Last 4 lectures Camera Structure HDR Image Filtering Image
Transform Slide 2 Today Camera Projection Camera Calibration Slide
3 Pinhole camera Slide 4 Pinhole camera model (optical center)
origin principal point P (X,Y,Z) p (x,y) The coordinate system We
will use the pin-hole model as an approximation Put the optical
center (Center Of Projection) at the origin Put the image plane
(Projection Plane) in front of the COP (Why?) Slide 5 Pinhole
camera model principal point Slide 6 Pinhole camera model (optical
center) origin principal point P (X,Y,Z) p (x,y) x y Slide 7
Pinhole camera model (optical center) origin principal point P
(X,Y,Z) p (x,y) x y Slide 8 Intrinsic matrix non-square pixels
(digital video) Is this form of K good enough? Slide 9 Intrinsic
matrix non-square pixels (digital video) skew Is this form of K
good enough? Slide 10 Intrinsic matrix non-square pixels (digital
video) skew radial distortion Is this form of K good enough? Slide
11 Distortion Radial distortion of the image Caused by imperfect
lenses Deviations are most noticeable for rays that pass through
the edge of the lens Slide 12 Barrel Distortion No distortion
Barrel Wide Angle Lens Slide 13 Pin Cushion Distortion Pin cushion
Telephoto lens No distortion Slide 14 Modeling distortion To model
lens distortion Use above projection operation instead of standard
projection matrix multiplication 2. Apply radial distortion 3.
Apply focal length translate image center 1. Project (X, Y, Z) to
normalized image coordinates Distortion-Free:With Distortion: Slide
15 Camera rotation and translation extrinsic matrix Slide 16 Two
kinds of parameters internal or intrinsic parameters: focal length,
optical center, skew external or extrinsic (pose): rotation and
translation: Slide 17 Other projection models Slide 18 Orthographic
projection Special case of perspective projection Distance from the
COP to the PP is infinite Image World Also called parallel
projection: (x, y, z) (x, y) Slide 19 Other types of projections
Scaled orthographic Also called weak perspective Affine projection
Also called paraperspective Slide 20 Fun with perspective Slide 21
Perspective cues Slide 22 Slide 23 Fun with perspective Ames room
Slide 24 Forced perspective in LOTR Elijah Wood: 5' 6" (1.68 m)Ian
McKellen 5' 11" (1.80 m) Slide 25 Camera calibration Slide 26
Estimate both intrinsic and extrinsic parameters Mainly, two
categories: 1.Using objects with known geometry as reference 2.Self
calibration (structure from motion) Slide 27 Camera calibration
approaches Directly estimate 11 unknowns in the M matrix using
known 3D points (Xi,Yi,Zi) and measured feature positions (ui,vi)
Slide 28 Linear regression Slide 29 Slide 30 Solve for Projection
Matrix M using least- square techniques Slide 31 Normal equation
(Geometric Interpretation) Given an overdetermined system the
normal equation is that which minimizes the sum of the square
differences between left and right sides Slide 32 Normal equation
(Differential Interpretation) n x m, n equations, m variables Slide
33 Normal equation Who invented Least Square? Carl Friedrich Gauss
Slide 34 Nonlinear optimization A probabilistic view of least
square Feature measurement equations Likelihood of M given {(u i,v
i )} Slide 35 Optimal estimation Log likelihood of M given {(u i,v
i )} It is a least square problem (but not necessarily linear least
square) How do we minimize C? Slide 36 Nonlinear least square
methods Slide 37 Least square fitting number of data points number
of parameters Slide 38 Nonlinear least square fitting Slide 39
Function minimization It is very hard to solve in general. Here, we
only consider a simpler problem of finding local minimum. Least
square is related to function minimization. Slide 40 Function
minimization Slide 41 Quadratic functions Approximate the function
with a quadratic function within a small neighborhood Slide 42
Function minimization Slide 43 Computing gradient and Hessian
Gradient Hessian Slide 44 Computing gradient and Hessian Gradient
Hessian Slide 45 Computing gradient and Hessian Gradient Hessian
Slide 46 Computing gradient and Hessian Gradient Hessian Slide 47
Computing gradient and Hessian Gradient Hessian Slide 48 Searching
for update h GradientHessian Idea 1: Steepest Descent Slide 49
Steepest descent method isocontourgradient Slide 50 Steepest
descent method It has good performance in the initial stage of the
iterative process. Converge very slow with a linear rate. Slide 51
Searching for update h GradientHessian Idea 2: minimizing the
quadric directly Converge faster but needs to solve the linear
system Slide 52 Recap: Calibration Directly estimate 11 unknowns in
the M matrix using known 3D points (Xi,Yi,Zi) and measured feature
positions (ui,vi) Camera Model: Slide 53 Recap: Calibration
Directly estimate 11 unknowns in the M matrix using known 3D points
(Xi,Yi,Zi) and measured feature positions (ui,vi) Linear Approach:
Slide 54 Recap: Calibration Directly estimate 11 unknowns in the M
matrix using known 3D points (Xi,Yi,Zi) and measured feature
positions (ui,vi) NonLinear Approach: Slide 55 Practical Issue is
hard to make and the 3D feature positions are difficult to measure!
Slide 56 A popular calibration tool Slide 57 Multi-plane
calibration Images courtesy Jean-Yves Bouguet, Intel Corp.
Advantage Only requires a plane Dont have to know
positions/orientations Good code available online! Intels OpenCV
library: http://www.intel.com/research/mrl/research/opencv/
http://www.intel.com/research/mrl/research/opencv/ Matlab version
by Jean-Yves Bouget:
http://www.vision.caltech.edu/bouguetj/calib_doc/index.html
http://www.vision.caltech.edu/bouguetj/calib_doc/index.html
Zhengyou Zhangs web site:
http://research.microsoft.com/~zhang/Calib/
http://research.microsoft.com/~zhang/Calib/ Slide 58 Step 1: data
acquisition Slide 59 Step 2: specify corner order Slide 60 Step 3:
corner extraction Slide 61 Slide 62 Step 4: minimize projection
error Slide 63 Step 4: camera calibration Slide 64 Slide 65 Step 5:
refinement
