Add `Projective` transformation

[hyrodium]

It's better to have Projective transformation to ImageTransformations.jl.

# Definition of projective transformation
"""
Projective transformation.
"""
struct Projective <: Transformation
    H::SArray{Tuple{3,3},Float64,2,9}
end
"""
Definition of projective transformation.
"""
function (p::Projective)(x)
    x_ = SA[x[1],x[2],1.0]
    x′_1, x′_2, x′_3 = p.H*x_
    return SA[x′_1/x′_3, x′_2/x′_3]
end
"""
Inverse of projective transformation.
"""
function Base.inv(p::Projective)
    Projective(inv(p.H))
end

Discussions in Slack: https://julialang.slack.com/archives/CB1R90P8R/p1626929588093000 https://julialang.slack.com/archives/CB1R90P8R/p1626983453103200

I feel the right place to add this is in CoordinateTransformations.jl

Or we could 1) first introduce a high-level API in ImageTransformations.jl with this Projective as internal type, and then 2) migrate it to upstream CoordinateTransformations, and finally 3) use the upstream version.

(The quoted words are by @johnnychen94)

[johnnychen94]

This issue can be seperated into two parts:

1. we want an invertible projective transformation so that warp's default inds = autorange(img, inv(tform)) is happy with it.
2. we might want a high-level API in ImageTransformations so that normal users don't need to talk to warp directly. We currently only have imrotate and imresize.

---

we want an invertible projective transformation

The Projective version you propose definitely works, but maybe the following is a better patch (since it doesn't introduce new concepts to the codebase):

function Base.inv(tform::CoordinateTransformations.ComposedTransformation{PerspectiveMap})
    PerspectiveMap() ∘ inv(tform.t2)
end

julia> tform = PerspectiveMap() ∘ inv(AffineMap(RotX(0.3), SVector((0, 0, 0))));

julia> inv(tform)
(PerspectiveMap() ∘ AffineMap([1.0 0.0 0.0; 0.0 0.955336489125606 -0.29552020666133955; 0.0 0.29552020666133955 0.955336489125606], [-0.0, 0.0, -0.0]))

I didn't yet check whether this is equivalent to the Projective homography, though...

---

we might want a high-level API in ImageTransformations

This is highly application-oriented so I don't know what your target usage is. Perhaps one just need to properly document it with some small examples #141.

---

FYI, I'm reaching @c42f for the privilege to CoordinateTransformations

[hyrodium]

Agreed!

I didn't yet check whether this is equivalent to the Projective homography, though...

Hmm, the tform seems ℝ³→ℝ², so the following error occours:

julia> warp(img, tform)
ERROR: "Dimension mismatch: cannot rotate a vector of length 2"
...

I don't know what your target usage is

I mainly use homography for image registration. (Sorry for Japanese, but this might be a good exaple: https://zenn.dev/hyrodium/articles/5dc951f378b46bedb211)

we might want a high-level API in ImageTransformations so that normal users don't need to talk to warp directly.

I think we can create impinch method:

• impinch(img, a1=>b1), this cause parallel translation.
• impinch(img, a1=>b1, a2=>b2), this cause similarity transformation.
• impinch(img, a1=>b1, a2=>b2, a3=>b3), this cause affine transformation.
• impinch(img, a1=>b1, a2=>b2, a3=>b3, a4=>b4), this cause projective transformation.

(I'm not sure "pinch" is an appropriate name for it.)