Advanced Arc Extraction Techniques for Computer Vision Applications

1. Computer and Robot Vision I Chapter 11 Arc Extraction and Segmentation Presented by: 傅楸善 & 何育哲 0937 960 615 r94922131@ntu.edu.tw 指導教授: 傅楸善 博士 Digital Camera and Computer Vision Laboratory Department of Computer Science and Information Engineering National Taiwan University, Taipei, Taiwan, R.O.C.

2. 11.1 Introduction • edge detection: labels each pixel as edge or no edge • additional properties of edge: direction, gradient magnitude, contrast • edge grouping: edge pixels in same region boundary grouped together • edge detector: typically produces short linear disjointed edge segments DC & CV Lab. CSIE NTU

3. 11.1 Introduction • each edge segment: has an orientation and a position • edge segments: of little use until aggregated into extended edges • global edge aggregation: upon proximity, relative orientation, contrast, … • local edge aggregation: extends edges by seeking compatible neighbor edges DC & CV Lab. CSIE NTU

4. 11.1 Introduction • global methods: can incorporate domain knowledge into cost functions • global methods: more robust to noisy edge data • global methods: bought at expense of relatively high computational cost • aggregation into extended edges and parametric description: new research area • extended edges: like other curves can be described at multiple scales DC & CV Lab. CSIE NTU

5. 11.1 Introduction • Fig. 3.18(a) DC & CV Lab. CSIE NTU

6. 11.1 Introduction • Fig. 3.18(b) DC & CV Lab. CSIE NTU

7. 11.2 Extracting Boundary Pixels from a Segmented Image • after segmentation/ connected components: region boundary may be extracted • border algorithm: to extract all region boundaries in left-right, top-bottom DC & CV Lab. CSIE NTU

8. 11.2.1 Concepts and Data Structures • input image: symbolic image whose pixel values denote region labels • current regions: borders partially scanned but not yet output • past regions: completely scanned and their borders output DC & CV Lab. CSIE NTU

9. 11.2.1 Concepts and Data Structures • future regions: not yet been reached by the scan • at most 2 * number_of_columns region labels may be active at a time • hash table: used to allow rapid access to chains of a region • each chain: linked list of pixel positions grow-able from beginning to end DC & CV Lab. CSIE NTU

10. 11.2.2 Border-Tracking Algorithm DC & CV Lab. CSIE NTU

11. 11.2.2 Border-Tracking Algorithm DC & CV Lab. CSIE NTU

12. 11.3 Linking One-Pixel-Wide Edges or Lines • segments: may consist of line, endpoint, corner, junctions DC & CV Lab. CSIE NTU

13. 11.3 Linking One-Pixel-Wide Edges or Lines DC & CV Lab. CSIE NTU

14. 11.3 Linking One-Pixel-Wide Edges or Lines • algorithm to track segments like these has to be concerned with • starting a new segment • adding an interior pixel to a segment • ending a segment • finding a junction • finding a corner DC & CV Lab. CSIE NTU

15. 11.3 Linking One-Pixel-Wide Edges or Lines • segments: lists of edge points representing straight or curved lines • pixeltype: isolated, starting, interior, ending, junction, corner DC & CV Lab. CSIE NTU

16. 11.3 Linking One-Pixel-Wide Edges or Lines DC & CV Lab. CSIE NTU

17. 11.4 Edge and Line Linking Using Directional Information • edge linking: labeled pixels with similar enough directions form chains • edge linking chains identified as arc segment with good fit to curvelike line DC & CV Lab. CSIE NTU

18. Take a Break DC & CV Lab. CSIE NTU

19. 11.5 Segmentation of Arcs into Simple Segments • extracted digital arc: sequence of row-column pairs 4-neighbor or 8-neighbor • arc segmentation: partitions extracted digital arc to fit straight/ curved line • endpoints of subsequences: corner points or dominant points DC & CV Lab. CSIE NTU

20. 11.5 Segmentation of Arcs into Simple Segments • basis for partitioning process identication of all locations with • sufficiently high curvature (high change in tangent angle to length) • enclosed by subsequences fitting different straight lines or curves DC & CV Lab. CSIE NTU

21. 11.5 Segmentation of Arcs into Simple Segments • simple arc segment: straight-line or curved-arc segment • techniques: from iterative endpoint fitting, splitting to using tangent angle deflection, prominence, or high curvature as basis of segmentation DC & CV Lab. CSIE NTU

22. 11.5.1Iterative Endpoint Fit and Split • one distance threshold d* • pixel with farthest distance to line AC is B and distance ≥d* then split DC & CV Lab. CSIE NTU

23. 11.5.1Iterative Endpoint Fit and Split DC & CV Lab. CSIE NTU

24. 11.5.2Tangential Angle Deflection • another approach: identify locations where two line segments meet • exterior angle between two line segments: change in angular orientation DC & CV Lab. CSIE NTU

