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Primitives and Transformations

  • inhomogeneous coordinate

    \(\tilde x = (\tilde x, \tilde y, \tilde z, \tilde w)^T\)

  • homogeneous coordinate

    \[x = (x, y, z)^T\]
  • Transformations

image-20240215182631979

  • Transformation relation of lines and pixels

    \(\tilde{x'} = \tilde H \tilde x\)

    \(\tilde {l'}\tilde x = \tilde{l'^T}\tilde H \tilde x = (\tilde {H^T} \tilde l' )^T \tilde x = \tilde l ^T \tilde x = 0\)

    \(\tilde l' = \tilde{H^{-T}} \tilde l\)

  • Direct Linear Transform for Homography Estimation

    ``` u, d, v_trans = np.linalg.svd(A)

    H is the homography function

约 114 个字

H = v_trans[-1].reshape(3,3) ```

Geometric Image Formation

  • Orthographic projection

    image-20240215194133790

  • Perspective projection

    image-20240215194228983

image-20240215194302476

K is the camera intrinsic, intrinsic matrix

s = 0 in practice

  • Chaining transformation

    image-20240215194438425

    image-20240215194451856

    Photometric Image formation

    • Rendering equation

      image-20240216100502459