translatioonivariant

Translation invariance refers to a property of a system or a function where its behavior or output remains unchanged when the input is translated (shifted) by a certain amount. In essence, if you shift the starting point of the input, the system's response is identically shifted but not altered in its pattern or characteristics.

This concept is particularly relevant in signal processing and image analysis. For example, a filter that is translation invariant will produce the same filtered output regardless of where a specific pattern or feature appears within the input signal or image. This means that if a dog is in the center of an image or on the edge, a translation-invariant edge detector would still detect the edges of the dog in the same way.

Mathematically, a function f(x) is translation invariant if f(x + a) = f(x) + b for some constant b, or more strictly, if the output pattern is identical when the input is shifted. A common example is the integral of a function over a fixed-length interval. If the function itself is constant, then shifting the interval by 'a' will result in the same integral value. In convolutional neural networks, the use of shared weights in convolutional layers inherently leads to a degree of translation equivariance, which is closely related to translation invariance. Equivariance means that if the input is translated, the output is also translated in a corresponding way, preserving the relationship between input and output features.