DTAM Real-time SFM (structure from motion) Narrow-baseline
frames Dense 3D surface model You Tube [Stuehmer+, 10] Energy
functional contribution camera-pose estimation [Lovegrove+, 10]
PTAM[Klein+ 07] follow
DTAM Regularized energy functional (2.2.0 - 2.2.1) Eq.
(6)robust spatial regularization term + photometric error data term
Eq. (2) photometric error data term Eq. (3)error function Eq.
(5)robust spatial regularization term Eq. (4)the Huber norm
Regularized energy functional (2.2.2 2.2.3) Eq. (7)introduction of
an auxiliary variable for alternating optimization Eq.
(8),(9),(10)replacement of the Huber norm Eq. (11),(12)optimization
of the regularization term Eq. (13),(14)optimization of the data
term (2.2.4 2.2.5) Dense tracking (2.3-) Camera pose estimation
(2.3.1-) (3.)
REGULARIZED ENERGY FUNCTIONAL (2.2.0 - 2.2.1)
311 3 3 2 ))(,(),( : : : Ruuu R R RI u R xdx r Inverse depth
map Inverse depth map 3D-2D RGB
Data term
Data term Key frame )(u 3 Rr m transfer 3 R RGB Narrow-baseline
: brightness constancy Error L1 occlusions occlusions
Data term RGB error (b) minimum (a) featureless minimum
featureless Regularization term
Regularized energy functional coarse-to-fine Total variation +
L2 non-convex exhaustive search 2.2.4 non-convex coarse-to-fine
[Stuehmer+, 10]
Regularized energy functional iteration 0 d=a A: G: Huber norm
|q|1 >1 indicator |q|1 = Huber norm regularization Data term
Coarse-to-fine Photometric error Regularization
: : f x f I x functional + I[ f (x, )] x, f (x, ), df dxa b dx
: 1[ ] http://hooktail.sub.jp/mathInPhys/variations1/ f df/dx I[ f
(x, )] x,(x )2 2 ,2xa b dx
: Legendre transformation y=f(x) {(x,y)} { , } p q px-f(x) x f
pxfpxpf x d d ,)(max)(* q px(p) f (x(p)) y f (x) y px q x x(p),y f
(x) y f (x) y px q x(p) px f (x)
: exponential map R d )0(d ),( d )(d R XXR R )exp()( )(log d )(
)(d d )( )(d )( d )(d XR XR X R R X R R XR R R(0) )exp(
)exp()exp()exp( )()()( zzyyxx zzyyxx zzyyxx AAA AAA RRR [ , ,
]
