Factorization methods for projective structure and motion
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Citations
PALM: portable sensor-augmented vision system for large-scene modeling
Optimal motion estimation from multiple images by normalized epipolar constraint
A factorization method for multiple perspective views via iterative depth estimation
Extending 3D Lucas–Kanade tracking with adaptive templates for head pose estimation
References
Numerical Recipes in C: The Art of Scientific Computing
Numerical Recipes in C: The Art of Scientific Computing
Shape and motion from image streams under orthography: a factorization method
Camera Self-Calibration: Theory and Experiments
Related Papers (5)
Shape and motion from image streams under orthography: a factorization method
Frequently Asked Questions (13)
Q2. What are the future works mentioned in the paper "Factorization methods for projective structure and motion" ?
Future work will expand on this. Summary: Projective structure and motion can be recovered from multiple perspective images of a scene consisting of points and lines, by estimating fundamental matrices and epipoles from the image data, using these to rescale the image measurements, and then factorizing the resulting rescaled measurement matrix using either SVD or a fast approximate factorization algorithm.
Q3. How are the fundamental matrices and epipoles estimated?
Fundamental matrices and epipoles are estimated using the linear least squares method with all the available point matches, followed by a supplementary SVD to project the fundamental matrices to rank 2 and find the epipoles.
Q4. What is the main reason for the expansion of the epipolar constraint?
As part of the current blossoming of interest in multiimage reconstruction, Shashua [14] recently extended the wellknown two-image epipolar constraint to a trilinear constraint between matching points in three images.
Q5. How can the authors recover projective structure and motion from multiple perspective images?
Summary: Projective structure and motion can be recovered from multiple perspective images of a scene consisting of points and lines, by estimating fundamental matrices and epipoles from the image data, using these to rescale the image measurements, and then factorizing the resulting rescaled measurement matrix using either SVD or a fast approximate factorization algorithm.
Q6. What is the key technical advance that makes this work possible?
The key technical advance that makes this work possible is a practical method for estimating these using fundamental matrices and epipoles obtained from the image data.
Q7. How many points can be recovered from a scene?
The authors need to recover 3D structure (point locations) and motion (camera calibrations and locations) from m uncalibrated perspective images of a scene containing n 3D points.
Q8. What are the key attractions of the factorization paradigm?
The factorization paradigm has two key attractions that are only enhanced by moving from the affine to the projective case: (i) All of the data in all of the images is treated uniformly — there is no need to single out ‘privileged’ features or images for special treatment; (ii) No initialization is required and convergence is virtually guaranteed by the nature of the numerical methods used.
Q9. How can the authors find the depths for each point p?
With such a non-redundant set of equations the depths for each point p can be found trivially by chaining together the solutions for each image, starting from some arbitrary initial value such as 1p = 1.
Q10. What is the way to estimate the r combinations of columns?
When the matrix is not exactly of rank r the guesses are not quite optimal and it is useful to include further sweeps (say 2r in total) and then SVD the matrix of extracted columns to estimate the best r combinations of them.
Q11. How can one factorize a rank r matrix?
Although SVD is probably near-optimal for full-rank matrices, rank r matrices can be factorized in ‘output sensitive’ time O(mnr).
Q12. What is the solution to the error modelling problem?
There is no obvious solution to the error modelling problem, beyond using the factorization to initialize a nonlinear least squares routine (as is done in some of the experiments below).
Q13. What is the underlying theory of projective depth recovery?
The full theory of projective depth recovery applies equally to two, three and four image matching tensors, but throughout this paper The authorwill concentrate on the two-image (fundamental matrix) case for simplicity.