Course-Relationship Between Principal Point and Image Dimensions

📌 Theoretical Relationship

In theory, the principal point refers to the intersection of the camera’s optical axis with the image plane. This point is typically located near the center of the image. Thus:

\[p_x \approx \frac{W}{2}\] \[p_y \approx \frac{H}{2}\]

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🎯 More Precise Expression

Let:

• W = image width
• H = image height

In a pixel coordinate system (origin at top-left corner):

• If pixel centers are at integer positions: \(p_x = \frac{W - 1}{2}\)

  \[p_y = \frac{H - 1}{2}\]

• If pixel centers are at half-integer positions: \(p_x = \frac{W}{2}\)

  \[p_y = \frac{H}{2}\]

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📸 Example

For a 1920×1080 image:

\[p_x \approx 960\] \[p_y \approx 540\]

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⚠️ Practical Considerations

• The actual principal point may deviate from the image center
• Reasons: lens manufacturing tolerances, sensor alignment, calibration accuracy
• Calibration often returns non-integer values for ( p_x, p_y )
• Offsets are more prominent in:
  • Wide-angle lenses
  • Mobile phone cameras
  • Industrial cameras

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📐 Camera Intrinsic Matrix

The camera intrinsic matrix ( K ) is defined as:

\[K = \begin{bmatrix} f_x & 0 & p_x \\ 0 & f_y & p_y \\ 0 & 0 & 1 \end{bmatrix}\]

Where:

• ( f_x = f \cdot s_x ): focal length scaled in x direction
• ( f_y = f \cdot s_y ): focal length scaled in y direction
• ( p_x, p_y ): principal point coordinates in pixels
• ( s_x, s_y ): number of pixels per unit distance in the x and y directions (i.e., pixel density). These are used to convert physical focal length (in mm) to pixel units.

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✅ Summary

Concept	Theoretical Value	Practical Observation
( p_x )	~ image width / 2	Slightly offset from center
( p_y )	~ image height / 2	May vary by a few pixels
Pixel origin	Top-left (0,0)	Standard in most systems
Deviation reason	Lens/sensor misalignment, distortion	Corrected via calibration