Jeremy S. De Bonet : Poxels: Probabilistic Voxel Reconstruction




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Introduction

In this paper we will be examining the problem of reconstructing a voxel representation of 3D space from a series of 2D or 1D projections of the space.
We reconstruct the 3D space by optimizing over the probability that each voxel is visible in each of the projections. An iterative algorithm is used to find the optimal probability distibution which jointly explains all the observed projections.
Previous research in this problem has been dichotomized. In computer vision problems common environments are typically considered in which all objects are assumed to be solid, and have definite positions. In medical imaging, volumes filled with semitransparent tissue and bone, in this case occlusion is ignored. In each case these simplifying assumtions can be exploited in the design of volume reconstruction algorithms. However, in reality these approimations are innacurate, the common environments contain transparent objects and because of data limitations, their exact locations can be uncertain; in medical images, bone and other solid tissue can cause occlusions.
The problem of reconstruting volumetric information from a collection of photographs is known as multiview stereo in the vision research community. Solutions have typically involved using a matching threhold to determine wether a position in space is filled. This decision is then used to the fullness of later voxels. Such a scheme results in two artifacts; first, errors in determining voxel fullness cannot be corrected, resulting in surfaces which teppnd to bulge toward the camera; second only completely opaque surfaces can be considered.
In medical imaging, this problem surfaces in computed tomography. Here solutions typically ignore occlusion, as most surfaces are only partially opaque, and reconstruction is performed with linear methods. Because it ignores the occlusion of one object by another, such reconstructions tend to have ``ghosts'' or ``shadows'' which are caused by contributions from regions which should be occluded.
Here we present a framework for the volume reconstruction problem so that both solid and transparent objects can be accurately represented. Furthermore, because the current approach formulates the problem as one of optimization over the probability distribution of the visibility of each region of space, uncertainty -- due to lack of data, or perhaps contradictory data -- can be captured as well.
From the recovered probability-visibility distributions images of the volume can be synthesized from novel points-of-view.
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Jeremy S. De Bonet
jsd@debonet.com
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