Particle-size distribution

2014-02-23T02:58:08Z

182.178.107.173: /* Significance in the collection of particulate matter */

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Magnification

2014-01-18T09:33:18Z

182.178.78.79: /* Magnification as a number (optical magnification) */

In [[algebraic geometry]], a '''proper morphism''' between [[scheme (mathematics)|schemes]] is a scheme-theoretic analogue of a [[proper map]] between [[Complex analytic variety|complex-analytic varieties]].

A basic example is a [[complete variety]] (e.g., [[projective variety]]) in the following sense: a ''k''-variety ''X'' is complete in the classical definition if it is universally closed. A proper morphism is a generalization of this to schemes.

A [[closed immersion]] is proper. A morphism is finite if and only if it is proper and quasi-finite.

== Definition ==

A [[morphism]] ''f'' : ''X'' → ''Y'' of [[algebraic variety|algebraic varieties]] or more generally of [[Scheme (mathematics)|schemes]], is called '''universally closed''' if for all morphisms ''Z'' → ''Y'', the projections for the [[fiber product]]
:<math>X \times_Y Z \to Z</math>
are [[closed map]]s of the underlying [[topological spaces]]. A [[morphism]] ''f'' : ''X'' → ''Y'' of [[algebraic variety|algebraic varieties]] is called '''proper''' if it is [[separated morphism|separated]] and universally closed. A morphism of schemes is called '''proper''' if it is separated, of [[morphism of finite type|finite type]] and universally closed ([EGA] II, 5.4.1 [http://modular.fas.harvard.edu/scans/papers/grothendieck/PMIHES_1961__8__5_0.pdf]). One also says that ''X'' is proper over ''Y''. A variety ''X'' over a [[field (mathematics)|field]] ''k'' is [[complete variety|complete]] when the structural morphism from ''X'' to the spectrum of ''k'' is proper.

== Examples ==
The [[projective space]] '''P'''''d'' over a field ''K'' is proper over a point (that is, Spec(''K'')). In the more classical language, this is the same as saying that projective space is a [[complete variety]]. [[Projective morphism]]s are proper, but not all proper morphisms are projective. For example, it can be shown that the scheme obtained by contracting two disjoint [[projective line]]s in some '''P'''3 to one is a proper, but non-projective variety.<ref>{{Citation | last1=Ferrand | first1=Daniel | title=Conducteur, descente et pincement | year=2003 | journal=[[Bulletin de la Société Mathématique de France]] | issn=0037-9484 | volume=131 | issue=4 | pages=553–585}}, 6.2</ref> [[Affine variety|Affine varieties]] of non-zero dimension are never complete. More generally, it can be shown that affine proper morphisms are necessarily finite. For example, it is not hard to see that the [[affine line]] '''A'''1 is not complete. In fact the map taking '''A'''1 to a point ''x'' is not universally closed. For example, the morphism
:<math>f \times \textrm{id}: \mathbb{A}^1 \times \mathbb{A}^1 \to \{x\} \times \mathbb{A}^1</math>
is not closed since the image of the hyperbola ''uv'' = 1, which is closed in '''A'''1 × '''A'''1, is the affine line minus the origin and thus not closed.

