Optimized stereo reconstruction of free-form space curves based on a nonuniform rational B-spline model

Yi Jun Xiao; Y. F. Li

doi:10.1364/JOSAA.22.001746

Journal of the Optical Society of America A
Vol. 22,
Issue 9,
pp. 1746-1762
(2005)
•https://doi.org/10.1364/JOSAA.22.001746

Optimized stereo reconstruction of free-form space curves based on a nonuniform rational B-spline model

Yi Jun Xiao and Y. F. Li

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Yi Jun Xiao and Y. F. Li, "Optimized stereo reconstruction of free-form space curves based on a nonuniform rational B-spline model," J. Opt. Soc. Am. A 22, 1746-1762 (2005)

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Abstract

Analytical reconstruction of 3D curves from their stereo images is an important issue in computer vision. We present an optimization framework for such a problem based on a nonuniform rational B-spline (NURBS) curve model that converts reconstruction of a 3D curve into reconstruction of control points and weights of a NURBS representation of the curve, accordingly bypassing the error-prone point-to-point correspondence matching. Perspective invariance of NURBS curves and constraints deduced on stereo NURBS curves are employed to formulate the 3D curve reconstruction problem into a constrained nonlinear optimization. A parallel rectification technique is then adopted to simplify the constraints, and the Levenberg–Marquardt algorithm is applied to search for the optimal solution of the simplified problem. The results from our experiments show that the proposed framework works stably in the presence of different data samplings, randomly posed noise, and partial loss of data and is potentially suitable for real scenes.

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Equations (51)

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Approach	Noise Levels
Approach	$σ_{t} = 0.1 t_{int}$	$σ_{t} = 0.2 t_{int}$	$σ_{t} = 0.3 t_{int}$
NURBS-based
	$e_{3 d} = (\begin{array}{l} 0.0072 \\ 0.0151 \\ 0.00027510 \\ 0.0038 \end{array})$ ^b	$e_{3 d} = (\begin{array}{l} 0.0075 \\ 0.0146 \\ 0.00015031 \\ 0.0037 \end{array})$	$e_{3 d} = (\begin{matrix} 0.0079 \\ 0.0160 \\ 0.00030896 \\ 0.0036 \end{matrix})$


	$e_{2 d l} = (\begin{array}{l} 0.1361 \\ 0.7078 \\ 0.0042 \\ 0.1192 \end{array})$	$e_{2 d l} = (\begin{array}{l} 0.1402 \\ 0.5967 \\ 0.0017 \\ 0.1214 \end{array})$	$e_{2 d l} = (\begin{array}{l} 0.1454 \\ 0.5976 \\ 0.0032 \\ 0.1188 \end{array})$


	$e_{2 d r} = (\begin{array}{l} 0.0966 \\ 0.4435 \\ 0.0052 \\ 0.0759 \end{array})$	$e_{2 d r} = (\begin{array}{l} 0.0990 \\ 0.3654 \\ 0.0022 \\ 0.0733 \end{array})$	$e_{2 d r} = (\begin{array}{l} 0.1044 \\ 0.3637 \\ 0.0037 \\ 0.0717 \end{array})$


Point-based

	$e_{3 d} = (\begin{array}{l} 0.0280 \\ 0.1018 \\ 0.0032 \\ 0.0260 \end{array})$	$e_{3 d} = (\begin{array}{l} 0.0573 \\ 0.1644 \\ 0.0064 \\ 0.0437 \end{array})$	$e_{3 d} = (\begin{array}{l} 0.0878 \\ 0.3347 \\ 0.0080 \\ 0.0823 \end{array})$


	$e_{2 d l} = (\begin{array}{l} 0.1626 \\ 0.5508 \\ 0.0133 \\ 0.1459 \end{array})$	$e_{2 d l} = (\begin{array}{l} 0.2257 \\ 1.0227 \\ 0.0355 \\ 0.2009 \end{array})$	$e_{2 d l} = (\begin{array}{l} 0.2861 \\ 1.2363 \\ 0.0389 \\ 0.2674 \end{array})$


	$e_{2 d r} = (\begin{array}{l} 0.1285 \\ 0.4368 \\ 0.0132 \\ 0.0953 \end{array})$	$e_{2 d r} = (\begin{array}{l} 0.1764 \\ 0.6986 \\ 0.0322 \\ 0.1278 \end{array})$	$e_{2 d r} = (\begin{array}{l} 0.2407 \\ 0.8416 \\ 0.0084 \\ 0.1998 \end{array})$

Noise Level	$e_{3 d}$	$e_{2 d l}$	$e_{2 d r}$
$σ = {[0.2 0.2]}^{T}$	$(\begin{matrix} 0.0119 \\ 0.0286 \\ 0.0020 \\ 0.0043 \end{matrix})$ ^b	$(\begin{matrix} 0.1952 \\ 0.7812 \\ 0.0083 \\ 0.1558 \end{matrix})$	$(\begin{matrix} 0.1354 \\ 0.5549 \\ 0.0041 \\ 0.0995 \end{matrix})$


$σ = {[0.4 0.4]}^{T}$	$(\begin{matrix} 0.0162 \\ 0.0721 \\ 0.0028 \\ 0.0198 \end{matrix})$	$(\begin{matrix} 0.1968 \\ 0.7687 \\ 0.0049 \\ 0.1486 \end{matrix})$	$(\begin{matrix} 0.1361 \\ 0.5596 \\ 0.0130 \\ 0.1042 \end{matrix})$


