When we have scanlines in rectified images, how can we find correspondances? What methods are there?
search along same scanline (1D search)
two methods:
straightforward search by pixelwise similarity
-> smallest inteisity difference -> correspondence
more reliable strategy: block-wise similarity (i.e. window around the pixel of interest)
What is the advantage of windowing around the point of interest to search point correpondances?
average effects of noise or mis-calibration
What are the effects of window size on the disparity map?
smaller window: (-> if window = 1 -> degenerates to pixel-wise…)
more detail
more noise
larger window:
smoother disparity maps
less detail
Does block matching suffice in real-world?
no, still too noisy!
How can we improve the correspondance search in dense correspondence establishment (i.e. stereo camera)?
there are additional “soft” constraints beside the epipolar constraint
-> use these to improve search
i.e. disparity gradient: disparity changes smoothly between points that lie om same surface
What is a homography?
transformation of point correspondances (typically, 2D-2D)
derived based on perspective projection (-> more general than affine transformation)
encodes co-planarity information
How can we derive a homography?
3D plane expression:
the normal of the plane the point lies on dot point on plane + disntance form origion = 0
projective geometry
point in right image = transformation of 3D point in left frame (i.e. P) and projection with K
-> here; we can add the term in the red box as according to first equaiton, equals 1…
bring P outside (distributive law)
and then, we can replace P by denormalizing of normalized image coords p1
-> thus, we achive a formula that sets p1 and p2 into relation
here, H is out homograpy…
What equation does a pair of points in homogenous coordinates satifsfy w.r.t. homography?
are up to scale…
=> 8 DOF (H33 -> scale)
How can we take the 8 DOF in the homography into consideration=?
without loss of generality
-> can set last element of Homography to 1…
How can we solve the homography?
-> each point correspondance provides two linear constraints
=> linear system w.r.t. elements of homography defined by four point correspondances
What are similiarities and differences in essential matrix and homography?
similarities:
both encode relative pose information
different point correspondences can be fitted by same matrix
differences:
essential matrix derived from arbitrary 3D points
homography derived form coplanar 3D points (vgl. zhang…)
essential matrix computation needs at least 5 point correspondances
homography computation needs at least 4 point correspondances
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