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Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
Digital Distance Geometry
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Digital Distance Geometry

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I presented this overview of my work in Digital Geometry at ICVGIP ’04. Science City, Kolkata on 18-Dec-04

I presented this overview of my work in Digital Geometry at ICVGIP ’04. Science City, Kolkata on 18-Dec-04

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  • 1. Digital Distance Geometry – Applications to Image Analysis Dr. P. P. Das [email_address] , [email_address] Interra Systems, Inc. www.interrasystems.com ICVGIP ’04. Science City. 18-Dec-04
  • 2. Dedication <ul><li>To the Memory of Professor Azriel Rosenfeld </li></ul><ul><li>Who taught us to “see” through the eyes of the machine </li></ul><ul><li>This photo is taken from http://www.cfar.umd.edu/~ar/ </li></ul>
  • 3. Agenda <ul><li>What is Distance? </li></ul><ul><li>Distance Geometry </li></ul><ul><li>Digital Distance Geometry </li></ul><ul><ul><li>Digital Distance in 2-D </li></ul></ul><ul><ul><li>Digital Distances in 3-D </li></ul></ul><ul><ul><li>Digital Distances in n-D </li></ul></ul><ul><li>Digital Distance Computation & Approximation </li></ul><ul><li>Glimpses of Applications </li></ul>
  • 4. What is Distance?
  • 5. What is Distance? <ul><li>Sense of being Near or Far </li></ul><ul><li>Distances in Space – As-the-crow-flies </li></ul><ul><ul><li>Normally follows the basic properties – well-defined, finite, positive, definite, symmetric & triangular. </li></ul></ul><ul><ul><li>Exception are not uncommon </li></ul></ul><ul><ul><ul><li>Not every place is reachable – violates being well-defined / finite </li></ul></ul></ul><ul><ul><ul><li>One-way roads do not have symmetry </li></ul></ul></ul><ul><ul><ul><li>Triangularity on sphere may not hold </li></ul></ul></ul><ul><li>Distances in Time </li></ul>
  • 6. Distances in Personal Space <ul><li>Proxemics </li></ul><ul><ul><li>People's use of personal space </li></ul></ul><ul><li>Four categories for informal space: </li></ul><ul><ul><li>Intimate distance </li></ul></ul><ul><ul><ul><li>for embracing or whispering (6-18 inches) </li></ul></ul></ul><ul><ul><li>Personal distance </li></ul></ul><ul><ul><ul><li>for conversations among good friends (1.5-4 feet) </li></ul></ul></ul><ul><ul><li>Social distance </li></ul></ul><ul><ul><ul><li>for conversations among acquaintances (4-12 feet) </li></ul></ul></ul><ul><ul><li>Public distance </li></ul></ul><ul><ul><ul><li>used for public speaking (12 feet or more) </li></ul></ul></ul>Edward T. Hall, 1963
  • 7. Distances in Other Spheres <ul><li>Psychological Distance </li></ul><ul><li>Emotive Distance </li></ul><ul><li>Separational Distance of Six-Degrees </li></ul><ul><ul><li>Everyone on Earth is separated by no more than six friends of friends of friends </li></ul></ul>
  • 8. Distance Geometry
