reflectioninvariant

Reflection invariance, or reflectioninvariant, is a property of an object, function, or descriptor that remains unchanged under a reflection transformation. In mathematics and physics, reflections are isometries that flip coordinates across a line (in two dimensions) or a plane (in three dimensions). An object is reflection-invariant if applying that reflection yields an object that is identical to the original under the considered equivalence.

Example: a real-valued function f is invariant under reflection about the origin if f(-x) = f(x); such functions are called even. In multiple variables, f(x, y) is invariant under reflection about the y-axis if f(-x, y) = f(x, y). A geometric shape is reflection-invariant if its set of points is unchanged by the reflection, e.g., a circle or a regular hexagon about certain axes.

In group theory, the set of all reflections generating a symmetry group acts on a space, and invariants are functions or quantities fixed by every group element. Invariant theory studies these polynomials and other objects. A widely used notion is that a feature is reflection-invariant if it remains the same when input data is mirrored; this is desirable in pattern recognition and computer vision.

Notes: Invariance under reflection is one aspect of symmetry; an object can be invariant under some reflections but not others. Achieving reflection invariance can be done by designing symmetric representations, data augmentation with mirrored samples, or architectures that pool symmetric features.