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

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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

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