Automatic Range Image Registration Using Mixed Integer Linear

Automatic Range Image Registration Using Mixed Integer Linear Programming Shizu Sakakubara1, Yuusuke Kounoike2, Yuji Shinano1, and Ikuko Shimizu1 1

Tokyo Univ. of Agri. & Tech. 2 Canon Inc.

Abstract. A coarse registration method using Mixed Integer Linear Programming (MILP) is described that finds global optimal registration parameter values that are independent of the values of invariant features. We formulate the range image registration problem using MILP. Our algorithm using MILP formulation finds the best balanced optimal registration for robustly aligning two range images with the best balanced accuracy. It adjusts the error tolerance automatically in accordance with the accuracy of the given range image data. Experimental results show that this method of coarse registration is highly effective.

1 Introduction Automatic 3D model generation of real world objects is an important technique in various fields. Range sensors are useful for generating 3D models because they directly measure the 3D shape of objects. The range image measured by a range sensor reflects the partial 3D shape of the object expressed in the coordinate system corresponding to the pose and position of the sensor. Therefore, to obtain the whole shape of the object, many range images measured from different viewpoints must be expressed in a common coordinate system. Estimation of relative poses and positions of sensors from range images is called range image registration. In general, the registration process is divided into two phases: coarse registration and fine registration. For the fine registration, the ICP (Iterative Closest Point) algorithm by Besl and McKay [1] and its extensions [11] are widely used. However, they need sufficiently good initial estimates to achieve fine registration because their error functions have a number of local minima. Therefore, many coarse registration methods have been developed for obtaining good initial estimates automatically. Many of these coarse registration methods [2] match invariant features under rigid transformation. For example, spin images [7] are matched based on their crosscorrelation. A splash [12], a point signature [4], and a spherical attribute image [6] are used as indices representing surface structures in a hash table for matching. For the modeling of buildings, planar regions [5] and circular features [3] are used for matching. These methods are effective if the features can be sufficiently discriminated and their values can be accurately calculated. However, they cannot guarantee global optimality. In addition, registration has been formalized as a discrete optimization task of finding the maximum strict sub-kernel in a graph [10]. While the globally optimal solution can be obtained with this method, the solution depends on the quality of the features. Y. Yagi et al. (Eds.): ACCV 2007, Part II, LNCS 4844, pp. 424–434, 2007. c Springer-Verlag Berlin Heidelberg 2007

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However, invariant features such as curvatures are difficult to calculate stably because they greatly affected by occlusion and discretization of the object surface. In this paper, we propose a novel coarse registration method using Mixed Integer Linear Programming (MILP) that guarantees the global optimality. It aligns two range images independent of the values of invariant features. Since MILP problems are NPhard problems, they cannot be solved to optimality in a reasonable amount of time when they exceed a certain size. However, progress in algorithms, software, and hardware has dramatically enlarged this size. Twenty years ago, a mainframe computer was needed to solve problems with a hundred integer variables. Now it is possible to solve problems with thousands of integer variables on a personal computer [8]. The advances related to MILP have made our proposed algorithm possible.

2 Definition of Best Balanced Optimal Registration Consider the registration of two sets of points in R3 . Let V 1 = {v11 , v21 , · · · , vn1 1 } be the source point set and V 2 = {v12 , v22 , · · · , vn2 2 } the target one. Assume that these points are corrupted by additive noises and they are independent random variables with zero mean and unknown variance. The task of registration is to estimate parameter values of a rigid transformation T such that the T aligns the two point sets. We define the rigid transformation as T (R, t; v) = Rv + t where R is a 3 × 3 rotation matrix and t is a translation vector. To deal with occlusion and discretization of range images, we introduce a more accurate point set 2 −T (R, t; vi1 )||∞ ≤ , MAPS(T, φ, )= vi1 : ||vφ(i) where the function φ(i) denotes the index of the corresponding target point for each source point and the constant is a threshold value to remove outliers. To obtain robust estimation of the rigid transformation, the number of elements of MAPS(T, φ, ), |MAPS(T, φ, )|, needs to be maximized. Because no a priori correspondence between the source and target points is given, the function φ must be determined. And, we make no assumption about the variance of the noise. Hence, the value cannot be estimated in advance. However, if a number N is given, we can find the minimal value of such that |MAPS(T, φ, )| ≥ N . Therefore, we define the N -accuracy optimal registration as the (TˆN , φˆN ) that minimizes (TˆN , φˆN ) = argmin{ : |MAPS(T, φ, )| ≥ N }, T,φ

