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Ray to Light Field Coordinates

MediumLight Field 📚 CV: Image-Based Rendering

Convert a 3D ray to light field (s, t, u, v) coordinates.

A light field parameterizes all rays in a scene using the two-plane representation:

$(s, t)$ : intersection with the first plane (camera plane)
$(u, v)$ : intersection with the second plane (focal plane)

Given a ray starting at origin $(x_0, y_0, z_0)$ on the first plane (at $z=0$ ) with direction $(d_x, d_y, d_z)$ , find where it intersects the second plane at $z = z_{plane}$ :

$t_{param} = \frac{z_{plane}}{d_z}$ $u = x_0 + t_{param} \cdot d_x$ $v = y_0 + t_{param} \cdot d_y$

Example:

Input:

ray_to_lightfield([0, 0, 0], [0, 0, 1], 1)

Output:

(0, 0, 0, 0)

Reasoning:

Ray from origin pointing straight ahead: s, t = ray_origin[0:2] = (0, 0)

t_param = 1 / 1 = 1
u = 0 + 1 × 0 = 0
v = 0 + 1 × 0 = 0 Result: (0, 0, 0, 0) - ray goes through both plane centers.

Constraints:

ray_origin: [x, y, z] point on first plane (z=0)
ray_direction: [dx, dy, dz] direction vector
plane_distance: z-coordinate of second plane
Return (s, t, u, v) coordinates, rounded to 4 decimal places

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Ray to Light Field Coordinates: Background & Strategy

Background Knowledge

Light Field Representation and the Two-Plane Model

A light field is a complete description of all light rays in a scene, capturing both the intensity and direction of light traveling through 3D space. The two-plane parameterization (also called the 4D light field or Lumigraph representation) is a practical way to encode this information using just four coordinates. The first plane (typically the camera plane at $z=0$ ) records where a ray originates or passes through, while the second plane (the focal plane at some distance $z_{plane}$ ) records where that same ray intersects further along its path. This representation is fundamental to image-based rendering because it allows you to synthesize novel viewpoints by resampling the light field at different $(s,t)$ positions.

Ray Parameterization in 3D Space

In computer graphics and computational imaging, rays are typically represented parametrically: a ray starts at a point $(x_0, y_0, z_0)$ and travels in direction $(d_x, d_y, d_z)$ . Any point along the ray can be expressed as: $\mathbf{p}(t) = (x_0, y_0, z_0) + t \cdot (d_x, d_y, d_z)$

where $t$ is a scalar parameter (often called the "time" or "distance" along the ray). To find where a ray intersects a plane perpendicular to the $z$ -axis at depth $z_{plane}$ , you solve for the parameter value where the $z$ -coordinate equals $z_{plane}$ , then use that parameter to compute the $x$ and $y$ coordinates at the intersection point.

Coordinate Mapping and Perspective Projection

The conversion from 3D ray space to light field coordinates is essentially a form of perspective projection. The $(s,t)$ coordinates represent the ray's position on the first plane, while $(u,v)$ represent its position on the second plane. This mapping encodes both the spatial location and the direction of the ray in a way that's efficient for rendering and analysis. Understanding this relationship is crucial for tasks like view synthesis, depth estimation, and light field reconstruction.

Algorithm/Approach

The solution follows a straightforward ray-plane intersection pattern:

Parameterize the ray using the given origin and direction
Solve for the intersection parameter by setting the ray's $z$ -coordinate equal to $z_{plane}$
Compute the intersection point by evaluating the ray equation at that parameter value
Extract the light field coordinates by reading off the $x$ and $y$ components of the intersection point

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solution.py

Python 3.13

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