gruvi Edge Aware Anisotropic Diffusion for 3D Scalar Data graphics + usability + visualization Simon Fraser University Burnaby, BC Canada.

Transcription

1 gruvi graphics + usability + visualization Edge Aware Anisotropic Diffusion for 3D Scalar Data Zahid Hossain Torsten Möller Simon Fraser University Burnaby, BC Canada.

2 Motivation Original Sheep s heart data page 2 of 32

3 Motivation Existing gradient based anisotropic diffusion Original Sheep s heart data page 2 of 32

4 Motivation Existing gradient based anisotropic diffusion Gradient magnitude map page 2 of 32

5 Motivation Existing gradient based anisotropic diffusion Gradient magnitude map Proposed method page 2 of 32

6 Motivation Existing gradient based anisotropic diffusion Note Gradient magnitude map Proposed method Our method yields a consistent smoothing over an iso-surface without undesirable artifacts. page 2 of 32

7 Previous Work Perona and Malik (PM) Model (TPAMI, 1990): = div(h( f ) f ) f t page 3 of 32

8 Previous Work Perona and Malik (PM) Model (TPAMI, 1990): = div(h( f ) f ) f t Carmona et al.(tip, 1998) generalized the PM model in 2D: SF f t = {}}{ h ( ) af nn + bf vv, SF = Stopping Function where v is the orthonormal direction of the normal n. page 3 of 32

9 Previous Work Perona and Malik (PM) Model (TPAMI, 1990): = div(h( f ) f ) f t Carmona et al.(tip, 1998) generalized the PM model in 2D: SF f t = {}}{ h ( ) af nn + bf vv, SF = Stopping Function where v is the orthonormal direction of the normal n. Extension in 3D by Gerig et al.(tmi, 1992): remains isotropic on the tangent plane of the gradient. page 3 of 32

10 Previous Work: Trouble in 3D Original page 4 of 32

11 Previous Work: Trouble in 3D PM model in 3D Original page 4 of 32

12 Previous Work: Trouble in 3D PM model in 3D Original Curvatures taken into account page 4 of 32

13 Previous Work: Trouble in 3D PM model in 3D Original Curvatures taken into account Therefore Weickert classified PM and higher order diffusion processes as non-linear rather than anisotropic. page 4 of 32

14 Previous Work: In 3D Krissian et al.(scale-space, 1997) developed a true anistropic diffusion, (KM model), taking curvatures into account as proposed by Whitaker (VBC, 1994). page 5 of 32

15 Previous Work: In 3D Krissian et al.(scale-space, 1997) developed a true anistropic diffusion, (KM model), taking curvatures into account as proposed by Whitaker (VBC, 1994). The stopping function had gradient magnitude based parameter. page 5 of 32

16 Previous Work: In 3D Krissian et al.(scale-space, 1997) developed a true anistropic diffusion, (KM model), taking curvatures into account as proposed by Whitaker (VBC, 1994). The stopping function had gradient magnitude based parameter. Level Set: The definition of edge is based on the curvature that is measured on the surface of the level set. page 5 of 32

17 Previous Work: In 3D Krissian et al.(scale-space, 1997) developed a true anistropic diffusion, (KM model), taking curvatures into account as proposed by Whitaker (VBC, 1994). The stopping function had gradient magnitude based parameter. Level Set: The definition of edge is based on the curvature that is measured on the surface of the level set. Our definition is based on the second derivative measured across the level sets. page 5 of 32

18 Previous Work: In 3D Krissian et al.(scale-space, 1997) developed a true anistropic diffusion, (KM model), taking curvatures into account as proposed by Whitaker (VBC, 1994). The stopping function had gradient magnitude based parameter. Level Set: The definition of edge is based on the curvature that is measured on the surface of the level set. Our definition is based on the second derivative measured across the level sets. De-noising: Bilateral filtering, mean-shift filtering and non-linear diffusion are equivalent: Barash et al., (IVC, 2004). page 5 of 32

