Redesign color arithmetic (#131)

This notably defines 3 multiplication operators for RGB colors.
It also un-defines `abs2`, because how that should work is a bit ambiguous.
Finally, it defines a new `varmult` function, which allows one to
compute variance using a specific multiplication operator.

There are some compatibility definitions for current releases of ColorTypes.

Closes #126

Co-authored-by: Johnny Chen <johnnychen94@hotmail.com>

Author: timholy

Project.toml

@@ -1,29 +1,27 @@
 name = "ColorVectorSpace"
 uuid = "c3611d14-8923-5661-9e6a-0046d554d3a4"
-version = "0.8.7"
+version = "0.9.0"
 
 [deps]
 ColorTypes = "3da002f7-5984-5a60-b8a6-cbb66c0b333f"
-Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
 FixedPointNumbers = "53c48c17-4a7d-5ca2-90c5-79b7896eea93"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
 Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
-StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
+TensorCore = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
 
 [compat]
-ColorTypes = "0.8, 0.9, 0.10"
-Colors = "0.9, 0.10, 0.11, 0.12"
-FixedPointNumbers = "0.6, 0.7, 0.8"
-SpecialFunctions = "0.7, 0.8, 0.9, 0.10, 1.0"
-StatsBase = "0.28, 0.29, 0.30, 0.31, 0.32, 0.33"
+ColorTypes = "0.10"
+FixedPointNumbers = "0.8"
+SpecialFunctions = "0.7, 0.8, 0.9, 0.10"
+TensorCore = "0.1"
 julia = "1"
 
 [extras]
+Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
-StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["Statistics", "StatsBase", "LinearAlgebra", "Test"]
+test = ["Colors", "Statistics", "LinearAlgebra", "Test"]

README.md

@@ -6,18 +6,27 @@
 This package is an add-on to [ColorTypes](https://github.com/JuliaGraphics/ColorTypes.jl), and provides fast
 mathematical operations for objects with types such as `RGB` and
 `Gray`.
+Specifically, with this package both grayscale and `RGB` colors are treated as if they are points
+in a normed vector space.
 
 ## Introduction
 
-Colorspaces such as RGB, unlike XYZ, are technically non-linear; the
-"colorimetrically correct" approach when averaging two RGBs is to
+Colorspaces such as RGB, unlike XYZ, are technically non-linear;
+perhaps the most "colorimetrically correct" approach when averaging two RGBs is to
 first convert each to XYZ, average them, and then convert back to RGB.
+Nor is there a clear definition of computing the sum of two colors.
+As a consequence, Julia's base color package,
+[ColorTypes](https://github.com/JuliaGraphics/ColorTypes.jl),
+does not support mathematical operations on colors.
 
 However, particularly in image processing it is common to ignore this
 concern, and for the sake of performance treat an RGB as if it were a
-3-vector.  This package provides such operations.
+3-vector.  The role of this package is to extend ColorTypes to support such mathematical operations.
+Specifically, it defines `+` and multiplication by a scalar (and by extension, `-` and division by a scalar) for grayscale and `AbstractRGB` colors.
+These are the requirements of a [vector space](https://en.wikipedia.org/wiki/Vector_space).
 
-If you're curious about how much difference it makes, the following
+If you're curious about how much the "colorimetrically correct" and
+"vector space" views differ, the following
 diagram might help. The first 10 `distinguishable_colors` were
 generated, and all pairs were averaged. Each box represents the
 average of the pair of diagonal elements intersected by tracing
@@ -27,16 +36,77 @@ represents the "RGB vector space" version.
 
 ![ColorVectorSpace](images/comparison.png "Comparison")
 
+This package also defines `norm(c)` for RGB and grayscale colors.
+This makes these color spaces [normed vector spaces](https://en.wikipedia.org/wiki/Normed_vector_space).
+Note that `norm` has been designed to satisfy equivalence of grayscale and RGB representations: if
+`x` is a scalar, then `norm(x) == norm(Gray(x)) == norm(RGB(x, x, x))`.
+Effectively, there's a division-by-3 in the `norm(::RGB)` case compared to the Euclidean interpretation of
+the RGB vector space.
+Equivalence is an important principle for the Colors ecosystem, and violations should be reported as likely bugs.
+
 ## Usage
 
 ```julia
 using ColorTypes, ColorVectorSpace
 ```
 
-That's it. Just by loading `ColorVectorSpace`, most basic mathematical
+For the most part, that's it; just by loading `ColorVectorSpace`, most basic mathematical
 operations will "just work" on `AbstractRGB`, `AbstractGray`
 (`Color{T,1}`), `TransparentRGB`, and `TransparentGray` objects.
 (See definitions for the latter inside of `ColorTypes`).
 
-If you discover missing operations, please open an issue, or better
-yet submit a pull request.
+However, there are some additional operations that you may need to distinguish carefully.
+
+### Multiplication
+
+Grayscale values are conceptually similar to scalars, and consequently it seems straightforward to define multiplication of two grayscale values.
+RGB values present more options.
+This package supports three different notions of multiplication: the inner product, the hadamard (elementwise) product, and the tensor product.
+
+```julia
+julia> c1, c2 = RGB(0.2, 0.3, 0.4), RGB(0.5, 0.3, 0.2)
+(RGB{Float64}(0.2,0.3,0.4), RGB{Float64}(0.5,0.3,0.1))
+
+julia> c1⋅c2     # \cdot<TAB> # or dot(c1, c2)
+0.09000000000000001
+
+# This is equivelant to `mapc(*, c1, c2)`
+julia> c1⊙c2     # \odot<TAB> # or hadamard(c1, c2)
+RGB{Float64}(0.1,0.09,0.08000000000000002)
+
+julia> c1⊗c2    # \otimes<TAB> # or tensor(c1, c2)
+RGBRGB{Float64}(
+ 0.1   0.06  0.04000000000000001
+ 0.15  0.09  0.06
+ 0.2   0.12  0.08000000000000002)
+```
+
+Note that `c1⋅c2 = (c1.r*c2.r + c1.g*c2.g + c1.b*c2.b)/3`, where the division by 3 ensures the equivalence `norm(x) == norm(Gray(x)) == norm(RGB(x, x, x))`.
+
+It is designed to not support the ordinary multiplication operation `*` because it is not obvious which one of these should be the default option.
+
+However, `*` is defined for grayscale since all these three multiplication operations (i.e., `⋅`, `⊙` and `⊗`) are equivalent in the 1D vector space.
+
+### Variance
+
+The variance `v = E((c - μ)^2)` (or its bias-corrected version) involves a multiplication,
+and to be consistent with the above you must specify which sense of multiplication you wish to use:
+
+```julia
+julia> cs = [c1, c2]
+2-element Array{RGB{Float64},1} with eltype RGB{Float64}:
+ RGB{Float64}(0.2,0.3,0.4)
+ RGB{Float64}(0.5,0.3,0.2)
+
+julia> varmult(⋅, cs)
+0.021666666666666667
+
+julia> varmult(⊙, cs)
+RGB{Float64}(0.045,0.0,0.020000000000000004)
+
+julia> varmult(⊗, cs)
+RGBRGB{Float64}(
+  0.045  0.0  -0.03
+  0.0    0.0   0.0
+ -0.03   0.0   0.020000000000000004)
+```