rwang@caip.rutgers.edu (Ruye Wang) (12/08/89)
Since I posted the problem of find the smallest enclosing sphere a week age,
I have received quite a few responses. I'd like to thank all the people who
helped me, especically professor Franklin from RPI who pointed out that this
is a well known and solved problem (see his posting).
Being encouraged by the good result (both me and some other people learned
something very useful), here I am posting another problem:
given two sets of 2D points, with known one-to-one correspondent relationship
{(Pi,Qi), i=1,2,...,n}, find out whether they could be two perspective
projections of a set of 3D points in the space. Put it in another way, given
two pictures each having a line drawing structure isomorphic to the other,
find out if they could be taken from a 3D object in the space.
This problem was discussed in the book by Duda and Hart (around page 400) and
the projective coordinate method was given for a special case where the 3D
points are known to be coplanar. I wonder if new methods have been found for
this problem since then. Dese anybody have some idea? Please send me email
if you have something to say. Thanks!tmb@wheaties.ai.mit.edu (Thomas M. Breuel) (12/08/89)
given two sets of 2D points, with known one-to-one correspondent relationship
{(Pi,Qi), i=1,2,...,n}, find out whether they could be two perspective
projections of a set of 3D points in the space. Put it in another way, given
two pictures each having a line drawing structure isomorphic to the other,
find out if they could be taken from a 3D object in the space.
This problem has been addressed in work on feature based visual object
recognition and structure-from-motion. I remember that Shimon Ullman
has proven a number of results about the number of views and number of
points in each view that are needed to determine the 3D coordinates of
the points given correspondences and the 2D positions of the points in
each view. Ullman and Basri have recently shown that the 2D
coordinates of points in new views can be written as the linear
combination of coordinates of points in three given 2D views. For
feature based object recognition, the standard problem is that the 3D
position of the points are known but the correspondences are unknown,
although some work has also been done on the 2D/2D case without known
correspondences.
Thomas.james@fungus.bsd.uchicago.edu (James Balter) (12/11/89)
The problem of finding the projections given two sets of matching points has been demonstrated solvable for radiographic projections in an article by Laura Fencil and Charles Metz published in MEDICAL PHYSICS recently. It states that, in optimal conditions on non-coplanar points, that 8 point pairs are needed to find the original projection information A more interesting problem, though, is how to find the correspondance between two arcs in different projections, where all that is known is that there is SOME overlap to the arcs. James Balter james@rover.bsd.uchicago.edu "If the hat fits, slice it!"