  • 9. Significance of Distance Geometry <ul><li>Distance is a Fundamental Concept in Geometry </li></ul><ul><ul><li>Shortest Paths </li></ul></ul><ul><ul><ul><li>Straight Lines </li></ul></ul></ul><ul><ul><li>Geodesic on Earth </li></ul></ul><ul><ul><li>Parallel Lines </li></ul></ul><ul><ul><ul><li>Equidistant Ever </li></ul></ul></ul><ul><ul><li>Circle </li></ul></ul><ul><ul><ul><li>Trajectory of a point equidistant from Center </li></ul></ul></ul><ul><ul><ul><li>Least Perimeter with Largest Area </li></ul></ul></ul><ul><ul><li>Conics are distance defined </li></ul></ul><ul><li>Geometries can be built on Distances </li></ul>
  • 10. Taxicab Geometry Krause 1975
  • 11. Digital Geometry <ul><li>Necessitated by Automated Analysis </li></ul><ul><li>Discrete Models are needed for Machine </li></ul><ul><ul><li>Image Analysis </li></ul></ul><ul><ul><li>Navigation </li></ul></ul><ul><ul><li>Robotics </li></ul></ul><ul><li>Digital Geometry – by Rosenfeld et al </li></ul>
  • 12. Euclidean :: Digital Geometry <ul><li>Jordan’s theorem in 2-D holds if 4-connectivity is maintained in the object (background) and 8-connectivity in the background (object) </li></ul><ul><li>A shortest path has a unique mid-point or a mid-point pair </li></ul><ul><li>Jordan’s theorem in 2-D </li></ul><ul><li>Every shortest path which connects two points has a unique mid-point </li></ul>Properties that hold after extension <ul><li>Euclidean distance is a metric Extendable to higher dimensions </li></ul><ul><li>Euclidean distance is a metric Extendable to higher dimensions </li></ul>Properties that hold   Digital Geometry Euclidean Geometry
  • 13. Euclidean :: Digital Geometry <ul><li>The shortest path between pair of points may not be unique </li></ul><ul><li>Lines may not intersect but may not be parallel </li></ul><ul><li>Angle is unlikely. Digital trigonometry has been ruled out (Rosenfeld 1983) </li></ul><ul><li>The shortest path between any pair of points is unique </li></ul><ul><li>Only parallel lines do not intersect </li></ul><ul><li>Two intersecting lines define an angle between them </li></ul>Properties that do not hold Digital Geometry Euclidean Geometry
  • 14. Significance of Digital Distance Geometry <ul><li>Divergence from Euclidean Geometry. </li></ul><ul><li>Preservation of “intuitive” Properties. </li></ul><ul><li>Preservation of Metric Properties. </li></ul><ul><li>Quality of Approximation </li></ul><ul><ul><li>How to work in digital domain with Euclidean accuracy? </li></ul></ul><ul><ul><li>“ Circularity” of Disks </li></ul></ul><ul><li>Computational Efficiency </li></ul><ul><ul><li>Distance Transformations </li></ul></ul><ul><ul><li>Medial Axis Transform </li></ul></ul>
  • 15. Digital Distance Geometry Basic Notions
  • 16. Model: Digitization of Space <ul><li>Digitization </li></ul><ul><ul><li>Partitioning through Cells </li></ul></ul><ul><ul><li>Covers the Space </li></ul></ul><ul><ul><li>No Overlap </li></ul></ul><ul><li>Homogeneous – uniformity around every vertex </li></ul><ul><li>Regular – identical tiles </li></ul><ul><ul><li>2D – 11 Homogeneous, 3 Regular </li></ul></ul><ul><ul><li>3D – 1 Regular </li></ul></ul><ul><ul><li>4D – 3 Regular </li></ul></ul><ul><ul><li>n-D – 1 Regular </li></ul></ul>
  • 17. Tessellations
  • 18. The Discrete Model <ul><li>Rectangular Tessellations </li></ul><ul><ul><li>Easy Algebraic Representation, Z n </li></ul></ul><ul><ul><li>Extensible to arbitrary dimensions </li></ul></ul><ul><li>In 2-D </li></ul><ul><ul><li>Hexagonal / Triangular is more “pleasant” </li></ul></ul><ul><ul><li>Melter did a mapping to Rectangular </li></ul></ul>
