twoprojection

Twoprojection is a term used across mathematics and computer science to describe methods, constructions, or analyses that involve two projection operations. It is not a single standardized technique, but a family of approaches that repeatedly apply two projection mappings in sequence or in combination to transform data, enforce constraints, or study relationships between spaces.

Definition and scope

In a typical setting, let P and Q denote projection operators related to two subspaces, sets, or constraint families. A common instantiation is the alternating projection scheme, where a point x is mapped successively by Q and then by P, yielding x1 = P(Q(x)), x2 = Q(P(x1)), and so on. In linear or convex contexts, these projections often target closed subspaces or convex sets, and the two-projection framework is used to seek a point in the intersection A ∩ B or to approximate that intersection.

Convergence and properties

If A and B are nonempty closed convex sets in a Hilbert space, the method of alternating projections typically converges to a point in A ∩ B, with the convergence behavior depending on the geometry of the sets. In nonconvex or more general settings, convergence can fail or may require additional safeguards. Variants include different sequencing, relaxation parameters, or replacing one projection with a more general proximal operator.

Applications

Two-projection methods appear in feasibility problems, signal and image reconstruction, computer vision, and multiview geometry, where the goal is to reconcile multiple constraints or observations by iteratively projecting onto each constraint set.

Variants and related concepts

Beyond two sets, extensions consider multiple projections, stochastic or weighted schemes, and non-orthogonal projections. Related topics include projection operators in linear algebra, convex optimization, and the broader area of projection methods for feasibility problems.

See also

Projection (linear algebra), Projection (geometry), Alternating projections, Convex optimization.