==Properties and characterizations of proper morphisms==
In the following, let ''f'' : ''X'' → ''Y'' be a morphism of schemes.
* Properness is a [[local property of a scheme morphism|local property]] on the base, i.e. if ''Y'' is covered by some open subschemes ''Yi'' and the restriction of ''f'' to all ''f-1(Yi)'' is proper, then so is ''f''.
* Proper morphisms are [[stable under base change]] and composition.
* [[Closed immersion]]s are proper.
* More generally, [[finite morphism]]s are proper. This is a consequence of the [[going up and going down|going up]] theorem.
* Conversely, every [[quasi-finite morphism|quasi-finite]], locally of finite presentation and proper morphism is finite. (EGA III, 4.4.2 in the noetherian case and EGA IV, 8.11.1 for the general case)
* [[Stein factorization]] theorem states that any proper morphism to a locally noetherian scheme can be factorized into <math>X\to Z\to Y</math>, where the first morphism has geometrically connected fibers and the second on is finite.
* Proper morphisms are closely related to [[projective morphism]]s: If ''f'' is proper over a [[noetherian scheme|noetherian]] base ''Y'', then there is a morphism: ''g'': ''X' '' →''X'' which is an isomorphism when restricted to a suitable open dense subset: ''g''-1(''U'') ≅ ''U'', such that ''f' '' := ''fg'' is projective. This statement is called [[Chow's lemma]].
*[[Nagata's compactification theorem]]<ref>B. Conrad, [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.190.9680&rep=rep1&type=pdf Deligne's notes on Nagata compactifications]</ref> says that a separated morphism of finite type between quasi-compact and quasi-separated schemes (e.g., noetherian schemes) factors as an open immersion followed by a proper morphism.
* Proper morphisms between locally noetherian schemes or complex analytic spaces preserve coherent sheaves, in the sense that the [[higher direct image]]s ''Rif''∗(''F'') (in particular the [[direct image]] ''f''∗(''F'')) of a [[coherent sheaf]] ''F'' are coherent (EGA III, 3.2.1). This boils down to the fact that the cohomology groups of [[projective space]] over some [[field (mathematics)|field]] ''k'' with respect to coherent sheaves are [[finitely generated module|finitely generated]] over ''k'', a statement which fails for non-projective varieties: consider '''C'''∗, the [[punctured disc]] and its sheaf of [[holomorphic function]]s <math>\mathcal O</math>. Its sections <math>\mathcal O(\mathbb C^*)</math> is the ring of [[Laurent polynomial]]s, which is infinitely generated over '''C'''.
*There is also a slightly stronger statement of this:{{harv|EGA III|loc=3.2.4}} let <math>f: X \to S</math> be a morphism of finite type, ''S'' locally noetherian and <math>F</math> a <math>\mathcal{O}_X</math>-module. If the support of ''F'' is proper over ''S'', then for each <math>i \ge 0</math> the [[higher direct image]] <math>R^i f_* F</math> is coherent.:
*{{harv|SGA 1|loc=XII}} If ''X'', ''Y'' are schemes of locally of finite type over the field of complex numbers <math>\mathbb{C}</math>, ''f'' induces a morphism of [[complex analytic space]]s
*:<math>f(\mathbb{C}): X(\mathbb{C}) \to Y(\mathbb{C})</math>
:between their sets of complex points with their complex topology. (This is an instance of [[Algebraic geometry and analytic geometry|GAGA]].) Then ''f'' is a proper morphism defined above if and only if <math>f(\mathbb{C})</math> is a proper map in the sense of Bourbaki and is separated.<ref>{{harvnb|SGA 1|loc=XII Proposition 3.2}}</ref>
* If ''f: X''→''Y'' and ''g:Y''→''Z'' are such that ''gf'' is proper and ''g'' is separated, then ''f'' is proper. This can for example be easily proven using the following criterion

=== Valuative criterion of properness ===

[[Image:Valuative criterion of properness.png|thumb|300px|Valuative criterion of properness]] There is a very intuitive criterion for properness which goes back to [[Claude Chevalley|Chevalley]]. It is commonly called the '''valuative criterion of properness'''. Let ''f'': ''X'' → ''Y'' be a morphism of finite type of [[noetherian scheme]]s. Then ''f'' is proper if and only if for all [[discrete valuation ring]]s ''R'' with [[field of fractions|fields of fractions]] ''K'' and for any ''K''-valued point ''x'' ∈ ''X''(''K'') that maps to a point ''f''(''x'') that is defined over ''R'', there is a unique lift of ''x'' to <math>\overline{x} \in X(R)</math>. (EGA II, 7.3.8). Noting that ''Spec K'' is the [[generic point]] of ''Spec R'' and discrete valuation rings are precisely the [[regular ring|regular]] [[local ring|local]] one-dimensional rings, one may rephrase the criterion: given a regular curve on ''Y'' (corresponding to the morphism ''s : Spec R → Y'') and given a lift of the generic point of this curve to ''X'', ''f'' is proper if and only if there is exactly one way to complete the curve.

Similarly, ''f'' is separated if and only if in all such diagrams, there is at most one lift <math>\overline{x} \in X(R)</math>.

For example, the [[projective line]] is proper over a field (or even over '''Z''') since one can always scale [[homogeneous co-ordinates]] by their [[least common denominator]].

