[Paper Review] Efficient Robust Mean Value Calculation of 1D Features

Erik Jonsson, Michael Felsberg|arXiv (Cornell University)|Jan 29, 2016

Advanced Statistical Methods and Models1 references3 citations

TL;DR

This paper presents an O(n) algorithm for computing the robust mean of 1D features using the truncated quadratic error norm, which efficiently identifies the global minimum by scanning maximal and boundary windows of sorted data. The method outperforms channel averaging by eliminating grid effects and enabling exact, globally optimal robust estimation in linear time.

ABSTRACT

A robust mean value is often a good alternative to the standard mean value when dealing with data containing many outliers. An efficient method for samples of one-dimensional features and the truncated quadratic error norm is presented and compared to the method of channel averaging (soft histograms).

Motivation & Objective

• Address the need for robust mean estimation in image processing where data contains many outliers.
• Develop a fast, exact method for computing the robust mean in one-dimensional feature spaces.
• Overcome limitations of existing approaches like channel averaging, which suffer from grid effects and implicit smoothing.
• Provide a globally optimal solution for the truncated quadratic error norm in 1D, ensuring robustness against outliers.
• Enable efficient edge-preserving image smoothing by leveraging the algorithm in sliding spatial windows.

Proposed method

• Sort the 1D feature data in ascending order to enable efficient window scanning.
• Define a feasible window [a,b] as one where |x_b - x_a| < 2c, with c as the truncation threshold.
• Identify maximal windows as feasible windows that cannot be extended without violating the feasibility condition.
• For each window, compute the candidate robust mean μ_w as the arithmetic mean of the window's values.
• Use the formula q_w = S₂ - S₁² / (b-a+1) to compute the within-window sum of squared deviations efficiently.
• Evaluate the approximate error Ê_w = q_w + n_o c², where n_o is the number of samples outside the window, and track the minimum Ê_w across all maximal and boundary windows.

Experimental results

Research questions

• RQ1 Can an exact, globally optimal robust mean be computed efficiently for 1D features under the truncated quadratic error norm?
• RQ2 How does the proposed method compare to channel averaging in terms of robustness and sensitivity to grid effects?
• RQ3 What is the computational complexity of finding the robust mean using window-based scanning of sorted data?
• RQ4 Does the method eliminate systematic biases introduced by discrete channel grids in soft histogram-based approaches?
• RQ5 Can the algorithm be efficiently adapted for use in sliding window operations for edge-preserving image smoothing?

Key findings

• The proposed algorithm computes the globally optimal robust mean in O(n) time by scanning only maximal and boundary windows of sorted 1D data.
• The method eliminates grid effects present in channel averaging, which cause position-dependent biases due to discrete kernel placement.
• The truncated quadratic error norm enables a hard decision on outlier inclusion, leading to sharper rejection of extreme values compared to the smooth decay of channel averaging.
• Channel averaging overcompensates for outliers at certain positions (e.g., around x=6.0 in the example), introducing systematic errors not present in the proposed method.
• The algorithm supports weighted samples by extending S₁, S₂, and n_o to weighted sums, preserving O(n) complexity.
• When applied in sliding windows, the method enables efficient edge-preserving image smoothing with minimal computational overhead due to partial sorting reuse.