Kruppa

                                                         kruppa [1]

Auto-calibration of a camera.

Characteristics:

• Overconstraint system. Generically, one solution.

Example 1:

var := [x1,x2,x3,x4,x5];
[-.665943977391716185500e-4*x5+.57446030378411210500e-6*x3
-.1751613950359005401700e-3*x4+.4196244389653656099100e-1*x2
+.790810468123839913500e-1*x1+.576239877677818343400e-6*x5*x1
+.2878669609051599434800e-9*x5*x3-.3793171623177442676000e-7*x5*x2
-.1413879606430073317000e-6*x3*x2+.2260715598278493751000e-6*x3*
x1+.1807977610453753238700e-4*x2^2-.436641385104830185600e-3*x2*
x1-.7611538299225383794000e-9*x4*x3+.850811727332718313800e-6*x4
*x2+.2115114651153454255600e-6*x4*x1-.6151155671580043684100e-4*
x1^2-.679218657484954924800e-9*x4*x5-.1484781695906232749900e-10
*x4^2+.2282679824198712676900e-11*x5^2-6.9769247755906706053,
 .2110060214660143967700e-5*x5+.4144285953595159155600e-4*x3
-.54394552730618051300e-5*x4-.3467463041748386549300e-2*x2
-.2140331089638663771600e-1*x1-.2719534201040163893200e-7*x5*x1
-.1702623115659556670500e-11*x5*x3+.2680086049409404086500e-9*x5*x2
-.1625848844223454788000e-9*x3^2+.4520633740597296727900e-7*x3*x2
-.3446099368558527309900e-6*x3*x1-.3087420810097024967200e-5*x2^2
+.140622165312776746000e-4*x2*x1+.304870430133298478900e-9*x4*x3
+.839472055445605435900e-7*x4*x2-.2590133019090686504200e-6*x4*x1
+.1305462899815279601400e-3*x1^2+.8760864140509743994200e-10*x4*x5
+.1576026908829047847300e-9*x4^2+1.1826782050651733260,
-.5512071046488164152300e-4*x5-.83143756564905307600e-7*x3
-.957036922257151281200e-4*x4+.3051426422684597984200e-1*x2
+.4686170555882289551700e-1*x1+.4826964540794779731400e-6*x5*x1
+.1625848844223454788000e-9*x5*x3-.2038492830775620589500e-7*x5*x2
-.7361898024912568609700e-7*x3*x2+.1069425676286599803300e-6*x3*x1
+.8881273513268696911800e-5*x2^2-.3014241254786833377300e-3*x2*x1
-.3396959730689738740600e-9*x4*x3+.546904606080249513600e-6*x4*x2
+.8883888149326932971700e-7*x4*x1-.2733386318298371620200e-4*x1^2
-.626570189478976739800e-9*x4*x5-.6399491108284453574900e-11*x4^2
+.1702623115659556670500e-11*x5^2-4.4014472669284116425,
 .3207696067928841112300e-5*x5+.487371291653174919400e-4*x3
+.44796228846808710100e-5*x4-.404932285692824059400e-2*x2
-.336358923748597430700e-1*x1-.4376017581335482685400e-7*x5*x1
-.2282679824198712676900e-11*x5*x3+.3389912578490747337300e-9*x5*x2

-.2878669609051599434800e-9*x3^2+.7956048280753772387500e-7*x3*x2
-.404587350244953551000e-6*x3*x1-.5466582902191680761100e-5*x2^2
+.156849356143918522500e-4*x2*x1+.339304230142815885200e-9*x4*x3
+.104006712183549433500e-6*x4*x2-.487305302729242293000e-6*x4*x1
+.2043225074084700103000e-3*x1^2+.1492222688503745521800e-9*x4*x5
+.3670883734196991159000e-9*x4^2+1.8076594872821129902,
.1530449695126830728600e-5*x5-.4457806485092739330300e-6*x3
-.4471368125552494678500e-5*x4+.509308631271440496400e-3*x2
+.1800887517012775260100e-2*x1-.609097372823032230000e-8*x5*x1
-.1258853259393682361600e-10*x5*x3+.3112699969451913220500e-8*x5*x2
+.4402988563609430296600e-8*x3*x2+.5488123987758462276800e-9*x3*x1
-.1147402523972623816600e-5*x2^2-.634938152177819890200e-5*x2*x1
-.1238847876744372048600e-11*x4*x3+.1942974533178491052600e-7*x4*x2
+.3608724108147413946000e-9*x4*x1-.1589643272453758478900e-6*x1^2
-.82383344534135310800e-12*x4*x5-.4601063463831239553400e-14*x4^2
+.4798265714190216903300e-12*x5^2-.17177290969169927671,
.2917641023893891547000e-6*x5-.9188332647086083948900e-5*x3
-.9491915829641256767100e-5*x4+.2085940028260375922400e-2*x2
+.6609292840561707240900e-2*x1-.2718337974873329238800e-8*x5*x1
-.4798265714190216903300e-12*x5*x3+.1062821257557842011900e-9*x5*x2
+.1258853259393682361600e-10*x3^2-.6068899144792232719100e-8*x3*x2
+.7757028881892643658900e-7*x3*x1+.7266399813916554957300e-6*x2^2
-.1772588602045364626700e-4*x2*x1-.1654976899960615756700e-9*x4*x3
+.3808014658443819642300e-7*x4*x2+.3890768116462328262900e-7*x4*x1
-.1824424946063856304500e-4*x1^2+.6325589727025528857800e-11*x4*x5
-.6163566382078377514700e-12*x4^2-.52449706997795274452];

Problem: Find a good approximation of the solution(s).

• Solution by Bernard Mourrain, INRIA, SAGA, 2004 Route des Lucioles, BP 93, 06902 Sophia-Antipolis (France), Bernard.Mourrain@sophia.inria.fr, 20 Feb. 1998

Compute the Macaulay matrix M constructed by \underline{http://www.inria.fr/saga/Bernard.Mourrain/ALP/}ALP, associated to these 6 exquations, of size 792 × 792, solve a linear system associated to this matrix. It yields one coordinate of the solution. The computation of one coordinate, using the \underline{http://math.nist.gov/tnt/}SparseLib solveur GMRES, with an ILU-preconditionner is performed in 0.698s on a Dec Alpha 500 AU workstation with 512M of local memory.

References

[1] D. Bini and B. Mourrain. Polynomial test suite. 1996.