== Proper morphism of formal schemes ==
Let <math>f: \mathfrak{X} \to \mathfrak{S}</math> be a morphism between [[locally noetherian formal scheme]]s. We say ''f'' is '''proper''' or <math>\mathfrak{X}</math> is '''proper''' over <math>\mathfrak{S}</math> if (i) ''f'' is an [[adic morphism]] (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map <math>f_0: X_0 \to Y_0</math> is proper, where <math>X_0 = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/I), S_0 = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/K), I = f^*(K) \mathcal{O}_\mathfrak{X}</math> and ''K'' is the ideal of definition of <math>\mathfrak{S}</math>.{{harv|EGA III|loc=3.4.1}} The definition is independent of the choice of ''K''. If one lets
<math>X_n = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/I^{n+1}), S_n = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/K^{n+1})</math>, then<math>f_n: X_n \to S_n</math> is proper.

For example, if <math>g: Y \to Z</math> is a proper morphism, then its extension <math>\widehat{g}: \widehat{Y} \to \widehat{Z}</math> between formal completions is proper in the above sense.

As before, we have the coherence theorem: let <math>f: \mathfrak{X} \to \mathfrak{S}</math> be a proper morphism between locally noetherian formal schemes. If ''F'' is a coherent <math>\mathcal{O}_\mathfrak{X}</math>-module, then the higher direct images <math>R^i f_* F</math> are coherent.

== See also ==
* [[Proper base change theorem]]
* [[Stein factorization]]

==References==
{{reflist}}
* {{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | last2=Dieudonné | first2=Jean | author2-link=Jean Dieudonné | title=Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes | url=http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1961__8_ | year=1961 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | volume=8 | pages=5–222 | doi=10.1007/BF02699291}}, section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)
* {{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | last2=Dieudonné | first2=Jean | author2-link=Jean Dieudonné | title=Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie | url=http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1966__28_ | year=1966 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | volume=28 | pages=5–255}}, section 15.7. (generalisations of valuative criteria to not necessarily noetherian schemes)
* {{Citation | last1=Hartshorne | first1=Robin | author1-link= Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | id={{MathSciNet | id = 0463157}} | year=1977}}

==External links==
*{{springer |id=P/p075450|title=Proper morphism|author=V.I. Danilov}}

[[Category:Morphisms of schemes]]

Photoelasticity

2013-11-11T21:51:45Z

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[[Image:Elephantsdream_vectorstill06.png|thumb|350px|Motion vectors that result from a movement into the <math>z</math>-plane of the image, combined with a lateral movement to the lower-right. This is a visualization of the motion estimation performed in order to compress an MPEG movie.]]'''Motion estimation''' is the process of determining [[motion vector]]s that describe the transformation from one 2D image to another; usually from adjacent [[video frame|frames]] in a video sequence. It is an [[well-posed problem|ill-posed problem]] as the motion is in three dimensions but the images are a projection of the 3D scene onto a 2D plane. The motion vectors may relate to the whole image (global motion estimation) or specific parts, such as rectangular blocks, arbitrary shaped patches or even per [[pixel]]. The motion vectors may be represented by a translational model or many other models that can approximate the motion of a real video camera, such as rotation and translation in all three dimensions and zoom.

Closely related to motion estimation is [[optical flow]], where the vectors correspond to the perceived movement of pixels. In motion estimation an exact 1:1 correspondence of pixel positions is not a requirement.

Applying the motion vectors to an image to synthesize the transformation to the next image is called [[motion compensation]]. The combination of motion estimation and motion compensation is a key part of [[video compression]] as used by [[MPEG]] 1, 2 and 4 as well as many other [[video codecs]].

==Algorithms==
The methods for finding motion vectors can be categorised into pixel based methods ("direct") and feature based methods ("indirect"). A famous debate resulted in two papers from the opposing factions being produced to try to establish a conclusion.<ref>Philip H.S. Torr and Andrew Zisserman: Feature Based Methods for Structure and Motion Estimation, ICCV Workshop on Vision Algorithms, pages 278-294, 1999</ref><ref>Michal Irani and P. Anandan: About Direct Methods, ICCV Workshop on Vision Algorithms, pages 267-277, 1999.</ref>

===Direct Methods===
* [[Block-matching algorithm]]
* [[Phase correlation]] and frequency domain methods
* Pixel recursive algorithms
* [[Optical flow]]

===Indirect Methods===
''Indirect methods'' use features, such as [[corner detection]], and match corresponding features between frames, usually with a statistical function applied over a local or global area. The purpose of the statistical function is to remove matches that do not correspond to the actual motion.

Statistical functions that have been successfully used include [[RANSAC]].

==References==
<references/>

[[Category:Video processing]]
[[Category:Motion]]
[[Category:Estimation theory]]

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Particle-size distribution

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Photoelasticity