Affine Reconstruction from Multiple Views using Singular Value Decomposition
Dr Peter Kovesi
- These pages contains documentation and scripts I wrote
for my project.
|Image source: The Geometry of Multiple Images by O Faugeras & Q T Luong|
An image with perspective projection gives us the ability to understand and interpret features in the 3D world. In a perspective image, parallel lines in the 3D world appear to converge to a vanishing point, providing information about the third dimension. An affine image has orthographic projection to preserve parallelism such that when parallel lines in the 3D world are projected on to the image plane, they remain parallel. In this thesis, I explain the reconstruc- tion of 3D affine structure from multiple affine images using the Singular Value Decomposition (SVD).
I use multiple affine images of the object ensuring that all images cover every feature in the object to be reconstructed. This poses a restriction on the reconstruction, since acquiring the complete object is not feasible. Hence, the reconstruction involves only the affine structure that is projected on the affine images. I identify feature points on affine images firstly, using a manual mechanism and later, using a tracking system. A measurement matrix is composed using the image points for all affine images. This matrix forms the basis for acquiring 3D affine coordinates.
The SVD decomposes a given matrix into three matrices: two orthogonal matrices and one singular matrix. I use the SVD to decompose the measurement matrix into three matrices to provide two solutions for the 3D affine coordinates. I employ a mechanism to form the metric structure from the two affine structures by using prior knowledge about the object in the 3D world.
I present my results using synthetic and real images. I analyse the results primarily on synthetic images and then use real images to examine the practicality of the method. In addition to using images with orthographic projection, I examine the behaviour of the reconstruction algorithm in the presence of noise, and images with perspective projection.
Keywords: Orthographic projection, factorization algorithm, singular value decomposition