Scale invariance Totally Explained

Everything about Scale Invariance totally explained

In physics and mathematics, scale invariance is a feature of objects or laws that don't change if length scales (or energy scales) are multiplied by a common factor. The technical term for this transformation is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry.

In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilatations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity.

In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale.

In quantum field theory, scale invariance has an interpretation in terms of particle physics. In a scale-invariant theory, the strength of particle interactions doesn't depend on the energy of the particles involved.

In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena. Such theories are scale-invariant statistical field theories, and are formally very similar to scale-invariant quantum field theories.

Universality is the observation that widely different microscopic systems can display the same behaviour at a phase transition. Thus phase transitions in many different systems may be described by the same underlying scale-invariant theory.

In general, dimensionless quantities are scale invariant. The analogous concept in statistics are standardized moments, which are scale invariant statistics of a variable, while the unstandardized moments are not.

Scale-invariant curves and self-similarity

In mathematics, one can consider the scaling properties of a function or curve

f(x)

under rescalings of the variable

x

. That is, one is interested in the shape of

f(lambda x)

for some scale factor

lambda

, which can be taken to be a length or size rescaling. The requirement for

f(x)

to be invariant under all rescalings is usually taken to be ť

f(x)=lambda^,lambda t)

are also solutions.

Computer vision

In computer vision, scale invariance refers to a local image description that remains invariant when the scale of the image is changed. A general framework for obtaining scale invariance in practice is by detecting local maxima over scales of normalized derivative responses -- see the article on scale-space for a brief introduction to the general theory and references. Examples of scale invariant blob detectors and ridge detectors are given in the articles on blob detection and ridge detection. An example of the application of scale invariance to object recognition is given in the article on the scale-invariant feature transform.

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