Image processing with comonads and phantom types

The page introduces comonads as the dual of monads, then uses a 2D array with indexes to model grayscale images and neighborhood filters. It shows that plain array-based composition can become slow, and that phantom types can separate raw and composed values at compile time.

Reading notes

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• A comonad is presented as the dual of a monad, with extract and extend as the corresponding operations.
• The text motivates comonads by cases where functions need extra context, like a runtime context or neighboring cells in an image.
• A grayscale image is represented as a 2D array of ints with indexes.
• The array wrapper defines get, extract, and extend, where extend behaves like a mapped neighborhood computation over the whole array.
• Filters such as laplace2d and gauss2D use values from the current cell and adjacent cells.
• Filter composition is built by applying extend to existing filters and combining intermediate results.
• The first array-based version can become slow because some compositions lead to O(N^2) behavior.
• A lazy array version avoids the complexity blowup, but the text says it duplicates underlying filter calls and is still not ideal.
• Phantom types are introduced as extra type parameters used only for compile-time checks.
• The final design marks raw arrays and composed arrays differently, so extend accepts only raw inputs and returns composed outputs.
• This type separation makes incorrect compositions fail at compile time.
• The final example rewrites contours and gaussLaplace so they typecheck under the phantom-type rules.
• The page reports runtime comparisons for the contour filter, with the final version faster than both the original and lazy versions.