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| The '''fast Kalman filter (FKF)''', devised by [[Antti Lange]] (1941- ), is an extension of the [[Helmert-Wolf blocking]]<ref name="fn_1"/> ('''HWB''') method from [[geodesy]] to real-time applications of [[Kalman filter]]ing (KF) such as satellite imaging of the Earth. Kalman filters are an important software technique for building fault-tolerance into a wide range of systems, including real-time imaging.
| | <br><br>It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you loved this post and you would like to receive additional details concerning [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist] kindly check out our webpage. |
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| ==Description==
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| The Fast Kalman filter applies only to systems with sparse matrices (Lange, 2001), since HWB is an inversion method to solve sparse linear equations (Wolf, 1978).
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| The ordinary Kalman filter is optimal for general systems. However, an '''optimal''' Kalman filter is probably stable only if Kalman's '''observability'''<ref name="fn_2"/> and '''controllability''' conditions<ref name="fn_3"/> are also satisfied (Kalman, 1960). These conditions are challenging to continuously maintain for a large system which means that even an optimal Kalman filter may diverge towards false solutions. Fortunately, the stability of an optimal Kalman filter can be controlled by monitoring its error variances if these can be reliably estimated. Their precise computation is, however, much more demanding than the optimal filtering itself but the FKF method may provide the required speed-up also in this respect.
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| == Optimum calibration ==
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| Calibration parameters are a typical example of those state parameters that may create serious observability problems if a narrow window of data (i.e. too few measurements) is continuously used by a Kalman filter (Lange, 1999). Observing instruments onboard orbiting satellites gives an example of '''optimal''' Kalman filtering where their calibration is done indirectly on ground (Olsson el al, 2001). There may also exist other state parameters that are hardly or not at all observable (estimable) if too small samples of data are processed (analysed) at a time by any sort of a Kalman filter.
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| === Inverse problem ===
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| The computing load of the [[inverse problem]] of an ordinary '''Kalman recursion''' is [[Cholesky decomposition#Computing the Cholesky decomposition|roughly proportional to the cube]] of the number of the measurements processed simultaneously, which can always be set to 1 by processing each scalar measurement independently and (if necessary) performing a simple pre-filtering algorithm to de-correlate these measurements.
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| Even when many measurements are processed simultaneously, it is not unusual that the linear equation system is sparse, because some measurements turn out to be independent of some state or calibration parameters.
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| In Satellite Geodesy problems (Brockmann, 1997), the computing load of the '''HWB'''
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| (and '''FKF''') method is only roughly proportional to the '''square''' of the number of the state parameters (and not of the measurements whose number may be billions).
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| === Reliable solution ===
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| Reliable operational Kalman filtering requires continuous fusion of data in real-time. Its optimality depends essentially on use of the error variances and covariances between all measurements and the estimated state and calibration parameters. This large error [[covariance matrix]] is obtained by [[matrix inversion]] from the respective system of '''[[Linear regression#Multiple linear regression|Normal Equations]]'''.<ref name="fn_4"/> Its coefficient matrix is usually sparse and the exact solution of all estimated parameters can be computed by using the '''HWB''' method.<ref name="fn_5"/> The optimal solution may also be obtained by Gauss elimination using other sparse-matrix techniques or iterative methods based e.g. on [[Calculus of variations|Variational Calculus]].
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| However, these latter methods can solve the large matrix of all error variances and covariances only approximately and it would thus be impossible to do the data fusion in a strictly '''optimal''' fashion. Consequently, the filter's stability may become uncertain even if the observability and controllability conditions were satisfied.
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| The sparse coefficient matrix to be inverted may often have either a '''bordered block- or band-diagonal''' (BBD) structure. If it is band-diagonal it can be transformed into a block-diagonal form e.g. by means of a generalised Canonical Correlation Analysis [[Generalized canonical correlation|('''gCCA''')]]. The large matrix can thus be most effectively inverted in a blockwise manner by using the following
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| '''[[Invertible matrix#Blockwise inversion|analytic inversion formula]]''':
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| :<math>\begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1} & -A^{-1}B(D-CA^{-1}B)^{-1} \\ -(D-CA^{-1}B)^{-1}CA^{-1} & (D-CA^{-1}B)^{-1} \end{bmatrix}</math>
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| of [[Ferdinand Georg Frobenius|Frobenius]] where
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| :<math> A = </math> a large '''block- or band-diagonal''' (BD) matrix to be easily inverted, and,
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| :<math> (D-CA^{-1}B) = </math> a much smaller matrix called the [[Issai Schur|Schur]] complement of <math>A</math>.
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| This is the '''FKF''' method that may make it computationally possible to estimate a much larger number of state and calibration parameters than an ordinary Kalman recursion can do. Their operational accuracies may also be reliably estimated from the theory of Minimum-Norm Quadratic Unbiased Estimation ([[Minque|MINQUE]]) of [[C. R. Rao]] (1920- ) and used for controlling the stability of optimal Kalman filtering.
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| == Applications ==
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| The '''FKF''' method extends the very high accuracies of Satellite Geodesy to Virtual Reference Station ('''VRS''') [[Real Time Kinematic]] ('''RTK''') surveying, mobile positioning and ultra-reliable navigation (Lange, 2003). First important applications will be real-time '''optimum calibration''' of global observing systems in Meteorology,<ref name="fn_6"/> Geophysics, Astronomy etc.
