OReX: Object Reconstruction from Planar Cross-sections Using Neural Fields

Haim Sawdayee, Amir Vaxman, Amit H. Bermano

Reconstruction from cross-sections

Parallel planes setting
Any orientation or distribution of planes

Problem Setting

[!tips] cross-sections

$\mathcal{P}=\left{P_1, \cdots, P_k\right}$ : 2D planes embedded in $\mathbb{R}^3$ with arbitrary offsets and orientations

Each plane $P_i$ contains an arbitrary set of (softly) non-intersecting oriented contours $C_i=\left{c_{i, 1}, \cdots, c_{i, l_i}\right}$ that consistently partition the plane into regions of “inside” and “outside” of an unknown domain $\Omega\subset\mathbb{R}^3$ with boundary $\partial \Omega$

[!tips] How to model it by INR
The target output of our method is an indicator function $Y:\mathbb{R}^3\rightarrow\mathbb{R}$ define $\Omega$ as :
$$Y(x)= \begin{cases}1 & x \notin \Omega \ 0 & x \in \Omega \ 0.5 & x \in \partial \Omega\end{cases}$$

CSLC # header
15 2  # number of planes, number of labels (should be at least 2 - inside and outside)

1 78 1 0.0 0.0 1.0 -0.86 # plane index (1-indexing, please state planes in order), number of vertices in the plane image (a hole is counted as another component), number of connected components, plane parameters A,B,C,D, such that Ax+By+Cz+D=0

0.10 0.08 0.86 # The vertices in x,y,z coordinates, should be on the plane.
0.09 0.08 0.86
0.08 0.09 0.86
0.07 0.09 0.86
[...] # rest of vertices

78 1 0 1 2 3 4 5 6 7 8 9 10 11 [...]  # image component: starts with the number of vertices, then label of the component (in case of a hole, h should be added and the index of the component contains the hole), then the indices of vertices that form a contour of the inside label, ordered CCW.
[...] # rest of components

gen1 by foxwarze on Sketchfab

gen2 by foxwarze on Sketchfab

gen3 by foxwarze on Sketchfab

gen4 by foxwarze on Sketchfab

gen5 by foxwarze on Sketchfab

Overview

Loss:
$$\mathcal{L}(x, \theta)=\sum_{i=0}^{N-1} \operatorname{BCE}\left(Y_i(x)\right)+\lambda \max \left(0,\left|\nabla f_{N-1}(x)\right|-\alpha\right)$$