functors, comonads, and digital image processing
TRANSCRIPT
Functors, Comonads, and Digital Image ProcessingJUSTIN LE, CHAPMAN UNIVERSITY SCHMID COLLEGE OF SCIENCE AND TECHNOLOGY
Categories
Objects Morphisms
Composition
ℤ ℝ
ℙ(people)
(integers)
(real numbers)
𝑟𝑜𝑢𝑛𝑑 :ℝ→ℤ
𝑎𝑔𝑒 :ℙ→ℝ
𝑟𝑜𝑢𝑛𝑑∘𝑎𝑔𝑒 :ℙ→ℤ(𝑟𝑜𝑢𝑛𝑑∘𝑎𝑔𝑒) (𝑝 )=𝑟𝑜𝑢𝑛𝑑 (𝑎𝑔𝑒 (𝑝 ))
𝒮Set
Ex: Infinite List Functor
𝐿 ( 𝑋 )=𝑋ℕ
X to infinite lists of things in X
92 4, 9, 8, 75, -3, ...∈ℤ ∈𝐿(ℤ)
𝑓 : 𝑋→𝑌𝐿 ( 𝑓 ) :𝐿(𝑋 )→𝐿(𝑌 )
Comonads1. Extract 2. Duplicate 3. Laws , etc.
𝜖 ([ 4 ,9 ,8 ,75 ,−3. .])=4 𝛿 ([4 ,9 ,8 ,75 ,−3. .])=?? ?
Infinite List Functor
Functor 1: “Image with Focus”
𝐼 ( 𝑋 )=ℤ2× 𝑋ℤ2
5 ((5,2 ) ,[⋱ ⋮ ⋮ ⋮ ⋰⋯ 7 3 19 ⋯⋯ 22 4 120 ⋯⋯ 8 79 1 ⋯⋰ ⋮ ⋮ ⋮ ⋱
])∈ℕ ∈ 𝐼 (ℕ)
Local/Relative Transformations
[ 0 −1 0−1 5 −10 −1 0 ]
Kernel/Convolution Matrix
[⋱ ⋮ ⋮ ⋮ ⋰⋯ 7 3 19 ⋯⋯ 2 5 6 ⋯⋯ 8 1 9 ⋯⋰ ⋮ ⋮ ⋮ ⋱
] 13ℤ𝐺 (ℤ)
Extensions of I are Decoded Filters
𝑓 : 𝐼 ( 𝑋 )→𝑌 𝑓 ∗ : 𝐼 (𝑋 )→ 𝐼 (𝑌 )
𝑓 ∗ : 𝑋ℕ2
→𝑌ℕ 2
Classical image filter
focus stays same
Commutation Abounds
𝑓 ,𝑔
𝑓 ∗ ,𝑔∗
𝑓 ∘𝑔
𝑓 ∗∘𝑔∗= ( 𝑓 ∘𝑔)∗
Compose Cokleisli
Compose Normally
Extend Extend
Decoded Neighborhoods
𝑓 :𝐺 ( 𝑋 )→𝑌
Γ ( 𝑓 ) : 𝐼 ( 𝑋 )→𝑌
“Globalization”,
Γ ( 𝑓 )∗ : 𝑋ℕ2
→𝑌ℕ2
Classical image filter
𝑓 ,𝑔
Γ ( 𝑓 )∗∘ Γ (𝑔 )∗
Γ ( 𝑓 ∘𝑔 )∗
(Γ ( 𝑓 )∘Γ (g ) )∗Globalize, extend, compose
Cokleisli compose, globalize, extend
Globalize, cokleisli compose, extend
Extension, Globalization are Cheap•Written once: less bugs
•Optimize once, unlimited return on performance
•Trivially parallelizable
•Globalization can handle boundary conditions, low-level
Generalization
𝐼𝑛 (𝑋 )=ℤ𝑛×𝑋ℤ𝑛
𝐺𝑛 ( 𝑋 )=𝑋ℤ𝑛
n Application1 Audio, Time signals
2 Images
3 Video
1000+ Difference Equations