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| For example, a [[Numerical weather prediction|Numerical Weather Prediction]] (NWP) system can now forecast observations with confidence intervals and their operational quality control can thus be improved. A sudden increase of uncertainty in predicting observations would indicate that important observations were missing (observability problem) or an unpredictable change of weather is taking place (controllability problem). Remote sensing and imaging from satellites may partly be based on forecast information. Controlling stability of such '''feedback''' between the forecast and satellite data calls for the theory of optimal Kalman filtering. No suboptimal solution would do a proper job as public safety is usually at stake.
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| The computational advantage of '''FKF''' is marginal for applications using only small amounts of data in real-time data. Therefore improved built-in calibration and data communication infrastructures need to be developed first and introduced to public use before personal gadgets and machine-to-machine (M2M) devices can make the best out of '''FKF'''.
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| == Notes ==
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| {{Reflist|refs=
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| <ref name="fn_1">see [http://www.ngs.noaa.gov/GRD/GPS/DOC/gpscom/mca.html GPScom Software Documentation] from Geoscience Research Division of NOAA.</ref>
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| <ref name="fn_2">see the [http://bass.gmu.edu/ececourses/ece521/lecturenote/chap3/chap3.html observability condition] of a Kalman filter as described by Dr. Hongxing Xia of George Mason University.</ref>
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| <ref name="fn_3">see the two stability conditions of an optimal Kalman filter as described e.g. by [http://www.bmva.ac.uk/bmvc/1998/papers/d043/h043.htm B. Southall, B. F. Buxton, J. A. Marchant (1998): "Controllability and Observability: Tools for Kalman Filter Design", ''On-Line Proceedings of the Ninth British Machine Vision Conference''].</ref> | |
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| <ref name="fn_4">see formulas (15.56-58) on pages 507-508 of Strang, G. and Borre, K. (1997): ''Linear Algebra, Geodesy, and GPS'', Wellesley-Cambridge Press.</ref> | |
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| <ref name="fn_5">see the HWB formula (unnumbered) at the end of page 508 of Strang, G. and Borre, K. (1997): ''Linear Algebra, Geodesy, and GPS'', Wellesley-Cambridge Press.</ref>
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| <ref name="fn_6">see Lange, A. A. (1988): "A high-pass filter for Optimum Calibration of observing systems with applications", '' Simulation and optimization of large systems'', edited by Andrzej. J. Osiadacz, Clarendon Press, Oxford, pp. 311-327.</ref> | |
| }}
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| ==References==
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| * [http://www.sgc.ethz.ch/sgc-volumes/sgk-55.pdf Brockmann, E. (1997)]: "Combination of solutions for geodetic and geodynamic applications of the Global Positioning System (GPS)", ''Geodätisch - geophysikalische Arbeiten in der Schweiz'', Volume 55, Schweitzerische Geodätische Kommission.
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| * Kalman, R. E. (1960): "A New Approach to Linear Filtering and Prediction Problems", ''Transactions of the ASME - Journal of Basic Engineering'', Vol. 82: pp. 35–45.
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| * Lange, A. A. (1999): "Statistical Calibration of Observing Systems", Academic Dissertation, ''Finnish Meteorological Institute Contributions'', No. 22, Helsinki, Finland.
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| * Lange, A. A. (2001): "Simultaneous Statistical Calibration of the GPS signal delay measurements with related meteorological data", ''Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy'', Vol. 26, No. 6-8, pp. 471–473.
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| * [http://www.fkf.net/LangeA2003.PDF Lange, A. A. (2003)]: "Optimal Kalman Filtering for ultra-reliable Tracking", ESA CD-ROM WPP-237, ''Atmospheric Remote Sensing using Satellite Navigation Systems'', Special Symposium of the URSI Joint Working Group FG, 13–15 October 2003, Matera, Italy.
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| * [http://www.ssc.se/data/content/DOCUMENTS/20049612942476Star%20Tracker,%20Gyro%20Calibration%20and%20Attitude%20Reconstruction.pdf Olsson, T. et al. (2001)]: "Star Tracker/Gyro Calibration and Attitude Reconstruction for the Scientific Satellite ODIN - In Flight Results."
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| * Wolf, H. (1978): "The Helmert block method, its origin and development", ''Proceedings of the Second International Symposium on Problems Related to the Redefinition of North American Geodetic Networks'', Arlington, Va. April 24–28, pp. 319–326.
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| ==External links==
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| * [http://portal.acm.org/citation.cfm?id=200979.200987 BBD] - software
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| * [http://www.fkf.net/FKFformula.html FKF] - formulas
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| * [http://www.fkf.net/Wolf.html HWB] - formulas
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| * [http://www.fkf.net/Langesformula.html The error covariance matrix of FKF] - formulas
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| * There are other Fast Kalman Algorithms designed for special signal processing purposes, see e.g. [http://ieeexplore.ieee.org/xpl/abs_free.jsp?arNumber=31289 Stabilizing the Fast Kalman Algorithms] on IEEE Xplore
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| *[http://portal.acm.org/citation.cfm?id=982359 Kalman filter recipes for real-time image processing]
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| {{DEFAULTSORT:Fast Kalman Filter}}
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| [[Category:Signal estimation]]
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| [[Category:Linear filters]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
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