  • 19. Neighborhood <ul><li>The neighborhood N (x) of a point x is the set of points defined to be neighbors of x. </li></ul><ul><li>A digital neighborhood is characterized by a set of difference vectors {e 1 ,e 2 , …, e k }  Z n such that </li></ul><ul><ul><li>y  N(x) or y  Z n is a neighbor of x  Z n </li></ul></ul><ul><ul><ul><li>iff (x-y)  {e 1 ,e 2 , … , e k }. </li></ul></ul></ul><ul><li>With every neighbor e i we also associate a cost  (e i ). Most often this cost is taken to be unity. </li></ul>
  • 20. Neighborhood: Examples <ul><li>Cityblock or 4-neighbors: </li></ul><ul><ul><ul><li>{( ± 1,0), (0, ± 1)} k = 4 </li></ul></ul></ul><ul><li>Chessboard or 8-neighbors: </li></ul><ul><ul><ul><li>{( ± 1, 0), (0, ± 1), ( ± 1, ± 1)}, k = 8 </li></ul></ul></ul><ul><li>Knight’s neighbors: </li></ul><ul><ul><ul><li>{( ± 1, ± 2), ( ± 2, ± 1)}, k = 8 </li></ul></ul></ul><ul><li>Neighborhoods in 3-D are 6, 18 and 26. </li></ul>
  • 21. Neighborhoods: 2D
  • 22. Neighborhoods: 3D
  • 23. Digital Neighborhood: 5 Factors <ul><li>Proximity : Any two neighbors are proximal sharing a common hyperplane. </li></ul><ul><ul><ul><li>That is, |e i j | ≤ 1, 1 ≤ i ≤ k , 1 ≤ j ≤ n </li></ul></ul></ul><ul><li>Separating dimension : The dimension d of the separating hyperplane is bounded by a constant r such that 0≤r≤d≤n-1. </li></ul><ul><ul><ul><li>4-neighbors have r = 1 – line separation </li></ul></ul></ul><ul><ul><ul><li>8-neighbors have r = 0 – both point- and line-separations. </li></ul></ul></ul><ul><ul><ul><li>That is, n - d =  1 n |e i j | ≤ n – r , 1 ≤ i ≤ k , </li></ul></ul></ul><ul><li>Separating cost : The cost between neighbors - usually unity. </li></ul><ul><ul><ul><li>That is,  (e i ) = 1, 1≤ i ≤ k . </li></ul></ul></ul>
  • 24. Digital Neighborhood: 5 Factors <ul><li>Isotropy and symmetry : The neighborhood is isotropic in all (discrete) directions. </li></ul><ul><ul><ul><li>That is, e j is a permutation and / or reflection of e i , 1 ≤ i, j ≤ k. </li></ul></ul></ul><ul><li>Uniformity : The neighborhood relation is identical at all points along a path and at all points of the space </li></ul><ul><li>Exceptions are not uncommon. </li></ul>
  • 25. Path – Graph-Theoretic Notion <ul><li>Given a neighborhood N (·), a digital path π( u , v ) connecting two points u and v is defined to be a sequence of points where all pairs of consecutive points are neighbors. </li></ul><ul><ul><ul><li>π(u, v) is {u = x 0 , x 1 , x 2 , …, x M = v} where x i  N (x i+1 ), 0≤ i < M . </li></ul></ul></ul><ul><li>The length of the path |π(u, v)| =   (x i +1 – x i ). </li></ul><ul><li>For unit cost this is the number of points on the path (excluding either u or v). </li></ul><ul><li>Of all paths that connect u to v the one having the smallest length is called the minimal path π*(u, v). </li></ul>
  • 26. 2D Paths
  • 27. 3D Paths
