Efficient Derivative Computation for Cumulative B-Splines on Lie Groups

1. Introduction

• Continuous-time trajectories represent pose as a function of time and are used in tasks including odometry, SLAM, tracking, and trajectory planning.B-splines are presented as a natural choice for this representation.Source: §1 · p. 1
• Existing Lie group spline derivative approaches have quadratic computational cost in spline order, limiting an important computation for continuous-time trajectories.The paper notes that differentiation is more involved for pose-valued trajectories than for trajectories in Euclidean space.Source: §1 · p. 1
• Recurrence relations yield time derivatives using a number of matrix operations that scales linearly with spline order k.The formulation also provides simple analytic Jacobians of SO(3) spline values and time derivatives with respect to knots.Source: §1 · pp. 1–2
• The proposed formulation speeds optimization compared with available implementations, as demonstrated in simulations and applications including camera-IMU calibration.Figure 1 depicts calibration-pattern observations combined with accelerometer and gyroscope measurements in a joint optimization.Source: Figure 1 · p. 1, §1 · p. 2

2. Related Work

• The paper reviews prior work on both Lie group spline theory and computer vision applications.The review is organized around these two parts.Source: §2 · p. 2
• Cumulative B-splines extend spline representations from Euclidean functions to Lie-group-valued functions, with prior work generalizing from SO(3) to arbitrary Lie groups.The cited arbitrary-group work includes derivative formulas that the authors describe as impractical to implement.Source: §2 · p. 2
• Computer vision and robotics applications include calibration, sensor odometry, 3D reconstruction, and trajectory planning, all of which require temporal spline derivatives.The authors state that earlier work had not explicitly investigated efficient computation and complexity of those derivatives.Source: §2 · p. 2
• Prior comparisons of SE(3) splines with split R3 and SO(3) representations found the split representation better on average, while the best choice depends on the use case.The reported advantage concerns both trajectory representation and computation time.Source: §2 · p. 2

3.1. Notation

• A Lie group is a differentiable manifold with differentiable multiplication and inversion; its Lie algebra is the tangent space at the identity.The paper identifies SO(3) and SE(3) as matrix Lie groups used in computer vision.Source: §3.1 · p. 2
• Trajectories take values in a Lie group, while the hat transform maps tangent vectors to the Lie algebra and the exponential maps algebra elements back to the group.The logarithm followed by the inverse hat transform maps group elements back to vectors.Source: §3.1 · p. 2
• The notation includes the adjoint, which equals the rotation matrix for SO(3), and the commutator of Lie-algebra elements.These constructions are introduced as tools used in the paper's Lie-group calculations.Source: §3.1 · pp. 2–3

3.2. Differentiation

• The right Jacobian describes how the logarithmic vector difference between Exp(v) and Exp(v + w) changes with perturbation w.For SO(3), the paper points to a reference for the right Jacobian and its inverse.Source: §3.2 · p. 3
• Optimization updates a Lie-group variable by left-multiplying it with Exp(δ), where δ is the increment supplied by the optimizer.This follows the paper's convention for differentiating functions with respect to a Lie-group element.Source: §3.2 · p. 3

4.1. Basics

• Uniform B-splines use control points and spline coefficients, and an order-k spline has C^(k−1) smoothness and local support.For linear splines with k = 2 the curve hits its control points, unlike higher-order splines in general.Source: §4.1 · p. 3, Figure 2 · p. 3
• The De Boor–Cox recurrence defines spline coefficients, and the standard representation can be transformed into a cumulative representation.The figure illustrates that changing one control point changes the curve only in its vicinity.Source: §4.1 · p. 3, Figure 2 · p. 3

4.2. Matrix Representation

• At time t in a segment, the spline depends only on control points p_i through p_(i+k−1), and normalized time u locates the position within that segment.This local segment representation supports evaluating the spline with a matrix form.Source: §4.2 · p. 3
• The cumulative matrix form uses control-point differences and cumulative coefficients, with its first matrix row equal to the unit vector e_0.That row makes λ_0(u) identically one and simplifies the resulting expression.Source: §4.2 · p. 4

5. Cumulative B-splines in Lie groups

• Lie-group cumulative splines replace vector addition with group multiplication and scale increments via Log, scalar multiplication, and Exp; multiplication order matters.The paper follows the multiplication convention most commonly used in related work.Source: §5 · p. 4

