Points, Distances, and Cellular Automata: Geometric and Spatial - PowerPoint PPT Presentation

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution Corrected Distance Field Evolution Corrected Distance Field  0 if x ∈ P t +1 else:   D [ P ] t +1 ( x ) = 0 . 5 if x ∈ P t else:  min { 1 + D [ P ] t ( y ) | y ∈ N ( x ) } .  .5 1 2 3 3 2 1 0 1 2 3 2 1 0 1 5 4 3 2 0 0 12 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution Corrected Distance Field Evolution Corrected Distance Field  0 if x ∈ P t +1 else:   D [ P ] t +1 ( x ) = 0 . 5 if x ∈ P t else:  min { 1 + D [ P ] t ( y ) | y ∈ N ( x ) } .  5 4 3 1 0 1 1 .5 3 3 0 3 0 12 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution Corrected Distance Field Evolution Corrected Distance Field  0 if x ∈ P t +1 else:   D [ P ] t +1 ( x ) = 0 . 5 if x ∈ P t else:  min { 1 + D [ P ] t ( y ) | y ∈ N ( x ) } .  5 4 2 1 0 1 2 2 .5 1 0 1 1 0 1 12 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution Corrected Distance Field Evolution Corrected Distance Field  0 if x ∈ P t +1 else:   D [ P ] t +1 ( x ) = 0 . 5 if x ∈ P t else:  min { 1 + D [ P ] t ( y ) | y ∈ N ( x ) } .  5 3 2 1 0 1 2 3 3 .5 2 1 0 1 2 2 1 0 1 12 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution Corrected Distance Field Evolution Corrected Distance Field  0 if x ∈ P t +1 else:   D [ P ] t +1 ( x ) = 0 . 5 if x ∈ P t else:  min { 1 + D [ P ] t ( y ) | y ∈ N ( x ) } .  4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 2 1 0 1 12 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field Checkpoint We have: distances locally, globally, and dynamically We don’t have: finite number of states 13 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field Checkpoint We have: distances locally, globally, and dynamically We don’t have: finite number of states Bounded information No bound on distances Bounded gradient (differences between neighboring sites) What about modulo ? 13 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field (Cont.) Modulo in action Particles maximal speed determines maximal gradient 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 14 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field (Cont.) Modulo in action Particles maximal speed determines maximal gradient .5 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 .5 0 2 2 2 2 2 2 2 2 2 2 0 0 14 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field (Cont.) Modulo in action Particles maximal speed determines maximal gradient .5 2 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 0 .5 .5 3 3 3 3 3 14 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field (Cont.) Modulo in action Particles maximal speed determines maximal gradient � 1 � 3 4 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 ☎ ☎ ☎ � ☎ � ☎ � 14 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field (Cont.) Modulo in action Particles maximal speed determines maximal gradient .5 0 1 .5 2 3 3 .5 4 5 5 5 5 5 5 5 5 4 3 .5 2 1 .5 1 0 .5 0 2 .5 3 5 5 5 5 14 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field (Cont.) Modulo in action Particles maximal speed determines maximal gradient ✁ ✂ 3 4 4 ✂ 5 6 6 6 6 6 6 5 4 .5 3 2 .5 2 1 .5 1 0 1 3 .5 4 6 6 4 2 ✁ ✂ ✁ ✂ ✁ 14 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field (Cont.) Modulo in action Particles maximal speed determines maximal gradient .5 2 .5 4 5 5 .5 6 7 7 7 7 6 .5 3 2 .5 2 .5 1 2 .5 5 7 3 2 .5 0 3 .5 3 14 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field (Cont.) Modulo in action Particles maximal speed determines maximal gradient In this case: 2 consecutive moves ⇒ gradient bound of 3 .5 3 .5 1 .5 5 6 6 .5 7 8 8 7 6 .5 5 4 .5 4 3 .5 3 1 0 .5 0 1 .5 2 3 5 .5 6 3 .5 14 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field From Infinite To Finite Field (Cont.) Modulo in action Particles maximal speed determines maximal gradient In this case: gradient bound of 3 ⇒ modulo 7 .5 3 .5 1 .5 5 6 .5 0 .5 3 .5 2 3 5 .5 6 3 .5 0 .5 .5 .5 5 .5 3 .5 14 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field Building on top of distances Distance fields as building blocks Moving according to the distance field Detecting patterns of distances and particles Case Study Density Uniformisation (unidimensional) Convex Hull (multidimensional) Gabriel Graph (multidimensional) 15 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Density Uniformisation 16 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Problem Statement Problem Statement Problem Definition Move the particles to a uniform distribution Input: Output: 17 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Problem Analysis Problem Analysis Intuition Each particle needs to occupy its space Boundary between individual spaces ⇔ middles Occupy its space ⇔ be at the middles 18 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Problem Analysis Application: 1D Uniformisation Solution 19 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting system Initial system state � p 0 ( x ) = x ∈ P w 0 ( x ) = x �∈ P System fields composition  dp = D [p]   dw = D [w]  p = M [p 0 , B [dp] ∧ Dir[dw , ≤ ]]   w = M [w 0 , B [dw] ∧ Dir[dp , ≤ ]]  20 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description The resulting evolution Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system 21 / 45