# A general imaging model and a method for finding its parameters

TL;DR: A novel calibration method is presented that uses structured light patterns to extract the raxel parameters of an arbitrary imaging system and experimental results for perspective as well as ion-perspective imaging systems are included.

Abstract: Linear perspective projection has served as the dominant imaging model in computer vision. Recent developments in image sensing make the perspective model highly restrictive. This paper presents a general imaging model that can be used to represent an arbitrary imaging system. It is observed that all imaging systems perform a mapping from incoming scene rays to photo-sensitive elements on the image detector. This mapping can be conveniently described using a set of virtual sensing elements called raxels. Raxels include geometric, radiometric and optical properties. We present a novel calibration method that uses structured light patterns to extract the raxel parameters of an arbitrary imaging system. Experimental results for perspective as well as ion-perspective imaging systems are included.

## Summary (2 min read)

### 1 Introduction

- After describing the general imaging model and its properties, the authors present a simple method for finding the parameters of the model for any arbitrary imaging system:.
- It is important to note that, given the non-perspective nature of a general device, conventional calibration methods based on known scene points [251 or self-calibration techniques that use unknown scene points [51, [lo] , [151, cannot be directly applied.
- Since the authors are interested in the mapping from rays to image points, they need a ruy-bused calibration method.
- The authors describe a simple and effective ray-based approach that uses structured light patterns.
- This method allows a user to obtain the geometric, radiometric, and optical parameters of an arbitrarily complex imaging system in a matter of minutes.

### 2 General Imaging Model: Geometry

- If the imaging system is perspective, all the incoming light rays are projected directly onto to the detector plane through a single point, namely, the effective pinhole of the perspective system.
- The goal of this section is to present a geometrical model that can represent such imaging systems.

### 2.1 Raxels

- Each raxel includes a pixel that measures light energy and imaging optics (a lens) that collects the bundle of rays around an incoming ray.
- The authors will focus on the geometric properties (locations and orientations) of raxels.
- Each raxel can posses its own radiometric (brightness and wavelength) response as well as optical (point spread) properties.
- These non-geometric properties will be discussed in subsequent sections.

### Figure 3: (a)

- It may be placed along the line of a principle ray of light entering the imaging system.
- In addition to location and orientation, a raxel may have radiometric and optical parameters.
- The notation for a raxel used in this paper.

### t f

- Multiple raxels may be located at the same point (p1 = p~ = p3), but have different directions.
- The choice of intersecting the incoming rays with a refer-3Many of the arrays of photo-sensitive elements in the imaging devices described in section 1 are one or two-dimensional.
- 41ntensities usually do not change much along a ray (particularly when the medium is air) provided the the displacement is small with respect to the total length of the ray.
- In 191 and [141, it was suggested that the plenoptic function could also be restricted to a plane.
- The important thing is to choose some reference surface so that each incoming ray intersects this surface at only one point.

### 3 Caustics

- 'For example, when light refracts through shallow water of a pool, bnght curves can be seen where the caustics intersect the bottom Figure 5 :.
- The caustic is a good candidate for the ray surface of an imaging system as it is closely related to the geometry of the incoming rays; the incoming ray directions are tangent to the caustic.

### 4.1 Local Focal Length and Point Spread

- An arbitrary imaging system cannot be expected to have a single global focal length.
- Each raxel may be modeled to have its own focal length.
- The authors can compute each raxel's focal length by measuring its point spread function for several depths.
- A flexible approach models the point spread as an elliptical Gaussian.

### 4.3 Complete Imaging Model

- In the case of perspective projection, the essential [51 or fundamental [ 101 matrix provides the relationship between points in one image and lines in another image (of the same scene).
- In the general imaging model, this correspondence need no longer be projective.

### 5 Finding the Model Parameters

- The major axis makes an angle 1c, with the x-axis in the image.
- Each raxel has two focal lengths, f a , hThe angle $ is only defined if the major and minor axis have different.
- In section the authors described how to compute The authors model for a known optical system.
- In contrast, their goal in this section lengths.

### Figure 7: (a)

- If these positions are known, the direction of the ray qf may be determined for each pixel.
- Now, the authors construct a calibration environment where the geometric and radiometric parameters can be efficiently estimated.
- If a display has N locations, the authors can make each point distinct in logN images using simple grey coding or bit coding.
- The authors may then compute the fall-off function across all the points.
- The authors compute both the radiometric response function and the falloff from seventeen uniform brightness levels.

### 5.1 Experimental Apparatus

- The laptop was oriented so as to give the maximum screen resolution along the axis of symmetry.
- Figure 8 (b) shows a sample binary pattern as seen from the parabolic catadioptric system.
- The perspective imaging system, consisting of just the camera itself, can be seen in Figure 1 l(a).

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