  • 28. Distance Function <ul><li>The distance d (u, v) between u, v (w.r.t. to a neighborhood N (·)) is the length of the shortest path connecting them. </li></ul><ul><ul><ul><li>That is d (u, v) = |π*(u, v)|. </li></ul></ul></ul><ul><li>Distance Function: </li></ul><ul><ul><li>d: R n x R n  R + </li></ul></ul><ul><li>Digital Distance Function: </li></ul><ul><ul><li>d: Z n x Z n  P </li></ul></ul><ul><li>Also, d (u, v) = d (0, u – v) = d (0, |u – v|), where 0 is the origin. That is, d (u, v) = d (x), where x = |u – v|. </li></ul>
  • 29. Metric <ul><li>A distance function d is said to be a metric if it is: </li></ul><ul><li>Total : d(u, v) is defined and finite; </li></ul><ul><ul><li>Note that for all u, v  Z n , π(u, v) may not exist, and hence d (u, v) may not be defined. Ref. super-knight’s distance. </li></ul></ul><ul><li>Positive : d (u, v) ≥ 0; </li></ul><ul><li>Definite : d (u, v) = 0, iff u = v; </li></ul><ul><li>Symmetric : d (u, v) = d (v, u), and; </li></ul><ul><li>Triangular : d (u, v) + d (v, z) ≥ d (u, z); for all u, v, z  Z n . </li></ul><ul><li>Euclidean distance : E n (u, v) = (  i ( u i – v i ) 2 ) 1/2 </li></ul><ul><li>Also E n (x) = E n (u, v) for x = |u – v|. </li></ul>
  • 30. Digital Distances in 2-D
  • 31. Basic Digital Distances <ul><li>For u, v  Z 2 and x = |u – v|, </li></ul><ul><li>Cityblock: d 4 (u, v) = x 1 + x 2 , </li></ul><ul><li>Chessboard: d 8 (u, v) = max(x 1 , x 2 ), </li></ul><ul><li>Octagonal ( uses non-uniform alternating neighborhoods ): </li></ul><ul><li>d oct (u, v) = max(x 1 , x 2 ,  2(x 1 + x 2 )/3  ) . </li></ul><ul><li>  </li></ul>Rosenfeld and Pfaltz ’68
  • 32. Octagonal Distance Neighborhood sequence {1,2} d(a,b) = 10 a b 2 2 2 2 2 1 2 2 2 1 * 1 2 2 2 1 2 2 2 2 2
  • 33. Generalized Octagonal Distances <ul><li>For u, v  Z 2 , B being a neighborhood sequence. </li></ul><ul><li>d (u, v; B ) = d (x; B ) = max( d 1 (x; B ), d 2 (x; B )), where x = |u – v|, </li></ul><ul><ul><li>ƒ( i ) =  1≤j≤i b ( j ), </li></ul></ul><ul><ul><li>g ( i ) = ƒ( p ) - ƒ( i - 1) – 1, p = | B |, </li></ul></ul><ul><ul><li>d 1 (x; B ) = max(x 1 , x 2 ), and </li></ul></ul><ul><ul><li>d 2 (x; B ) =  1≤j≤ p  ((x 1 + x 2 ) + g ( j ))/ƒ( p )  . </li></ul></ul><ul><li>  </li></ul><ul><li>Example: Let B = {1, 2}, So, p = 2, ƒ(0) = 0, ƒ(1) = 1, ƒ(2) = 3 and g (1) = 2, g (2) = 1. </li></ul><ul><li>Thus </li></ul><ul><li>d 1 (x) = max(x 1 , x 2 ) and </li></ul><ul><li>d 2 (x) =  (x 1 + x 2 + 2)/3  +  (x 1 + x 2 + 1)/3  =  2(x 1 + x 2 )/3  . </li></ul><ul><li>  </li></ul>Das et al 1987; Das & Chatterji 1990
  • 34. Properties of Octagonal Distances <ul><li>  </li></ul><ul><li>Metric: </li></ul><ul><li>d ( B ) is a metric iff B is well-behaved, that is, </li></ul><ul><li>ƒ( i ) + ƒ( j ) ≤ ƒ( i + j ), if i + j ≤ p ; </li></ul><ul><li> ≤ ƒ( p ) + ƒ( i + j - p ), if i + j ≥ p . </li></ul><ul><li>Lattice: </li></ul><ul><li>Octagonal Distances form a Lattice where the Partial Order is defined between B 1 & B 2 iff d(x; B 1 ) ≤ d(x; B 2 ) for all x  Z 2 . </li></ul><ul><li>  </li></ul>Das 1990
  • 35. Simple Octagonal Distances <ul><li>An Octagonal Distance with a single Integer Function is Simple . </li></ul><ul><li>d (x; B ) is Simple iff </li></ul><ul><ul><li>b(j) =  j.f( p )/ p  -  (j-1).f( p )/ p  , gcd(p, f(p)) = 1. </li></ul></ul><ul><li>d (x; B ) = max(| x 1 |, | x 2 |,  (|x 1 |+ |x 2 |)/m  ), </li></ul><ul><ul><li>where m = f(p)/p, called the effective neighborhood. </li></ul></ul><ul><li>  </li></ul><ul><li>Example: Let p = 5, f(p) = 7. Then B = {1, 1, 2, 1, 2}, </li></ul><ul><li>d (x; B ) = max(| x 1 |, | x 2 |,  5(|x 1 |+ |x 2 |)/7  ) </li></ul><ul><li>Simple Octagonal Distances are always Metric.   </li></ul>Das 1992
