[Paper Review] Remarks on the O(N) Implementation of the Fast Marching Method

Christian Rasch, Thomas Satzger|ArXiv.org|Mar 15, 2007

Computational Fluid Dynamics and Aerodynamics6 references4 citations

TL;DR

This paper provides a rigorous error analysis for the O(N) implementation of the fast marching method using an untidy priority queue, which replaces the traditional heap-based queue to achieve linear time complexity. The authors derive a sharp bound on the relative error introduced by the untidy queue, showing it scales as √2·Fmax/(Fmin·nB), where nB is the number of buckets, and demonstrate through numerical experiments that the error grows with Fmax/Fmin when nB is fixed, validating the theoretical bound.

ABSTRACT

The fast marching algorithm computes an approximate solution to the eikonal equation in O(N log N) time, where the factor log N is due to the administration of a priority queue. Recently, Yatziv, Bartesaghi and Sapiro have suggested to use an untidy priority queue, reducing the overall complexity to O(N) at the price of a small error in the computed solution. In this paper, we give an explicit estimate of the error introduced, which is based on a discrete comparison principle. This estimates implies in particular that the choice of an accuracy level that is independent of the speed function F results in the complexity bound O(Fmax /Fmin N). A numerical experiment illustrates this robustness problem for large ratios Fmax /Fmin .

Motivation & Objective

• To rigorously analyze the error introduced by using an untidy priority queue in the fast marching method to achieve O(N) complexity.
• To quantify the trade-off between computational efficiency (reduced to O(N)) and solution accuracy due to the untidy queue's inexact minimization.
• To establish a theoretical error bound that depends on the speed function's dynamic range Fmax/Fmin and the number of buckets nB.
• To validate the theoretical error estimate with numerical experiments showing the dependence on Fmax/Fmin and the linear scalability of the modified algorithm.

Proposed method

• The method uses a bucket-based untidy priority queue that groups trial values within intervals of width δ, where δ = h/(Fmin·nB), to avoid the O(log N) cost of a heap.
• The algorithm proceeds by selecting any point from the bucket with the smallest index, rather than the globally smallest value, introducing a controlled error.
• A discrete comparison principle is developed to compare the approximate solution ˆT with the exact solution T of the eikonal equation.
• The key technical step involves analyzing the residual of the discrete eikonal equation for the approximate solution ˆT and bounding the deviation from the exact solution using the quantization error δ.
• Theoretical bounds are derived by considering the worst-case deviation in the update step, leading to a relative error bound involving Fmax/Fmin and nB.
• Numerical experiments validate the theoretical bound by measuring the relative error as a function of Fmax/Fmin for different nB values.

Experimental results

Research questions

• RQ1 What is the precise bound on the relative error introduced by using an untidy priority queue in the fast marching method?
• RQ2 How does the error depend on the ratio Fmax/Fmin of the speed function's maximum and minimum values?
• RQ3 Can the error be made independent of Fmax/Fmin by adjusting the number of buckets nB, and what is the resulting complexity trade-off?
• RQ4 Does the O(N) implementation maintain solution accuracy in practice, especially for large Fmax/Fmin ratios?
• RQ5 How does the performance of the untidy queue variant scale with grid size compared to the original O(N log N) fast marching method?

Key findings

• The relative error in the solution ˆT compared to the exact solution T is bounded by √2·Fmax/(Fmin·nB), showing a direct dependence on the speed ratio Fmax/Fmin.
• When nB is chosen proportional to Fmax/Fmin, the error bound becomes independent of Fmax/Fmin, but the complexity increases to O(Fmax/Fmin·N).
• Numerical experiments confirm the linear dependence of the relative error on Fmax/Fmin for fixed nB, validating the theoretical bound.
• The modified algorithm achieves asymptotically linear runtime with grid size, as shown in timing comparisons with the original fast marching method.
• The error bound is sharp and scales with the square root of the spatial dimension, generalizing to higher dimensions.
• The method remains robust for practical Fmax/Fmin ratios when nB is chosen appropriately, but performance degrades significantly when Fmax/Fmin is large and nB is small.