Taylor and Kriegman [1995] show that on further examination of Equation 5 two implicit constraints become apparent:
(7) (8)
The geometric interpretation of Equation 7 is that the vectors m and v; = RjVi' are orthogonal.
When the parameters of the model are correct, this must always be true as
v:
lies on the plane to which m is normal. Similarly, the interpretation of Equation 8 is that. when the model parameters are correct, the vectors m and d; = Rj (di - tj) must. also. be orthogonal.On further examination, one notes that when given an observed edge, defined by the endpoints {(XI.YI), (X2,Y2)}, and assuming a unit focal length, m', the observed normal to the plane defined by the camera centre and the observed edge is defined by:
(9)
By replacing m with m' in Equation 7, it becomes apparent that any model edges of known ori-entation act as constraints on Rj . As architectural scenes usually contain a number of horizontal or vertical edges, it is generally possible to determine an estimate for Rj independent of any other parameters in the model. Minimizing the objective function
over only the camera rotation matrix Rj limits the extent to which that matrix violates the constraint expressed by Equation 7.
Once the estimates of the camera rotation matrices have been computed, they can be treated as constants, and the constraint expressed by Equation 8 can be used to obtain estimates of the rest of the model parameters. INhere Pi and Qi are expressions for the vertices of model edge i, minimizing the objective function
O2 =
L2)mT
Rj(Pi - tj))2+
(mT Rj(Qi - tj))2j
over all parameters of the model apart from the camera rotation matrices, results in good initial estimates for those parameters.
Both of the initial estimation objective functions can be optimized efficiently using the :'-Jewton's method variant described above, and can be run at any stage by the user to quickly compute a reasonably accurate model. When the initial estimates of the parameters have been computed, the nOll-lillear refinement can be used to calculate their final values.
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3.0.4
The
l\oll-1ille~,' Optimiz~tionWhere Ihc twu initi.l ~",ti","t.;" o Jlr"",lu"", (0, ""d 0..) co<:qutcd i"lat"d ,;et, oj l,"m"""'~" loc
,poc~1 ~ml ,in,pli.,;ry, rh~ nOL_li""a! objocti,." hmctim" "ptiIlli,."j ''''e,. rh~ ~ot.i", >,"rom.",,,r
=
to ",",u ...
pc"";"",, . )"
rhi. way .• ny eHOC i, ",;>r""u" ''''e, ,u. "r tt., l""'"'f't"' •. ,nd ,he ro,ult" ~ ,"I~'"I"i"lly n.o", acru,ate ,,,odol Rcc,.llinp, tho onW """,ie l""."o",d l>l Lqua'ion 6, tbe r'OL l:rlC~' "hjecti "0 :ll""tio" can be r!ef."c" as
Ago.in th~ ol'tlmi,atiun "'~te1,Y pre""n'"" ~IK'V' i, 11",,1 to mini,,,;"e O. and gcn<r~lly "o<,verl"" [(, a ",,:,ltiotl wit!-.in ter. ile ... t'"n,.
:3. i Heal-world Example
,.\ ".)
F:g:uro 1.1 T .. ·u photO!(,"pl"
"f.
I~'f:" I);",:ot. Six ro'I,I~o"". Tb, rch""'am edge. u: tl,e;c ;>t<o-'O£"'pb, ;,a,'o ",,-,n markoc out. in rod by ;iLe U>C".To illu,tr~tc Ihe furl<ti,,".~li:y "I ,be h",,\e .... ork, con'iic.oT the t""'O ima~.,. of • laTP,~ Di,crict Six rc,idctlCC. d'.o.m OIl FiR"r".J C' "lOg "<l.y I to,,", ' .... 0 p;wtO£ropiB, ~ model ofl!-.i. oouoc i, ",,,"'"tely roc()(]"trocted, ~I •. I tl;<,>\ I"'er 10rllrcrL In t!-,,'if' two poot"'i':,"pk tjoe cd);e, "f Ite 'o,,:I([;og bvo 1-""'<1 Ill..-lrod (~lt:o
"oc
I,,· Ibe ll"" ""r! ,il<''lO I""", ,HO hler linW tu tbe I:IOOOI;n nnk" \0nJ,,,,,
tn. "hje<;;.:"" 111",--ion (~tion .U) ",-"c t!-.~: one OI:'.!\" pre",n" ~ ,"0',)' """"'''l' v:o .... of on:y ~
;>ol'tiotl of :he huilGing, ~·tile d,e "Ii"" p,,,,,,nr, , "il'~' 01 'b" oot;", I"ild:n:>:, hu: :rOI:! ~ great
d,"~nce ,''''~y,
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CH .. IPTER..1 REC'OSSTR[-'CTWN OVIORVW1\
for Ihe ohimney, o.nd ~ cnb0id and a pd."" for ~",ch vi the ~able, (~e"tivn 3_~), Figure 14 ,110"'"
the ,,,,ono ,,-.-,dol ..,xi tao hioTucny of the pIimitiyes Ihat <'OHlPIi"" the lJl'Jdd, The Ioc<. Ilode ill Ihe
"""'" nioc.rchy i, It. boo< cuooid, It, children ",e the wof p,ism, the chilUm'.', alld ti,e t,,'U l'l't.c.id>
1:"".l in moJdling the gable< Ea.cn of these cuboids h..,. fu"n'~' child - a pIi,nL that oompH,"
tho gaN""
~ ~ :. : ,: ,I . : ,: ._ :,": . ~ '---~'I G<ru: F~~ I
,.,
(b)rig""" 1,',0 n." Hl<Xi<. l'",d ;" ~r""
,w ..
",,,,,,,tioo of t.lw I~,g<' f);,,,[ct Six ,c,iJ~""" ~"d i" ""<;oci",cd scene ~Taph, J.I(a) is the model that w'-" u""dw
rK<Jnstr"l't tI., b"ildcng 11(\,} ilb,;""a'''' ,he primitive hi<Yacchy of the pIimitLycs wed in tho model.Fi~u'~ 15 The
''''0
i"p'" j>butug"aj>h< uf [b~ la,~~ ,c,;J~,,,,, ~,itb the r(~o.\"'I'l,(,t"j ""xlel pm:<'C1.cJ unto tl~·m. 'Ii", ;>rujecwd liw~' nll't' h th,· rna,:m Jine, and b,,;ldill~ c,lg<e' "'''l' '-' ,-or"t<"l)'_lncil:din): the two cam"",., (vhich oontrib1:te six p.ram~IO" oocnj. the to .. : m:mbco: of fro< lXll'am·
eV-e< u...,d to IUud.1 the' buildillg i< l~, Twu "re '''''trib"ted b.'i the b .. ...-- ""b0id fur it< dqxh and
heighI one '''''
the t.ighI oftne ba.<o cuboid. on" for the hoight of tn~ pci,,,,,, MId one foc tao ~'idth of tao g~bh,
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Since the two gables are identical, both share the same parameters.
The reconstruction of the model was successful, matching closely to the user observations. Figure 15 illustrates the model projected onto the original photographs, from the perspectives of the recon-structed cameras. The projected lines match up very closely to the building edges and the marked lines; however, in the second image, the base of the model appears to be lower than the base of the building. This problem is due to the fact that in the second photo it was impossible to mark out where the base of the building joined the ground, and thus this was not explicitly taken into account during the optimization. The general problem of foundations is discussed further in Chapter 6. Irrespective of this, the reconstruction was otherwise very accurate.