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==Brillouin Function==
[[Image:ev26221 KlyuchevskayaSopka.A2004012.0035.500m.jpg|thumb||250px|Ash plumes on Kamchatka Peninsula, eastern Russia. A [[MODIS]] image.]]


The '''Brillouin function'''<ref name=Kittel>C. Kittel, ''Introduction to Solid State Physics'' (8th ed.), pages 303-4 ISBN 978-0-471-41526-8</ref><ref>{{Cite journal
'''Imaging spectroscopy''' (also '''[[Hyperspectral imaging|hyperspectral]]''' or '''[[spectral imaging]]''' or [[chemical imaging]])
  | last = Darby
is similar to [[color photography]], but each pixel acquires many bands of light intensity data from the spectrum, instead of just the three bands of the [[RGB color model]]. More precisely, it is the simultaneous acquisition of spatially [[Image registration|coregistered images]] in many [[electromagnetic spectrum|spectrally]] contiguous [[frequency band|bands]].
  | first = M.I.
  | author-link =
  | title = Tables of the Brillouin function and of the related function for the spontaneous magnetization
  | journal = Brit. J. Appl. Phys.
  | volume = 18
  | issue = 10
  | pages = 1415–1417
  | year = 1967
  | doi =10.1088/0508-3443/18/10/307
  | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->  |bibcode = 1967BJAP...18.1415D }}</ref> is a [[special function]] defined by the following equation:
<blockquote style="border: 1px solid black; padding:10px;">
:<math>B_J(x) = \frac{2J + 1}{2J} \coth \left ( \frac{2J + 1}{2J} x \right )
                - \frac{1}{2J} \coth \left ( \frac{1}{2J} x \right )</math>
</blockquote>
The function is usually applied (see below) in the context where ''x'' is a real variable and ''J'' is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as <math>x \to +\infty</math> and -1 as <math>x \to -\infty</math>.


The function is best known for arising in the calculation of the [[magnetization]] of an ideal [[paramagnet]]. In particular, it describes the dependency of the magnetization <math>M</math> on the applied [[magnetic field]] <math>B</math> and the [[total angular momentum quantum number]] J of the microscopic [[magnetic moment]]s of the material. The magnetization is given by:<ref name=Kittel/>
Some spectral images contain only a few [[image plane]]s of a spectral [[data cube]], while others are better thought of as full spectra at every location in the image.  For example, [[solar physics|solar physicist]]s use the [[spectroheliograph]] to make images of the [[Sun]] built up by scanning the slit of a spectrograph, to study the behavior of surface features on the Sun; such a spectroheliogram may have a spectral resolution of over 100,000 (<math>\lambda / \Delta \lambda</math>) and be used to measure local motion (via the [[Doppler shift]]) and even the [[magnetic field]] (via the [[Zeeman splitting]] or [[Hanle effect]]) at each location in the image plane.  The [[multispectral image]]s collected by the [[Opportunity rover]], in contrast, have only four wavelength bands and hence are only a little more than [[color photography|3-color image]]s.
:<math>M = N g \mu_B J \cdot B_J(x)</math>


where
To be scientifically useful, such measurement should be done using an internationally recognized system of units.
*<math>N</math> is the number of atoms per unit volume,
*<math>g</math> the [[g-factor (physics)|g-factor]],
*<math>\mu_B</math> the [[Bohr magneton]],
*<math>x</math> is the ratio of the [[Zeeman effect|Zeeman]] energy of the magnetic moment in the external field  to the thermal energy <math>k_B T</math>:
::<math>x = \frac{g \mu_B J B}{k_B T}</math>
*<math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> the temperature.


