|
|
| Line 1: |
Line 1: |
| The '''orthogonal Procrustes problem''' <ref>{{Citation | last1=Gower | first1=J.C | last2=Dijksterhuis | first2=G.B. | year=2004 | title=Procrustes Problems | publisher = Oxford University Press}}</ref> is a [[matrix approximation]] problem in [[linear algebra]]. In its classical form, one is given two [[Matrix (mathematics)|matrices]] <math>A</math> and <math>B</math> and asked to find an [[orthogonal matrix]] <math>R</math> which most closely maps <math>A</math> to <math>B</math>. <ref>{{Citation | last1=Hurley | first1=J.R. | last2=Cattell | first2=R.B. | year=1962 | title=Producing direct rotation to test a hypothesized factor structure | journal = Behavioral Science | volume = 7 | pages=258–262 | doi=10.1002/bs.3830070216 | issue=2}}</ref> Specifically,
| | Alyson Meagher is the title her parents gave her but she doesn't like when people use her full title. Office supervising is where her primary income arrives from but she's currently utilized for an additional 1. The favorite pastime for him and his children is to perform lacross and he would never give it up. For many years he's been living in Alaska and he doesn't strategy on online psychic readings; [http://www.familysurvivalgroup.com/easy-methods-planting-looking-backyard/ www.familysurvivalgroup.com], changing it.<br><br>Feel free to surf to my site; online [http://help.ksu.edu.sa/node/65129 real psychic] chat ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 chorokdeul.co.kr]) |
| | |
| :<math>R = \arg\min_\Omega \|A\Omega-B\|_F \quad\mathrm{subject\ to}\quad \Omega^T
| |
| \Omega=I,</math>
| |
| | |
| where <math>\|\cdot\|_F</math> denotes the [[Frobenius norm]].
| |
| | |
| The name [[Procrustes]] refers to a bandit from Greek mythology who made his victims fit his bed by either stretching their limbs or cutting them off. | |
| | |
| == Solution ==
| |
| This problem was originally solved by [[Peter Schonemann]] in a 1964 thesis. The individual solution was later published. <ref>{{Citation | last=Schonemann | first=P.H. | authorlink = Peter Schonemann | year=1966 | title=A generalized solution of the orthogonal Procrustes problem | journal=Psychometrika | volume=31 | pages=1–10 | url=http://www2.psych.purdue.edu/~phs/pdf/3.pdf | doi=10.1007/BF02289451 | postscript=.}}</ref> A proof is also given in <ref>{{Citation | last = Zhang | first = Z. | contribution = A Flexible New Technique for Camera Calibration | series = MSR-TR-98-71 | url=http://research.microsoft.com/en-us/um/people/zhang/Papers/TR98-71.pdf | year = 1998}}
| |
| </ref>
| |
| | |
| This problem is equivalent to finding the nearest orthogonal matrix to a given matrix <math>M=A^{T}B</math>. To find this orthogonal matrix <math>R</math>, one uses the [[singular value decomposition]]
| |
| :<math>M=U\Sigma V^*\,\!</math>
| |
| to write | |
| :<math>R=UV^*.\,\!</math>
| |
| | |
| == Generalized/constrained Procrustes problems ==
| |
| There are a number of related problems to the classical orthogonal Procrustes problem. One might generalize it by seeking the closest matrix in which the columns are [[orthogonal]], but not necessarily [[orthonormal]]. <ref>{{Citation| last=Everson| first=R| year=1997| title=Orthogonal, but not Orthonormal, Procrustes Problems| url=http://citeseer.ist.psu.edu/everson97orthogonal.html}}</ref>
| |
| | |
| Alternately, one might constrain it by only allowing [[rotation matrix|rotation matrices]] (i.e. orthogonal matrices with [[determinant]] 1, also known as [[Orthogonal matrix|special orthogonal matrices]]). In this case, one can write (using the above decomposition <math>M=U\Sigma V^*</math>)
| |
| | |
| :<math>R=U\Sigma'V^*,\,\!</math> | |
| | |
| where <math>\Sigma'\,\!</math> is a modified <math>\Sigma\,\!</math>, with the smallest singular value replaced by <math>\operatorname{sign}(\det(UV^*))</math> (+1 or -1), and the other singular values replaced by 1, so that the determinant of R is guaranteed to be positive. <ref>{{Citation|last1=Eggert|first1=DW| last2=Lorusso|first2=A| last3=Fisher|first3=RB| title=Estimating 3-D rigid body transformations: a comparison of four major algorithms| journal=Machine Vision and Applications| volume=9| issue=5| pages=272–290| year=1997|doi=10.1007/s001380050048}}</ref> For more information, see the [[Kabsch algorithm]].
| |
| | |
| == See also ==
| |
| * [[Procrustes analysis]]
| |
| * [[Procrustes transformation]]
| |
| | |
| == References ==
| |
| <references/>
| |
| | |
| [[Category:Linear algebra]]
| |
| [[Category:Matrix theory]]
| |
| [[Category:Singular value decomposition]]
| |
Alyson Meagher is the title her parents gave her but she doesn't like when people use her full title. Office supervising is where her primary income arrives from but she's currently utilized for an additional 1. The favorite pastime for him and his children is to perform lacross and he would never give it up. For many years he's been living in Alaska and he doesn't strategy on online psychic readings; www.familysurvivalgroup.com, changing it.
Feel free to surf to my site; online real psychic chat (chorokdeul.co.kr)