25. 11.5.2Tangential Angle Deflection DC & CV Lab. CSIE NTU

26. 11.5.2Tangential Angle Deflection • caution: spatial quantization at small distances can completely mask direction DC & CV Lab. CSIE NTU

27. 11.5.3Uniform Bounded-Error Approximation • segment arc sequence into maximal pieces whose points deviate ≤ given amount • optimal algorithms: excessive computational complexity DC & CV Lab. CSIE NTU

28. 11.5.3Uniform Bounded-Error Approximation DC & CV Lab. CSIE NTU

29. 11.5.4Breakpoint Optimization • after initial segmentation: shift breakpoints to produce better segmentation • first: shift odd final point i.e. even beginning point • then: shift even final point i.e. odd beginning point DC & CV Lab. CSIE NTU

30. 11.5.5Split and Merge • first: split arc into segments with error sufficiently small • then: merge successive segments if resulting segment sufficiently small error • third: try to adjust breakpoints to obtain better segmentation • repeat: until all three steps produce no further change DC & CV Lab. CSIE NTU

31. 11.5.6Isodata Segmentation • iterative isodata line-fit clustering procedure: determines line-fit parameter • then each point assigned to cluster whose line fit closest to the point DC & CV Lab. CSIE NTU

32. 11.5.7Curvature • geometry of circular arc segment defined by and 􀀀 DC & CV Lab. CSIE NTU

33. 11.5.7Curvature DC & CV Lab. CSIE NTU

34. 11.5.7Curvature • : curve represented parametrically, • : arc length going from to DC & CV Lab. CSIE NTU

35. 11.5.7Curvature DC & CV Lab. CSIE NTU

36. 11.5.7Curvature • : unit length tangent vector at measured clockwise from column axis DC & CV Lab. CSIE NTU

37. 11.5.7Curvature • : unit normal vector at DC & CV Lab. CSIE NTU

38. 11.5.7Curvature • : curvature defined at point of arc length s along curve • : change of arc length • : change in tangent angle DC & CV Lab. CSIE NTU

39. 11.5.7Curvature • natural curve breaks: curvature maxima and minima • curvature passes: through zero local shape changes from convex to concave • surface elliptic: when limb in line drawing is convex • surface hyperbolic: when its limb is concave DC & CV Lab. CSIE NTU

40. 11.5.7Curvature • surface parabolic: wherever curvature of limb zero • cusp singularities of projection: occur only within hyperbolic surface DC & CV Lab. CSIE NTU

41. 11.5.7Curvature • Nalwa, A Guided Tour of Computer Vision, Fig. 4.14 􀀀 DC & CV Lab. CSIE NTU

42. 11.6 Hough Transform • Hough transform: method for detecting straight lines and curves on images • Hough transform: template matching DC & CV Lab. CSIE NTU

43. 11.6.1Hough Transform Technique • The Hough transform algorithm requires an accumulator array whose dimension corresponds to the number of unknown parameters in the equation of the family of curves being sought. DC & CV Lab. CSIE NTU

44. Finding Straight-Line Segments • : for straight lines does not work for vertical lines • : slope, : intercept • : where and used • : perpendicular distance from the line to the origin • : the angle the perpendicular makes with the x-axis DC & CV Lab. CSIE NTU

45. Finding Straight-Line Segments DC & CV Lab. CSIE NTU

46. Finding Straight-Line Segments • use , and , to determine if point falls into cell an accumulator array quantized in this fashion DC & CV Lab. CSIE NTU

47. Finding Straight-Line Segments DC & CV Lab. CSIE NTU

48. Finding Straight-Line Segments • Fig. 7.18(a) • Fig. 7.18(b) • Fig. 7.18(c) • Fig. 7.18(d) • Gonzalez and Woods Digital Image Processing Fig. 7.18 􀀀 • Fig. 7.19(a) • Fig. 7.19(b) • Fig. 7.19(c) • Fig. 7.19(d) Gonzalez and Woods Digital Image Processing Fig. 7.19 􀀀 DC & CV Lab. CSIE NTU

49. Finding Straight-Line Segments • Fig. 7.18(a) • Fig. 7.18(b) • Fig. 7.18(c) • Fig. 7.18(d) • Gonzalez and Woods Digital Image Processing Fig. 7.18 􀀀 • Fig. 7.19(a) • Fig. 7.19(b) • Fig. 7.19(c) • Fig. 7.19(d) Gonzalez and Woods Digital Image Processing Fig. 7.19 􀀀 DC & CV Lab. CSIE NTU

50. Finding Circles • : row • : column • : row-coordinate of the center • : column-coordinate of the center • : radius • : implicit equation for a circle DC & CV Lab. CSIE NTU