Geometry from images --------------------------- IBM = geometry + images what geometric info? raysets -> camera pose depth/disparity based reprojection -> camera pose + depth/disparity model + texture (hybrid) -> approximate model of the scene + IB texture Goal: automatically extract geometric information about the scene/ camera position from a set of images Assumptions: scene does not change a set of primitive have been tracked in image set (correspondence) here: point correspondence ---------------------------------- What kind of geometric information? Euclidean model + describes the real world object/scene + natural to think of for most people - angles, distances make sense - difficult to extract: camera is not a measurement device (nonlinear problem) Is it really necessary? - rendering - needs only projection of the model on the image plane - robotics - visual servoing - alignment can be done with image features Other models - Projective Affine ---------------------------------------------------- Stratification Projective > Affine > Metric > Euclidean ex 2D ex 3D -transformation (collineation): group of transformations from a stratum to itself -invariant: property of a geometric configuration that does not change when a transformation of a given group is applied to that geometric configuration Group transformation DOF invariants distortion ---------------------------------------------------------------- Euclidean Metric Affine Projective - Projective stratum ---------------------- Intro to projective geometry Projective geometry was invented by the French mathematician Desargues (1591-1661) `Two triangles are in perspective from a point if and only if they are in perspective from a line'' Theorem: Let A, B, C and A', B, C' be two triangles in the (projective) plane. The lines AA', BB', CC' intersect in a single point if and only if the intersections of corresponding sides (AB, A'B'), (BC, B'C'), (CA, C'A') lie on a single line. 2D (projective plane): point: (same point is scale ~, relation to R: R3-(0 0 0) R2+points at infinity) line ideal point, line at infinity duality Duality Principle: For any projective result established using points and hyperplanes, a symmetrical result holds in which the roles of hyperplanes and points are interchanged: points become planes, the points in a plane become the planes through a point, etc. (Sheriff's applets) lTm = 0 point on line; line of 2 pts; inters of lines model of P2 collineation -map between planes H line->line (incidence of points is preserved) lTH-1Hx=lTx=0 cross ratio topology ?? 3D (projective 3D space) point plane point/plane at inf conics and quadrics conic, dual conic - quadric summary - Affine stratum --------------------- transformations invariants summary from projective to affine (fix plane at infinity) - Metric stratum -------------------- transformations invariants summary from affine to metric (fix absolute conic) ************************************************* How is this connected to imaging process ?? Camera models projective, orthographic, weak perspective, paraperspective ... projection connection to previous theory (vanishing points = ...) image of a plane calibration (grid, vanishing points) * APPL:raysets ----------------------------------- Multi view geometry Pb - resection intersection SFM projective structure affine structure Euclidean structure - Recovering projective structure 2 views: epipolar constraint, F, stereo (calibrated cameras) * APPL: depth/disparity based reprojection (Laveau& Faugeras, Warping eq, Layer depth images, Plenoptic modeling ... multiviews: affine factorization projective pb: occlusions, depth for projective fact, noise bundle adjustment - Recovering Affine/Metric autocalibration degenerate configurations A complete system: from image points to 3D structure and camera motions * APPL: hybrid IB-G models: Debevec, P Allen, Pollyfyes, Zisserman... ------------------------------------- References