Note that in the SI system of units <math>B</math> given in Tesla stands for [[magnetic induction]], <math>B=\mu_0 H</math>, where <math>H</math> is the applied [[magnetic field]] given in A/m and <math>\mu_0</math> is the [[permeability of vacuum]].
One application is spectral [[geophysical imaging]], which allows quantitative and qualitative characterization of the surface and of the [[Earth's atmosphere|atmosphere]], using geometrically [[Coherence (physics)|coherent]] spectrodirectional{{Huh?|date=February 2013}} [[Radiometry|radiometric]] measurements. These measurements can then be used for unambiguous direct and indirect identification of surface materials and atmospheric trace gases, the measurement of their relative concentrations, subsequently the assignment of the proportional contribution of mixed pixel signals (e.g., the spectral unmixing problem), the derivation of their spatial distribution (mapping problem), and finally their study over time (multi-temporal analysis). The [[Moon Mineralogy Mapper]] on [[Chandrayaan-1]] was an [[imaging spectrometer]].<ref>{{cite news|title=Large quantities of water found on the Moon|url=http://www.telegraph.co.uk/science/space/6224974/Large-quantities-of-water-found-on-the-Moon.html|newspaper=The Telegraph|date=24 Sep 2009}}</ref>


:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
== Background ==
!Click "show" to see a derivation of this law:
In 1704, [[Isaac Newton|Sir Isaac Newton]] demonstrated that white light could be split up into component colours. The subsequent [[history of spectroscopy]] led to precise measurements and provided the empirical foundations for atomic and [[molecular physics]] (Born & Wolf, 1999). Significant achievements in imaging spectroscopy are attributed to airborne instruments, particularly arising in the early 1980s and 1990s (Goetz et al., 1985; Vane et al., 1984). However, it was not until 1999 that the first imaging spectrometer was launched in space (the [[MODIS|NASA Moderate-resolution Imaging Spectroradiometer]], or MODIS).
|-
|A derivation of this law describing the magnetization of an ideal paramagnet is as follows.<ref name=Kittel/> Let '''z''' be the direction of the magnetic field. The z-component of the angular momentum of each magnetic moment (a.k.a. the [[azimuthal quantum number]]) can take on one of the 2J+1 possible values -J,-J+1,...,+J. Each of these has a different energy, due to the external field '''B''': The energy associated with quantum number ''m'' is
:<math>E_m = -mg \mu_B B = -k_BTxm/J</math>
(where ''g'' is the [[g-factor (physics)|g-factor]], μ<sub>B</sub> is the [[Bohr magneton]], and ''x'' is as defined in the text above). The relative probability of each of these is given by the [[Boltzmann factor]]:
:<math>P(m)=e^{-E_m/(k_BT)}/Z=e^{xm/J}/Z</math>
where ''Z'' (the [[Partition function (statistical mechanics)|partition function]]) is a normalization constant such that the probabilities sum to unity. Calculating ''Z'', the result is:
:<math>P(m) = e^{xm/J}/\left(\sum_{m'=-J}^J e^{xm'/J}\right)</math>.
All told, the [[expectation value]] of the azimuthal quantum number ''m'' is
:<math>\langle m \rangle = (-J)\times P(-J) + \cdots + J\times P(J) = \left(\sum_{m=-J}^J m e^{xm/J}\right)/ \left(\sum_{m=-J}^J e^{xm/J}\right)</math>.
The denominator is a [[geometric series]] and the numerator is a type of [http://planetmath.org/encyclopedia/ArithmeticGeometricSeries.html arithmetic-geometric series], so the series can be explicitly summed. After some algebra, the result turns out to be
:<math>\langle m \rangle = J B_J(x)</math>
With ''N'' magnetic moments per unit volume, the magnetization density is
:<math>M = Ng\mu_B\langle m \rangle = NgJ\mu_B B_J(x)</math>.
|}


==Langevin Function==
Terminology and definitions evolve over time.  At one time, >10 spectral bands sufficed to justify the term "[[imaging spectrometer]]" but presently the term is seldom defined by a total minimum number of spectral bands, rather by a contiguous (or redundant) statement of [[spectral bands]].


[[File:Langevin function.png|300px|thumb|right|Langevin function (red line), compared with
The term [[hyperspectral]] imaging is sometimes used interchangeably with imaging spectroscopy. Due to its heavy use in military related applications, the civil world has established a slight preference for using the term imaging spectroscopy.
<math>\tanh(x/3)</math> (blue line).]]


