Functors, Comonads, and Digital Image Processing

Functors, Comonads, and Digital Image ProcessingJustin Le, Chapman University Schmid College of Science and TechnologyLets talk about Filters

getglasses.com.auCategoriesObjectsMorphismsComposition(people)(integers)(real numbers)SetFunctorsObjects to ObjectsMorphisms to MorphismsEx: Infinite List FunctorX to infinite lists of things in X924, 9, 8, 75, -3, ...ComonadsInfinite List FunctorCokleisli ArrowsComonads only!ExtensionFunctor 1: Image with Focus

Position-Aware Transformation

Affine Transformation MatrixCompose AffineCompose CokleisliEncodeEncodeFunctor 2: Local NeighborhoodLocal/Relative Transformations

Kernel/Convolution MatrixConvolve KernelsCompose CokleisliEncodeEncodeExtensions of I are Decoded FiltersClassical image filterfocus stays sameCommutation AboundsCompose CokleisliCompose NormallyExtendExtendDecoded NeighborhoodsClassical image filterGlobalize, extend, composeCokleisli compose, globalize, extendGlobalize, cokleisli compose, extendExtension, Globalization are CheapWritten once: less bugsOptimize once, unlimited return on performanceTrivially parallelizableGlobalization can handle boundary conditions, low-levelGeneralizationnApplication1Audio, Time signals2Images3Video1000+Difference EquationsIn ConclusionBetter mathBetter engineeringBetter development processBetter world