Linear Diffusion. E9 242 STIP- R. Venkatesh Babu IISc

Transcription

1 Linear Diffusion

2 Derivation of Heat equation Consider a 2D hot plate with Initial temperature profile I 0 (x, y) Uniform (isotropic) conduction coefficient c Unit thickness (along z) Problem: What is temperature at point (x, y) at time t? Find governing equation of temperature distribution and solve for I(x, y, t) Heat conduction law: Rate of heat flow proportional to temperature gradient and length of interface Heat flows opposite to temperature gradient Heat required to increase temperature by I: m x I

3 Derivation of Heat equation (contd)

4 Derivation of Heat equation (contd)

5 Solution of Heat Equation 2

6 Solution of HE

7 Solution of HE

8 Solution of HE

9 LOG Vs DOG

10 Discretization

11 Discretization

12 Discretization

13 Nonlinear Diffusion

14 Topics Anisotropic Diffusion Discretization Adaptive Smoothing Adaptive Smoothing Vs Anisotropic Diffusion Bilateral Filter

15 Scale Space Filtering I(x,y,t)=I 0 (x,y)*g(x,y;t) Family of Derived Images Original Image Gaussian Kernel The Family of Derived Images I(x,y,t) can be equivalently viewed as the Solution of the heat equation with initial condition I(x,y,0)= I 0 (x,y) I t = I=(I xx +I yy )

16 Heat propagation with linear diffusion Equation

17 Diffusion Eqn Motivation Causality: Feature at coarse level is cause at final level. (No spurious detail should be generated when when the resolution is reduced) Homogeneity and Isotropy Space Invarient. (for simplicity only!)

18 Zeros of Laplacian through linear Scale Space [Witkin 83]

19 Gaussian blurring is a local averaging operation. It does not respect natural boundaries.

20 New Meaningful Multi-scale Description [Perona-Malik 90] Causality: No spurious detail should be generated when when the resolution is reduced Immediate Localization: At each localization the region boundaries should be sharp. Piecewise Smoothing: Intra region smoothing preferred over inter region smoothing.

21 Anisotropic Diffusion Where is gradient operator. c(x,y,t)=constant I t =c Given the region boundaries appropriate for that scale, we can set conduction coeff. c=1 inside each region and set c=0 at boundaries

22 The Problem : Region boundaries Estimate boundary locations appropriate to that scale. Let E(x,y,t) be such estimate. Properties of E(x,y,t): E(x,y,t)=0 in interior region E(x,y,t)=k e(x,y,t), where e is unit vector normal to the edge and k local contrast at that point. (k=i(x-1)-i(x+1) for a vertical edge).

23 Conduction Coefficient Choose c(x,y,t)=g( E ) g(.) non negative, monotonically decreasing function with g(0)=1 Here, diffusion process is encouraged at interior regions ( E ~0) and Does not affect boundaries where E is large

24 E= gives excellent results (Perona, Malik) Hence, choose C(x,y,t)=g( (x,y,t) ) Properties of Anisotropic Diffusion: The Maximum Principle All maxima of the solution of ADE in space and time belong to the initial condition (the original image) and boundaries of domain of interest given c>0. Edge Enhancement: Appropriate choice of function for c enhancement of edges (in forward time)

25 Edge Enhancement (cont ) For an edge aligned with y-axis Let c(x,y,t)=g(i x (x,y,t)) Flux Φ (I x )= c.i x =g(i x ).I x 1-D version of HE Modified Step Edge and Its Derivatives

26 Edge Enhancement (cont ) How the slope changes with time? ( / t(i x ) ) If c(.)>0, the function I(.) is smooth and the order of differentiation may be inverted Suppose I x >0, at the point of inflection, I xx =0 and I xxx <<0 Then in neighborhood of pt of inflection / t(i x ) has sign opposite to Φ (I x ). If Φ (I x ).>0 slope of edge will decrease If Φ (I x ).<0 slope of edge will increase

27 Choices of Φ(.) g(i x )=C/(1+(I x /K) 1+a ) with a>0 (recall Φ(I x )=g(i x ).I x ) [0 K] Φ >0 edge slope decreases [K inf] Φ <0 edge slope increases

28 The function g(s) Perona and Malik suggest two possible functions

29 Effect of varying K on g(.)

30 Effect of varying α on g(.)

31 Effect of varying K and α on c()

32 Effect of varying K on c() K=3,5,10 and 100

33 Effect of varying α on c() α= 1,2,3 and 5

34 Heat Equation Discretization

35 Discretized Heat Equation

36 ADE- Discretization Anisotropic Equation: Using centered differences for the Laplacian and gradients: (4 NN discretization)

37 Discretization (cont ) By splitting the Laplacian and averaging the forward and backward differences in the gradient:

38 Discretization (cont )

39 Discretization (cont ) Where,

40 Discretization (cont )

41 Compute g(.) using the projection of the gradient along one direction. Let, Computing s x,y using forward differences, and s x+1,y using backward differences, So,

42 Discretization (cont )

43 Image Neighborhood System

44 Discretization (cont ) The Previous formulation, Can be written as, Range of λ is [0-1/4] for stable numerical scheme.

45 Causality Proof Since all c s are positive and between 0,1,

46 Set K every Iteration

47 Original Anisotropic Diffusion after 10, 20, 50 and 100 iterations

48 Adaptive Smoothing Given an image I(x,y), the adaptive smoothing yields, [Saint-Marc et al., PAMI 91, vol.13, no.6] (x 1 +i,x 2 +j) (x 1 +i,x 2 +j) Where, Here, Where,

49 AD and Adaptive Smoothing For a 1-D signal, With, Let c 1 =c 3 =c,

50 When the weights are space variant, with, which is an implementation of anisotropic diffusion proposed by Perona and Malik. Where diffusion coeff is chosen as,

51 Bilateral Filtering [Tomasi & Manduchi, ICCV 98] Smooths images while preserving edges, by means of a nonlinear combination of nearby image values. Low-pass filter: Range filter:

52 Bilateral Filter Combine low pass filter & Range filter Where, This filter replaces the pixel value at x with an average of similar and nearby pixel values.

53 Bilateral Filter

54 Bilateral Filter (a) A 100-gray-level step perturbed by Gaussian noise with σ= 10 gray levels. (b) Combined similarity weights. (c) after bilateral filtering with r = 50 gray levels and d = 5pixels.

55 Bilateral Filter - Results

56 Original After 1 & 5 iterations

57 Spatial Support Anisotropic diffusion cannot diffuse across edges Bilateral filtering can Larger support more reliable estimator Support of anisotropic diffusion Support of bilateral

58 Bilateral Filtering and Adaptive Smoothing Discrete version of bilateral filtering With, Exercise: Now show that bilateral filter for 3x3 neighborhood Is equivalent to anisotropic diffusion. [Ref: Danny Barash, PAMI, vol. 24, No 6, June 2002 ]

59 Anisotropic Vs Adaptive smoothing