Title: CS4277_3CV / Single View Metrology
On every note of this deck, unless a note says otherwise:
Title: CS4277_3CV::Single View Metrology
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Front: What is a forward projection?
Back: A line in 3-space projects to a line in the image.

Front: Describe the following picture:
Back: Forward projection of line in 3-space to a line in the image.

Front: Given two 3-space points [$]\bold{A}, \bold{B}[/$], where [$]\bold{a}, \bold{b}[/$] are their images under P, then a point [$]\bold{X}(\mu) = \bold{A} + \mu\bold{B}[/$] on L projects to a point ...
Back: [$$]x(\mu) = P(\bold{A} + \mu\bold{B}) = P\bold{A} + \muP\bold{B} = \bold{a} + \mu \bold{b}[/$$]

Front: What is the back-projection of lines?
Back: The set of points in space which map to a line in the image is a plane in space defined by the camera centre and image line.

Front: The set of points in space mapping to a line l via the camera matrix P is the plane...
Back: [$$]\pi = P^Tl[/$$]

Front: What is the back-projection of conics under the camera P?
Back: Conic C back-projects to a cone: [$$]Q_\text{co} = P^TCP

Front: What is a cone?
Back: A degenerate quadric: The 4x4 matrix Q does not have full rank.

Front: What is the forward-projection of a quadric under the camera matrix P?
Back: [$$]C^* = PQ^*P^T[/$$]

Front: What effect does the moving of the image plane by increasing the focal length have?
Back: Zooming

Front: What is the magnification factor, when moving the image plane?
Back: [$]f'/f[/$]

Front: How to manipulate K to account for a zooming by a factor k?
Back: Multiply the calibration matrix K on the right by [$]\text{diag}(k, k, 1)[/$].

Front: Consider the camera is rotated about its centre with no change in the internal parameters.
How are x and x' related?
Back: [$$]x' = Hx, \ H = KRK^{-1}[/$$]

Front: How do we call the homography [$]H = KRK^{-1}[/$], which is a pure rotation of the camera?
Back: Conjugate rotation.

Front: How can the angle of rotation between views computed by only having the conjugate rotation matrix H?
Back: From the phase of the complex eigenvalues of H

Front: How is moving the camera centre (motion parallax) often used?
Back: To partially determine the structure.

Front: What are synthetic views?
Back: New images corresponding to different camera orientations (same camera centre).

Front: How can synthetic views be generated?
Back: From an existing image by warping with planar homographies

Front: What is the algorithm for creating a synthetic view?
Back: 1. Compute the homography H which maps the image quadrilateral to a rectangle with correct aspect ratio.
2. Projectively warp the source image with this homography.

Front: Outline the Planar Panoramic Mosaicing algorithm.
Back: 1. Choose one image of the set as a reference
2. Compute the homography H (4-point) which maps one of the other images of the set to this reference image
3. Projectively warp the images with this homography, and augment the refgerence image with the non-overlapping part of the warped image
4. Repeat step 2 and 3 for the remaining images

Front: Points on [$]\pi_\infty[/$] may be written as [$]X_\infty = (\bold{d}^T, 0)^T, and are imaged by a general camera as ...
Back: [$$]x = PX_\infty = KR[I | -\tilde{C}] \begin{pmatrix} \bold{d} \\ 0\end{pmatrix} = KR\bold{d}[/$$]

Front: What is the image of the absolute conic (IAC)?
Back: The conic [$]\omega = (KK^T)^{-1} = K^{-T}K^{-1}[/$]

Front: What is the dual image of the absolute conic (DIAC)?
Back: [$$]\omega^* = w^{-1} = KK^T[/$$]

Front: How can K be identified, once the IAC is known?
Back: Via Cholesky factorization, i.e. [$]\omega^* = KK^T[/$]