and define the optimal registration error ˆN by N -accuracy optimal registration as ˆN = min{ : |MAPS(T, φ, )| ≥ N }. If 1 ≥ 2 , then max |MAPS(T, φ, 1 )| ≥ max |MAPS(T, φ, 2 )|. On the other hand, if N1 ≤ N2 , then min{ : |MAPS(T, φ,)| ≥ N1 } ≤ min{ : |MAPS(T, φ, )| ≥ N2 }. A well-balanced registration with a large number of elements in MAPS(T, φ, ) with a small value is desired. Therefore, we define best balanced optimal registration ˆ that attains N ˆ such that as the (Tˆ, φ) ˆ = argmax{|MAPS(TˆN , φˆN , max )| : (TˆN , φˆN ) = argmin{ : |MAPS(T, φ, )| ≥ N }}, N 3≤N≤κ

T,φ

(1)

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where κ is the maximum value for N and max = max{ˆ N }. The lower bound for N is three because a rigid transformation can be estimated using no less than three point pairs. To avoid that max might be too large, we give an upper bound ¯. In best balanced optimal registration, the N -accuracy optimal registration (TˆN , φˆN ) ˆ and for each N from three to κ is evaluated using |MAPS(TˆN , φˆN , max )|. We call N ˆNˆ the best balanced values. They depend on the accuracy of the original point sets. In general, this accuracy is unknown in advance. Our proposed method finds the best balanced optimal registration automatically. We assume that similarity sij between any pair of points vi1 and vj2 is given. Because the feature values of the corresponding points are similar, we assume that point pairs that have explicitly low similarity are not corresponding and remove them from the candidates for corresponding point pairs. Note that the precise values of the features are not needed because feature values are used only to remove obviously noncorresponding pairs.

3 Mixed Integer Linear Programming Based Registration In this section, we will first introduce the general form of Mixed Integer Linear Programming(MILP) problems. Next, we will give an MILP formulation of N -accuracy optimal registration. We also give an ILP formulation to evaluate each N -accuracy registration by counting |MAPS(TˆN , φˆN , max )|. Note that this step is not necessary to be formalized by ILP because it can be done by many other ways. Then, we will describe our algorithm for obtaining the best balanced optimal registration. Finally, we will address the problem size issue and our approach to overcoming it. 3.1 Mixed Integer Linear Programming Problems An MILP problem is the minimization or maximization of a linear objective function subject to linear constraints and to some of the variables being integers. Here, without loss of generality, we consider a maximization problem. Explicitly, the problem has the form (MILP) max c x sub. to Ax ≤ b, l ≤ x ≤ u, xi : integer, i = 1, . . . , p,

where A is an m×n matrix, c, x, l, and u are n-vectors, and b is an m-vector. Elements of c, l, u, b, and A are constant, and x is a variable vector. In this MILP problem, we assume 1 ≤ p ≤ n. If p = n, then the problem is called an Integer Linear Programming(ILP) problem. If all of the variables need not be integers, then the problem is called a Linear Programming(LP) problem. The set S = {x : Ax ≤ b, l ≤ x ≤ u, xi : integer, i = 1, . . . , p} is called the feasible region and an x ∈ S is called a feasible solution. A feasible solution x∗ that satisfies c x∗ ≥ c x, ∀x ∈ S is called an optimal solution. The value of c x∗ is called the optimal value.


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MILP problems are NP-hard in general. Therefore, heuristic methods for solving them are widely used. However, in this paper, we apply exact methods. Owing to recent progress in MILP techniques, relatively large-scale MILP problems can be solved to optimality with commercial solvers. Throughout this paper, the solutions for MILP problems are optimal ones. 3.2 MILP Formulation of the N -Accuracy Optimal Registration MILP formulation of N -accuracy optimal registration requires that the constraints on the rigid transformation be written in a linear form. Since this cannot be done straightforwardly, we introduce pseudo rotation matrix R and pseudo translation vector t and give linear constraints so that R and t are close to R and t. Symbols denoted with prime ( ) are obtained using R and t . def Initially, to simplify the notation, we define an index set for point set V as I(V ) = def

