1,132 research outputs found
On Computing the Translations Norm in the Epipolar Graph
This paper deals with the problem of recovering the unknown norm of relative
translations between cameras based on the knowledge of relative rotations and
translation directions. We provide theoretical conditions for the solvability
of such a problem, and we propose a two-stage method to solve it. First, a
cycle basis for the epipolar graph is computed, then all the scaling factors
are recovered simultaneously by solving a homogeneous linear system. We
demonstrate the accuracy of our solution by means of synthetic and real
experiments.Comment: Accepted at 3DV 201
Revisiting Viewing Graph Solvability: an Effective Approach Based on Cycle Consistency
In the structure from motion, the viewing graph is a graph where the vertices correspond to cameras (or images) and the edges represent the fundamental matrices. We provide a new formulation and an algorithm for determining whether a viewing graph is solvable, i.e., uniquely determines a set of projective cameras. The known theoretical conditions either do not fully characterize the solvability of all viewing graphs, or are extremely difficult to compute because they involve solving a system of polynomial equations with a large number of unknowns. The main result of this paper is a method to reduce the number of unknowns by exploiting cycle consistency. We advance the understanding of solvability by (i) finishing the classification of all minimal graphs up to 9 nodes, (ii) extending the practical verification of solvability to minimal graphs with up to 90 nodes, (iii) finally answering an open research question by showing that finite solvability is not equivalent to solvability, and (iv) formally drawing the connection with the calibrated case (i.e., parallel rigidity). Finally, we present an experiment on real data that shows that unsolvable graphs may appear in practice
Multicast Network Coding and Field Sizes
In an acyclic multicast network, it is well known that a linear network
coding solution over GF() exists when is sufficiently large. In
particular, for each prime power no smaller than the number of receivers, a
linear solution over GF() can be efficiently constructed. In this work, we
reveal that a linear solution over a given finite field does \emph{not}
necessarily imply the existence of a linear solution over all larger finite
fields. Specifically, we prove by construction that: (i) For every source
dimension no smaller than 3, there is a multicast network linearly solvable
over GF(7) but not over GF(8), and another multicast network linearly solvable
over GF(16) but not over GF(17); (ii) There is a multicast network linearly
solvable over GF(5) but not over such GF() that is a Mersenne prime
plus 1, which can be extremely large; (iii) A multicast network linearly
solvable over GF() and over GF() is \emph{not} necessarily
linearly solvable over GF(); (iv) There exists a class of
multicast networks with a set of receivers such that the minimum field size
for a linear solution over GF() is lower bounded by
, but not every larger field than GF() suffices to
yield a linear solution. The insight brought from this work is that not only
the field size, but also the order of subgroups in the multiplicative group of
a finite field affects the linear solvability of a multicast network
Definability of linear equation systems over groups and rings
Motivated by the quest for a logic for PTIME and recent insights that the
descriptive complexity of problems from linear algebra is a crucial aspect of
this problem, we study the solvability of linear equation systems over finite
groups and rings from the viewpoint of logical (inter-)definability. All
problems that we consider are decidable in polynomial time, but not expressible
in fixed-point logic with counting. They also provide natural candidates for a
separation of polynomial time from rank logics, which extend fixed-point logics
by operators for determining the rank of definable matrices and which are
sufficient for solvability problems over fields. Based on the structure theory
of finite rings, we establish logical reductions among various solvability
problems. Our results indicate that all solvability problems for linear
equation systems that separate fixed-point logic with counting from PTIME can
be reduced to solvability over commutative rings. Moreover, we prove closure
properties for classes of queries that reduce to solvability over rings, which
provides normal forms for logics extended with solvability operators. We
conclude by studying the extent to which fixed-point logic with counting can
express problems in linear algebra over finite commutative rings, generalising
known results on the logical definability of linear-algebraic problems over
finite fields
"Graph Entropy, Network Coding and Guessing games"
We introduce the (private) entropy of a directed graph (in a new network coding sense) as well as a number of related concepts. We show that the entropy of a directed graph is identical to its guessing number and can be bounded from below with the number of vertices minus the size of the graph’s shortest index code. We show that the Network Coding solvability of each specific multiple unicast network is completely determined by the entropy (as well as by the shortest index code) of the directed graph that occur by identifying each source node with each corresponding target node. Shannon’s information inequalities can be used to calculate up- per bounds on a graph’s entropy as well as calculating the size of the minimal index code. Recently, a number of new families of so-called non-shannon-type information inequalities have been discovered. It has been shown that there exist communication networks with a ca- pacity strictly ess than required for solvability, but where this fact cannot be derived using Shannon’s classical information inequalities. Based on this result we show that there exist graphs with an entropy that cannot be calculated using only Shannon’s classical information inequalities, and show that better estimate can be obtained by use of certain non-shannon-type information inequalities
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