Algorithm: how calculate INVERSE of bilinear interpolation? INVERSE of mapping on to an arbitrary quadrilateral?

Question

UPDATE: My terminology below is wrong. The \"forward\" algorithm I describe in \"Lerp2D\" (which I need inverse-of) takes 4 arbitrary corners. It is linear

Answer 1

To simplify, let's begin by just considering a single intepolated value z.
Assume four values z₀₀, z₀₁, z₁₀, z₁₀, and two weights w₀ and w₁ applied to the first and second index, giving

z₀ = z₀₀ + w₀ × (z₁₀ - z₀₀)
z₁ = z₀₁ + w₀ × (z₁₁ - z₀₁)

and finally

z = z₀ + w₁ × (z₁ - z₀)
   = z₀₀ + w₀ × (z₁₀ - z₀₀) + w₁ × (z₀₁ - z₀₀) + w₁ × w₀ × (z₁₁ - z₁₀ - z₀₁ + z₀₀)

So, for your problem you will have to invert a two dimensional quadratic equation

x = x₀₀ + w₀ × (x₁₀ - x₀₀) + w₁ × (x₀₁ - x₀₀) + w₁ × w₀ × (x₁₁ - x₁₀ - x₀₁ + x₀₀)
y = y₀₀ + w₀ × (y₁₀ - y₀₀) + w₁ × (y₀₁ - y₀₀) + w₁ × w₀ × (y₁₁ - y₁₀ - y₀₁ + y₀₀)

Unfortunately, there isn't a simple formula to recover w₀ and w₁ from x and y. You can, however, treat it as a non-linear least squares problem and minimise

(x_w(w₀,w₁) - x)² + (y_w(w₀,w₁) - y)²

which you can do efficiently with the Levenberg–Marquardt algorithm.

Edit: Further Thoughts

It has occurred to me that you might be satisfied with an interpolation from (x, y) to (w₀, w₁) rather than the actual inverse. This will be less accurate in the sense that rev(fwd(w₀, w₁)) will likely be further from (w₀, w₁) than the actual inverse.

The fact that you're interpolating over an irregular mesh rather than a regular grid is going to make this a trickier proposition. Ideally you should join up your (x, y) points with non-overlapping triangles and use barycentric coordinates to linearly interpolate.
For numerical stability you should avoid shallow, pointy triangles. Fortunately, the Delaunay triangulation satifies this requirement and isn't too difficult to construct in two dimensions.

If you would like your reverse interpolation to take a similar form to your forward interpolation you can use the basis functions

1
x
y
x × y

and compute coefficients a_i, b_i, c_i and d_i (i equal to 0 or 1) such that

w₀ = a₀ + b₀ × x + c₀ × y + d₀ × x × y
w₁ = a₁ + b₁ × x + c₁ × y + d₁ × x × y

By substituting the relevant known values of x, y, w₀ and w₁ you'll get four simultaneous linear equations for each w that you can solve to get its coefficients.
Ideally you should use a numerically stable matrix inversion algorithm that can cope with near singular matrices (e.g. SVD), but you may be able to get away with Gaussian elimination if you're careful.

Sorry I can't give you any simpler options, but I'm afraid that this really is a rather tricky problem!