{i : vi ∈ V }, and define ABS(x) = (|x1 |, |x2 |, |x3 |) for a vector x = (x1 , x2 , x3 ) . We introduce a corresponding point pair vector p = (p11 , · · · , p1n2 , p21 , · · · , pn1 1 , · · · , pn1 n2 ) as one of the decision variables. Each element pij is a 0-1 integer variable that is designed to be 1 if source point vi1 corresponds to target point vj2 , that is, j = φ(i), and is designed to be zero otherwise. The other decision variables are the elements of R and t . All elements of these variables are continuous. We denote the ij-th element of R as rij (i, j = 1, 2, 3) and the i-th element of t as ti (i = 1, 2, 3). We assume that ≤ r¯ij , and ti ≤ ti ≤ t¯i . There are trivial upper and lower bounds are given: r ij ≤ rij bounds: −1 ≤ rij ≤ 1, (i, j = 1, 2, 3). As mentioned above, the constraints on the elements of the rotation matrix cannot be straightforwardly written in linear form. Therefore, we formulate this condition indirectly. Let vi1 , vk1 ∈ V 1 and vj2 , vl2 ∈ V 2 and d(vi1 , vk1 ) be the distance between vi1 and vk1 and and d(vj2 , vl2 ) is the distance between vj2 and vl2 . If |d(vi1 , vk1 ) − d(vj2 , vl2 )| > l is valid, then either j = φ(i) or l = φ(k) needs to be satisfied. This condition can be formulated as pij + pkl ≤ 1, (i, k ∈ I(V 1 ), j, l ∈ I(V 2 ), |d(vi1 , vk1 )−d(vj2 , vl2 )| > l ). This condition can be further rewritten with fewer pkl ≤ 1, (i, k ∈ I(V 1 ), j ∈ I(V 2 )). constraints as pij + 1 )−d(v 2 ,v 2 )|> l∈I(V 2 ),|d(vi1 ,vk l j l

Here, √ l depends on . Since a rigid transformation √ error within ± is allowed, l less than 2 3 should be allowed. Therefore, l ≥ 2 3 needs to be satisfied. 2 We assume one-to-one correspondence between subsets of V 1 and V . That is, two 1 conditions need to be satisfied: pij ≤ 1, (i ∈ I(V )) and pij ≤ 1, (j ∈ j∈I(V 2 )

i∈I(V 1 )

I(V 2 )). The minimal number of elements of MAPS(T, φ, ) is given as a constant N .Hence, pij ≥ N. i∈I(V 1 ) j∈I(V 2 )

For the points in MAPS(T, φ, ), only when pij = 1, ABS(vj2 − R vi1 − t ) ≤ e(i ∈ I(V 1 ), j ∈ I(V 2 )), where e = (, , ) . When pij is set to zero, these conditions need to be eliminated. They can be written by introducing a continuous vector

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M = (m1 , m2 , m3 ) that is determined so as to always satisfy ABS(vj2 −R vi1 −t ) ≤ M, (i ∈ I(V 1 ), j ∈ I(V 2 )). Using this vector, we can formulate the condition for the points in MAPS(T, φ, ) as ABS(vj2 − R vi1 − t ) ≤ M(1 − pij ) + e, (i ∈ I(V 1 ), j ∈ pij vj2 − R vi1 − t ) ≤ M(1 − pij )+ I(V 2 )) and further rewrite it as ABS( j∈I(V 2 )

j∈I(V 2 )

e, (i ∈ I(V )). If several variables could be fixed in advance, the MILP problem could be solved more quickly. To fix pij to zero in advance, we use the similarity: pij = 0, (i ∈ I(V 1 ) j ∈ I(V 2 ), sij < s ), where s is a given parameter. If similarity sij is small enough, the pair vi1 and vj2 is removed from the putative corresponding point pairs. The objective of N -accuracy optimal registration is to minimize . Therefore, we can give an MILP formulation as follows. 1

(P1 ) min pij ≤ 1, (i ∈ I(V 1 )),

sub. to j∈I(V 2 )

pij ≤ 1, (j ∈ I(V 2 )), i∈I(V 1 )

pij ≥ N, i∈I(V 1 ) j∈I(V 2 )

pi,j vj2 − R vi1 − t ≥ −M(1 − j∈I(V 2 )

j∈I(V 2 )

pij vj2 j∈I(V

−R

vi1

− t ≤ M(1 −

2)

l∈I(V

pij ) + e, (i ∈ I(V 1 )),

j∈I(V 2 )

pkl ≤ 1, (i, k ∈ (I(V 1 ), j ∈ I(V 2 )),

pij + r ij

pij ) −e, (i ∈ I(V 1 )),

2 ),|d(v1 ,v1 )−d(v2 ,v2 )|> l i j k l

≤ rij ≤ r¯ij , (i = 1, 2, 3, j = 1, 2, 3), ti ≤ ti ≤ t¯i ,

(i = 1, 2, 3),

pij = 0 (i ∈ I(V ), j ∈ I(V ), sij < s ), pij ∈ {0, 1} (i ∈ I(V 1 ), j ∈ I(V 2 )). 1