In the classical limit, the moments can be continuously aligned in the field and <math>J</math> can assume all values (<math>J \to \infty</math>). The Brillouin function is then simplified into the '''Langevin function''', named after [[Paul Langevin]]:
== Unmixing ==
Hyperspectral data is often used to determine what materials are present in a scene.  Materials of interest could include roadways, vegetation, and specific targets (i.e. pollutants, hazardous materials, etc.).  Trivially, each pixel of a hyperspectral image could be compared to a material database to determine the type of material making up the pixel.  However, many hyperspectral imaging platforms have low resolution (>5m per pixel) causing each pixel to be a mixture of several materials. The process of unmixing one of these 'mixed' pixels is called hyperspectral image unmixing or simply hyperspectral unmixing.
=== Models ===


<blockquote style="border: 1px solid black; padding:10px; width:230px">
A solution to hyperspectral unmixing is to reverse the mixing process.  Generally, two models of mixing are assumed: linear and nonlinear.
:<math>L(x) = \coth(x) - \frac{1}{x}</math>
Linear mixing models the ground as being flat and incident sunlight on the ground causes the materials to radiate some amount of the incident energy back to the sensor.  Each pixel then, is modeled as a linear sum of all the radiated energy curves of materials making up the pixel. Therefore, each material contributes to the sensor's observation in a positive linear fashion.  Additionally, a conservation of energy constraint is often observed thereby forcing the weights of the linear mixture to sum to one in addition to being positive. The model can be described mathematically as follows:
</blockquote>
For small values of {{math|''x''}}, the Langevin function can be approximated by a truncation of its [[Taylor series]]:
:<math>
  L(x) = \tfrac{1}{3} x - \tfrac{1}{45} x^3 + \tfrac{2}{945} x^5 - \tfrac{1}{4725} x^7 + \dots
  </math>
An alternative better behaved approximation can be derived from the
[[Gauss's continued fraction#The series 0F1 2|Lambert's continued fraction]] expansion of {{math|tanh(''x'')}}:
:<math>
L(x) = \frac{x}{3+\tfrac{x^2}{5+\tfrac{x^2}{7+\tfrac{x^2}{9+\ldots}}}}
</math>
For small enough {{math|''x''}}, both approximations are numerically better than a direct evaluation of the actual analytical expression, since the latter suffers from [[Loss of significance]].


The inverse Langevin function can be approximated to within 5% accuracy
:<math>p = A*x\,</math>
by the formula<ref name="Cohen">{{cite journal |title=A Padé approximant to the inverse Langevin function |last=Cohen |first=A. |journal=[[Rheologica Acta]] |volume=30 |issue=3 |pages=270–273 |year=1991 |doi=10.1007/BF00366640 }}</ref>
:<math>
  L^{-1}(x) \approx x \frac{3-x^2}{1-x^2},
</math>
valid on the whole interval (-1, 1).
For small values of x, better approximations are the [[Padé approximant]]
:<math>
  L^{-1}(x) = 3x \frac{35-12x^2}{35-33x^2} + O(x^7)
</math>
and the
[[Taylor series]]<ref name="Johal">{{cite journal |title=Energy functions for rubber from microscopic potentials |last=Johal |first=A. S. |last2=Dunstan |first2=D. J. |journal=[[Journal of Applied Physics]] |volume=101 |issue=8 |page=084917 |year=2007 |doi=10.1063/1.2723870 |bibcode = 2007JAP...101h4917J }}</ref>
:<math>
  L^{-1}(x) = 3 x + \tfrac{9}{5} x^3 + \tfrac{297}{175} x^5 + \tfrac{1539}{875} x^7 + \dots
</math>


==High Temperature Limit==
where <math>p</math> represents a pixel observed by the sensor, <math>A</math> is a matrix of material reflectance signatures (each signature is a column of the matrix), and <math>x</math> is the proportion of material present in the observed pixel.  This type of model is also referred to as a [[simplex]].


