Single-view geometry Odilon Redon, Cyclops, 1914

Our goal: Recovery of 3D structure Recovery of structure from one image is inherently ambiguous x X?

Our goal: Recovery of 3D structure Recovery of structure from one image is inherently ambiguous

Our goal: Recovery of 3D structure Recovery of structure from one image is inherently ambiguous

Ames Room

Our goal: Recovery of 3D structure We will need multi-view geometry

Recall: Pinhole camera model Principal axis: line from the camera center perpendicular to the image plane Normalized (camera) coordinate system: camera center is at the origin and the principal axis is the z-axis

Recall: Pinhole camera model

Principal point Principal point (p): point where principal axis intersects the image plane Normalized coordinate system: origin of the image is at the principal point Image coordinate system: origin is in the corner How to go from normalized coordinate system to image coordinate system? pxpx pypy

Principal point offset principal point: pxpx pypy

Principal point offset calibration matrix principal point:

Pixel coordinates m x pixels per meter in horizontal direction, m y pixels per meter in vertical direction Pixel size: pixels/m mpixels

Camera rotation and translation In general, the camera coordinate frame will be related to the world coordinate frame by a rotation and a translation coords. of point in camera frame coords. of camera center in world frame coords. of a point in world frame Conversion from world to camera coordinate system (in non-homogeneous coordinates):

Camera rotation and translation Note: C is the null space of the camera projection matrix (PC=0)

Camera parameters Intrinsic parameters Principal point coordinates Focal length Pixel magnification factors Skew (non-rectangular pixels) Radial distortion

Camera parameters Intrinsic parameters Principal point coordinates Focal length Pixel magnification factors Skew (non-rectangular pixels) Radial distortion Extrinsic parameters Rotation and translation relative to world coordinate system

Camera calibration Source: D. Hoiem

Camera calibration Given n points with known 3D coordinates X i and known image projections x i, estimate the camera parameters XiXi xixi

Camera calibration: Linear method Two linearly independent equations

Camera calibration: Linear method P has 11 degrees of freedom One 2D/3D correspondence gives us two linearly independent equations Homogeneous least squares: find p minimizing ||Ap|| 2 Solution given by eigenvector of A T A with smallest eigenvalue 6 correspondences needed for a minimal solution

Camera calibration: Linear method Note: for coplanar points that satisfy Π T X=0, we will get degenerate solutions (Π,0,0), (0,Π,0), or (0,0,Π)

Camera calibration: Linear method Advantages: easy to formulate and solve Disadvantages Doesn’t directly tell you camera parameters Doesn’t model radial distortion Can’t impose constraints, such as known focal length and orthogonality Non-linear methods are preferred Define error as squared distance between projected points and measured points Minimize error using Newton’s method or other non-linear optimization Source: D. Hoiem