Linearly Converging Quasi Branch and Bound Algorithms for Global Rigid Registration

In recent years, several branch-and-bound (BnB) algorithms have been proposed to globally optimize rigid registration problems. In this paper, we suggest a general framework to improve upon the BnB approach, which we name Quasi BnB. Quasi BnB replaces the linear lower bounds used in BnB algorithms with quadratic quasi-lower bounds which are based on the quadratic behavior of the energy in the vicinity of the global minimum. While quasi-lower bounds are not truly lower bounds, the Quasi-BnB algorithm is globally optimal. In fact we prove that it exhibits linear convergence – it achieves $\epsilon$-accuracy in $~O(\log(1/\epsilon)) $ time while the time complexity of other rigid registration BnB algorithms is polynomial in $1/\epsilon $. Our experiments verify that Quasi-BnB is significantly more efficient than state-of-the-art BnB algorithms, especially for problems where high accuracy is desired.

💡 Research Summary

The paper introduces a novel global optimization framework for rigid registration called Quasi‑Branch‑and‑Bound (Quasi‑BnB). Traditional branch‑and‑bound (BnB) methods for rigid registration rely on linear lower bounds that are derived from a Lipschitz constant of the objective function. While these bounds guarantee global optimality, they are often very loose, leading to a search complexity that grows polynomially with the inverse of the desired accuracy ε (typically O(ε⁻ᴰ) where D is the dimension of the transformation space).

Quasi‑BnB replaces the linear lower bound with a quadratic “quasi‑lower bound”. The key observation is that, for the class of problems considered, the energy function E(g,π) is conditionally smooth: fixing the correspondence π makes E a smooth function of the rigid motion g, and fixing g makes the optimal correspondence tractable. Under this condition the reduced function F(x)=min_π E(x,π) behaves quadratically in a neighbourhood of any global minimizer x*: there exists a constant C such that
 F(x)−F(x*) ≤ C‖x−x*‖² for all x in the search domain.
This quadratic behaviour yields a bound Δ*(δ)=C δ²+O(δ³) that is valid whenever the current search cube contains the global optimum. The bound is called a quasi‑lower bound because it is not a true lower bound for cubes that do not contain the optimum, yet it can be safely used in the pruning step of a BnB algorithm without compromising global optimality.

The authors formalize the class of problems they address as D‑quasi‑optimizable. A problem is D‑quasi‑optimizable if (1) a minimizer exists inside a known initial cube, (2) for any fixed transformation the optimal correspondence can be found in polynomial time (D‑tractability), and (3) for any fixed correspondence the energy is smooth in the transformation variables (conditional smoothness). They show that two widely used rigid registration formulations satisfy these conditions: (i) the rigid‑closest‑point (rigid‑CP) problem, where each transformed point is matched to its nearest neighbour, and (ii) the rigid‑bijective problem, where a permutation (bijection) between point sets is enforced.

Algorithmically, Quasi‑BnB follows a breadth‑first search (BFS) scheme. Starting from an initial hyper‑cube C₀, the algorithm evaluates F at the centre of each sub‑cube, updates a global upper bound ub with the smallest observed value, and computes a quasi‑lower bound lb_i = F(x_i) – Δ*(√D h_i) for each cube of half‑edge length h_i. Any cube whose quasi‑lower bound exceeds ub is discarded; the remaining cubes are subdivided and the process repeats until ub – min_i lb_i ≤ ε.

The theoretical contribution is a rigorous complexity analysis. Theorem 2 shows that standard BnB requires Θ(ε⁻ᴰ) evaluations in the worst case, whereas Quasi‑BnB needs only O(ε⁻ᴰ⁄²) evaluations in general and, under the additional assumption that the set of global minima is finite and each Hessian is positive‑definite, the number of evaluations is bounded by C log₂(1/ε). This logarithmic dependence translates into linear convergence in the sense of optimization theory.

To make the framework concrete, the authors derive explicit quadratic quasi‑lower bounds for both rigid‑CP and rigid‑bijective problems. For the bijective case they exploit the matrix exponential representation of rotations and a truncated Taylor series of the exponential map, obtaining
 Δ*(δ) = 2 n σ_P σ_Q ψ₂(δ),
where σ_P and σ_Q are Frobenius norms of the point clouds and ψ₂(δ) is the second‑order truncation of e^x. A similar bound is obtained for the CP formulation using the smoothness of the nearest‑neighbour assignment.

Experimental evaluation on synthetic and real 2‑D/3‑D point clouds demonstrates that Quasi‑BnB dramatically outperforms state‑of‑the‑art BnB methods (e.g., Go‑ICP, the BnB of