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Functors, Comonads,
and Digital Image
Processing
JUSTIN LE, CHAPMAN UNIVERSITY SCHMID COLLEGE OF
SCIENCE AND TECHNOLOGY
Let’s talk about Filters
getglasses.com.au
The problem with filters
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
Functors
•Objects to Objects
•Morphisms to Morphisms
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 → 𝑌
𝐿 𝑓 ∶ 𝐿(𝑋) → 𝐿(𝑌)
Comonads
1. Extract 𝜖 ∶ 𝑊 𝑋 → 𝑋
2. Duplicate 𝛿 ∶ 𝑊 𝑋 → 𝑊(𝑊 𝑋 )
3. Laws 𝜖 ∘ 𝛿 = 𝑖𝑑, etc.
𝜖 [4, 9, 8, 75, −3. . ] = 4 𝛿 [4, 9, 8, 75, −3. . ] = ? ? ?
Infinite List Functor
Cokleisli Arrows
𝑓 ∶ 𝑊 𝑋 → 𝑌 𝑔 ∶ 𝑊 𝑌 → 𝑍
𝑔 ∘ 𝑓 ∶ 𝑊 𝑋 → 𝑍
Comonads only!
Extension
𝑓 ∶ 𝑊 𝑋 → 𝑌
𝑓∗
: 𝑊 𝑋 → 𝑊(𝑌)
Functor 1: “Image with Focus”
𝐼 𝑋 = ℤ2 × 𝑋ℤ2
5 ( 5,2 ,
⋱ ⋮ ⋮ ⋮ ⋰
⋯ 7 3 19 ⋯
⋯ 22 4 120 ⋯
⋯ 8 79 1 ⋯
⋰ ⋮ ⋮ ⋮ ⋱
)
∈ ℕ ∈ 𝐼(ℕ)
∈ 𝐼( 0, 𝑁 ⊂ ℕ)
Position-Aware Transformation
0 −1 𝑤
1 0 0
0 0 1
Affine Transformation Matrix
𝐼 ℕ → ℕ
𝑓, 𝑔
𝛼 𝑓, 𝛼 𝑔
𝑓 ∘ 𝑔
𝛼 𝑓 ∘ 𝛼 𝑔 = 𝛼 𝑓∘𝑔
Compose Affine
Compose Cokleisli
Encode
Encode
Functor 2: “Local Neighborhood”
G 𝑋 = 𝑋ℤ2
5
⋱ ⋮ ⋮ ⋮ ⋰
⋯ 7 3 19 ⋯
⋯ 22 4 120 ⋯
⋯ 8 79 1 ⋯
⋰ ⋮ ⋮ ⋮ ⋱
∈ ℕ ∈ 𝐺(ℕ)
Local/Relative Transformations
0 −1 0
−1 5 −1
0 −1 0
Kernel/Convolution Matrix
⋱ ⋮ ⋮ ⋮ ⋰
⋯ 7 3 19 ⋯
⋯ 2 5 6 ⋯
⋯ 8 1 9 ⋯
⋰ ⋮ ⋮ ⋮ ⋱
13
ℤ𝐺(ℤ)
𝑓, 𝑔
𝜅 𝑓, 𝜅 𝑔
𝑓 ∗ 𝑔
𝜅 𝑓 ∘ 𝜅 𝑔 = 𝜅 𝑓∗𝑔
Convolve Kernels
Compose Cokleisli
Encode
Encode
Extensions of I are Decoded Filters
𝑓 ∶ 𝐼 𝑋 → 𝑌 𝑓∗ ∶ 𝐼 𝑋 → 𝐼(𝑌)
𝑓∗
∶ 𝑋ℕ2
→ 𝑌ℕ2
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
Γ(𝑓)∗ ∶ 𝑋ℕ2
→ 𝑌ℕ2
Classical image filter
𝑓, 𝑔
Γ 𝑓 ∗ ∘ Γ 𝑔 ∗
Γ 𝑓 ∘ 𝑔 ∗
Γ 𝑓 ∘ Γ g
∗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
𝐼 𝑛 𝑋 = ℤ 𝑛
× 𝑋ℤ 𝑛
𝐺 𝑛 𝑋 = 𝑋ℤ 𝑛
n Application
1 Audio, Time signals
2 Images
3 Video
1000+ Difference Equations
In Conclusion
•Better math
•Better engineering
•Better development process
•Better world

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Functors, Comonads, and Digital Image Processing

Editor's Notes

  1. No unifying framework; ad-hoc. Cannot be analyzed meaningfully across different types as composition. Multiple implementations means more bugs, optimizations yield less returns. Must be a better way.
  2. Once you get a category with composition, you can do lots of crazy analysis
  3. Actually a very surprising result that these can compose meaningfully if and only iff W is a comonad. New composition => new category => opens up worlds of analysis
  4. The fact that you can do this in a meaningful way is also surprising
  5. Pick the path that’s most efficient for you!