Honeycomb Geometry Rigid Motions on the Hexagonal Grid by Kacper Pluta, Pascal Romon, Yukiko Kenmochi and Nicolas Passat The figure comes from "Insects The Yearbook of Agriculture 1952" United States Dept. of Agriculture." Published by the US Government Printing Office. Deemed to be in the Public Domain under US Law.
Motivations We came to agree with Nouvel & Rémila that digitized rigid motions de fi ned on the square grid are burdened with a fundamental incompatibility between rotations and the geometry of the grid.
Agenda Introduction to the Bees' Point of View Quick Introduction to Rigid Motions Neighborhood Motion Maps Contributions Conclusions & Perspectives The beehive figure's source and author unknown (if you recognize it, please let me know). The image of the bee comes from http://karenswhimsy.com/public-domain-images (public domain)
Introduction to the Bees' Point of View Or why bees are right The fi gure comes from http://thegraphicsfairy.com/vintage-clip-art-bees-with-honeycomb
Pros and Cons Square grid Hexagonal grid + Memory addressing + Uniform connectivity + Sampling is easy to de fi ne + Equidistant neighbors - Sampling is not optimal (ask bees) + Sampling is optimal - Neighbors are not equidistant - Memory addressing is not trivial - Connectivity paradox - Sampling is di ffi cult to de fi ne
Pros and Cons Square grid Hexagonal grid + Memory addressing + Uniform connectivity + Sampling is easy to de fi ne + Equidistant neighbors - Sampling is not optimal (ask bees) + Sampling is optimal - Neighbors are not equidistant ~ Memory addressing is not trivial ~ Connectivity paradox ~ Sampling is di ffi cult to de fi ne The fi gure by Pearson Scott Foresman, Wikimedia.
Hexagonal Grid Λ = Z ϵ ⊕ Z ϵ The hexagonal lattice: and the hexagonal grid H 1 2
Digitization Model The digitization operator is de fi ned as a function D : R → Λ such that ∀ x ∈ R , ∃ ! D ( x ) ∈ Λ 2 2 and x ∈ C ( D ( x )).
Digitization Model The digitization operator is de fi ned as a function D : R → Λ such that ∀ x ∈ R , ∃ ! D ( x ) ∈ Λ 2 2 and x ∈ C ( D ( x )). This is a de fi nition for digital geometers not for computer vision guys...
How many digital balls do you see? The fi gure of bumble bee comes from http://www.ase.org.uk (public domain)
Quick Lesson on Rigid Motions Or how to become a beekeeper. Part I - Equipment The fi gure comes from Wikimedia. Original source The New Student's Reference Work (public domain)
Rigid Motions on R 2 Properties ∣ U R 2 R 2 : → Isometry map - distance ∣ ∣ preserving map x Rx + t ↦ ∣ Bijective
Rigid Motions on Λ Properties U = D ∘ U ∣Λ - Do not preserve distances - Non-injective - Non-surjective U ( Λ )
Related Studies Nouvel, B., Rémila, E.: On colorations induced by discrete rotations. In: DGCI, Proceedings. Volume 2886 of Lecture Notes in Computer Science ., Springer (2003) 174–183 Pluta, K., Romon, P., Kenmochi, Y., Passat, N.: Bijective digitized rigid motions on subsets of the plane. Journal of Mathematical Imaging and Vision (2017)
Contributions in Short Pure extracted honey Extension of the former framework to the hexagonal grid Comparison of the loss of information between the hexagonal and square grids Complete set of neighborhood motion maps Source code of a tool to study digitized rigid motions on the hexagonal grid
Neighborhood Motion Maps Or a manual of instructions in apiculture The fi gure comes from Wikimedia. The original source The honey bee: a manual of instruction in apiculture (public domain)
Neiborhood The neighbourhood of κ ∈ Λ (of squared radius r ∈ R + ) 2 N ( κ ) = κ + δ ∈ Λ ∣ ∥ δ ∥ ≤ r { } r
Neiborhood Motion Maps The neighbourhood motion map of κ ∈ Λ with respect to U r ∈ R + and is the function ∣ G r : N (0) → N (0) U ∣ r ′ r ∣ ↦ U ( κ + δ ) − U ( κ ). δ ∣
Remainder Map step-by-step
Remainder Map step-by-step U ( κ + δ ) = R δ + U ( κ )
Remainder Map step-by-step Without loss of generality, U ( κ ) is an origin, then U ( δ ) = R δ + U ( κ ) − U ( κ )
Remainder Map step-by-step Remainder map de fi ned as F ( κ ) = U ( κ ) − U ( κ ) ∈ C ( 0 ) where the range C ( 0 ) is called the remainder range.
Remainder Map step-by-step
Remainder Map and Critical Rigid Motions Critical cases can be observed via the relative positions of F ( κ ) which are formulated by the translation H − R δ that is to say C ( 0 ) ∩ ( H − R δ ).
Remainder Map and Critical Rigid Motions ( H − R δ ) ∩ C ( 0 ) H = ⋃ δ ∈ N ( 0 ) r
Frames Each region bounded by critical lines is called a frame.
Frames Each region bounded by critical lines is called a frame. Proposition if and only if F ( κ ) and F ( λ ) For any κ , λ ∈ Λ , G ( κ ) = G ( λ ) U U r r are in the same frame.
Remainder Range Partitioning
Contributions Or extracting the pure, organic honey The fi gure comes from Wikimedia. The original comes from A practical treatise on the hive and honey-bee (public domain)
Rational Rotations F ( Λ ) For what kind of parameters has the mapping a fi nite number of images?
Rational Rotations F ( Λ ) Corollary 2 a − b and sin θ = √ 3 where ( a , b , c ) ∈ Z , gcd( a , b , c ) = 1 and If cos θ = b 3 2 c 2 c then the mapping has a fi nite number of images. 0 < a < c < b ,
Non-injective Digitized Rigid Motions
Loss of Information
Conclusions & Perspectives An extension of a framework to study digitized rigid motions Characterization of rational rotations We have showed that the loss of information is relativly lower for digitized rigd motions defained on the hexagonal grid Our tools on BSB-3 license: https://github.com/copyme/NeighborhoodMotionMapsTools
hal.archives-ouvertes.fr/hal-01540772
Homework If you want to get into the honey business, then this book is an obligatory lecture: Middleton, Lee, and Jayanthi Sivaswamy. Hexagonal image processing: A practical approach . Springer Science & Business Media, 2006.
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