Marija Dodig

Factorization for non-rigid and articulated structure using metric projections

This paper describes a new algorithm for recovering the 3D shape and motion of deformable and articulated objects purely from uncalibrated 2D image measurements using an iterative factorization approach. Most solutions to non-rigid and articulated structure from motion require metric constraints to be enforced on the motion matrix to solve for the transformation that upgrades the solution to metric space. While in the case of rigid structure the metric upgrade step is simple since the motion constraints are linear, deformability in the shape introduces non-linearities. In this paper we propose an alternating least-squares approach associated with a globally optimal projection step onto the manifold of metric constraints. An important advantage of this new algorithm is its ability to handle missing data which becomes crucial when dealing with real video sequences with self-occlusions. We show successful results of our algorithms on synthetic and real sequences of both deformable and articulated data.

Optimal Metric Projections for Deformable and Articulated Structure-from-Motion

This paper describes novel algorithms for recovering the 3D shape and motion of deformable and articulated objects purely from uncalibrated 2D image measurements using a factorisation approach. Most approaches to deformable and articulated structure from motion require to upgrade an initial affine solution to Euclidean space by imposing metric constraints on the motion matrix. While in the case of rigid structure the metric upgrade step is simple since the constraints can be formulated as linear, deformability in the shape introduces non-linearities. In this paper we propose an alternating bilinear approach to solve for non-rigid 3D shape and motion, associated with a globally optimal projection step of the motion matrices onto the manifold of metric constraints. Our novel optimal projection step combines into a single optimisation the computation of the orthographic projection matrix and the configuration weights that give the closest motion matrix that satisfies the correct block structure with the additional constraint that the projection matrix is guaranteed to have orthonormal rows (i.e. its transpose lies on the Stiefel manifold). This constraint turns out to be non-convex. The key contribution of this work is to introduce an efficient convex relaxation for the non-convex projection step. Efficient in the sense that, for both the cases of deformable and articulated motion, the proposed relaxations turned out to be exact (i.e. tight) in all our numerical experiments. The convex relaxations are semi-definite (SDP) or second-order cone (SOCP) programs which can be readily tackled by popular solvers. An important advantage of these new algorithms is their ability to handle missing data which becomes crucial when dealing with real video sequences with self-occlusions. We show successful results of our algorithms on synthetic and real sequences of both deformable and articulated data. We also show comparative results with state of the art algorithms which reveal that our new methods outperform existing ones.

Combinatorics of Column Minimal Indices and Matrix Pencil Completion Problems

In this paper we describe the possible Kronecker invariants of a matrix pencil with a prescribed subpencil, when the prescribed subpencil has only column (row) minimal indices as Kronecker invariants. This is done by exploiting combinatorics of column minimal indices and by introducing some novel labels. The result is given over arbitrary fields.

ON PROPERTIES OF THE GENERALIZED MAJORIZATION

C ‡ Abstract. In this paper, a complete solution of a problem involving generalized majorization of partitions is given: for two pairs of partitions (d,a) and (c,b) necessary and sufficient conditions for the existence of a partition g that is majorized by both pairs is determined. The obtained conditions are explicit, the solution is constructive and it uses novel techniques and indices. Although the problem is motivated by the applications in matrix pencil completions problems, all results are purely combinatorial and they give a new perspective on comparison of partitions.

Similarity class of a matrix with a prescribed submatrix

In this article we study the possible similarity classes of a square matrix when an arbitrary submatrix is prescribed. As the main result, we solve the even more general problem of describing the possible strict equivalence classes of a regular pencil when a subpencil is prescribed. This result improves the result by [Cabral and Silva, Similarity invariants of completions of submatrices, Lin. Alg. Appl. 169 (1992), 151–161.], since an explicit solution is obtained without any existential quantifiers involved, over an algebraically closed field. In fact, the sufficiency of the conditions obtained by [Gohberg, Kaashoek and Van Schangen, Eigenvalues of completions of submatrices, Lin. Multilin. Alg. 25 (1989), 55–70.] is proved. In the proof, we use various results and techniques including matrix pencils completions, Kronecker canonical form, Littlewood–Richardson coefficients, Young diagrams, the solution of the Carlson problem and localization techniques.

Rank distance problem for pairs of matrices

In this article we give a complete description of the possible feedback invariants of a completely controllable pair of matrices submitted to an additive perturbation of low rank. This result is valid over an arbitrary field.

Pole placement problem for singular systems

In this paper the pole placement problem for singular systems via state feedback is studied. We give a complete solution to this problem for systems without row minimal indices. As a corollary, the eigenvalue assignment problem is solved for singular systems in the case they are regularizable. Copyright

Eigenvalues of partially prescribed matrices

In this paper, loop connections of two linear systems are studied. As the main result, the possible eigenvalues of a matrixof a system obtained as a result of these connections are determined.

CONTROLLABILITY OF SERIES CONNECTIONS

In this paper the controllability of series connections of arbitrary many linear systems is studied. As the main result, necessary and sufficient conditions are given, under which the system obtained as a result of series connections of arbitrary many linear systems is controllable.

Descriptor Systems Under Feedback and Output Injection

In this paper we study simultaneous feedback and output injection on descriptor linear system described by a quadruple of matrices (E,A,B,C). We describe the possible Kronecker invariants of the resulting pencil λE−(A+ BF + KC), when F and K vary, in the case when the pencil corresponding to the system (E,A,B,C) has no infinite elementary divisors of the second, third and fourth type. The solution is constructive and explicit, and is given over algebraically closed fields.