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The inverse Gaussian distribution is a two-parameter exponential family with natural parameters −λ/(2μ 2) and −λ/2, and natural statistics X and 1/X. For λ > 0 {\displaystyle \lambda >0} fixed, it is also a single-parameter natural exponential family distribution [ 3 ] where the base distribution has density
This has the convenient implication for 2 × 2 and 3 × 3 rotation matrices that the trace reveals the angle of rotation, θ, in the two-dimensional space (or subspace). For a 2 × 2 matrix the trace is 2 cos θ, and for a 3 × 3 matrix it is 1 + 2 cos θ. In the three-dimensional case, the subspace consists of all vectors perpendicular to the ...
a concatenation of a 3D rotation matrix and a 3-dimensional translation vector. This type of camera matrix is referred to as a normalized camera matrix , it assumes focal length = 1 and that image coordinates are measured in a coordinate system where the origin is located at the intersection between axis X3 and the image plane and has the same ...
Given 3D point = (,,) with world coordinates in a reference frame (,,), observed from different views, the inverse depth parametrization of is given by: = (,,,,,) where the first five components encode the camera pose in the first observation of the point, being = (,,) the optical centre, the azimuth, the elevation angle, and = ‖ ‖ the inverse depth of at the first observation.
p ↦ q p for q = 1 + i + j + k / 2 on the unit 3-sphere. Note this one-sided (namely, left) multiplication yields a 60° rotation of quaternions. The length of is √ 3, the half angle is π / 3 (60°) with cosine 1 / 2 , (cos 60° = 0.5) and sine √ 3 / 2 , (sin 60° ≈ 0.866). We are therefore dealing with a ...
Similarly, the inverse image (or preimage) of a given subset of the codomain is the set of all elements of that map to a member of . The image of the function f {\displaystyle f} is the set of all output values it may produce, that is, the image of X {\displaystyle X} .