doi:10.1016/j.cviu.2003.09.003 
Copyright © 2003 Elsevier Inc. All rights reserved.
Depth distortion under calibration uncertainty
Loong-Fah Cheong
, 
and Chin-Hwee Peh
Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore
Received 1 March 2002;
accepted 16 September 2003. ;
Available online 27 October 2003.
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Abstract
There have been relatively little works to shed light on the effects of errors in the intrinsic parameters on motion estimation and scene reconstruction. Given that the estimation of the extrinsic and intrinsic parameters apts to be imprecise, it is important to study the resulting distortion on the recovered structure. By making use of the iso-distortion framework, we explicitly characterize the geometry of the distorted space recovered from 3D motion with freely varying focal length. This characterization allows us: (1) to investigate the effectiveness of the visibility constraint in disambiguating errors in calibration parameters by studying the negative distortion regions and (2) to make explicit those ambiguous error situations under which the visibility constraint is not effective. An important finding is that under these ambiguous situations, the direction of heading can be accurately recovered and the structure recovered experienced a well-behaved distortion. The distortion is given by a relief transformation which preserves ordinal depth relations. Thus in the case where the change in focal length is not well estimated, structure information in the form of depth relief can be obtained. Experiments were presented to support the use of the visibility constraint in obtaining such partial motion and structure solutions.
Author Keywords: Structure from motion; Depth distortion; Space perception; Calibration parameters
Fig. 1. D=±∞ iso-distortion surfaces in the xyZ-space for δe=−0.01 : (A) 2D and (B) 3D view (□, true FOE and 
, estimated FOE).
Fig. 2.
D=±∞ iso-distortion surfaces in the
xyZ-space for δ
e=0.01 ): (A) 2D and (B) 3D view (□, true FOE and
, estimated FOE).
Fig. 3. Iso-distortion surfaces for δ
e=−0.01 at (A)
Z=0 and (B)
Z=200 with different
D: .
Fig. 4. Iso-distortion surfaces for δ
e=0.01 at (A)
Z=0 and (B)
Z=200 with different
D: .
Fig. 5.
D=±∞ surface with second-order effect for δ
e>0 ). Left: 2D and right: 3D.
Fig. 6.
D=±∞ surface with second-order effect for large focal length. Left: 2D and right: 3D.
Fig. 7. 2D representations of iso-distortion volume sliced at
y=10 : (A) δ
e=0.005, ; (B) δ
e=−0.005, ; and (C)δ
e=−0.01, (shaded region represents negative distortion region and darker shaded region represents scene in view that falls under the negative distortion region).
Fig. 8. Iso-distortion surfaces when and (
x0=30,
y0=30,
Oxe=0,
Oye=0,Γ
e=0.3,Λ
e=0.3): when (A) δ
e=0.01 and (B) δ
e=−0.01.
Fig. 9. Slant profiles for the three synthetic planes in
x–
Z-plane: solid lines, original profiles and broken lines, reconstructed profiles.
Fig. 10. Minimum negative depth distribution with coarse sampling of FOE: (A) scatter plot and (B) contour plot in plane scene.
Fig. 11. Image Sequences I. Left: image (‘+’, true FOE and ‘X’, estimated FOE) and right: scatter plot of minimum negative depth distribution.
Fig. 12. Image Sequences II. Left: image (‘+’, true FOE and ‘X’, estimated FOE) and right: scatter plot of minimum negative depth distribution.
Fig. 13. Image Sequences III. Left: image (‘+’, true FOE and ‘X’, estimated FOE) and right: scatter plot of minimum negative depth distribution.
Fig. 14. Image Sequences IV. Left: image (‘+’, true FOE and ‘X’, estimated FOE) and right: scatter plot of minimum negative depth distribution.
Fig. 15. Reconstructed depth map. Top left: Image Sequence I; top right: Image Sequence II; bottom left: Image Sequence III; and bottom right: Image Sequence IV. (Color representation from near to far: red, magenta, yellow, green and blue, respectively.)
Table 1. Intrinsic and extrinsic parameters
Table 2. Boundaries and step-sizes of searched parameters (subscripts i, f, and s denote initial, final, and step-size, respectively)
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