19 Previous Work: In 3D Krissian et al.(scale-space, 1997) developed a true anistropic diffusion, (KM model), taking curvatures into account as proposed by Whitaker (VBC, 1994). The stopping function had gradient magnitude based parameter. Level Set: The definition of edge is based on the curvature that is measured on the surface of the level set. Our definition is based on the second derivative measured across the level sets. De-noising: Bilateral filtering, mean-shift filtering and non-linear diffusion are equivalent: Barash et al., (IVC, 2004). A recent methods namely SRNRAD and ORNRAD (TIP, 2009). page 5 of 32

20 Previous Work: In 3D Krissian et al.(scale-space, 1997) developed a true anistropic diffusion, (KM model), taking curvatures into account as proposed by Whitaker (VBC, 1994). The stopping function had gradient magnitude based parameter. Level Set: The definition of edge is based on the curvature that is measured on the surface of the level set. Our definition is based on the second derivative measured across the level sets. De-noising: Bilateral filtering, mean-shift filtering and non-linear diffusion are equivalent: Barash et al., (IVC, 2004). A recent methods namely SRNRAD and ORNRAD (TIP, 2009). We will compare our method with these two. page 5 of 32

21 Contributions 1 Removal of the gradient magnitude based parameter. page 6 of 32

22 Contributions 1 Removal of the gradient magnitude based parameter. 2 Dynamic adaptation of anisotropy based on local curvatures. page 6 of 32

23 Contributions 1 Removal of the gradient magnitude based parameter. 2 Dynamic adaptation of anisotropy based on local curvatures. 3 Application in de-noising and visualization. page 6 of 32

24 The Proposed Method page 7 of 32

25 Proposed Method: Edge The red solid curve is the error function while the dashed black curve is the second derivative of it. page 8 of 32

26 Proposed Method: Edge The red solid curve is the error function while the dashed black curve is the second derivative of it. A common technique to detect edges in 2D image processing. Basis of 2D transfer functions, Kindlmann and Durkin (VIS 1998). page 8 of 32

27 Four Objectives of Our Diffusion Objectives 1 No diffusion will be performed along the gradient direction. Note This one is different from the classical methods. This prevents blurring across an edge. page 9 of 32

28 Four Objectives of Our Diffusion Objectives 1 No diffusion will be performed along the gradient direction. 2 Diffusion will be stopped around the edge locations. Note Check for f nn = 0. Note it does not create any problem in constant homogeneous regions. page 9 of 32

29 Four Objectives of Our Diffusion Objectives 1 No diffusion will be performed along the gradient direction. 2 Diffusion will be stopped around the edge locations. 3 Diffusion will be performed anisotropically along the direction of the minimum curvature. Note Similar to Krissian et al.(1997), i.e the KM model. page 9 of 32

30 Four Objectives of Our Diffusion Objectives 1 No diffusion will be performed along the gradient direction. 2 Diffusion will be stopped around the edge locations. 3 Diffusion will be performed anisotropically along the direction of the minimum curvature. 4 Diffuse isotropically on the tangent plane of the normal n in regions where the local iso-surface has similar principal curvatures. Note For example, on the surface of a sphere or a slab. page 9 of 32

31 Modeling the PDE Consider the 1D heat, also known as diffusion, equation: f t = cf xx page 10 of 32

32 Modeling the PDE Consider the 1D heat, also known as diffusion, equation: f t = cf xx A 3D extension of the above: f t = hf r 1 r 1 + gf r2 r 2 + wf nn Where the orthonormal bases [r 1,r 2,n] are min curvature, max curvature and normal directions respectively. page 10 of 32

33 Modeling the PDE Consider the 1D heat, also known as diffusion, equation: f t = cf xx A 3D extension of the above: f t = hf r 1 r 1 + gf r2 r 2 + wf nn Where the orthonormal bases [r 1,r 2,n] are min curvature, max curvature and normal directions respectively. Setting g = τh and w = ηh we can simplify the above f ( t = h f r1 r 1 + τf r2 r 2 + ηf nn ), τ,η [0,1] page 10 of 32

34 Modeling the PDE: Objective 1 Objectives 1 No diffusion will be performed along the gradient direction. 2 Diffusion will be stopped around the edge locations. 3 Diffusion will be performed anisotropically along the direction of the minimum curvature. 4 Diffuse isotropically on the tangent plane of the normal n in regions where the local iso-surface has similar principal curvatures. f ( t = h f r1 r 1 + τf r2 r 2 + ηf nn ), τ,η [0,1] page 11 of 32