When <math>x \ll 1</math> i.e. when <math>\mu_B B / k_B T</math> is small, the expression of the magnetization can be approximated by the [[Curie's law]]:
With <math>x</math> satisfying the two constraints:
1. Abundance Nonnegativty Constraint (ANC) - each element of x is positive.
2. Abundance Sum-to-one Constraint (ASC) - the elements of x must sum to one.


:<math>M = C \cdot \frac{B}{T}</math>
Non-linear mixing results from multiple scattering often due to non-flat surface such as buildings and vegetation.


where <math>C = \frac{N g^2 J(J+1) \mu_B^2}{3k_B}</math> is a constant. One can note that <math>g\sqrt{J(J+1)}</math> is the effective number of Bohr magnetons.
===Unmixing (Endmember Detection) Algorithms===


==High Field Limit==
There are many algorithms to unmix hyperspectral data each with their own strengths and weaknesses.  Many algorithms assume that pure pixels (pixels which contain only one materials) are present in a scene.
Some algorithms to perform unmixing are listed below:


When <math>x\to\infty</math>, the Brillouin function goes to 1. The magnetization saturates with the magnetic moments completely aligned with the applied field:
* Pixel Purity Index (PPI) - Works by projecting each pixel onto one vector from a set of random vectors spanning the reflectance space.  A pixel receives a score when it represent an extremum of all the projections.  Pixels with the highest scores are deemed to be spectrally pure.
* NFINDR
* Gift Wrapping Algorithm
* Independent Component Analysis Endmember Extraction Algorithm (ICA-EEA) - Works by assuming that pure pixels occur independently than mixed pixels.  Assumes pure pixels are present.
* Vertex Component Analysis (VCA) - Works on the fact that the affine transformation of a simplex is another simplex which helps to find hidden (folded) verticies of the simplex. Assumes pure pixels are present.
* Principal component analysis -(PCA) could also be used to determine endmembers, projection on principal axes could permit endmember selection [ Smith,Johnson et Adams (1985), Bateson et Curtiss (1996) ]
* Multi Endmembers Spatial Mixture Analysis (MESMA) based on the SMA algorithm


:<math>M = N g \mu_B J</math>
* Spectral Phasor Analysis (SPA) based on Fourier transformation of spectra and plotting them on a 2D plot.
Non-linear unmixing algortithm also exist ([[support vector machine]]s (SVM)) or Analytical Neural Network (ANN).
 
Probabilistics methods have also been attempted to unmix pixel through Monte Carlo Unmixing (MCU) algorithm.
 
===Abundance Maps===
 
Once the fundamental materials of a scene are determined, it is often useful to construct an abundance map of each material which displays the fractional amount of material present at each pixel.  Often [[linear programming]] is done to observed ANC and ASC.{{Disambiguation needed|date=February 2013}}
 
==Sensors==
*[[MODIS]] &mdash; on board [[Earth Observing System|EOS]] [[Terra (satellite)|Terra]] and [[Aqua (satellite)|Aqua]] platforms
*[[MERIS]] &mdash; on board [[Envisat]]
*Several commercial manufacturers for laboratory, ground based, aerial, or industrial imaging spectrographs
 
== See also ==
 
* [[Remote sensing]]
* [[Hyperspectral imaging]]
* [[Full Spectral Imaging]]
* [[List of Earth observation satellites]]
* [[Chemical Imaging]]
* [[Imaging spectrometer]]
* [[Microscopy#Infrared microscopy|Infrared Microscopy]]
* [[Phasor approach to fluorescence lifetime and spectral imaging]]


== References ==
== References ==
<references/>
{{Reflist}}
* Goetz, A.F.H., Vane, G., Solomon, J.E., & Rock, B.N. (1985) Imaging spectrometry for earth remote sensing. Science, 228, 1147.
* Schaepman, M. (2005) Spectrodirectional Imaging: From Pixels to Processes. Inaugural address, Wageningen University, Wageningen (NL).
* Vane, G., Chrisp, M., Emmark, H., Macenka, S., & Solomon, J. (1984) Airborne Visible Infrared Imaging Spec-trometer ([[AVIRIS]]): An Advanced Tool for Earth Remote Sensing. European Space Agency, (Special Publication) ESA SP, 2, 751.
 
