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2013 IEEE International Conference on Computer Vision Enhanced Continuous Tabu Search for Parameter Estimation in Multiview Geometry Guoqing Zhou Qing Wang School of Computer Science and Engineering Northwestern Polytechnical University, Xi’an 710072, P. R. China {zhouguoqing,qwang}@nwpu.edu.cn Abstract lems in general is difficult due to the inherent non-convexity and the presence of local optima. Optimization using the L∞ norm has been becoming To remedy these problems, a number of literatures have an effective way to solve parameter estimation problems in shown that many multiview geometric problems are quasi- multiview geometry. But the computational cost increases convex under L∞ norm [8][13]. A particularly fruitful line rapidly with the size of measurement data. Although some of work has been the development of methods that mini- strategies have been presented to improve the efficiency of mize the maximal of reprojection errors (L∞ norm) across L∞ optimization, it is still an open issue. In the paper, we observations, instead of the sum of squared reprojection er- propose a novel approach under the framework of enhanced rors. It has been proven that many multiview problems have continuous tabu search (ECTS) for generic parameter es- a single local optimum under the framework of L∞ nor- timation in multiview geometry. ECTS is an optimization m. The existence of globally optimal solution enables it ef- method in the domain of artificial intelligence, which has fective to use convex optimization in parameter estimation an interesting ability of covering a wide solution space by [11]. However, this kind of algorithm is too complicated to promoting the search far away from current solution and solve large-scale geometric problems efficiently. consecutively decreasing the possibility of trapping in the Recently, researchers proposed a new strategy, that by local minima. Taking the triangulation as an example, we giving a simple sufficient condition for global optimality propose the corresponding ways in the key steps of ECTS, that can be used to verify that a solution obtained from any diversification and intensification. We also present theoret- local methods is indeed global [15][17]. This algorithm ical proof to guarantee the global convergence of search returns either a certificate of optimality for local solution with probability one. Experimental results have validated or global solution. Agarwal et al. [1] discovered that Ols- that the ECTS based approach can obtain global optimum son’s method[17] is a special case for generalized fractional efficiently, especially for large scale dimension of param- programming. Dai et al. [5] found the sequence of convex eter. Potentially, the novel ECTS based algorithm can be problems are highly related and proposed a method to de- applied in many applications of multiview geometry. rive a Newton-like step from any given point. The efficiency of L∞ algorithm has been improved obviously. All of the above mentioned algorithms are still on the 1. Introduction ways of traditional optimization, and fewer modern opti- mization methods are considered to solve these problems so Parameter estimation is one of the most fundamental far. In recent years, Tabu search (TS), a meta-heuristic opti- problems in multiview geometry. The typical measurements mization method originally proposed by Glover [6][7], has of error function include algebraic distance, geometric dis- extensively attracted attentions of researchers. It enhances tance, reprojection error, and Sampson error [9]. Traditional the performance of a local search method by using memory optimization algorithms have been dominated by local opti- structures that describe the visited solutions: once a poten- mization techniques based on the L2 norm, such as Newton tial solution has been determined, it is marked as ‘tabu’ so or Levenberg-Marquardt iterations [9] or bundle adjustment that the algorithm does not visit that possibility repeated- [20] for finding a local optimum. While some of method- ly. However, the basic TS is proposed for combinatorial s except iterative nonlinear optimization yield closed-form optimization problems primitively. Chelouah et al. [4] pro- solutions, they are quite efficient and relatively easy to im- posed a variant of TS for the global continuous optimization plement. However, solving multiple view geometry prob- problems (GCOPs), called enhanced continuous tabu search 1550-5499/13 $31.00 © 2013 IEEE 32333240 DOI 10.1109/ICCV.2013.402 (ECTS). This scheme divides the optimization process into to minimize the following cost function subject to the con- two sequential phases, namely diversification and intensifi- straint of bi x + ˜bi > 0, cation. As a common drawback in GCOPs, meta-heuristic N approaches cannot guarantee finding the global optimum. 