Geometric Problems with Imprecise Input Points Maarten L offler - PowerPoint PPT Presentation

Geometric Problems with Imprecise Input Points Maarten L¨ offler Marc van Kreveld Center for Geometry, Imaging and Virtual Environments Utrecht University 1-1

Overview • Introduction • Geometric problems • Imprecise input points • Overview of problems and results • Algorithms • Largest diameter of squares • Smallest diameter of squares • Concluding remarks 2-1 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets • Given a set P of n points in the plane P • Geometric structures: 3-1 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets • Given a set P of n points in the plane P • Geometric structures: • Bounding box 3-2 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets • Given a set P of n points in the plane P • Geometric structures: • Bounding box • Diameter 3-3 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets • Given a set P of n points in the plane P • Geometric structures: • Bounding box • Diameter • Convex hull 3-4 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets • Given a set P of n points in the plane P • Geometric structures: • Bounding box • Diameter • Convex hull • Minimum spanning tree 3-5 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets • Given a set P of n points in the plane P • Geometric structures: • Bounding box • Diameter • Convex hull • Minimum spanning tree • Many others • Optimal algorithms are known 3-6 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Imprecision • Traditional algorithms assume exact input • In practice, input data is often not exact • Measured from the real world • Stored with limited precision • Computed by inexact algorithms • Output of traditional algorithms is unreliable • Given exact description of input imprecision, we can exactly predict output imprecision 4-1 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Imprecise Points • Unknown location • Known region of possible locations • Regions are simple geometric objects • Disc • Square • Rectangle • Convex polygon 5-1 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Imprecise Points • Unknown location • Known region of possible locations • Regions are simple geometric objects • Disc • Square • Rectangle • Convex polygon 5-2 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Imprecise Points • Unknown location • Known region of possible locations • Regions are simple geometric objects • Disc • Square • Rectangle • Convex polygon 5-3 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Imprecise Points • Unknown location • Known region of possible locations • Regions are simple geometric objects • Disc • Square • Rectangle • Convex polygon 5-4 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Imprecise Points • Unknown location • Known region of possible locations • Regions are simple geometric objects • Disc • Square • Rectangle • Convex polygon 5-5 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Imprecision in Geometric Structures • Given a set L of n imprecise points • Consider the same geometric structures • Multiple possibilities • True structure is unknown • We want to capture the imprecision in the output 6-1 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Imprecision in Geometric Structures • Given a set L of n imprecise points • Consider the same geometric structures • Multiple possibilities • True structure is unknown • We want to capture the imprecision in the output 6-2 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Imprecision in Geometric Structures • Given a set L of n imprecise points • Consider the same geometric structures • Multiple possibilities • True structure is unknown • We want to capture the imprecision in the output 6-3 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Imprecision in Geometric Structures • Given a set L of n imprecise points • Consider the same geometric structures • Multiple possibilities • True structure is unknown • We want to capture the imprecision in the output 6-4 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision 7-1 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area 7-2 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area 7-3 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area 7-4 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area • Diameter 7-5 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area • Diameter 7-6 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area • Diameter 7-7 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area • Diameter • Convex hull perimeter 7-8 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area • Diameter • Convex hull perimeter 7-9 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area • Diameter • Convex hull perimeter 7-10 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area • Diameter • Convex hull perimeter • MST weight 7-11 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area • Diameter • Convex hull perimeter • MST weight 7-12 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007

Bounds on Measures • Measure function µ : F ( R 2 ) → R • Largest and smallest possible values of µ • Output imprecision • For example • Bounding box area • Diameter • Convex hull perimeter • MST weight 7-13 Geometric Problems with Imprecise Input Points Maarten L¨ offler and Marc van Kreveld, July 4, 2007