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A standard 4-D light field is represented with a two-plane parameterization, in which each light ray is uniquely determined by its intersection with two predefined planes parallel to each other. The 4-D index of each light ray is represented with the coordinates of the two intersections on these planes, respectively.

Here we demonstrate the rendering problem with the 2-D light field and then the two parallel planes become two lines. The 3-D space also becomes a 2-D plane that can be seen as a top view of the original scene. The most simple setup to capture a 2-D light field is to use a 1-D pinhole camera that moves on a line as shown in Figure 5.1. Both the pinhole camera’s field of view and its moving range determine the field of view of the acquired 2-D light field.

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Figure 5.2: The novel views and the corresponding slicing lines in the 2-D light field. We use the blue view in the scene to represent a novel viewing position that is closer to the scene, compared with the original camera plane. The novel view is created by slicing the 2-D light field with the blue line, as shown on the right. The slope of the line is determined by how far the novel view moves from the original camera plane. We use the red view and a red slicing line to represent a novel viewing position behind the original camera plane.

As shown in Figure 5.2, by using the acquired 2-D light field, we can render novel views, and we can create a walk-through for users to interactively view the captured scene. A user can move freely on the line and has the same field of view as the pinhole camera. The user can also move towards the scene to observe the details (illustrated by the blue view in the scene and blue slices in the 2-D light field), or move away from the scene to see the whole scene (illustrated by the red view in the scene and red slices in the 2-D light field). When the user is close to the scene, the field of view becomes narrower compared with the pinhole camera; whereas when the user is far away from the scene, the field of view becomes wider compared with the pinhole camera.

In conclusion, with the standard light field, the novel views can be rendered for areas in front of the scene within a limited range. However, this rendering technique only provides a limited view that is clearly smaller than 180 degree. The walk-through path is heavily constrained by the range of the line on which the pinhole camera moves. Therefore, to address the rendering problem, we propose a novel representation in which novel views with a 360-degree viewing angle can be rendered at any chosen location in a given area.

5.2.2

Definitions and notations

Our goal is to design a representation for light rays such that we can render novel views at various positions with a 360-degree viewing angle. Hence, we move away from the standard approach that uses two lines to parameterize the 2-D light field or uses two planes to parameterize the 4-D light field. Instead, we turn to an omnidirectional representation and propose a novel representation, the circular light-field in which we represent a given light ray with its intersections with two concentric circles.

Although using circles to represent a light field seems to be quite similar to the creation of a panorama photo, there are fundamental differences between our proposed representation and

Figure 5.3: Creating a panorama photo by rotating a pinhole camera around the pinhole. The rotation is around the pinhole that is the optical center of the camera such that the depth of scene does not affect the registration and stitching process. Note that as we use the pinhole camera here, the optical center locates at the pinhole.

the panorama technique. When creating a panorama photo, the camera is rotated around its optical center such that the depth of the scene does not affect the registration and stitching process, as shown in Figure 5.3. Hence, a panorama image can not specify the directions of the recorded light rays. But by using the two concentric circles, the direction of each light rays can be uniquely specified.

The scene is a 2-D space that is represented with the x− y plane. The object or light origin in the 2-D space can be either represented with (x, y) or (z, ψ) where

z = (x2+ y2), ψ = arccos x x2+ y2. The index (z, ψ) is actually the polar coordinates of the 2-D space.

By using the two-concentric-circles parameterization, as shown in Figure 5.4, each light ray is represented by its two intersections φ and θ with the two concentric circles, respectively. To be more specific, φ is the angle of the intersection that is determined by the light ray and the outer circle, and θ is the relative angle of the intersection on the inner circle to the line that connects the intersection φ on the outer circle and the circle center, as shown in Figure 5.9. Thus, each light ray in 2-D space is represented with a circular light-field L(φ, θ).

Note that the variables x, y, z and ψ are all used to describe the locations in the 2-D space, and θ and φ are used to represent the index of the circular light-field.

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Figure 5.4: Illustration of the 2-D circular light-field representation of a red light ray. For the sake of clarity, we use the blue dashed-lines to illustrate the information of the origin of the red light ray whereas we use the green dashed-lines to illustrate the intersections of the light ray with the concentric circles. The red light ray comes from the position (x, y), or (z, ψ) in the polar coordinates. It intersects with the outer circle at the angle φ. To record its intersection with the inner circle, we use θ to represent the angle between the line that connects the two intersections and the line that connects the circle center and the intersection on the outer circle.

We draw a comparison between the standard light-field L(x, p) and the circular light-field L(φ, θ). In the standard 2-D light field L(x, p), we see the index x as the intersection on the camera plane, whereas we see the index p as the direction of the corresponding light ray. In the 2-D circular light-field L(φ, θ), the index φ is the intersection of the light ray with the outer circle, whereas tan θ is the direction of the recorded light ray. Thus the circular light-field L(φ, θ) is much more flexible for rendering novel views because circles, unlike the planes or lines, do not face towards a specific direction.