2

Note that l is a given parameter and affects the optimal value of (P1 ). Here, we define Si and ˆi as the feasible region and the optimal value of the (P1 ) with a given parameter li . If l1 ≥ l2 , S1 ⊃ S2 , because the smaller l value forces the more elements of p to be zero, that is, it restricts the feasible region. Therefore, if l1 ≥ l2 , ˆ1 ≤ ˆ 2 . ˆN , max )| 3.3 ILP Formulation to Count |MAPS(TˆN , φ In this section, we describe ILP problem for evaluating (TˆN , φˆN ) by counting |MAPS(TˆN , φˆN , max )|. Its formulation uses almost the same components described in the previous section, except for the following ones. – The rigid transformation is given. In the formulation in this subsection, rotation matrix R and translation vector t with constant elements are used. They are calculated using a corresponding point pair vector introduced by R and t in our algorithm. – max is not a variable but a constant. – N is not given: the number of correspondences is maximized in this formulation.


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We can give an ILP formulation for counting |MAPS(TˆN , φˆN , max )| as follows using emax = (max , max , max ) . (P2 ) max

pij i∈I(V 1 ) j∈I(V 2 )

pij ≤ 1, (i ∈ I(V 1 )),

sub. to j∈I(V 2 )

pij ≤ 1, (j ∈ I(V 2 )), i∈I(V 1 )

pi,j vj2

−

pij vj2

−

Rvi1

− t ≥ −M(1 −

j∈I(V 2 )

j∈I(V

pij ) − emax (i ∈ I(V 1 )),

j∈I(V 2 )

Rvi1

− t ≤ M(1 −

2)

j∈I(V

pij ) + emax (i ∈ I(V 1 )), 2)

pij = 0 (i ∈ I(V 1 ), j ∈ I(V 2 ), sij < s ),

pij ∈ {0, 1} (i ∈ I(V 1 ), j ∈ I(V 2 )).

3.4 Algorithm for Best Balanced Optimal Registration Now, we describe our algorithm for obtaining the parameter values for the best balanced optimal registration. Equation (1) is solved in two phases. ˆ N . The p ˆ N that attains Phase 1. For N from 5 to κ, find ˆN and obtain (TˆN , φˆN ) as p ˆN ≤ ¯ is stored in list L. The minimum N value is set to five to avoid trivial solutions. ˆ N and count Phase 2. For each (ˆ pN , ˆN ) ∈ L, make (TˆN , φˆN ) from p ˆ N : (ˆ pN , ˆN ) ∈ L}. Then, select the N |MAPS(TˆN , φˆN , max )|, max = max{ˆ ˆ ˆ that attains argmax{|MAPS(TN , φN , max )| : N such that(ˆ pN , ˆN ) ∈ L}. N

In phase 1, To solve the problem P1 , an l needs to be given. However, the value of l affects the optimal value of the problem. The minimum ˆN value is found by solving P1 repeatedly by narrowing its range. ˆ by We find the corresponding point pair vector that attains the best balanced value N a two-phase algorithm as follows. ) [Algorithm] MILP-based Registration(¯ L = ∅; /* Phase 1 */ for(N = 5; N ≤ κ; N ++){ N = ¯; /* N : upper bound for ˆN */ N = 0.0; /* N : lower bound for ˆ N */ N = ¯; while(N > N ){ /* Narrowing process to find ˆ N */ √ l = 2 3N ; Solve problem (P1 ) with given parameters N and l ; ˆ N be the optimal solution and ˆ /* Let p N be the optimal value of problem (P1 ) */ if ( No feasible solution is found ) {break;} if ( ˆN < N ){ if ( ˆN > N ){ N = ˆ N ; } N = N ;

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S. Sakakubara et al. }else{ if ( ˆN > ¯) {break; } if ( ˆN < N ){ N = ˆ N ; } N = N ; } N = (N − N )/2 + N ;

} L = L ∪ {(ˆ pN , ˆN )} ;

