02. Motion and Scene representation

• the global frame / coordinate systrem we use

• -> allows us to provide absolute positions for individual cameras / local coordinate systems

• in relation to their global position

• only in the relative positions from two or more cameras

• but not in a global context

• we transform local frames

• -> change the coordinate systems

• but usually fix the points (static)

• basis * coefficients -> point in space

• meaning the point is the same but on the right hand side with new basis (and thus also new coefficients)

• special orthogonal group

• a = R a’

• RR^T = I (rotation matrix * translated rotaiotn matrix = identity matrix)

• det(R) = 1

• a’ = R^-1 a = R^T a

• a’ = R a + t

• rotation matrix R

• point a

• translation t

• Transformatoin matrix T

• requires a, b, c to be (from 3 to) 4 dimensional…

• rotation and translation matrix (set)

• matrix multiplicatoin (operator)

• have point p in origional frame

• apply translation and rotation to compute same point

• in new frame (coordinate system)

=>

• provide the rotaiotn and translation

• from camera to world

  • not from world to camera…

• get the inverse/transpose from camera to world…

• 7

• 2D -> 4

  • rotation along each axis: 2

  • translation along each axis: 2

• 3D -> 6

  • ‘‘ -> 3

  • ‘‘ -> 3

• euclidean (6)

• plus scale (1)

• -> 7

• to express lines in R^3 (3D lines…)

• projection plane is based on origin of camera frame

• and 3D line

• =>

• v: direcotin of 3D line (typically unit vector)

• n: normal of projection plane

  • -> we have V

  • -> we have Q (perpendicular to v through origin)

  • n: normal of projection plane (n = Q x v)

• v gives directoin

• n gives distance and position to origin

• Q is the point in the image frame (w.r.t. coordinate system of image frame) that lies on the 3D line

  • -> Q gives point, v gives direction…

• use definintion of cross product

  • || a x b || = ||a|| ||b|| sin theta

• n = Q x v

• as Q x v orthogonal -> sin (90) = 1…

• -> || n || = d * ||v||

• with d = ||Q|| = ||n||/||v||

  • distance from origin to image plane…

• no -> usually use segments (3D lines with start and endpoint…)

• has 9 elements

• but actually only 3 degrees of freedom…

  • -> x,y,z axis rotation…

• => redundancy…!

• => there are better ways to express rotations

• roation matrix

• euler angles

• angle-axis (rotation vector)

• quaternion

• cayley’s representation

=> no ideal rotatoin representation for all purposes

-> are equivalent in some sense as they express the same

• roll axis

• pitch axis

• yaw axis

• intrinsic rotation

• extrinsic rotation

• extrinsic:

  • the initial axis stay fixed when we conduct the invididual rotations

  • -> comparable to world space

• intrinsic:

  • axis change with the rotation (dynamic)

  • -> rotation on one axis change palcement of other axis thus the following rotation…

  • -> compared to object space / camera frame

• degenerate case of euler rotations

• -> first two rotations (along z, y axis)

• -> lead to last rotation along x asix being along initial z axis

• give rotation axis e

• give rotation angle theta

-> rotation vector is theta e

• based on polar coordinates

-> similar to axis-angle representaiton

but: actual representation with 4-dim vector q

• q = (q0,q1,q2,q3)

• with

(n = [x,y,z]…)

• denote as quaternion with real coordinate equal to 0

• p = [0, x, y, z] = [0,v]

• Point cloud

• Voxel grid

• Implicit surface

• Mesh

• discrete set of data points in space

• may represent shape or object

• each point position has its set of cartesian coordinates and independent of each other

• similar to pixel grid but in 3D

• voxel: 3D cube

• => grid of values organized into layers, rows and columns (resulting in these 3D cubes…)

-> collectoin of vertices, edges and faces

• that define shape of polyhedral object

• triangle mesh

  • preffered if gemoetry function is easy and less complex

  • mostly for regular geometrical shapes

• quad mesh

  • (relatively) accurate results

  • more used in complex systems

• previous: explicit

• SDF: implicit

• create voxel grid

• calculate distance of actual surface to each voxel

• -> creates zeru-value SDF isosurface

  • voxel lies outside: distance positive

  • voxel lies inside: negative value

• based on this, regress a decision boundary

• => creating water-thight surface

• take voxel corners

• determin sign of these corners (positive: inside; negatrive: outside)

• create iso-surface delimiting these edges

• similar to point clouds

• but with lines…

• integrate infromation

• -> i.e. parallelism and orthogonality

  • man-made structures usually orthogonal and paralell

  • -> enforce this in line-clouds…

• -> enforce co-planarity

• assign semantic meaning to

• points of pointcloud

• Rotation and translation