[Paper Review] Multiple View Reconstruction of Calibrated Images using Singular Value Decomposition

Ayan Chaudhury, Abhishek Kumar Gupta+3 more|arXiv (Cornell University)|Nov 2, 2010

Optical measurement and interference techniques6 references3 citations

TL;DR

This paper presents a novel approach to multiple view reconstruction in calibrated multi-camera systems using Singular Value Decomposition (SVD) to unify image coordinates across different camera positions and orientations. By calibrating each camera in its local coordinate system and transforming all data into a common global coordinate system, the method ensures that a fixed 3D world point projects to the same image coordinate regardless of camera viewpoint, achieving consistent reconstruction with a key result of (158, 88) pixel coordinates across all cameras for a single calibration point.

ABSTRACT

Calibration in a multi camera network has widely been studied for over several years starting from the earlier days of photogrammetry. Many authors have presented several calibration algorithms with their relative advantages and disadvantages. In a stereovision system, multiple view reconstruction is a challenging task. However, the total computational procedure in detail has not been presented before. Here in this work, we are dealing with the problem that, when a world coordinate point is fixed in space, image coordinates of that 3D point vary for different camera positions and orientations. In computer vision aspect, this situation is undesirable. That is, the system has to be designed in such a way that image coordinate of the world coordinate point will be fixed irrespective of the position & orientation of the cameras. We have done it in an elegant fashion. Firstly, camera parameters are calculated in its local coordinate system. Then, we use global coordinate data to transfer all local coordinate data of stereo cameras into same global coordinate system, so that we can register everything into this global coordinate system. After all the transformations, when the image coordinate of the world coordinate point is calculated, it gives same coordinate value for all camera positions & orientations. That is, the whole system is calibrated.

Motivation & Objective

• To address the challenge of inconsistent image coordinates for a fixed 3D world point across multiple camera views in multi-camera systems.
• To unify multiple camera views into a single global coordinate system for accurate multiple view reconstruction.
• To demonstrate that SVD-based transformation enables consistent image coordinates regardless of camera position or orientation.
• To validate the method using a single calibration point across multiple camera positions, achieving identical image coordinates.

Proposed method

• Calibrate each camera in its local coordinate system using 3D-2D point correspondences from a calibration pattern with 19 points.
• Extract camera parameters (rotation and translation) via standard calibration models, assuming no lens distortion for simplicity.
• Use SVD to solve the system of equations for coordinate transformation from local to global coordinate systems.
• Apply morphological image processing to detect feature points in each image, enabling accurate pixel coordinate extraction.
• Transform all local camera data into a common global coordinate system to unify image projections.
• Verify consistency by computing image coordinates of the same world point across all cameras and confirming identical results.

Experimental results

Research questions

• RQ1 Can a multi-camera system achieve consistent image coordinates for a fixed 3D world point across varying camera positions and orientations?
• RQ2 How can SVD be effectively used to unify multiple camera views into a single global coordinate system?
• RQ3 What is the impact of coordinate system transformation on image coordinate consistency in stereo reconstruction?
• RQ4 Can a simplified calibration model with negligible lens distortion still achieve high consistency in image projection?

Key findings

• After transformation into the global coordinate system, the image coordinate of a fixed 3D world point was consistently (158, 88) across all camera views.
• The use of SVD enabled accurate and stable solution of the transformation system, ensuring alignment across multiple camera views.
• Morphological image processing successfully extracted and localized calibration points, enabling precise pixel coordinate computation.
• The method achieved consistent reconstruction without requiring complex non-linear optimization or advanced distortion models.
• The image center was estimated as the midpoint of the sensor array (500, 550) for a 1000×1100 image, simplifying calibration without sacrificing accuracy.