5.1. Time derivatives

• Recurrences replace prior quadratic first-derivative and cubic second-derivative computations with schemes whose operation counts scale linearly with spline order.First derivatives use k−1 matrix-vector multiplications and one matrix-matrix multiplication; second derivatives use 2k matrix-matrix and k−1 matrix-vector multiplications, plus additions. The approach also extends to higher derivatives, with third-order computation remaining linear in spline order.Source: §5.1 · pp. 4–5

5.2. Jacobians w.r.t. control points in SO(3)

• For SO(3), a backward recurrence computes knot Jacobians of the spline value ρ, angular velocity ω, and acceleration ˙ω with linear-in-k multiplications and additions.Here ρ maps the rotation R to R^d via the Log map.Source: §5.2 · p. 6

6.1. Simulated Sequence

• The proposed formulation outperformed the baseline in every simulated case, with a larger time difference for acceleration measurements because baseline computation is cubic in spline order.Both formulations converged to the same result in the same number of iterations.Source: §6.1 · p. 7, Table 1 · p. 7

6.2. Camera-IMU calibration

• Calibration converged to the same values after 12 iterations across the tested formulations, so switching formulations did not affect solution accuracy.The comparison included SO(3) × R3 and SE(3) spline representations.Source: §6.2 · p. 8, Table 2 · p. 8
• The custom solver with analytic SO(3) Jacobians completed camera–IMU calibration in less than 6 seconds, 2.6 times faster than the baseline split representation.With Ceres Jacobians, the proposed formulation reduced computation time by 6% for the split representation and 36.7% for SE(3).Source: §6.2 · p. 8, Table 3 · p. 8
• The authors note that reduced derivative computation could have a larger effect in multiple-IMU calibration, where rotational acceleration is also required with the split formulation.Source: §6.2 · p. 8

• Time derivatives and Jacobians can both be computed in O(k) matrix operations, versus traditional schemes that are up to cubic in k.The authors mathematically prove the correctness of these claims.Source: §7 · p. 8
• Choosing a suitable representation together with the derivative computation produced speedups of up to 4.4x in practical computer-vision applications.Source: §7 · p. 8
• The authors identify real-time continuous-time trajectory applications as a relevant scope for the proposed method.Source: §7 · p. 8

8.1.1 Symbols with specific meaning

• The appendix uses p(u) and pi for Euclidean spline values and control points, X(u) and Xi for Lie-group values and control points, and R(u) and Ri for SO(3).Source: §8.1.1 · p. 10

8.1.2 Indices

• The index k denotes spline order, i indexes control points or time intervals, and l, m, n, and s are dummy indices.Source: §8.1.2 · p. 10

8.2. Right Jacobian for SO(3)

• For L = SO(3), the appendix points to the right Jacobian and its inverse in [3, p. 40].Source: §8.2 · p. 10

8.3. Third order time derivatives

• The appendix gives third-order time derivatives for a general Lie group and identifies ¨ω as jerk, with a simpler expression for SO(3).Source: §8.3 · p. 10

• The proposed temporal derivatives require O(k) matrix operations, while prior temporal-derivative formulations can require more.The authors state that they are the first to compute Lie-group spline temporal derivatives in O(k) matrix operations.Source: §8.4 · p. 10
• The proposed SO(3) Jacobian computation also requires no more than O(k) matrix operations.Source: §8.4 · p. 10
• Table 4 compares the numbers of matrix multiplications and additions for temporal derivatives in related work and the proposed formulation.Source: §8.4 · p. 10

8.5.1 Proof of (20) (cumulative blending matrix)

• The cumulative blending matrix identity is proved by induction over spline order, using binomial-sum properties and index shifts.Source: §8.5.1 · pp. 10–11

8.5.2 Proof of (51) (Jacobian of Exp(−λd) multiplied by a vector)

• The proof of the Jacobian of Exp(−λd) multiplied by a vector uses the right Jacobian property (10).Source: §8.5.2 · p. 11

8.5.3 Proof of (59) (Jacobian of acceleration)

• The acceleration Jacobian proof establishes equivalence of two formulas by induction, with the base case at l = j + 2 and cancellation advancing the induction step.Source: §8.5.3 · pp. 11–12