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Camera Intrinsics Matrix

The camera intrinsics matrix K (also called the calibration matrix) encodes the optical properties of a camera that map 3D world points onto a 2D image plane. Every point cloud produced by vizion3d is built from these four numbers.

The matrix

        | fx    0   cx |
K  =    |  0   fy   cy |
        |  0    0    1 |

In full projection form — taking a 3D point (X, Y, Z) in camera coordinates and projecting it to image pixel (u, v):

| u |       | fx    0   cx |   | X/Z |
| v |  =    |  0   fy   cy | × | Y/Z |
| 1 |       |  0    0    1 |   |  1  |

Expanded:

u  =  fx × (X / Z)  +  cx
v  =  fy × (Y / Z)  +  cy

Inverted (what vizion3d does when building a point cloud from a depth map):

Z  =  d
X  =  (u - cx) × d / fx
Y  =  (v - cy) × d / fy

where d is the depth value at pixel (u, v).

Each element explained

fx — horizontal focal length (pixels)

Position in K: row 0, col 0.

The horizontal focal length is the product of the physical focal length of the lens and the horizontal pixel density of the sensor. It describes how strongly the camera compresses horizontal depth into horizontal pixel distance.

  • Large fx → narrow horizontal field of view; objects appear wider in pixel space.
  • Small fx → wide horizontal field of view; objects appear narrower in pixel space.

Relation to horizontal field of view FoV_h:

fx  =  (image_width / 2) / tan(FoV_h / 2)

Read from a calibration matrix: K[0][0].

fy — vertical focal length (pixels)

Position in K: row 1, col 1.

The vertical focal length. For cameras with square pixels fy ≈ fx. Non-square sensors (rare in modern hardware) have fy ≠ fx.

  • Large fy → narrow vertical field of view.
  • Small fy → wide vertical field of view.

Relation to vertical field of view FoV_v:

fy  =  (image_height / 2) / tan(FoV_v / 2)

Read from a calibration matrix: K[1][1].

cx — horizontal principal point (pixels)

Position in K: row 0, col 2.

The x-coordinate of the optical axis on the image sensor — the pixel column where a ray travelling straight through the centre of the lens hits the sensor. Ideally the exact horizontal centre of the image.

  • For a 640-wide image: cx ≈ 319.5
  • For a 1920-wide image: cx ≈ 959.5

A miscalibrated cx shifts the entire point cloud left or right, making the scene appear viewed from an off-centre position.

Read from a calibration matrix: K[0][2].

cy — vertical principal point (pixels)

Position in K: row 1, col 2.

The y-coordinate of the optical axis on the sensor. Ideally the exact vertical centre of the image.

  • For a 480-tall image: cy ≈ 239.5
  • For a 720-tall image: cy ≈ 359.5

A miscalibrated cy shifts the entire point cloud up or down.

Read from a calibration matrix: K[1][2].

Off-diagonal zeros and the bottom row

        | fx    0   cx |
K  =    |  0   fy   cy |
        |  0    0    1 |
  • K[0][1] = 0 — the skew coefficient. Zero for all modern digital cameras (pixels are rectangular).
  • K[1][0] = 0 — symmetric constraint; always zero.
  • K[2][0] = K[2][1] = 0, K[2][2] = 1 — homogeneous row. Required by the projective geometry convention; never changes.

You will never need to set these values — they are always fixed.

Where to get K for your camera

Source How
Camera SDK Most SDKs expose intrinsics directly. e.g. intr.fx, intr.fy, intr.ppx, intr.ppy on Intel RealSense.
OpenCV calibration cv2.calibrateCamera returns a camera_matrix; read K[0,0], K[1,1], K[0,2], K[1,2].
Camera datasheet Look for focal length in mm and sensor pixel pitch; fx = f_mm / pixel_size_mm.
Field of view approximation fx = (W/2) / tan(FoV_h/2), cx = W/2 − 0.5. Accurate enough for a first test.

Using K in vizion3d

Supply the four values via advanced_config on any depth or stereo command:

from vizion3d.lifting import DepthEstimationAdvanceConfig

config = DepthEstimationAdvanceConfig(
    fx=909.15,   # K[0,0]
    fy=908.48,   # K[1,1]
    cx=640.0,    # K[0,2]
    cy=360.0,    # K[1,2]
)

Without advanced_config, vizion3d defaults to the PrimeSense / Kinect v1 intrinsics at 640×480:

        | 525.0    0.0   319.5 |
K  =    |   0.0  525.0   239.5 |
        |   0.0    0.0     1.0 |

For a different camera or resolution, always supply calibrated values — intrinsics that do not match your camera produce correct topology but geometrically distorted metric scale.

See the full field reference and per-entry-point usage examples in the Advanced Config pages: