Pinhole Camera Model

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Camera Models

Camera models describe how a camera projects points in the 3D world onto a 2D image plane. Each model makes different assumptions. Common camera models include:

1. The Pinhole Camera Model: The pinhole model is the simplest camera model. It assumes that light rays pass through a single point, called the pinhole, before reaching the image plane. Its parameters include the focal length and the position of the pinhole.
2. The Perspective Camera Model: The perspective model extends the pinhole model to include lens distortion and other optical effects. It is used in computer vision and graphics to model camera behavior.
3. The Orthographic Camera Model: In the orthographic model, parallel lines in the 3D world remain parallel in the 2D image. This property is useful for technical drawings and architectural visualizations that require accurate measurements.
4. The Spherical Camera Model: The spherical model uses a spherical image sensor to capture a 360-degree view of a scene. It is used in virtual reality and panoramic photography.
5. The Omnidirectional Camera Model: The omnidirectional model uses multiple lenses or a fisheye lens to capture a wide field of view (FOV). It is used in robotics and surveillance.

Each model defines equations and parameters for projecting 3D points onto the 2D image plane. These models are used for image rectification, 3D reconstruction, and camera calibration. This section focuses on the pinhole camera model. Pinhole camera model. The image is formed on the image plane by light rays passing through a small aperture (the pinhole) at the center of projection. The image is inverted and smaller than the object.

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Camera Model Fundamentals

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Pinhole Camera Model

The functions in this section use the pinhole camera model. A perspective transformation projects a 3D scene point $P_w$ onto the image plane at pixel $p$. The variables $P_w$ and $p$ are represented in homogeneous coordinates as 3D and 2D vectors, respectively. The pinhole camera model gives the following distortion-free projective transformation: $\lambda \; p = K \begin{bmatrix} R|t \end{bmatrix} P_w$ where:

• $P_w$ is a 3D point expressed with respect to the world coordinate system
• $p$ is a 2D pixel in the image plane
• $K$ is the camera intrinsic matrix
• $R$ and $t$ are the rotation and translation that describe the change of coordinates from world to camera coordinate systems
• $\lambda$ is the projective transformation’s arbitrary scaling

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Camera Intrinsic Matrix

The camera intrinsic matrix $K$ maps 3D points in the camera coordinate system to 2D pixel coordinates, up to a scale factor: $\lambda\, p = K P_c, \qquad p = \begin{bmatrix} u \\ v \\ 1 \end{bmatrix}, \quad \lambda = Z_c$ Dividing by $\lambda$ is the perspective division that turns $K P_c$ into pixel coordinates. The matrix $K$ contains the focal lengths $f_x$ and $f_y$, expressed in pixels, and the principal point $(c_x, c_y)$, which is usually near the image center: $K = \begin{bmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{bmatrix}$ Therefore: $\lambda \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \begin{bmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} X_c \\ Y_c \\ Z_c \end{bmatrix}$

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Coordinate Transformations

The joint rotation-translation matrix $[R|t]$ factors into a 3-by-4 projection and a 4-by-4 homogeneous transformation (Hartley & Zisserman, 2004; Szeliski, n.d.): $[R|t] = \begin{bmatrix} I & 0 \end{bmatrix} \begin{bmatrix} R & t \\ 0 & 1 \end{bmatrix}$ The homogeneous transformation, built from the extrinsic parameters, maps a world point to camera coordinates $(X_c, Y_c, Z_c)$. The projection $\begin{bmatrix} I & 0 \end{bmatrix}$ keeps these three coordinates and drops the homogeneous one. Read as a homogeneous 2D point, the result gives the normalized camera coordinates $x' = X_c / Z_c$ and $y' = Y_c / Z_c$ on the image plane. The extrinsic parameters $R$ and $t$ define the homogeneous transformation from the world coordinate system $w$ to the camera coordinate system $c$: $P_c = \begin{bmatrix} R & t \\ 0 & 1 \end{bmatrix} P_w$ The complete transformation is: $\lambda \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \begin{bmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_x \\ r_{21} & r_{22} & r_{23} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \end{bmatrix} \begin{bmatrix} X_w \\ Y_w \\ Z_w \\ 1 \end{bmatrix}$ If $Z_c \neq 0$, this is equivalent to: $\begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} f_x X_c/Z_c + c_x \\ f_y Y_c/Z_c + c_y \end{bmatrix}$

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Lens Distortion Model

Real lenses introduce radial and tangential distortion. Distortion examples: barrel, pincushion, and tangential distortions. The extended camera model includes this distortion: $\begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} f_x x'' + c_x \\ f_y y'' + c_y \end{bmatrix}$ where: $\begin{bmatrix} x'' \\ y'' \end{bmatrix} = \begin{bmatrix} x' \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} + 2 p_1 x' y' + p_2(r^2 + 2 x'^2) + s_1 r^2 + s_2 r^4 \\ y' \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} + p_1 (r^2 + 2 y'^2) + 2 p_2 x' y' + s_3 r^2 + s_4 r^4 \end{bmatrix}$ with $r^2 = x'^2 + y'^2$ and $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} X_c/Z_c \\ Y_c/Z_c \end{bmatrix}$ if $Z_c \neq 0$. Distortion parameters:

• Radial coefficients: $k_1$, $k_2$, $k_3$, $k_4$, $k_5$, $k_6$
• Tangential coefficients: $p_1$, $p_2$
• Thin prism coefficients: $s_1$, $s_2$, $s_3$, $s_4$

The distortion coefficients are passed as: $(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$ Types of distortion: The radial part of the model scales each point by the full radial factor $\rho(r) = \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6},$ so the type of radial distortion depends on how the whole ratio changes with $r$, not on the numerator alone:

• Barrel distortion: $\rho(r)$ decreases as $r$ grows, so points far from the center move inward.
• Pincushion distortion: $\rho(r)$ increases as $r$ grows, so points far from the center move outward.

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References

• Hartley, R., Zisserman, A. (2004). Multiple View Geometry in Computer Vision.
• Szeliski, R. (n.d.). Computer Vision: Algorithms and Applications.

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