NeuS: LearningNeuralImplicitSurfaces byVolumeRenderingforMulti-viewReconstruction-SupplementaryMaterial-ADerivationforComputingOpacityαi

Next consider the case where[ti,ti+1] lies in a range[t`,tr] over which the camera ray is exiting the surface, i.e. the signed distance function is increasing onp(t) over [t`,tr]. Then we have ( f(p(t)) v) < 0 in [ti,ti+1]. Then, according to Eqn. 1, we haveρ(t) = 0. Therefore, by Eqn.12ofthepaper,wehave αi=1 exp Recall that our S-density fieldφs(f(x)) is defined using the logistic density functionφs(x) = se sx/(1+e sx)2, which is the derivative of the Sigmoid functionΦs(x) = (1+e sx) 1, i.e. φs(x)=Φ0s(x). As a first-order approximation of signed distance functionf, suppose that locally the surface is tangentially approximated byasufficiently small planar patch with itsoutwardunitnormal vector denotedas n. Nowsupposep(t)isapoint on the surfaceS,that is, f(p(t)) = 0. Next we will examine the value ofdwdt(t) at t = t . Thesigneddistancefunction f ismodeledbyanMLP that consists of 8hidden layers with hidden size of 256.