35 Modeling the PDE: Objective 1 Objectives 1 No diffusion will be performed along the gradient direction. 2 Diffusion will be stopped around the edge locations. 3 Diffusion will be performed anisotropically along the direction of the minimum curvature. 4 Diffuse isotropically on the tangent plane of the normal n in regions where the local iso-surface has similar principal curvatures. Set η = 0 f ( t = h f r1 r 1 + τf r2 r 2 + ηf nn ), τ,η [0,1] f ( t = h f r1 r 1 + τf r2 r 2 ), τ [0,1] page 11 of 32

36 Modeling the PDE: Objective 2 Objectives 1 No diffusion will be performed along the gradient direction. 2 Diffusion will be stopped around the edge locations. 3 Diffusion will be performed anisotropically along the direction of the minimum curvature. 4 Diffuse isotropically on the tangent plane of the normal n in regions where the local iso-surface has similar principal curvatures. page 12 of 32

37 Modeling the PDE: Objective 2 Define h h(α) = 1 (0.9) ( α σ ) 2, σ R h(α) α Plot of h(α) page 13 of 32

38 Modeling the PDE: Objective 2 Define h h(α) = 1 (0.9) ( α σ ) 2, σ R h(α) h can take, as argument, the directional second derivative f nn along the normal n α Plot of h(α) page 13 of 32

39 Modeling the PDE: Objective 2 Define h h(α) = 1 (0.9) ( α σ ) 2, σ R h(α) h can take, as argument, the directional second derivative f nn along the normal n. ) So far: f t = h(f nn) (f r1 r 1 + τf r2 r α Plot of h(α) page 13 of 32

40 Modeling the PDE: Objective 2 Define h h(α) = 1 (0.9) ( α σ ) 2, σ R h(α) h can take, as argument, the directional second derivative f nn along the normal n. ) So far: f t = h(f nn) (f r1 r 1 + τf r2 r α Plot of h(α) Note Diffusion stops when f nn 0, i.e. around the edge locations. Constant homogeneous regions, where f nn 0, would not be affected by diffusion anyway. page 13 of 32

41 Modeling the PDE: Objectives 3 and 4 Objectives 1 No diffusion will be performed along the gradient direction. 2 Diffusion will be stopped around the edge locations. 3 Diffusion will be performed anisotropically along the direction of the minimum curvature. 4 Diffuse isotropically on the tangent plane of the normal n in regions where the local iso-surface has similar principal curvatures. page 14 of 32

42 Modeling the PDE: Objectives 3 and 4 f ) t = h(f nn) (f r1 r 1 + τf r2 r 2 page 15 of 32

43 Modeling the PDE: Objectives 3 and 4 f ) t = h(f nn) (f r1 r 1 + τf r2 r 2 ( ) 2λ κ1 τ = κ 2 where κ 2 > 0,λ Z 1 κ 2 = 0 where, κ 1 : minimum curvature κ 2 : maximum curvature such that, κ 1 κ 2. page 15 of 32

44 Modeling the PDE: Objectives 3 and 4 where, such that, κ 1 κ 2. Note f ) t = h(f nn) (f r1 r 1 + τf r2 r 2 ( ) 2λ κ1 τ = κ 2 where κ 2 > 0,λ Z 1 κ 2 = 0 κ 1 : κ 2 : minimum curvature maximum curvature Both the curvatures are measured from a smoothed version of the data f ρ with a Gaussian filter having a small variance ρ 2. So we will use the notation τ ρ for the rest of the presentation to depict this and call it isotropy function. page 15 of 32

45 Simplification So far we have f ( IF t = h(f {}}{ nn) f r1 r 1 + τ ρ f r2 r 2 ), IF=Isotropy Function page 16 of 32

46 Simplification So far we have f ( IF t = h(f {}}{ nn) f r1 r 1 + τ ρ f r2 r 2 ), IF=Isotropy Function Which can be simplied to. f t = h(f nn) f (κ 1 + τ ρ κ 2 ) page 16 of 32

47 Simplification So far we have f ( IF t = h(f {}}{ nn) f r1 r 1 + τ ρ f r2 r 2 ), IF=Isotropy Function Which can be simplied to. f t = h(f nn) f (κ 1 + τ ρ κ 2 ) Note Has a similar form to Mean Curvature Motion (MCM), yet essentially different. f MCM: t = f (κ 1 + κ 2 ) page 16 of 32