== External links ==


[[Category:Articles with inconsistent citation formats]]
* About imaging spectroscopy (USGS): http://speclab.cr.usgs.gov/aboutimsp.html
[[Category:Magnetism]]
* Link to resources (OKSI): http://www.techexpo.com/WWW/opto-knowledge/IS_resources.html
* Special Interest Group Imaging Spectroscopy (EARSeL): http://www.op.dlr.de/dais/SIG-IS/SIG-IS.html
* Applications of Spectroscopic and Chemical Imaging in Research: http://www3.imperial.ac.uk/vibrationalspectroscopyandchemicalimaging/research
* Analysis tool for spectral unmixing : http://www.spechron.com


[[de:Langevin-Funktion]]
[[Category:Spectroscopy]]
[[fr:Fonction de Langevin]]
[[ja:ブリルアン関数とランジュバン関数]]

Revision as of 21:40, 14 August 2014

Ash plumes on Kamchatka Peninsula, eastern Russia. A MODIS image.

Imaging spectroscopy (also hyperspectral or spectral imaging or chemical imaging) is similar to color photography, but each pixel acquires many bands of light intensity data from the spectrum, instead of just the three bands of the RGB color model. More precisely, it is the simultaneous acquisition of spatially coregistered images in many spectrally contiguous bands.

Some spectral images contain only a few image planes of a spectral data cube, while others are better thought of as full spectra at every location in the image. For example, solar physicists use the spectroheliograph to make images of the Sun built up by scanning the slit of a spectrograph, to study the behavior of surface features on the Sun; such a spectroheliogram may have a spectral resolution of over 100,000 () and be used to measure local motion (via the Doppler shift) and even the magnetic field (via the Zeeman splitting or Hanle effect) at each location in the image plane. The multispectral images collected by the Opportunity rover, in contrast, have only four wavelength bands and hence are only a little more than 3-color images.

To be scientifically useful, such measurement should be done using an internationally recognized system of units.

One application is spectral geophysical imaging, which allows quantitative and qualitative characterization of the surface and of the atmosphere, using geometrically coherent spectrodirectionalTemplate:Huh? radiometric measurements. These measurements can then be used for unambiguous direct and indirect identification of surface materials and atmospheric trace gases, the measurement of their relative concentrations, subsequently the assignment of the proportional contribution of mixed pixel signals (e.g., the spectral unmixing problem), the derivation of their spatial distribution (mapping problem), and finally their study over time (multi-temporal analysis). The Moon Mineralogy Mapper on Chandrayaan-1 was an imaging spectrometer.[1]

Background

In 1704, Sir Isaac Newton demonstrated that white light could be split up into component colours. The subsequent history of spectroscopy led to precise measurements and provided the empirical foundations for atomic and molecular physics (Born & Wolf, 1999). Significant achievements in imaging spectroscopy are attributed to airborne instruments, particularly arising in the early 1980s and 1990s (Goetz et al., 1985; Vane et al., 1984). However, it was not until 1999 that the first imaging spectrometer was launched in space (the NASA Moderate-resolution Imaging Spectroradiometer, or MODIS).

Terminology and definitions evolve over time. At one time, >10 spectral bands sufficed to justify the term "imaging spectrometer" but presently the term is seldom defined by a total minimum number of spectral bands, rather by a contiguous (or redundant) statement of spectral bands.

The term hyperspectral imaging is sometimes used interchangeably with imaging spectroscopy. Due to its heavy use in military related applications, the civil world has established a slight preference for using the term imaging spectroscopy.

