A First Order Predicate Logic Formulation of the 3D Reconstruction Problem and its Solution Space

Robinson, M., Kubik, K. and Lovell, B. (2005) A First Order Predicate Logic Formulation of the 3D Reconstruction Problem and its Solution Space. International Journal of Pattern Recognition and Artificial Intelligence, 19 1: 45-62.

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Author Robinson, M.
Kubik, K.
Lovell, B.
Title A First Order Predicate Logic Formulation of the 3D Reconstruction Problem and its Solution Space
Journal name International Journal of Pattern Recognition and Artificial Intelligence   Check publisher's open access policy
ISSN 0218-0014
Publication date 2005-01-01
Sub-type Article (original research)
DOI 10.1142/S0218001405003910
Volume 19
Issue 1
Start page 45
End page 62
Total pages 18
Editor P. Shen Pei Wang
X. Jiang
Place of publication Singapore
Publisher World Scientific Publishing Co. PTE LTD
Collection year 2005
Language eng
Subject C1
280207 Pattern Recognition
280208 Computer Vision
700199 Computer software and services not elsewhere classified
Abstract This paper defines the 3D reconstruction problem as the process of reconstructing a 3D scene from numerous 2D visual images of that scene. It is well known that this problem is ill-posed, and numerous constraints and assumptions are used in 3D reconstruction algorithms in order to reduce the solution space. Unfortunately, most constraints only work in a certain range of situations and often constraints are built into the most fundamental methods (e.g. Area Based Matching assumes that all the pixels in the window belong to the same object). This paper presents a novel formulation of the 3D reconstruction problem, using a voxel framework and first order logic equations, which does not contain any additional constraints or assumptions. Solving this formulation for a set of input images gives all the possible solutions for that set, rather than picking a solution that is deemed most likely. Using this formulation, this paper studies the problem of uniqueness in 3D reconstruction and how the solution space changes for different configurations of input images. It is found that it is not possible to guarantee a unique solution, no matter how many images are taken of the scene, their orientation or even how much color variation is in the scene itself. Results of using the formulation to reconstruct a few small voxel spaces are also presented. They show that the number of solutions is extremely large for even very small voxel spaces (5 x 5 voxel space gives 10 to 10(7) solutions). This shows the need for constraints to reduce the solution space to a reasonable size. Finally, it is noted that because of the discrete nature of the formulation, the solution space size can be easily calculated, making the formulation a useful tool to numerically evaluate the usefulness of any constraints that are added.
Keyword Computer Science, Artificial Intelligence
3d Reconstruction
Solution Space
Uniqueness
Voxels
Logic
reconstruction algorithms
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