# The Conformal Camera in Modeling Active Binocular Vision

## Abstract

**:**

## 1. Introduction

#### 1.1. Prior and Recent Work

#### 1.2. Original Contributions

#### 1.3. Related Work

## 2. The Conformal Camera

#### 2.1. Stereographic Projection

#### 2.2. The Group of Image Projective Transformations

#### 2.3. Geometry of the Image Plane

## 3. Fourier Analysis on the Projective Group

#### 3.1. Group Representations and Fourier Analysis

#### 3.2. Projective Fourier Transform

#### 3.3. Non-Compact and Compact Realizations of PFT

#### 3.4. Discrete Projective Fourier Transform

## 4. Discussion: Biologically-Mediated Vision

#### 4.1. Imaging with the Conformal Camera

#### 4.2. Intermediate-Level Vision

#### 4.3. DPFT in Modeling Retinotopy

#### 4.4. Numerical Implementation of DPFT

**Example 1.**

#### 4.5. Visual Information during Robotic Eye Movements

#### 4.5.1. Visual Information during Smooth Pursuit

#### 4.5.2. Visual Information during Saccades

## 5. Binocular Vision and the Conformal Camera

## 6. The Asymmetric Conformal Camera

## 7. Discussion: Modeling Empirical Horopters

## 8. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PFT | projective Fourier transform |

DPFT | discrete projective Fourier transform |

FFT | fast Fourier transform |

SC | superior colliculus |

LGN | lateral geniculate nucleus |

V1 | primary visual cortex |

ECT | exponential chirp transform |

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**Figure 1.**The conformal camera as the eye model. The points in the object space are centrally projected into the sphere. The sphere and the center of projection N represent the eye’s retina and nodal point. The ‘image’ entity is given by stereographic projection σ from the sphere to the plane $\mathbb{C}$. Because σ is conformal and maps circles to circles (see the text), it preserves the ‘retinal illuminance’, that is the pixels. This image representation is appropriate for efficient computational processing.

**Figure 2.**(

**a**) When the camera gaze is rotated by ϕ, the image projective transformation is given here in the rotated image plane by the g-transformation that results from the composition of two basic image transformations. The first involves the image that is translated by the vector ($\mathbf{b}$) and projected back into the image plane. The second transformation is given in terms of the image projected into the sphere, rotated by $-2\varphi $, and projected back into the image plane. The image transformation adds the conformal distortions, schematically shown by the transformed image’s back projection into the plane containing the planar object. (

**b**) The camera and scene are shown in the view seen when looking from above. Here, the sequence of transformations $q\mapsto z\mapsto {q}^{1}\mapsto {q}^{2}\mapsto {q}^{3}\mapsto {z}^{\prime}=g\xb7z$ explains the image projective transformation.

**Figure 3.**When the camera gaze is rotated by ϕ, the image projective transformation is given by the g-transformation where $g=kh$ is the result of the composition of the line segment $PQ$ relative movements. Since the gaze rotation induces the nodal point translation by $\mathbf{t}$, the object $PQ$ relative movements in the scene are composed of the translation by $-\mathbf{t}$ corresponding to the translation of the image by $\mathbf{b}$ (i.e., the h-transformation) and the rotation by $-\varphi $ corresponding to the sphere rotation by $-2\varphi $ (i.e., the k-transformation).

**Figure 4.**Empirical longitudinal horopters are shown schematically for symmetric convergence points. Abathic distance is defined here as the distance from the line connecting the eye centers to the fixation point F at which the horopter is a straight line.

**Figure 5.**The eye model with an asymmetrically-displaced fovea and a tilted and decentered lens outlined with thin (1 pt) curves. The asymmetric conformal camera is outlined with thick (2 pt) curves. The stereo system is obtained if the left camera is reflected in the head axis ${X}_{3}$. Each camera’s image plane, shown here for the left eye, is perpendicular to the head axis, such that the horopter curve is a straight line passing through the fixation point F at the abathic distance from origin O.

**Figure 6.**The graphs of the three horizontal horopters, the ellipse for fixation ${F}_{1}\left(163.7,79.5\right)$, the straight line for fixation $F\left(126.8,161.4\right)$ and the hyperbola for fixation ${F}_{2}\left(53.4,211.5\right)$, with coordinates given in millimeters. The eye radius is 7.9 mm; the interocular distance is 78.0 mm, $\alpha ={16}^{\xb0}$ and $\beta ={10}^{\xb0}$. For each of the fixations ${F}_{i}$, $i=1,2$, the six points were obtained by back projecting the corresponding points. These six points included the fixation point, the two nodal points, the point $Pinft{y}_{i}$ that projects to ∞ and two additional points, $P{1}_{i}$ and $P{2}_{i}$. Five of these points were used to obtain the conics, and the sixth point was used for the verification. The straight-line horopter is for the fixation point at the abathic distance and is given by three points, F, ${P}_{1}$ and ${P}_{2}$.

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**MDPI and ACS Style**

Turski, J.
The Conformal Camera in Modeling Active Binocular Vision. *Symmetry* **2016**, *8*, 88.
https://doi.org/10.3390/sym8090088

**AMA Style**

Turski J.
The Conformal Camera in Modeling Active Binocular Vision. *Symmetry*. 2016; 8(9):88.
https://doi.org/10.3390/sym8090088

**Chicago/Turabian Style**

Turski, Jacek.
2016. "The Conformal Camera in Modeling Active Binocular Vision" *Symmetry* 8, no. 9: 88.
https://doi.org/10.3390/sym8090088