Robotics (IT-802 (C)) - Important Questions

Derive the pinhole camera projection equation and explain the roles of intrinsic and extrinsic parameters. Show how a 3D world point $X_\text{w}=\left[X\;Y\;Z\;1\right]^T$ is mapped to an image point $x=\left[u\;v\;1\right]^T$ using the camera matrix $P$. Explicitly state $P$ in terms of $K$, $R$, and $t$.

Describe the steps of camera calibration using a planar checkerboard (Zhang's method). Explain how intrinsic parameters, distortion coefficients, and extrinsic parameters are estimated and how reprojection error is computed.

Explain radial and tangential lens distortion models. Give the equations for correcting distorted image coordinates and describe how distortion coefficients are included in the reprojection function during calibration.

Explain homogeneous coordinates and show how perspective projection can be represented as a linear mapping in homogeneous coordinates. Why are homogeneous coordinates necessary for projective transformations?

Compare Euclidean, similarity, affine, and projective transformations in 2D. For each transformation type give its $3\times 3$ homogeneous matrix form, list its degrees of freedom, and state one geometric invariant preserved by that transformation.

Derive the Direct Linear Transform (DLT) algorithm for estimating a planar homography $H$ from point correspondences. State the minimum number of correspondences required and explain the normalization step and why it is important.

Define the fundamental matrix $F$ and the essential matrix $E$. State and explain the epipolar constraint. Show the relation between $E$ and $F$ when camera intrinsics $K$ and $K'$ are known, i.e. express $E$ in terms of $F$, $K$, and $K'$.

Given the essential matrix $E$, explain how to recover the relative rotation $R$ and translation direction $t$ between two calibrated views. Outline the SVD-based decomposition and describe how four possible solutions are disambiguated using cheirality (positive depth) constraints.

Explain the RANSAC algorithm for robust estimation of a homography or fundamental matrix. Describe the algorithm steps and discuss how model verification and inlier selection are performed.

Define reprojection error for camera calibration. How is mean reprojection error computed over multiple calibration views? Discuss why minimizing geometric (reprojection) error is preferable to minimizing algebraic error.

Explain the thin lens model and derive the relationship between object distance, image distance, and focal length. Describe how focal length is represented in the intrinsic matrix $K$ and its effect on perspective projection.

What is stereo image rectification and why is it used? Describe the steps to compute rectifying homographies for a calibrated stereo pair and explain how rectification simplifies stereo matching.