This paper brings a new method for line reconstruction from many perspective images by factorization of a matrix containing line correspondences. No point correspondences are used. We formulate the reconstruction from line correspondences in the language of Pl?cker line coordinates. The reconstruction is posed as the factorization of 3m ? n matrix S into the product S = QL of 3m ? 6 projection matrix Q and 6 ? n line matrix L, both satisfying Klein identities. The matrix S contains coordinates of lines detected in perspective images. Similarly to reconstruction from point correspondences in perspective images, the matrix S has to be properly rescaled before it can be factorized. We propose a scaling of image line coordinates based on trifocal tensors that is analogical to the scaling proposed by Sturm and Triggs for points. We propose an SVD based factorization enforcing Klein identities on Q and L in a noise-free situation. We show experiments on real data that suggest that a good reconstruction may be obtained even if data is noisy and the identities are not enforced exactly. We also discuss an extension of the method for images with occlusions.