$σ = {[0.6 0.6]}^{T}$	$(\begin{matrix} 0.0260 \\ 0.0892 \\ 0.0028 \\ 0.0198 \end{matrix})$	$(\begin{matrix} 0.1924 \\ 0.8090 \\ 0.0164 \\ 0.1272 \end{matrix})$	$(\begin{matrix} 0.1777 \\ 0.5555 \\ 0.1009 \\ 0.0260 \end{matrix})$


$σ = {[0.8 0.8]}^{T}$	$(\begin{matrix} 0.0483 \\ 0.1575 \\ 0.0029 \\ 0.0398 \end{matrix})$	$(\begin{matrix} 0.1901 \\ 0.8186 \\ 0.0105 \\ 0.1498 \end{matrix})$	$(\begin{matrix} 0.1839 \\ 0.5280 \\ 0.0179 \\ 0.0987 \end{matrix})$


$σ = {[1.0 1.0]}^{T}$	$(\begin{matrix} 0.0674 \\ 0.4081 \\ 0.0091 \\ 0.0639 \end{matrix})$	$(\begin{matrix} 0.2069 \\ 0.7834 \\ 0.0255 \\ 0.1704 \end{matrix})$	$(\begin{matrix} 0.2086 \\ 0.5885 \\ 0.0267 \\ 0.1470 \end{matrix})$

Approach	$e_{3 d}$	$e_{2 d l}$	$e_{2 d r}$
NURBS-based

	$(\begin{matrix} 0.0077 \\ 0.0148 \\ 4.9422 \times 10^{- 0.005} \\ 0.0039 \end{matrix})$ ^b	$(\begin{matrix} 0.1999 \\ 0.5455 \\ 0.0014 \\ 0.1805 \end{matrix})$	$(\begin{matrix} 0.1330 \\ 0.5226 \\ 0.0012 \\ 0.1070 \end{matrix})$


B-Spline-based
	$(\begin{matrix} 0.5812 \\ 1.1926 \\ 0.0087 \\ 0.4052 \end{matrix})$	$(\begin{matrix} 2.3153 \\ 6.3277 \\ 0.0450 \\ 1.7830 \end{matrix})$	$(\begin{matrix} 1.2288 \\ 2.4278 \\ 0.0199 \\ 0.6562 \end{matrix})$

Approach	Curve 1	Curve 2	Curve 3
NURBS-based


	$e_{2 d l} = (\begin{matrix} 0.2095 \\ 0.8384 \\ 0.0016 \\ 0.1872 \end{matrix})$ ^b	$e_{2 d l} = (\begin{matrix} 0.2129 \\ 1.0026 \\ 0.0054 \\ 0.1700 \end{matrix})$	$e_{2 d l} = (\begin{matrix} 0.0592 \\ 0.4045 \\ 0.00070460 \\ 0.0677 \end{matrix})$


	$e_{2 d r} = (\begin{matrix} 0.1957 \\ 0.6801 \\ 0.0023 \\ 0.1541 \end{matrix})$	$e_{2 d r} = (\begin{matrix} 0.2304 \\ 0.8392 \\ 0.0028 \\ 0.1813 \end{matrix})$	$e_{2 d r} = (\begin{matrix} 0.0696 \\ 0.3640 \\ 0.00030307 \\ 0.0744 \end{matrix})$


B-Spline based

	$e_{2 d l} = (\begin{matrix} 2.2089 \\ 4.9437 \\ 0.0095 \\ 1.7267 \end{matrix})$	$e_{2 d l} = (\begin{matrix} 1.8797 \\ 4.3470 \\ 0.0130 \\ 1.9866 \end{matrix})$	$e_{2 d l} = (\begin{matrix} 1.8849 \\ 9.5069 \\ 0.0268 \\ 2.2086 \end{matrix})$


	$e_{2 d r} = (\begin{matrix} 1.9709 \\ 4.5874 \\ 0.0158 \\ 1.5445 \end{matrix})$	$e_{2 d r} = (\begin{matrix} 1.7997 \\ 4.5549 \\ 0.0037 \\ 2.0025 \end{matrix})$	$e_{2 d r} = (\begin{matrix} 1.7864 \\ 9.5556 \\ 0.0140 \\ 2.2680 \end{matrix})$

Approach	Curve 1	Curve 2
NURBS-based

	$e_{2 d l} = (\begin{matrix} 0.3079 \\ 0.8992 \\ 0.0007 \\ 0.1993 \end{matrix})$ ^b	$e_{2 d l} = (\begin{matrix} 0.3663 \\ 1.4299 \\ 0.0014 \\ 0.2815 \end{matrix})$


	$e_{2 d r} = (\begin{matrix} 0.2657 \\ 0.8835 \\ 0.0019 \\ 0.2013 \end{matrix})$	$e_{2 d r} = (\begin{matrix} 0.3467 \\ 1.9214 \\ 0.0004 \\ 0.2849 \end{matrix})$


B-spline based

	$e_{2 d l} = (\begin{matrix} 4.8211 \\ 44.9963 \\ 0.0447 \\ 8.9624 \end{matrix})$	$e_{2 d l} = (\begin{matrix} 2.5231 \\ 8.8661 \\ 0.0040 \\ 1.6836 \end{matrix})$


	$e_{2 d r} = (\begin{matrix} 3.4315 \\ 34.3148 \\ 0.0409 \\ 4.8338 \end{matrix})$	$e_{2 d r} = (\begin{matrix} 2.4030 \\ 11.8237 \\ 0.0139 \\ 1.9952 \end{matrix})$

Abstract

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Figures (17)

Tables (5)

Equations (51)

Journal of the Optical Society of America A