Pin Hole Cameras & Warp Functions
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1 Pin Hole Cameras & Warp Functions Instructor - Simon Lucey Designing Computer Vision Apps
2 Today Pinhole Camera. Homogenous Coordinates. Planar Warp Functions.
3 Example of SLAM for AR Taken from: H. Liu et al. Robust Keyframe-based Monocular SLAM for Augmented Reality, ISMAR 2016.
4 Example of SLAM for AR Taken from: H. Liu et al. Robust Keyframe-based Monocular SLAM for Augmented Reality, ISMAR 2016.
5 Example of SLAM for AR Taken from: H. Liu et al. Robust Keyframe-based Monocular SLAM for Augmented Reality, ISMAR 2016.
6 Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince Motivation
7 Taken from: Pinhole Camera
8 Pinhole Camera Real camera image is inverted Instead model impossible but more convenient virtual image Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
9 Pinhole Camera Terminology Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
10 Normalized Camera By similar triangles: Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
11 Focal length parameters Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
12 Focal length parameters Can model both the effect of the distance to the focal plane the density of the receptors with a single focal length parameter φ In practice, the receptors may not be square: So use different focal length parameter for x and y dims Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
13 Offset parameters Current model assumes that pixel (0,0) is where the principal ray strikes the image plane (i.e. the center) Model offset to center Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
14 Skew parameter Finally, add skew parameter Accounts for image plane being not exactly perpendicular to the principal ray Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
15 Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince Radial distortion
16 Camera & World Coordinates w o world coordinate frame u
17 Camera & World Coordinates w w 0 u 0 camera coordinate frame o 0 o world coordinate frame apple u 0 w 0 u apple!1! = 2! 3! 4 apple u w + apple x z
18 Camera & World Coordinates Rotation Matrix Translation Vector apple u 0 apple u w + apple x w 0 = apple!1! 2! 3! 4 z
19 Position and orientation of camera Position w=(u,v,w) T of point in the world is generally not expressed in the frame of reference of the camera. Transform using 3D transformation or Point in frame of reference of camera Point in frame of reference of world
20 Constraints on As is a rotation matrix it is constrained by the following, T = I det( ) =1 We refer to these matrices as belonging to the Special Orthogonal Group - SO(3). How many degrees of freedom do you think has?
21 Something to try In MATLAB type, >> R1 = orth(randn(3,3)); >> R1(:,end) = det(r1)*r1(:,end); >> R2 = orth(randn(3,3)); >> R2(:,end) = det(r2)*r2(:,end); If you form a new matrix as a linear combination of R1 & R2, >> R3 = 0.5*R *R2; Does R3 lie in SO(3)?
22 Reminder: Convex Set 18
23 Reminder: Convex Set 18
24 Reminder: Convex Set 18
25 Reminder: Non-Convex Set 19
26 Reminder: Non-Convex Set 19
27 Reminder: Non-Convex Set 19
28 Complete pinhole camera model Intrinsic parameters (stored as intrinsic matrix) Extrinsic parameters Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
29 Complete pinhole camera model For short: Question: is a linear function? Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
30 Perspective Transform
31 Learning extrinsic parameters ˆ, ˆ =min, NX n=1 {x n pinhole[w n,,, ]} e.g. {x} = x 2 2 Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
32 Learning intrinsic parameters ˆ =min [min, NX n=1 {x n pinhole[w n,,, ]}] e.g. {x} = x 2 2 Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
33 Camera Calibration Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince Use 3D target with known 3D points.
34 For you to try.. There exists camera calibration tools in MATLAB, see Bouget s Calibration Toolbox in MATLAB. Or if you prefer, you can use OpenCV s tutorial. What are the intrinsics of your device? How sensitive are vision algorithms to the correct intrinsics?
35 Today Pinhole Camera. Homogenous Coordinates. Planar Warp Functions.
36 Homogeneous Coordinates Convert 2D coordinate to 3D To convert back Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
37 Geometric interpretation Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
38 Pinhole camera Camera model: In homogeneous coordinates: (linear!) Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
39 Pinhole camera Writing out these three equations Eliminate λ to retrieve original equations Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
40 Adding in extrinsics Or for short: Or even shorter: Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
41 Today Pinhole Camera. Homogenous Coordinates. Planar Warp Functions.
42 Planar Warp Functions Consider viewing a planar scene There is now a 1 to 1 mapping between points on the plane and points in the image We will investigate models for this 1 to 1 mapping Euclidean Similarity Affine Homography
43 Piecewise planarity Many scenes are not planar, but are nonetheless piecewise planar Can we match all of the planes to one another?
44 Euclidean warp Consider viewing a fronto-parallel plane at a fixed distance D. In homogeneous coordinates, the imaging equations are: 3D rotation matrix becomes 2D (in plane) Plane at known distance D Point is on plane (w=0) Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
45 Euclidean warp Simplifying Rearranging the last equation Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
46 Euclidean warp Homogeneous: Cartesian: For short: Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince
47 Euclidean warp Homogeneous: Cartesian: For short: Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince How many unknowns?
48 More to read Prince et al. Chapter 14, Section 1 & 3. Chapter 15, Section 1.
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