  • 36. Simple Octagonal Distances
  • 37. Properties of Simple Octagonal Distances <ul><li>  </li></ul><ul><li>Chamfer Computable </li></ul><ul><ul><li>repeated iterations of forward and reverse scans </li></ul></ul><ul><li>Simple in Functional Form </li></ul><ul><li>Simple in Neighborhood </li></ul><ul><li>Unity in Cost </li></ul><ul><li>Has good Circularity for Disks </li></ul><ul><li>Works as Good Approximations for Euclidean Distance </li></ul><ul><li>  </li></ul>
  • 38. Best Simple Octagonal Distances <ul><li>A Simple Octagonal Distance that “Best” approximates the Euclidean Distance. </li></ul><ul><li>Best Distance is: B = {1, 1, 2, 1, 2} </li></ul><ul><li>d (x; B ) = max(| x 1 |, | x 2 |,  5(|x 1 |+ |x 2 |)/7  ) </li></ul><ul><li>Other Good Candidates: </li></ul><ul><ul><li>B = {1, 1, 2} </li></ul></ul><ul><ul><li>B = {1, 2} </li></ul></ul><ul><ul><li>B = {2} </li></ul></ul>Das 1992
  • 39. Fun Distances 
  • 40. Knight’s Distance in 2D <ul><li>  </li></ul><ul><li>d knight (x) = max(  x 1 /2  ,  (x1 + x2)/3  ) + </li></ul><ul><li>((x 1 + x 2 ) – </li></ul><ul><li>max(  x 1 /2  ,  (x 1 + x 2 )/3  )) mod 2, </li></ul><ul><li> if x  (1, 0), (2, 2), </li></ul><ul><li> = 3, if x = (1, 0), </li></ul><ul><li> = 4, if x = (2, 2). </li></ul><ul><li>  </li></ul>Das & Chatterji 1988
  • 41. Knight’s Circle & Disk
  • 42. Properties of Knight’s Distance <ul><li>A Metric </li></ul><ul><li>A non-Proximal Distance </li></ul><ul><li>Results in Porous Disks </li></ul><ul><li>Generalizations – Super-Knight's Distances may not be well-defined </li></ul>
  • 43. Digital Distances in 3-D
  • 44. Digital Distances: <ul><li>For u, v  Z 3 and x = |u – v|, </li></ul><ul><li>d 6 (u, v) = x 1 + x 2 + x 3 , (grid distance) </li></ul><ul><li>d 18 (u, v) = max(x 1 , x 2, x 3 ,  (x 1 + x 2 + x 3 )/2  ) </li></ul><ul><li>d 26 (u, v) = max(x 1 , x 2, x 3 ) (lattice distance). </li></ul>Yamashita and Ibaraki ‘86
  • 45. Digital Distances in n-D
  • 46. n-D Extension of Neighborhood <ul><li>m-neighborhood distances </li></ul><ul><li>Type – m neighbor: </li></ul>Das, Chakrabarti and Chatterji ‘87
  • 47. m-Neighbor Distance <ul><li>All are metric </li></ul><ul><li>n metrics in n-D </li></ul><ul><li>Generalizes simple distances for 2-D & 3-D </li></ul>
  • 48. t-cost Distance <ul><li>Cost between neighbors: </li></ul><ul><ul><li> (u – v) = min i (t,  |u(i) – v(i)|) </li></ul></ul><ul><li>Cost bound: t, 1 ≤ t ≤ n </li></ul><ul><li>i th maximum function </li></ul>Das, Mukherjee and Chatterji ‘92
  • 49. 2D 3D t-cost neighbors 1 1 1 1 o 1 1 1 1 2 1 2 1 o 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2 2 2 3 2 3 2 2 2 3 2 3 1 1 1 1 o 1 1 1 1 2 1 2 1 o 1 2 1 2 2 1 2 1 o 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2 2 2 3 2 3 2 1 2 3 2 3
  • 50. Path by t-Cost Distance
  • 51. Properties of t-Cost Distance <ul><li>t-Cost Distances are Metrics </li></ul><ul><li>D n 1 = d n n </li></ul><ul><li>D n n = d n 1 </li></ul><ul><li>New Distance in 3-D: </li></ul><ul><ul><ul><li>D 3 2 = max(x 1 , x 2 , x 3 )+ </li></ul></ul></ul><ul><ul><ul><li>max(min(x 1 , x 2 ), min(x 1 , x 3 ), min(x 2 , x 3 )) </li></ul></ul></ul>
