Revisiting the Linear Programming Relaxation Approach to Gibbs Energy Minimization and Weighted Constraint Satisfaction

We present a number of contributions to the LP relaxation approach to weighted constraint satisfaction (= Gibbs energy minimization). We link this approach to many works from constraint programming, which relation has so far been ignored in machine vision and learning. While the approach has been mostly considered only for binary constraints, we generalize it to n-ary constraints in a simple and natural way. This includes a simple algorithm to minimize the LP-based upper bound, n-ary max-sum diffusion – however, we consider using other bound-optimizing algorithms as well. The diffusion iteration is tractable for a certain class of higharity constraints represented as a black-box, which is analogical to propagators for global constraints CSP. Diffusion exactly solves permuted n-ary supermodular problems. A hierarchy of gradually tighter LP relaxations is obtained simply by adding various zero constraints and coupling them in various ways to existing constraints. Zero constraints can be added incrementally, which leads to a cutting plane algorithm. The separation problem is formulated as finding an unsatisfiable subproblem of a CSP

Joint Orientation of Epipoles

It is known that epipolar constraint can be augmented with orientation by formulating it in the oriented projective geometry. This oriented epipolar constraint requires knowing the orientations (signs of overall scales) of epipoles and fundamental matrix. The current belief is that these orientations cannot be obtained from the fundamental matrix only and that additional information is needed, typically, a single correct point correspondence. In contrary to this, we show that fundamental matrix alone encodes orientation of epipoles up to their common scale sign. We present two formulations of this fact. The algebraic formulation gives a closed formula to compute the second epipole from fundamental matrix and the first epipole. The geometric formulation is in terms of the conic formed by intersections of corresponding epipolar lines in the common image plane; we show that the epipoles always lie on different antipodal components of the spherical interpretation of this conic. Further, we show that, under mild assumptions, fundamental matrix can discriminate between two classes of mutual position of a pair of directional cameras

Two-view Geometry Estimation Unaffected by a Dominant Plane

A RANSAC-based algorithm for robust estimation of epipolar geometry from point correspondences in the possible presence of a dominant scene plane is presented. The algorithm handles scenes with (i) all points in a single plane, (ii) majority of points in a single plane and the rest off the plane, (iii) no dominant plane. It is not required to know a priori which of the cases (i)-(iii) occurs. The algorithm exploits a theorem we proved, that if five or more of seven correspondences are related by a homography then there is an epipolar geometry consistent with the seven-tuple as well as with all correspondences related by the homography. This means that a seven point sample consisting of two outliers and five inliers lying in a dominant plane produces an epipolar geometry which is wrong and yet consistent with a high number of correspondences. The theorem explains why RANSAC often fails to estimate epipolar geometry in the presence of a dominant plane. Rather surprisingly, the theorem also implies that RANSAC-based homography estimation is faster when drawing nonminimal samples of seven correspondences than minimal samples of four correspondences

Marginal Consistency: Unifying Constraint Propagation on Commutative Semirings

We generalise the linear programming relaxation approach to Weighted CSP by Schlesinger and the max-sum diffusion algorithm by Koval and Kovalevsky twice: from Weighted CSP to Semiring CSP, and from binary networks to networks of arbitrary arity. This generalisation reveals a deep property of constraint networks on commutative semirings: by locally changing constraint values, any network can be transformed into an equivalent form in which all corresponding marginals of each constraint pair coincide. We call this state marginal consistency. It corresponds to a local minimum of an upper bound on the Semiring CSP. We further show that a hierarchy of gradually tighter bounds is obtained by adding neutral constraints with higher arity. We argue that marginal consistency is a fundamental concept to unify local consistency techniques in constraint networks on commutative semirings

The Cytokinin Status of the Epidermis Regulates Aspects of Vegetative and Reproductive Development in Arabidopsis thaliana

The epidermal cell layer of plants has important functions in regulating plant growth and development. We have studied the impact of an altered epidermal cytokinin metabolism on Arabidopsis shoot development. Increased epidermal cytokinin synthesis or breakdown was achieved through expression of the cytokinin synthesis gene LOG4 and the cytokinin-degrading CKX1 gene, respectively, under the control of the epidermis-specific AtML1 promoter. During vegetative growth, increased epidermal cytokinin production caused an increased size of the shoot apical meristem and promoted earlier flowering. Leaves became larger and the shoots showed an earlier juvenile-to-adult transition. An increased cytokinin breakdown had the opposite effect on these phenotypic traits indicating that epidermal cytokinin metabolism can be a factor regulating these aspects of shoot development. The phenotypic consequences of abbreviated cytokinin signaling in the epidermis achieved through expression of the ARR1-SRDX repressor were generally milder or even absent indicating that the epidermal cytokinin acts, at least in part, cell non-autonomously. Enhanced epidermal cytokinin synthesis delayed cell differentiation during leaf development leading to an increased cell proliferation and leaf growth. Genetic analysis showed that this cytokinin activity was mediated mainly by the AHK3 receptor and the transcription factor ARR1. We also demonstrate that epidermal cytokinin promotes leaf growth in a largely cell-autonomous fashion. Increased cytokinin synthesis in the outer layer of reproductive tissues and in the placenta enhanced ovule formation by the placenta and caused the formation of larger siliques. This led to a higher number of seeds in larger pods resulting in an increased seed yield per plant. Collectively, the results provide evidence that the cytokinin metabolism in the epidermis is a relevant parameter determining vegetative and reproductive plant growth and development