Implement a function to apply radial distortion to normalized image coordinates, a crucial step in correcting lens distortions in computer vision. This process involves modeling the distortion that occurs when light passes through a camera lens, causing images to appear warped.

Radial distortion is a type of distortion that occurs when light rays bend differently as they pass through a lens, resulting in a distorted image. The distortion can be modeled using the distortion coefficients $k_1$ and $k_2$ , which describe the amount of distortion present in the lens. The relationship between the original and distorted coordinates can be described using the equations $r^2 = x^2 + y^2$ , $x_{distorted} = x(1 + k_1 r^2 + k_2 r^4)$ , and $y_{distorted} = y(1 + k_1 r^2 + k_2 r^4)$ .

To apply radial distortion, follow these steps:

Calculate the squared distance $r^2$ from the optical center.
Compute the distortion factor using the coefficients $k_1$ and $k_2$ .
Apply the distortion factor to the original coordinates.

r^2 = x^2 + y^2

x_{distorted} = x(1 + k_1 r^2 + k_2 r^4)

y_{distorted} = y(1 + k_1 r^2 + k_2 r^4)

This technique is widely used in camera calibration to correct for lens distortions and produce more accurate images.

First, compute $r^2 = x^2 + y^2 = 0.5^2 + 0.5^2 = 0.25 + 0.25 = 0.5$ .
Then compute the distortion factor: $1 + k_1 r^2 + k_2 r^4 = 1 + (-0.1)(0.5) + 0.01(0.5^2) = 1 - 0.05 + 0.01 \cdot 0.25 = 0.9525$ .
Apply this factor to each coordinate: $x_{distorted} = 0.5 \cdot 0.9525 = 0.47625$ , $y_{distorted} = 0.5 \cdot 0.9525 = 0.47625$ , which rounds (with the problem’s precision/rounding) to approximately $[0.4688, 0.4688]$ .

Input point is [x, y] (normalized coordinates)
k1, k2 are distortion coefficients
Return distorted [x', y'] rounded to 4 decimal places

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.

📘

Radial Distortion

MediumCamera Model

To apply radial distortion, follow these steps:

Calculate the squared distance $r^2$ from the optical center.
Compute the distortion factor using the coefficients $k_1$ and $k_2$ .
Apply the distortion factor to the original coordinates.

r^2 = x^2 + y^2

x_{distorted} = x(1 + k_1 r^2 + k_2 r^4)

y_{distorted} = y(1 + k_1 r^2 + k_2 r^4)

This technique is widely used in camera calibration to correct for lens distortions and produce more accurate images.

Example:

Input:

radial_distort([0.5, 0.5], -0.1, 0.01)

Output:

[0.4688, 0.4688]

Reasoning:

First, compute $r^2 = x^2 + y^2 = 0.5^2 + 0.5^2 = 0.25 + 0.25 = 0.5$ .
Then compute the distortion factor: $1 + k_1 r^2 + k_2 r^4 = 1 + (-0.1)(0.5) + 0.01(0.5^2) = 1 - 0.05 + 0.01 \cdot 0.25 = 0.9525$ .
Apply this factor to each coordinate: $x_{distorted} = 0.5 \cdot 0.9525 = 0.47625$ , $y_{distorted} = 0.5 \cdot 0.9525 = 0.47625$ , which rounds (with the problem’s precision/rounding) to approximately $[0.4688, 0.4688]$ .

Constraints:

Input point is [x, y] (normalized coordinates)
k1, k2 are distortion coefficients
Return distorted [x', y'] rounded to 4 decimal places

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.