  • 52. Generalized Octagonal Distance in n-D <ul><li>Cyclic neighborhood sequence: </li></ul><ul><li>All sorted (in non decreasing order) sequences yield to metric. </li></ul><ul><li>There exists a necessary as well as sufficient condition on B for metricity. </li></ul><ul><li>Relative proportions of different neighborhood types in the sequence influence the shape of the discs. </li></ul>Das and Chatterji ‘ 90
  • 53. Generalized Octagonal Distance in n-D: Special Cases <ul><li>d({m}) = d m n , |B| = 1. </li></ul><ul><li>d(u, v; {m, m + 1}), |B| = 2 </li></ul><ul><ul><ul><li>= d(x) </li></ul></ul></ul><ul><ul><ul><li>= max(max i x i ,  2  i x i /(2m + 1)  ) </li></ul></ul></ul><ul><ul><ul><ul><ul><li>(recollect d oct where n = 2 and m = 1). </li></ul></ul></ul></ul></ul>Das and Chatterji ‘ 90
  • 54. Generalized Octagonal Distances in 3-D
  • 55. Digital Distance Computation & Approximation
  • 56. Chamfering for computing Distance Transform o a b a Forward Scanning From Left to Right and Top to Bottom Backward Scanning From Right to Left and Bottom to Top b Distance at o = min (Distance value at Neighboring pixel + local distance between them) 1. Initialize all distance values to a Maximum Value. 2. At every point o compute the distance value from its visited neighbors as follows: Extend this concept with larger neighborhood and dimension. o a b a b
  • 57. Chamfer/Weighted distances Borgefors and her colleagues 1984-2004
  • 58. 2D 3D Weighted Distance Templates b a b a o a b a b c b c b a b c b c b a b a o a b a b c b c b a b c b c
  • 59. Benchmarking with Euclidean Metric: Analytical Approach <ul><li>Maximum Absolute Error(MAE) </li></ul><ul><li>Mean Square Error(MSE) </li></ul>Borgefors ’84-04, Das and Chatterji ’92
  • 60. Benchmarking Euclidean Metric: Geometric Approach <ul><li>Comparing Geometric Properties of hyper-spheres. </li></ul><ul><li>Perimeter, area, shape-descriptors in 2-D </li></ul><ul><li>Surface Area, Volume, Shape-descriptors in 3-D. </li></ul>Danielsson’93, Kumar et al’95, Butt and Maragos’98, Mukherjee et al’2000
  • 61. Optimal m-neighbor Distance in Bounded Images <ul><li>Solution of the following equation: </li></ul><ul><li>m : neighborhood type </li></ul><ul><li>M : maximum size along a dimension </li></ul><ul><li>n : dimension </li></ul>
  • 62. Relative Error: t-cost distance … 3 2 … 2 2 1 1 t opt … 56 55 … 4 3 2 1 n
  • 63. Finding Best Octagonal Distance <ul><li>Compare the area, perimeter, volume, surface etc. with the Euclidian Discs. </li></ul><ul><li>Best in 2D: {1,1,2} </li></ul><ul><li>Best in 3D: {1,1,3} </li></ul><ul><li>They also minimize the MAE & MSE in a finite space </li></ul>Mukherjee et al’2000
  • 64. Benchmarking 2D Distances 0.030 0.073 <8,11> 0.035 0.083 <5,7> 0.025 0.081 <3,4> 0.068 0.134 <2,3> 0.043 0.118 {1,2} 0.056 0.187 {1,1,1,2} 0.026 0.087 {1,1,2} MSE MAE Distance Function
  • 65. Benchmarking 3D Distances 0.034 0.107 <13,17,23> 0.043 0.107 <13,17,22> 0.043 0.107 <8,11,13> 0.043 0.118 <3,4,5> 0.070 0.269 {1,2} 0.034 0.146 {1,1,1,2,3} 0.027 0.105 {1,1,3} MSE MAE Distance Function
