Camera Intrinsics and Extrinsics - 5 Minutes with Cyrill's Video - 10 Minutes with Shannon's Note

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Camera Intrinsics and Extrinsics - 5 Minutes with Cyrill's Video - 10 Minutes with Shannon's Note

original video: https://www.youtube.com/watch?v=ND2fa08vxkY

markdown note: https://github.com/shannon112/5_Minutes_with_Cyrill_Notes/blob/main/Camera_Intrinsics_and_Extrinsics.md

what are intrinsics and extrinsics?

  • intrinsics and extrinsics are parametersof a camera model
  • a camera model is a mathematical description of how a camera works
  • aka. interior and exterior orientation of a camera
  • they cover the whole process of getting a 3D point in world coordinate onto the image plane

extrinsics

  • basically describe where is my camera in the 3d world
  • it's basically xyz location where your camera is and a 3d orientation where your camera is actually looking to
  • aka. position/translation and orientation/rotation, or simply pose
  • it's a 6 degree of freedom vector or 6 dimensional vector

camera localization

  • if you want to perform camera localization, we want to compute the extrinsics of our camera
  • typically we use the projection center of our camera as the pose of our camera
  • the point according to the pinhole model where all rays intersect

extrinsices v.s. transformation matrix

  • extrinsics is a world to camera transformation matrix, if defined as [R|t]
  • the intuitive camera pose is camera to world transformation matrix, can be defined as [R^T|-R^Tt]
  • camera origin in camera coordinate camera to world transformation matrix
    = [I|(0,0,0)^T]     [R^T|-R^Tt] = camera pose in world coordinate

intrinsics

  • the intrinsics are the parameters that basically sit inside the camera
  • basically describe how a point in the 3D world (camera coordinate) is mapped onto the 2d image plane
  • to describe the intrinsics we typically use at least 4 or 5 parameters (it depends if you have a digital camera or an analog camera)

so what are those intrinsic parameters? focal length

  • the camera constant which is basically the distance of your image plane to your production center
  • a scale difference in x and y and depending on the lens
  • focal lengths fx and fy as two parameters but they are basically equivalent one can be very trivially mapped into the other

so what are those intrinsic parameters? principle point

  • principal point is basically the pixel in your image through which the optical axis of your camera passes
  • it basically describes where that point sits in your image typically it's somewhere near the center of your image
  • but of course not precisely because your chip is not precisely glued into your camera

so what are those intrinsic parameters? shear

  • and if you have an analog camera then you also have a shear parameter aka. axis skew s
  • in most digital cameras this sheer parameter should be very close to zero
  • s/fx = tan(alpha)

direct linear transform (DLT)

  • described how a point from the 3d world is actually mapped onto the image plane
  • the direct linear transform is a 11 degree of freedom
  • transformation taking the 6DoF from the extrinsics and the 5 DoF from the intrinsics into account
  • describe a so-called a fine camera model or the model of the fine camera
  • a camera which has a perfect lens so there are no lens distortions or other distortions involved in here

6 control points to solve

  • dlt can be computed using 6 or more control points so points with known coordinates in the environment
  • known extrinsics, 5 equations can used to solve 5 unknown intrinsics
  • known intrinsics, 6 equations can used to solve 6 unknown extrinsics
  • so 6 control points can first solve intrinsics and then solve extrinsics

distortion

  • in practice we have a few more parameters involved for example for lens distortion
  • for example if you have a barrel distortion or a cushion distortion or any other form of distortion in your lens
  • which adds additional parameters so-called non-linear parameters to your model and that you need to estimate additionally

x = PX

  • once you have all those parameters you can actually map any point from the 3d world onto the 2d image plane
  • X is in the 3d world
  • P is a projection matrix, all the intrinsics and extrinsics sit inside
  • x is the point in your image plane here expressed in something which is called homogeneous coordinates

camera calibration

  • we can estimate those intrinsics and extrinsics using calibration patterns, typically call camera calibration
  • if you want to compute only the extrinsics then the something which would typically refer to as camera localization
  • you do them if you want to perform measurements or perform geometric estimations with your camera

References

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