2 f(x)= ui − Pix (P1) In this paper, we proposed a novel method under the i=1 ECTS framework for parameter estimation in multiview ge- ometry. The procedure takes the result of linear method After a simple expansion, (P1) could be rewritten as, as initial estimation, and utilizes the ECTS to attain the x N x global optimum. In the phase of diversification, we pro- E( )= i=1 fi( ) 2 (a x+˜a )2 x j=1 ij ij pose a non-iterative way to obtain an initial bounding con- s.t.fi( )= ˜ 2 (1) (bi x+bi) vex hull that contains the global optimum. At the stage b x + ˜bi > 0 of intensification, we propose a new approach to attain the i best neighbor set according to the characteristics of mul- n where x, aij , bi ∈ R and a˜ij , ˜bi ∈ R and tiview geometric problems. Finally, we prove the conver- gence of ECTS method in multiview geometry from the aij = pij − uij pi3, a˜ij =˜pij − uij p˜i3,j =1, 2, (2) viewpoint of probability. The algorithm tends to achieve bi = pi3, ˜bi =˜pi3,i=1, 2,...,N, the global estimation within an arbitrary small tolerance. For the reason, we can prove that the proposed ECTS where pij is the j-th row of Pi concatenated with the scaler method converges with probability one to the global op- p˜ij . timum. Comparing with L∞ algorithm[11] and its variants or improvements[1][15][5], our method not only obtains ac- 3. ECTS in multiview geometry curate estimations, but also decreases computational cost ECTS is a variant of traditional tabu search for the glob- dramatically. al continuous optimization [6]. It is consisted of five stages, including setting of parameters, diversification, search for 2. Problem formulation the most promising area, intensification, and output of the best point found. The key stages of ECTS are diversifi- The geometric vision problems we are considering in cation and intensification. At the stage of diversification, this paper are the ones where the reprojection error can be the algorithm scans the whole solution space and detects written as affine functions composed with a projection, i.e., the promising areas, which are likely to contain the global quotients of affine functions. These problems can be repre- minimum. The centers of these promising areas are stored sented as (P0) based on the squared reprojection error, in a so-called promising list. The aim of diversification is x N x to determine the most promising area from the promising min f( )= i=1 fi( ) 2 (a x+˜a )2 list. When the diversification ends, the step of intensifica- x j=1 ij ij bx ˜ s.t.fi( )= ˜ 2 , i + bi > 0(P0) tion will start. It searches inside the most promising area (bi x+bi) for a more optimal result. In this phase, the search is con- where x ∈ Rn is the unknowns to be solved for, and centrated on the most promising area by making the search n aij , bi ∈ R and a˜ij , ˜bi ∈ R are differentiated with geo- domain smaller and gradually reducing the neighborhood metric problems. The constraint bi x + ˜bi > 0 reflects the structure. This strategy improves the performance of the al- fact that the reconstructed points should be located in front gorithm and allows exploiting the most promising area with of the cameras. The dimension of the problem (P0) is n, more accuracy. which is often fixed and intrinsic to particular application. 3.1. The diversification For example, n =3for multi-view triangulation, n =6for 2D affinity, n =7for fundamental matrix, n =8for planar Now, we present how to carry out the diversification for homography, and n =11for camera calibration, etc. In or- multiview geometry problems. In order to facilitate discus- der to facilitate the following discussions, here, we take the sion, we also take the triangulation as the example. At first, N-view triangulation as an example. we show the way how to determine the most promising area, Consider a set of camera matrixes Pi and correspond- which should contain the global optimum xopt.Westart ing image points ui =(ui1,ui2) of x =(x1,x2,x3) . with a convex hull and an initial point xinit found by linear The objective of triangulation is to recover x.Thesim- algebraic method. If xopt is the true global optimum of L2 plest way is based on a linear algebraic method. Though norm minimization, it follows this method may seem attractive, the cost function that it N is minimized has no particular meaning and the method is 2 E(xopt)=min fi(xopt) ≤ E(xinit)=δ , (3) not reliable. Under the framework of L2 norm, we are led i=1 32343241 ∗ ∗ where δ is a positive value. This means that each fi(xopt) Lemma 1. Let f(x)=maxfi(x), x solves μ = 2 i term is less than δ . According to Eq.(1), Eq.(3) can be minf(x), S = {x|bi x + ˜bi > 0, ∀i}, if and only if there rewritten in the following form. For each i, x∈S exists λ∗ such that, 2 2 (a xopt +˜ai1) +(a xopt +˜ai2) i1 i2 ≤ N 2 δ.

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