  • 66. Good Weighted distances 0.184 0.167 0.167 0.155 0.143 - - - - - <3,4,5,6> <6,8,10,11> <6,9,10,11> <7,10,12,13> <8,11,13,15> 4 0.118 0.107 0.107 0.107 0.103 0.103 0.043 0.043 0.043 0.034 0.036 0.043 <3,4,5> <8,11,13> <13,17,22> <13,17,23> <16,21,27> <16,21,28> 3 0.134 0.081 0.083 0.073 0.068 0.025 0.035 0.03 <2,3> <3,4> <5,7> <8,11> 2 MAE MSE Weights Dimension
  • 67. Neighborhood :: Digital Distance Position Independent Isotropic Unity Undefined Non-Proximal Others (Knight’s / Super-Knight’s) Position Dependent Isotropic Unity Graded Proximal Sequence-based (Octagonal) Position Independent Isotropic Unity Graded Proximal m-Neighbor Position Independent Position Independent Uniformity Isotropic Isotropic Isotropy & Symmetry F n of Separating Dimension Non-Unity Separating Cost Maximal Maximal Separating Dimension Proximal Proximal Proximity Chamfer / Weighted t-Cost
  • 68. Glimpses of Applications
  • 69. Distance Transform Minimum distance of a feature point from the back ground.
  • 70. Medial Axis Transform A set of maximal blocks contained in the pattern.
  • 71. Computation of Medial Axis Transform <ul><li>Compute the distance transform. </li></ul><ul><li>Compute local maxima in the distance transformed image. </li></ul>
  • 72. Computation of minimal set of maximal disks 1. Compute Local Maximum Blocks from the distance transformed image. Form a relational table expressing the relationships between boundary pixels and individual disks. The problem is mapped to the covering of the list of boundary pixels with the optimal set of maximal blocks. 2 . 3. Nilson-Danielsson’96
  • 73. Digital discs 2D 3D d 6 d 26 d 18 d 4 d 8 R R
  • 74. Vertices of octagonal discs:
  • 75. Application of MAT in Image Analysis <ul><li>Geometric Transformation ( Kumar et al ’96 ) </li></ul><ul><li>Computation of Normals ( Mukherjee et al ’02 ) </li></ul><ul><li>Thinning of binary pattern ( Costa ’00, Pudney ’98 ) </li></ul><ul><li>Computation of cross-sections of 3D objects ( Mukherjee et al ’00 ) </li></ul><ul><li>Visualization of 3D objects ( Mukherjee et al ’99, Prevost and Lucas ‘00 ) </li></ul><ul><li>Image compression ( Kumar et al ’95 ). </li></ul><ul><li>Shape Description ( Baja and Svensson ’02 ) </li></ul>
  • 76. Thinning from Distance Transform Compute the set of Maximal Blocks. Use them as anchor points while iteratively deleting boundary points preserving the topology. Vincent ’91, Ragnemalm ’93, Svensson-Borgefors-Nystrom ’99
  • 77. Normal Computation <ul><li>Normal at a point p computed by computing the resultant vector from that point to the neighboring medial circles. </li></ul>
  • 78. Normal Computations: Examples
  • 79. Discrete Shading <ul><li>Render individual medial sphere independently using Z-buffering. </li></ul>
  • 80. Discrete Shading: Examples
  • 81. Discrete Shading: Examples
  • 82. Discrete Shading: Examples
  • 83. Decomposition of 3D Objects Identification of seed of a component from inner layers of Distance Transformed Image. Seed-fusion by expansion and shrinking Region growing by reversed DT. Surface smoothing and merging. Svensson-Saniti di Baja’02
  • 84. Cross-sectioning
  • 85. Cross-sectioning with different distance functions.
