Closed-Form Solutions to Multiple-View Homography Estimation Pierre Schroeder 1 schroedp@in.tum.de Adrien Bartoli 2 adrien.bartoli@gmail.com Pierre Georgel 3 pierre.georgel@gmail.com Nassir Navab 1 navab@in.tum.de 1 Chair for Computer Aided Medical Procedures & Augmented Reality, Technische Universität München, Germany 2 Image Science for Interventional Techniques, Université d’Auvergne, Clermont-Ferrand, France 3 UNC Chapel Hill, Department of Computer Science, USA Abstract The quality of a mosaic depends on the projective align- ment of the images involved. After point-correspondences between the images have been established, bundle adjust- ment finds an alignment considered optimal under certain hypotheses. This procedure minimizes a nonlinear cost and has to be initialized with care. It is very common to compose inter-frame homographies which have been computed with standard methods in order to get an initial global alignment. This technique is suboptimal if there is noise or missing ho- mographies as it typically uses a small part of the available data. We propose four new closed-form solutions. They all provide non-heuristic initial alignments using all the known inter-frame homographies. Our methods are tested with synthetic and real data and are compared to the standard method. These experiments reveal that our methods are more accurate, taking advantage of the redundant informa- tion available in the set of inter-frame homographies. 1. Introduction In computer vision, many applications require panoramic stitching [18] from a collection of images or frames from a video. This technique allows one to create for instance wide-angle recordings without using special hardware such as fish-eye lenses. Panoramic stitching is also used in more advanced applications such as super-resolution imaging [4, 15], video compression [10], and camera auto-calibration [8, 14]. Creating a mosaic from only two overlapping images is a relatively easy task and standard techniques provide very good results [3]. Unfortunately aligning multiple images is more complicated, particularly if some input images do not overlap. We propose a means to linearly extract and entirely use the redundantly contained information from inter-frame ho- mographies in order to better initialize the final bundle ad- justment. Paper organization. A brief introduction to the mathe- matical background of stitching is given in Section 2. We discuss prior work (threading and batch methods) in Section 3. We then introduce in Section 4 our proposed methods. They fall into two categories: those which require all the inter-frame homographies and those which handle missing information. We experimentally show the improvements brought by our methods in Section 5 and provide a con- clusion in Section 6. Notation. P 2 represents the 2D-projective space and ∼ equality up to scale. Vectors are denoted using bold fonts. In general but not exclusively, small letters (e.g. q) refer to homogeneous point coordinates in images and capitals (Q) represent point coordinates in 3D; matrices are denoted with type-writer capitals (e.g. M). ‖·‖ refers to the standard two-norm when used for vectors and to the Frobenius norm if used for matrices. 2. Theoretical Background In order to create a mosaic, it is necessary to warp an image from its own to the mosaic’s coordinate frame. A projective camera projects a point Q ∈ R 3 in space to a point q ∈ P 2 in the image plane as: q ∼ KRQ, (1) with K the matrix of intrinsic parameters and R the rotation matrix specifying the camera orientation. (We assume that the origin coincides with the centre of projection; hence there is no translational component.) A point q i from im- age I i is related to its corresponding point q j in image I j by: q j ∼ KR j R T i K −1 q i . (2) 1