CSE445Computer Vision
ModulesGeometry & CalibrationPinhole Camera
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Pinhole Camera and Projection

World, camera, and image frames tied together by K, R, and t, with perspective's scale ambiguity.

L10CO2M03 · Geometry & CalibrationMidterm

Key Terms

Pinhole ModelIntrinsics KExtrinsics R,tPerspective DivisionProjection MatrixScale Ambiguity

Python Tools

NumPyOpenCVMatplotlib

Textbook Reference

Szeliski Ch.2Hartley & Zisserman Ch.6

Topics Covered

  1. 1Coordinate frames: world, camera, image, and pixel
  2. 2Intrinsic matrix K: focal length, principal point, and skew
  3. 3Extrinsics R and t and the camera center
  4. 4The 3x4 projection matrix P = K[R|t] and perspective division
  5. 5Depth-scale ambiguity and the loss of metric information under projection

Learning Outcomes

  • Assemble the full projection matrix from intrinsic and extrinsic parameters
  • Project 3D points to pixels and explain perspective division
  • Explain why a single view cannot recover absolute scale

Lecture Infographics

1 image file

Camera Projection Pipeline

World > Camera > Image > Pixel frame - all four coordinate transformations at once

1

World Frame

3D global coordinates

A 3D point P_w is expressed in the world coordinate frame — a fixed global reference shared by all objects. Measured in real-world units (e.g. metres).

X_wY_wZ_wO_wP_w

Formula

Pw=[XwYwZw1]TP_w = \begin{bmatrix} X_w & Y_w & Z_w & 1 \end{bmatrix}^{\mathsf T}

Homogeneous 3D point in the fixed world frame.

Next: R, t (extrinsic) →
2

Camera Frame

Extrinsic [R | t]

The extrinsic matrix [R|t] is a rigid-body transform that converts world coordinates into the camera's local frame. The camera optical axis points along +Z_c.

X_cY_cZ_cO_cP_cpZ_c = depth

Formula

Pc=RPw+tP_c = R P_w + t
[R∣t]\begin{bmatrix} R \mid t \end{bmatrix}

Extrinsics map world coordinates into the camera frame.

Next: ÷ Z_c (perspective) →

Extrinsic [R | t]

[R11R12R13txR21R22R23tyR31R32R33tz0001]\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 \\ 0 & 0 & 0 & 1\end{bmatrix}
3

Image Plane

Perspective divide

Dividing by depth Z_c performs perspective projection onto the normalised image plane at z = 1. Points on the same ray through the optical centre map to the same image point.

xyO (0,0)r=1p(x, y)yx

Formula

x=XcZc,y=YcZcx = \frac{X_c}{Z_c}, \qquad y = \frac{Y_c}{Z_c}

Depth divides the camera point, creating perspective shrinkage.

Next: K (intrinsic) →

Projection P = K [R | t]

[fx0cx00fycy00010]\begin{bmatrix}f_x & 0 & c_x & 0 \\ 0 & f_y & c_y & 0 \\ 0 & 0 & 1 & 0\end{bmatrix}

p~ = P * P~_w

4

Pixel Frame

Intrinsic matrix K

The intrinsic matrix K scales by focal lengths (f_x, f_y) and offsets by the principal point (c_x, c_y) to produce discrete pixel coordinates (u, v) on the image sensor.

(c_x, c_y)p(u, v)u →↓ v

Formula

u=fxx+cxu = f_x x + c_x
v=fyy+cyv = f_y y + c_y

Intrinsic K

[fx0cx0fycy001]\begin{bmatrix}f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1\end{bmatrix}

Pinhole Camera Model — Ray Projection

Scene (World)CameraImage Sensoroptical axisimage planepinhole(optical centre)f (focal length)P₁P₂P₃p₁p₂p₃inverted& flipped

Each scene point P projects through the optical centre onto the image plane. The projection is perspective: a ray from P through the pinhole hits the image plane at p = (f·X_c/Z_c, f·Y_c/Z_c). The image is inverted and depth information is lost.

Lecture PDFs

Pinhole Camera and Projection

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