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

Regularization term Featureless Inverse depth map smoothness smooth featureless Regularization RGB gradient regularization Inverse depth map gradient Huber norm Total variation staircase effect L1

REGULARIZED ENERGY FUNCTIONAL (2.2.2 2.2.3)

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 [ , , ]