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Transitive consistency is an intrinsic property for collections of linearinvertible transformations between Euclidean coordinate frames. In practice,when the transformations are estimated from data, this property is lacking.This work addresses the problem of synchronizing transformations that are nottransitively consistent. Once the transformations have been synchronized, theysatisfy the transitive consistency condition - a transformation from frame $A$to frame $C$ is equal to the composite transformation of first transforming Ato B and then transforming B to C. The coordinate frames correspond to nodes ina graph and the transformations correspond to edges in the same graph. Twodirect or centralized synchronization methods are presented for different graphtopologies; the first one for quasi-strongly connected graphs, and the secondone for connected graphs. As an extension of the second method, an iterativeGauss-Newton method is presented, which is later adapted to the case of affineand Euclidean transformations. Two distributed synchronization methods are alsopresented for orthogonal matrices, which can be seen as distributed versions ofthe two direct or centralized methods; they are similar in nature to standardconsensus protocols used for distributed averaging. When the transformationsare orthogonal matrices, a bound on the optimality gap can be computed.Simulations show that the gap is almost right, even for noise large inmagnitude. This work also contributes on a theoretical level by providinglinear algebraic relationships for transitively consistent transformations. Oneof the benefits of the proposed methods is their simplicity - basic linearalgebraic methods are used, e.g., the Singular Value Decomposition (SVD). For awide range of parameter settings, the methods are numerically validated.

On Transitive Consistency for Linear Invertible Transformations between Euclidean Coordinate Systems