x in U and suppose u is a solution of the above PDE, we would like to compute u(x) by finding some curve lying within U connecting x with a point on Γ and along which we can compute u. • Suppose the curve is parameterized as

  43. Method of Characteristics • Define: • Differentiating the second equation of (*) w.r.t. s • Differentiating the original PDE w.r.t. • Evaluating the above equation at x(s) (*)

  44. Method of Characteristics • Letting • Then • Differentiating the first equation of (*) w.r.t. s • Finally Defines the characteristics Characteristic ODEs

  45. The Osher and Rudin Shock Filters • Consider the simplified PDE with • Convert to the general formulation

  46. The Osher and Rudin Shock Filters • First case: . Then and • Thus • For s=0, we have and . Thus

  47. The Osher and Rudin Shock Filters • Determine : • Since • Using the PDE we have • Thus, we obtain the characteristic curve and solution Constant alone characteristic curve

  48. The Osher and Rudin Shock Filters • Characteristic curves and solution for case I

  49. The Osher and Rudin Shock Filters • Characteristic curves and solution for case II

  50. The Osher and Rudin Shock Filters • Observe • Discontinuity (shock) alone the vertical line at • Solution not defined in the white area • To not introduce further discontinuities, we set their values to 1 and -1 respectively • Final solution

More Related