Justin Le Chapman University Schmid College of Science and Technology Lets talk about Filters getglassescomau The problem with filters Categories Objects Morphisms Composition ID: 649841
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Slide1
Functors, Comonads, and Digital Image Processing
Justin Le, Chapman University Schmid College of Science and TechnologySlide2
Let’s talk about FiltersSlide3
getglasses.com.au
The problem with filtersSlide4
Categories
Objects
Morphisms
Composition
(people)
(integers)
(real numbers)
SetSlide5
Functors
Objects to Objects
Morphisms to MorphismsSlide6
Ex: Infinite List Functor
X to infinite lists of things in X
92
4, 9, 8, 75, -3, ...
Slide7
Comonads
Extract
Duplicate
Laws
, etc.
Infinite List FunctorSlide8
Cokleisli Arrows
Comonads only!Slide9
Extension
Slide10
Functor 1: “Image with Focus”
Slide11
Slide12
Position-Aware Transformation
Affine Transformation Matrix
Slide13
Compose Affine
Compose Cokleisli
Encode
EncodeSlide14
Functor 2: “Local Neighborhood”
G
Slide15
Local/Relative Transformations
Kernel/Convolution Matrix
Slide16
Convolve Kernels
Compose Cokleisli
Encode
EncodeSlide17
Extensions of I are Decoded Filters
Classical image filter
f
ocus stays sameSlide18
Commutation Abounds
Compose Cokleisli
Compose Normally
Extend
ExtendSlide19
Decoded Neighborhoods
“Globalization”,
Classical image filterSlide20
Globalize, extend, compose
Cokleisli compose, globalize, extend
Globalize,
cokleisli
compose, extendSlide21
Extension, Globalization are Cheap
Written once:
less bugs
Optimize once,
unlimited return
on performance
Trivially
parallelizable
Globalization can handle boundary conditions, low-levelSlide22
Generalization
n
Application
1
Audio,
Time signals
2
Images
3
Video
1000+
Difference EquationsSlide23
In Conclusion
Better
math
Better
engineering
Better
development process
Better
world