} if(L = ∅){return “No solution”; /* ¯ value is too small */ } /* Phase 2 */ ˆ = 0; max = max{ˆ N : (ˆ pN , ˆ N ) ∈ L}; N while(L = ∅){ Extract (ˆ pN , ˆN ) from L; /* L = L\{(ˆ pN , ˆ N )} */ ˆ ˆ ˆ N and count |MAPS(TˆN , φˆN , max )| by solving problem (P2 ) with Make (TN , φN ) from p a given parameter max ; /* Let Ntp = |MAPS(TˆN , φˆN , max )| */ ˆ = Ntp ; p ˆ < Ntp ){ N ˆ Nˆ = p ˆ N ;} if(N } ˆ, p ˆ Nˆ ); return (N

This algorithm has three particular advantages. – Although ¯ is initially given, l is adjusted to an appropriate value in accordance with the accuracy of the point sets automatically. – If the value of ¯ is too small to find a solution, it outputs “No solution.” If ¯ is too big, it outputs a feasible solution at the cost of time. – Because the optimal values of (P1 ) are used, the narrowing process is completed faster than that in a binary search. 3.5 Problem Size Issue and How to Overcome It Unfortunately, the MILP solver and computers still do not have enough power to solve the MILP problem (P1 ) for all points sets obtained from normal size range images. Therefore, we preselect the feature points from V 1 and V 2 . Since our algorithm evaluates only the distances between corresponding point pairs, it is robust against noise and the preselection method. When the algorithm is applied to the preselected point sets V˙ 1 ⊂ V 1 and V˙ 2 ⊂ V 2 , the robustness of the registration can be improved to solve the ILP problem (P2 ) with V˙ 1 and V 2 . The current MILP solver and computers have enough power to solve this size of problem (P2 ) within a reasonable time.

4 Experiments We tested the robustness of our method by applying it to three synthetic datasets and one real dataset. We used the ILOG CPLEX (ver.10.1) MILP solver installed on a PC


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with a Pentium D 950 CPU (3.4 GHz), 2 GB of RAM, and linux 2.6. We used GNU GLPK to generate files for our formalization. We use the algorithm of Umeyama [13] to estimate the rigid transformation from the point correspondences. We used 3D models of “Stanford Bunny”[14], “Horse” [16], and “Armadillo”[14] to generate synthetic range images. We generated 18 synthetic range images for each model with the size 200 × 200 pixels by rotating 20-degree rotation steps around the Y axis. Then, to generate five noisy datasets for each model, the Z coordinate of each point was perturbed by adding Gaussian noise with zero mean and a standard deviation of σ = 0.02, 0.04, 0.06, 0.08, or 1.00. We also applied our method to the real range images of “Pooh” [15]. The selected feature points had curvedness [9] values that were the maximal with a constraint on thedistance between feature points. The curvedness c of a point is calculated by c = κ21 + κ22 /2 where κ1 and κ2 are two main curvatures of the point. The number of feature points for the each range image was 50 for “Stanford Bunny” and “Horse”, and 70 for “Armadillo” and “Pooh”. The similarity sij between feature points vi1 and vj2 was defined using curvatures values of the local surfaces around the points. It is set to −100 if the shape of the surfaces, such as convex or concave, was not identical with each other, or else it is calculated based on the curvedness of points ci and cj as sij = 1/ ci − cj . We applied our method to all adjacent pairs in the range image sequences. For all = 1 (i, j = 1, 2, 3). For the synthetic datasets, κ = 10, s = 10, rij = −1, and r¯ij images, ¯ = 0.15, M = (100, 100, 100) , ti = −10, and t¯i = 10 (i = 1, 2, 3). We also used ¯ = 0.25 for the pairs whose solutions could not be obtained by ¯ = 0.15. For the real images, ¯ = 0.25, M = (1000, 1000, 1000), ti = −100, and t¯i = 100 (i = 1, 2, 3). We also applied ¯ = 0.50 for the pairs whose solutions could not be obtained by ¯ = 0.25. In order to improve accuracy of calculation, each points vi = (xi , yi , zi ) of “Pooh” dataset are translated near the origin of the coordinates ◦ ◦ by following: vir − ((min{xj : j ∈ I(V 0 )} + max{xj : j ∈ I(V 0 )})/2, (min{yj : ◦ ◦ ◦ j ∈ I(V 0 )} + max{yj : j ∈ I(V 0 )})/2, (min{zj : j ∈ I(V 0 )} + max{zj : j ∈ ◦ ◦ I(V 0 )})/2) , where V 0 are point set of 0◦ . The experimental results for the synthetic range image datasets are shown in Table 1( “b”, “h”, and “a” in the column “Dataset” indicate “Stanford Bunny”, “Horse” and “Armadillo”, respectively, followed by the numbers indicating the standard deviation of the added noise). The parameter ¯ = 0.25 was used to calculate the results for pairs 180◦ –200◦ of h00, 180◦ –200◦ and 280◦ –300◦ of h04, 180◦ –200◦ of h06, 280◦ –300◦ of h08, 20◦ –40◦ and 280◦ –300◦ of h10, and 60◦ –80◦ of all “Armadillo” datasets, and ¯ = 0.15 for the other pairs. Table 1 shows that the rigid transformation errors did not always increase with the standard deviation σ of the added Gaussian noise. There are two main reasons that our method is robust against the measurement noise: First, our method does not need accurate values for the invariant features. We use them to only select the feature points and to reduce the putative corresponding point pairs. Second, the rigid transformation is estimated using more than five corresponding pairs of feature

