Undistort Point

Implement a function to remove radial distortion from a given point in an image. This process is crucial in computer vision to correct for distortions introduced by camera lenses.

The concept of radial distortion arises when a camera's lens bends light rays, causing straight lines to appear curved. This distortion can be modeled using a polynomial expression, where the distorted coordinates $x_d$ and $y_d$ are related to the undistorted coordinates $x$ and $y$ through a distortion factor.

To correct for this distortion, we can use an iterative refinement process, such as Newton's method or fixed-point iteration, which refines an initial estimate of the undistorted coordinates until convergence. The steps involved are:

Initialize the undistorted coordinates with the distorted coordinates
Compute the distortion factor using the current estimate of the undistorted coordinates
Update the estimate of the undistorted coordinates using the distortion factor
Repeat steps 2-3 until convergence

r^2 = x^2 + y^2

\text{factor} = 1 + k_1r^2 + k_2r^2^2

This technique is widely used in image processing pipelines.

Example:

Input:

undistort_point([0.4688, 0.4688], -0.1, 0.01)

Output:

[0.5, 0.5]

Reasoning:

Start with distorted point $\mathbf{x}_d = [0.4688, 0.4688]$ and radial model $r^2 = x^2 + y^2$ , $x_d = x_u(1 + k_1 r^2 + k_2 r^4)$ (applied similarly to $y$ ) with $k_1=-0.1$ , $k_2=0.01$ .
Assume the undistorted point lies on the same ray, so try $\mathbf{x}_u = [0.5, 0.5]$ and compute $r^2 = 0.5^2 + 0.5^2 = 0.5$ .
Compute the radial factor: $1 + k_1 r^2 + k_2 r^4 = 1 + (-0.1)\cdot 0.5 + 0.01 \cdot 0.5^2 = 1 - 0.05 + 0.0025 = 0.9525$ , and then the distorted coordinate: $0.5 \cdot 0.9525 = 0.47625 \approx 0.4688$ (the small mismatch is corrected by iteration).
Using Newton’s method or fixed-point iteration, the algorithm refines this guess until the forward distortion of $\mathbf{x}_u$ matches $[0.4688, 0.4688]$ , converging to the undistorted output $\mathbf{x}_u = [0.5, 0.5]$ .

Constraints:

Use 10 iterations of fixed-point iteration
Return undistorted [x, y] rounded to 4 decimal places

Example:

Constraints:

Test Results