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### Transcript of 1 Ch. 3: Geometric Camera Calibration Objective: Estimates the intrinsic and extrinsic parameters of...

• Ch. 3: Geometric Camera Calibration Objective: Estimates the intrinsic and extrinsic parameters of a camera.Idea: Formulate camera calibration as an optimization process, in which the discrepancy between the theoretical and observed image features is minimized w.r.t. the cameras parameters.Steps: (1) Evaluate the perspective projection matrix M of the camera, (2) Estimate the intrinsic and extrinsic parameters of the camera from M.

• Perspective Projection (Imaging Process)practically,whereideally,

• Evaluate MLetMeasure n pairs of correspondingimage and scene points.

• For each pair we obtainFor all pairs,

• In matrix form,where

• Solve for x using optimization techniques

• 3.1 Least-Squares Parameter Estimation 3.1.1 Linear Least-Squares Methods Consider a system of p linear equations in q unknowns:Idea: Find the solution x by minimizing the squared deviation ( ) from theoretical (Ux) to observed (y) image features

• ,In vector-matrix form, , where

• Let Consider the over-constrained case (p > q) Find x that minimizes the errorThe normal equations: the pseudoinverse of U.

• Homogenous systems: Two issues: (i) By equation , we obtain trivial(ii) If x is a solution, x is also a solution. The least squares error solution of is the eigenvector of to the smallest eigenvalue.solution To resolve the issues, impose corresponding

• Find the least squares error solution by the method of Lagrange multipliers Error: We obtain The solution x is an eigenvector of with eigenvalue : Lagrange multiplierLet ConstraintMinimizewhere

• The associated error The least squares error solution to is the eigenvector of to the smallest eigenvalue.correspondingExample:

Fit a line to a set of data points in the 2D space

• Line equation: Let

• The perpendicular distance from pointto lineError measure: Minimize E w.r.t. (a, b, d)Letis

• where Recall , whose squared error The solution of min w.r.t. n under is the unit eigenvectorconstraint with the minimum eigenvalue of

• where 3.1.2 Nonlinear Least-Squares Methods

• f(x): f(x) :e.g.,e.g.,

• : the Jacobian of f where

• Taylor expansion ofaround pointf(x): f(x) :

• Newtons Method (Gradient Descent) (i) Square Systems (p = q) Idea: Given an initial x, find s.t..Since .When: nonsingular,LetRepeat until f(x) stabilizes at some x Drawbacks: i) Square system, ii) Nonsingulariii) Locally optimal.

• Finding x s.t.Finding x s.t. F(x) = 0 (square system)Since: p by q matrix, f(x): p by 1, : q by 1

• where: q by q matrix

• (g) Proof: From

• Fromand

• Degenerated Point Configurations e.g., points lie on a line or a plane, may cause failure of camera calibration.3.3. Shape Distortions Types of distortions: (a) Tangential distortion (b) Radial distortion Barrel distortion, Pincushion distortion

• Radial distortion: (a) Changes the distance between the image center and the image point (b) Does not affect the direction joining the image center and the image pointd: actual distance: distorted distance: distortion function

• : coefficientswhere Polynomial model: FOV model: Logarithmic model, Fisheye model, Radial model, Rational function model where : distortion coefficient

• Given an image point (u,v), determine its actual d Consider Polynomial model

• Determine the distortion functioni.e., determine its coefficients

• 3.5 Application: Mobile Robot Localization -- Calibrate a static camera for monitoring a robot

• 20 images of the planar rectangular gridImage resolution: 576 by 768Camera: height = 4m, focal length = 4.5mm, Skew = 0, precision = 0.1 pixel 3 radial distortion coefficients.Experimental results:Localization error: 2 cm in position and 1 degree in orientationMaximum error: 5 cm in position and 5 degrees in orientation

• 3.4. A Nonlinear Approach

• and