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Two-view geometry

Invite you to consult the lecture content "Two-view geometry" below. Contents of lectures introduce to you the content: Epipolar geometry, the epipole, epipolar geometry, epipolar constraint, problem with eight point algorithm. Hopefully document content to meet the needs of learning, work effectively. | Two-view geometry Epipolar Plane – plane containing baseline (1D family) Epipoles = intersections of baseline with image planes = projections of the other camera center Baseline – line connecting the two camera centers Epipolar geometry X x x’ The Epipole Photo by Frank Dellaert Epipolar Plane – plane containing baseline (1D family) Epipoles = intersections of baseline with image planes = projections of the other camera center Epipolar Lines - intersections of epipolar plane with image planes (always come in corresponding pairs) Baseline – line connecting the two camera centers Epipolar geometry X x x’ Example: Converging cameras Example: Motion parallel to image plane e e’ Example: Forward motion Epipole has same coordinates in both images. Points move along lines radiating from e: “Focus of expansion” Epipolar constraint If we observe a point x in one image, where can the corresponding point x’ be in the other image? x x’ X Potential matches for x have to lie on the corresponding epipolar line l’. Potential matches for x’ have to lie on the corresponding epipolar line l. Epipolar constraint x x’ X x’ X x’ X Epipolar constraint example X x x’ Epipolar constraint: Calibrated case Assume that the intrinsic and extrinsic parameters of the cameras are known We can multiply the projection matrix of each camera (and the image points) by the inverse of the calibration matrix to get normalized image coordinates We can also set the global coordinate system to the coordinate system of the first camera X x x’ Epipolar constraint: Calibrated case R t The vectors x, t, and Rx’ are coplanar = RX’ + t X’ is X in the second camera’s coordinate system We can identify the non-homogeneous 3D vectors X and X’ with the homogeneous coordinate vectors x and x’ of the projections of the two points into the two respective images Essential Matrix (Longuet-Higgins, 1981) Epipolar constraint: Calibrated case X x x’ The vectors x, t, and Rx’ . | Two-view geometry Epipolar Plane – plane containing baseline (1D family) Epipoles = intersections of baseline with image planes = projections of the other camera center Baseline – line connecting the two camera centers Epipolar geometry X x x’ The Epipole Photo by Frank Dellaert Epipolar Plane – plane containing baseline (1D family) Epipoles = intersections of baseline with image planes = projections of the other camera center Epipolar Lines - intersections of epipolar plane with image planes (always come in corresponding pairs) Baseline – line connecting the two camera centers Epipolar geometry X x x’ Example: Converging cameras Example: Motion parallel to image plane e e’ Example: Forward motion Epipole has same coordinates in both images. Points move along lines radiating from e: “Focus of expansion” Epipolar constraint If we observe a point x in one image, where can the corresponding point x’ be in the other image? x x’ X Potential matches for x .