Advanced Computer Vision Structure from Motion Structure from

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Advanced Computer Vision Structure from Motion

Advanced Computer Vision Structure from Motion

Structure from Motion l Geometric structure-from-motion problem: using image matches to estimate: l l

Structure from Motion l Geometric structure-from-motion problem: using image matches to estimate: l l The 3 D positions of the corresponding scene points in some fixed coordinate (the scene structure) The projection matrices associated with the camera observing them (the motion of the points relative to the cameras)

Two Types l l Affine structure from motion (Chapter 12) Projective structure from motion

Two Types l l Affine structure from motion (Chapter 12) Projective structure from motion (Chapter 13)

Basic Assumption l l l The cameras’ positions and possibly their intrinsic parameters are

Basic Assumption l l l The cameras’ positions and possibly their intrinsic parameters are a priori unknown and may change over time. Different from, say, stereopsis, where the camera parameters are known so we can focus on the correspondence problem. Chapters 12 and 13 ignore the correspondence problem, assuming that the projections of n points have been matched across m pictures.

Applications l l Image-based rendering: a video clip recorded by a hand-held camcorder, possibly

Applications l l Image-based rendering: a video clip recorded by a hand-held camcorder, possibly zooming during the shoot, is used to capture the shape of an object and render it under new viewing conditions. (Chapter 26). Active vision system: robot probes.

Affine Structure from Motion Problem: estimate the m 2 x 4 matrices Mi i

Affine Structure from Motion Problem: estimate the m 2 x 4 matrices Mi i and the n positions Pjj from the mxn correspondences pij. ij 2 mn equations in 8 m+3 n unknowns Overconstrained problem, that can be solved using (non-linear) least squares!

Ambiguity of Solution If M i and P are solutions, j So are M’i

Ambiguity of Solution If M i and P are solutions, j So are M’i and P’ where j and Q is an affine transformation.

The Projective Structure-from-Motion Problem Given m perspective images of n fixed points P we

The Projective Structure-from-Motion Problem Given m perspective images of n fixed points P we can write j Problem: estimate the m 3 x 4 matrices M iand the n positions P j from the mn correspondences p. ij 2 mn equations in 11 m+3 n unknowns Overconstrained problem, that can be solved using (non-linear) least squares!