Unmixing

Hyperspectral data is often used to determine what materials are present in a scene. Materials of interest could include roadways, vegetation, and specific targets (i.e. pollutants, hazardous materials, etc.). Trivially, each pixel of a hyperspectral image could be compared to a material database to determine the type of material making up the pixel. However, many hyperspectral imaging platforms have low resolution (>5m per pixel) causing each pixel to be a mixture of several materials. The process of unmixing one of these 'mixed' pixels is called hyperspectral image unmixing or simply hyperspectral unmixing.

Models

A solution to hyperspectral unmixing is to reverse the mixing process. Generally, two models of mixing are assumed: linear and nonlinear. Linear mixing models the ground as being flat and incident sunlight on the ground causes the materials to radiate some amount of the incident energy back to the sensor. Each pixel then, is modeled as a linear sum of all the radiated energy curves of materials making up the pixel. Therefore, each material contributes to the sensor's observation in a positive linear fashion. Additionally, a conservation of energy constraint is often observed thereby forcing the weights of the linear mixture to sum to one in addition to being positive. The model can be described mathematically as follows:

where represents a pixel observed by the sensor, is a matrix of material reflectance signatures (each signature is a column of the matrix), and is the proportion of material present in the observed pixel. This type of model is also referred to as a simplex.

With satisfying the two constraints: 1. Abundance Nonnegativty Constraint (ANC) - each element of x is positive. 2. Abundance Sum-to-one Constraint (ASC) - the elements of x must sum to one.

Non-linear mixing results from multiple scattering often due to non-flat surface such as buildings and vegetation.

Unmixing (Endmember Detection) Algorithms

There are many algorithms to unmix hyperspectral data each with their own strengths and weaknesses. Many algorithms assume that pure pixels (pixels which contain only one materials) are present in a scene. Some algorithms to perform unmixing are listed below:

  • Pixel Purity Index (PPI) - Works by projecting each pixel onto one vector from a set of random vectors spanning the reflectance space. A pixel receives a score when it represent an extremum of all the projections. Pixels with the highest scores are deemed to be spectrally pure.
  • NFINDR
  • Gift Wrapping Algorithm
  • Independent Component Analysis Endmember Extraction Algorithm (ICA-EEA) - Works by assuming that pure pixels occur independently than mixed pixels. Assumes pure pixels are present.
  • Vertex Component Analysis (VCA) - Works on the fact that the affine transformation of a simplex is another simplex which helps to find hidden (folded) verticies of the simplex. Assumes pure pixels are present.
  • Principal component analysis -(PCA) could also be used to determine endmembers, projection on principal axes could permit endmember selection [ Smith,Johnson et Adams (1985), Bateson et Curtiss (1996) ]
  • Multi Endmembers Spatial Mixture Analysis (MESMA) based on the SMA algorithm
  • Spectral Phasor Analysis (SPA) based on Fourier transformation of spectra and plotting them on a 2D plot.

Non-linear unmixing algortithm also exist (support vector machines (SVM)) or Analytical Neural Network (ANN).

Probabilistics methods have also been attempted to unmix pixel through Monte Carlo Unmixing (MCU) algorithm.

Abundance Maps

Once the fundamental materials of a scene are determined, it is often useful to construct an abundance map of each material which displays the fractional amount of material present at each pixel. Often linear programming is done to observed ANC and ASC.Template:Disambiguation needed

Sensors

  • MODIS — on board EOS Terra and Aqua platforms
  • MERIS — on board Envisat
  • Several commercial manufacturers for laboratory, ground based, aerial, or industrial imaging spectrographs

See also

References

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  • Goetz, A.F.H., Vane, G., Solomon, J.E., & Rock, B.N. (1985) Imaging spectrometry for earth remote sensing. Science, 228, 1147.
  • Schaepman, M. (2005) Spectrodirectional Imaging: From Pixels to Processes. Inaugural address, Wageningen University, Wageningen (NL).
  • Vane, G., Chrisp, M., Emmark, H., Macenka, S., & Solomon, J. (1984) Airborne Visible Infrared Imaging Spec-trometer (AVIRIS): An Advanced Tool for Earth Remote Sensing. European Space Agency, (Special Publication) ESA SP, 2, 751.

External links

  1. ↑ Template:Cite news
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