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Table 1. Error evaluation of MILP-based registration. “Error of angle” is the absolute angle between the true rotation angle and the estimated one in degrees. “Error of axis” is the deviation in the estimated rotation axis in degrees. “Error of translation” is the norm of the error of the translation vector. Dataset σ

Time [sec.]

b00 b02 b04 b06 b08 b10 h00 h02 h04 h06 h08 h10

0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10

4578 4587 4906 5406 4700 4877 5334 5213 4273 4264 3202 3525

Error Error Error of Dataset σ of angle of axis translation [degree]

[degree]

0.343 0.321 0.466 0.247 0.366 0.421 0.281 0.272 0.354 0.364 0.417 0.344

1.89 1.74 2.00 1.30 1.67 1.47 2.14 1.94 1.97 2.19 2.55 3.04

Time [sec.]

0.328 0.311 0.390 0.298 0.364 0.291 0.587 0.552 0.507 0.485 0.625 0.592

a00 a02 a04 a06 a08 a10

0.00 119514 0.02 118699 0.04 124356 0.06 127393 0.08 118821 0.10 109337

Error Error Error of of angle of axis translation [degree]

[degree]

0.215 0.231 0.205 0.320 0.244 0.393

0.83 0.88 0.95 0.94 0.84 0.96

0.244 0.227 0.224 0.282 0.219 0.262

Table 2. Error evaluation of data set “Pooh”. “Error of angle” is the absolute angle between the true rotation angle and the estimated one in degrees. “Error of axis” is the angle in degrees between the rotation axis estimated from the given pair and the rotation axis estimated from all pairs. “Error of translation” is the norm of the error between the translation vector estimated from the given pair and the translation vector estimated from all pairs. Angle Time[sec.] Error of angle[degree] Error of axis[degree] Error of translation average 19496 1.950 0.072 0.719

points. If we can find only five pairs of points that are not so noisy by chance, we can accurately estimate rigid transformation. The computational time for “Armadillo” was higher than that of others because the numbers of MILP variables and constraints are proportional to the number of feature points. Table 2 shows the results for the real range image dataset of “Pooh”. The parameter ¯ = 0.50 was used to calculate the results for pairs 20◦ -40◦ , 60◦ –80◦ , 100◦ –120◦, 220◦ –240◦, and 320◦ –140◦, and ¯ = 0.25 was used for the other pairs. The registration results for “Pooh” are shown in Figure 1. While the errors for some pairs were relatively large, the results are good enough for coarse registration.

Fig. 1. Results for “Pooh”


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5 Concluding Remarks Our proposed coarse registration method using Mixed Integer Linear Programming (MILP) can find global optimal registration without using the values of the invariant features. In addition, it automatically adjusts the error tolerance depending on the accuracy of the given range image data. Our method finds the best consistent pairs from all possible point pairs using an MILP solver. While such solvers are powerful tools, all of the constraints should be written in linear form. This means that constraints on the rotation matrix cannot be applied directly. Therefore, we selected a relevant number of consistent pairs in the sense of the distances between point pairs, which gives the constraints on the rotation matrix indirectly. The number of corresponding point pairs and the distances between them are automatically balanced by our algorithm using two different MILP formulations. Future work will focus on reducing the computational time and improving the selection of the feature points.

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14. Stanford 3D Scanning Repository, http://www-graphics.stanford.edu/data/3Dscanrep/ 15. The Ohio State University Range Image Repository, http://sampl.ece.ohio-state.edu/data/3DDB/RID/minolta/ 16. Georgia Institute of Technology Large Geometric Models Archive, http://www-static.cc.gatech.edu/projects/large models/