48 Results: Significance of σ Original h( ) = 1 σ = 40 Our: ( ) fnn 2 f t = h(f nn) f (κ 1 + τ ρ κ 2 ), h(f nn ) = 1 (0.9) σ, σ R MCM: f t = f (κ 1 + κ 2 ) page 17 of 32

49 Results page 18 of 32

50 Results: Consistent Smoothing KM method: k = 40 Original Our method: σ = 1 ( ) fnn 2 f t = h(f nn) f (κ 1 + τ ρ κ 2 ), h(f nn ) = 1 (0.9) σ, σ R page 19 of 32

51 Results: Consistent Smoothing KM method: k = 60 Original Our method: σ = 10 ( ) fnn 2 f t = h(f nn) f (κ 1 + τ ρ κ 2 ), h(f nn ) = 1 (0.9) σ, σ R page 19 of 32

52 De-noising Properties page 20 of 32

53 De-noising Properties Note We used the same parameter settings for all our de-noising experiments across all datasets and noise types. With a unit grid spacing in all dimensions we empirically found that t 0.4 is stable for most practical purposes. page 20 of 32

54 Results: De-noising (Gaussian) Noisy, SNR=12.89 Original Diffused, SNR=26.12 page 21 of 32

55 Results: De-noising (Gaussian) Noisy, SNR=12.89 Original Diffused, SNR=26.12 Original Noisy Diffused Profile page 21 of 32

56 Results: De-noising (Gaussian) Noisy, SNR=12.89 Original Diffused, SNR=26.12 Original Noisy Diffused Profile page 21 of 32

57 Results: De-noising (Salt and Pepper) Noisy, SNR=12.89 Original Diffused, SNR=26.12 Original Noisy Diffused Profile page 22 of 32

58 De-noising Comparison page 23 of 32

59 De-noising Comparison Note We compared our anisotropic diffusion with two recent de-noising methods: SRNRAD and ORNRAD proposed by Krissian and Aja-Fernández (2009). We did not change any parameter settings. page 23 of 32

60 Quantitative Measures Mean Squared Error (MSE). Structural Similarity Index (SSIM). Quality Index Based on Local Variance (QILV). Original Blurred Gaussian noise SSIM: (Good) SSIM: (Bad) QILV: (Bad) QILV: (Good) page 24 of 32

61 De-noising: Comparison Gaussian noise Original MSE SSIM QILV Noise SRNRAD ORNRAD Our Noisy Our SRNRAD ORNRAD page 25 of 32

62 De-noising: Comparison Rician noise Original MSE SSIM QILV Noise SRNRAD ORNRAD Our Noisy Our SRNRAD ORNRAD page 25 of 32

63 De-noising: Comparison Note Our method performs similarly if not better in many cases. page 25 of 32

64 De-noising: Comparison Note Our method performs similarly if not better in many cases. And our Matlab implementation is 14 times faster than a multi-threaded C/C++ based implementation of ORNRAD. page 25 of 32

65 Impact on Visualization page 26 of 32

66 2D Transfer Function (With Noise) Gaussian noise Original Noisy Diffused page 27 of 32

67 2D Transfer Function (With Noise) Speckle noise Original Noisy Diffused page 27 of 32

68 2D Transfer Function (Without Noise) Tooth Dataset Original Diffused with our method page 28 of 32

69 2D Transfer Function Sheep s Heart Dataset KM method Original Our method page 29 of 32

70 Conclusion We developed an anistropic diffusion that: Smoothes consistently over an iso-surface Requires much less effort to choose a parameter value. Performs similarly, if not better, to a recent de-noising method. Is easy to implement and fast. Enhances 2D transfer functions. page 30 of 32

71 Acknowledgements We would like to thank: Dr. Karl Krissian and Dr. Santiago Aja-Fernández for providing supports in various occasions. Dr. Philippe Thévenaz for the Sphere phantom data. Natural Science and Engineering Research Council of Canada (NSERC) for funding this project. page 31 of 32

72 Thank you for your attention :) Contact Information Zahid Hossain Dr. Torsten Möller page 32 of 32