1. A grey-level image may be seen as a topographic relief,topographic where the grey level of a pixel is interpreted as its altitude in the relief. A drop of water falling on a topographic relief flows along a path to finally reach a local minimum. Intuitively, the watershed of a relief correspond to the limits of the adjacent catchment basins of the drops of water. Watershed transform Watershed of the gradient Watershed of the gradient (relief) Relief of the gradient Gradient image Cardiac MRI image

13. Skeletonization

14. Skeleton by distance transforms Maxima of distance transform

22. Distance Transform

23. Skeleton Reconstruction: the original object can be reconstructed by given knowledge of the skeleton subsets S i (F), the SE K, and i : Examples of skeleton:

33. The Distance Transform on Curved Space (DTOCS)

35. Distance Transform on Curved Space (DTOCS) Calculates minimal distances between 2 points along a curved surface Calculates minimal distances between areas and areas/points on curved surface Uses a 3x3 calculation kernel with different metrics: Chessboard City block Is a gray-level extension to the Rosenfeldt-Pfaltz-Lay algorithm (which calculates a distance transform for binary images) Presented by Toivanen and Vepsäläinen in 1991 and 1993. Applications: Texture feature extraction and classification (e.g. paper roughness) (Kuparinen and Toivanen 2006, 2007) Shortest distance calculations (Ikonen and Toivanen 2006) Image compression

36. Weighted Distance Transform on Curved Space (WDTOCS) Calculates minimal distances between 2 points along a curved surface Calculates minimal distances between areas and areas/points on curved surface Uses a 3x3 calculation kernel with different metrics: Chessboard City block Measures the differences between adjacent pixels by their Euclidean distance + (1 or 1,4 for the xy-surface displacement) Presented by Toivanen and Vepsäläinen in 1991 and 1993. Applications: Texture feature extraction and classification (e.g. paper roughness) (Kuparinen and Toivanen 2006, 2007) Shortest distance calculations (Ikonen and Toivanen 2006) Image compression

37. Definition of the Distance Transform On Curved Space (DTOCS)

38. p ne pnpn pwpw pcpc pepe p sw psps p se The 3x3 kernel used in DTOCS algorithm

39. p ne pnpn pwpw pcpc pepe p sw psps p se The 3x3 kernel used in DTOCS algorithm

40. The Distance Transform on Curved Space (DTOCS)

42. Original imageDistance image after forward pass

43. Distance image after backward pass Distance image after 2nd iteration (= forward+backward pass second time)

45. Original Lena image 521 x 521 x 8 bits Curves in which DTOCS distance > binary distance Control points chosen along the curves

47. (a) LWC(b) SC(c) Cardboard (d) LWC(e) SC(f) Cardboard

49. Shortest route calculation with Route DTOCS.

50. Original image a b

51. a b Shortest route between a and b

52. Fig. 2. a) Original image, b) distance from source point, c) distance from destination point, d) sum of distance images, e) route by DTOCS, f) route by WDTOCS.

53. Original labyrinth Shortest routes by DTOCS

54. Original labyrinth Shortest routes by DTOCS

55. Fig. 3. a) Shortest routes from corner to corner of an ”eggbox” surface, b) 3D-visualization of the routes on the surface a) a)b)

56. Quantization Thresholding

57. Original image Constant additionNegative image Addition + contrast strecthing Histogram equalization

58. Original imageRange compressionAddition + contrast strecthing Original image Contrast stretching to [0, 128] Histogram equalization

59. Filtering

60. Figure 2.12: General mask for filtering with a 3 x 3 window.

65. Original image "airplane" (5125128) Smoothed by 3 x 3 averaging filter Smoothed by 5 x 5 averaging filter Smoothed by 7 x 7 averaging filter Smoothed by 15 x 15 averaging filter

66. The End