  • 86. A set of objects for experimentation
  • 87. Cross-sectioning: Voxel data, MAT & Sphere Approx. Voxel Data MAT Euclidean Sphere Approximation
  • 88. Acknowledgement <ul><li>Azriel Rosenfeld </li></ul><ul><li>B N Chatterji </li></ul><ul><li>Gunilla Borgefors </li></ul><ul><li>Hanan Samet </li></ul><ul><li>Jayanta Mukhopadhayay </li></ul><ul><li>P P Chakrabarti </li></ul><ul><li>R A Melter </li></ul><ul><li>Y V Venkatesh </li></ul><ul><li>Innumerable others whose results have been used in this presentation and all other works in the area. </li></ul><ul><li>And the ICVGIP 2004 Committee for the opportunity to present. </li></ul>
  • 89. Thank you !
  • 90. O(1) neighbors O(2) neighbors O(3) neighbors O(1) neighbors O(2) neighbors 2D 3D M-neighbors o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o
  • 91. Digital Distances: <ul><li>u, v Z 2 and x = |u – v|, </li></ul><ul><li>d 4 (u, v) = x 1 + x 2 , </li></ul><ul><li>d 8 (u, v) = max(x 1 , x 2 ) ( see figure 4 for an example of d 8 ), </li></ul><ul><li>d oct (u, v) = max(x 1 , x 2 ,  2(x 1 + x 2 )/3  . </li></ul><ul><li>  </li></ul><ul><li>************ </li></ul><ul><li>  </li></ul><ul><li>For u, v Z 2 , d (u, v; B ) = d (x; B ) = max( d 1 (x; B ), d 2 (x; B )), where x = |u – v|, </li></ul><ul><li>ƒ( i ) = 1≤j≤i b ( j ), g ( i ) = ƒ( p ) - ƒ( i - 1) – 1, p = | B |, </li></ul><ul><li>d 1 (x; B ) = max(x 1 , x 2 ), and </li></ul><ul><li>d 2 (x; B ) = 1≤j≤ p  ((x 1 + x 2 ) + g ( j ))/ƒ( p )  . </li></ul><ul><li>  </li></ul><ul><li>Example 3: Let B = {1, 2}, So, p = 2, ƒ(0) = 0, ƒ(1) = 1, ƒ(2) = 3 and g (1) = 2, g (2) = 1. </li></ul><ul><li>Thus d 1 (x) = max(x 1 , x 2 ) and d 2 (x) =  (x 1 + x 2 + 2)/3  +  (x 1 + x 2 + 1)/3  =  2(x 1 + x 2 )/3  . </li></ul><ul><li>  </li></ul><ul><li>********* </li></ul><ul><li>  </li></ul><ul><li>d ( B ) is a metric iff B is well-behaved, that is, </li></ul><ul><li>ƒ( i ) + ƒ( j ) ≤ ƒ( i + j ), if i + j ≤ p ; </li></ul><ul><li>≤ ƒ( p ) + ƒ( i + j - p ), if i + j ≥ p . </li></ul><ul><li>  </li></ul><ul><li>**************** </li></ul><ul><li>  </li></ul><ul><li>d knight (x) = max(  |x 1 /2  ,  (x1 + x2)/3  ) + ((x 1 + x 2 ) – max(  x 1 /2  ,  (x 1 + x 2 )/3  )) mod 2, </li></ul><ul><li>if x (1, 0), (2, 2), </li></ul><ul><li>= 3, if x = (1, 0), </li></ul><ul><li>= 4, if x = (2, 2). </li></ul><ul><li>  </li></ul><ul><li>************* </li></ul><ul><li>  </li></ul><ul><li>u, v Z 3 and x = |u – v|, </li></ul><ul><li>d 6 (u, v) = x 1 + x 2 + x 3 , (grid distance) </li></ul><ul><li>d 18 (u, v) = max(x 1 , x 2, x 3 ,  (x 1 + x 2 + x 3 )/2  ) </li></ul><ul><li>d 26 (u, v) = max(x 1 , x 2, x 3 ) (lattice distance). </li></ul><ul><li>  </li></ul><ul><li>*********** </li></ul>Rosenfeld and Pfaltz ’68, Yamashita and Ibaraki ‘86
  • 92. Digital Distances: <ul><li>2D: </li></ul><ul><li>3D: </li></ul>Rosenfeld and Pfaltz ’68, Yamashita and Ibaraki ‘86
  • 93. Closed form expression:
  • 94. Cross-sectioning using MAT
  • 95. Cross-sectioning using Euclidean Sphere Approximation

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