Editing Scale invariance (section)

==Other examples of scale invariance==

===Newtonian fluid mechanics with no applied forces===

Under certain circumstances, [[fluid mechanics]] is a scale-invariant classical field theory. The fields are the velocity of the fluid flow, <math>\mathbf{u}(\mathbf{x},t)</math>, the fluid density, <math>\rho(\mathbf{x},t)</math>, and the fluid pressure, <math>P(\mathbf{x},t)</math>. These fields must satisfy both the [[Navier–Stokes equation]] and the [[continuity equation#Fluid dynamics|continuity equation]]. For a [[Newtonian fluid]] these take the respective forms

<math display="block">\rho\frac{\partial \mathbf{u}}{\partial t}+\rho\mathbf{u}\cdot\nabla \mathbf{u} = -\nabla P+\mu \left(\nabla^2 \mathbf{u}+\frac{1}{3}\nabla\left(\nabla\cdot\mathbf{u}\right)\right)</math>
:<math>\frac{\partial \rho}{\partial t}+\nabla\cdot \left(\rho\mathbf{u}\right)=0</math>
where <math>\mu</math> is the [[dynamic viscosity#Viscosity .28dynamic viscosity.29: .CE.BC|dynamic viscosity]].

In order to deduce the scale invariance of these equations we specify an [[equation of state]], relating the fluid pressure to the fluid density. The equation of state depends on the type of fluid and the conditions to which it is subjected. For example, we consider the [[isothermal]] [[ideal gas]], which satisfies
:<math>P=c_s^2\rho,</math>
where <math>c_s</math> is the speed of sound in the fluid. Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformations
:<math>x\rightarrow\lambda x,</math>
:<math>t\rightarrow\lambda^2 t,</math>
:<math>\rho\rightarrow\lambda^{-1} \rho,</math>
:<math>\mathbf{u}\rightarrow\lambda^{-1}\mathbf{u}.</math>
Given the solutions <math>\mathbf{u}(\mathbf{x},t)</math> and <math>\rho(\mathbf{x},t)</math>, we automatically have that
<math>\lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t)</math> and <math>\lambda\rho(\lambda\mathbf{x},\lambda^2 t)</math> are also solutions.

===Computer vision===
{{Main article|Scale space}}
In [[computer vision]] and [[biological vision]], scaling transformations arise because of the perspective image mapping and because of objects having different physical size in the world. In these areas, scale invariance refers to local image descriptors or visual representations of the image data that remain invariant when the local scale in the image domain is changed.<ref name=Lin13PONE>[https://dx.doi.org/10.1371/journal.pone.0066990 Lindeberg, T. (2013) Invariance of visual operations at the level of receptive fields, PLoS ONE 8(7):e66990.]</ref>  
Detecting local maxima over scales of normalized derivative responses provides a general framework for obtaining scale invariance from image data.<ref name=Lindeberg1998>{{cite journal
 | author = Lindeberg, Tony
 | year = 1998
 | title = Feature detection with automatic scale selection
 | journal = International Journal of Computer Vision
 | volume = 30
 | issue = 2
 | pages = 79–116
 | doi = 10.1023/A:1008045108935
 | s2cid = 723210
 | url = https://kth.diva-portal.org/smash/get/diva2:453064/FULLTEXT01
| url-access = 
 | url-status = 
 | archive-url = 
 | archive-date = 
 }}</ref><ref name=Lin14CompVis>T. Lindeberg (2014) [http://www.csc.kth.se/~tony/abstracts/Lin14-ScSel-CompVisRefGuide.html "Scale selection", Computer Vision: A Reference Guide, (K. Ikeuchi, Editor), Springer, pages 701-713.]</ref>
Examples of applications include [[blob detection]], [[corner detection]], [[ridge detection]], and object recognition via the [[scale-invariant feature transform]].