Bias-Free Mean Plane Measurement

• The paper demonstrates bias-corrected mean plane estimation using intrinsic and Ziezold means, ensuring manifold stability against systematic errors.
• It highlights the 1:3 ratio property that links intrinsic, extrinsic, and Procrustes means, providing an internal consistency check for estimator reliability.
• By incorporating bias-correction via independent selection functions and autocorrelation adjustments, the method enhances statistical power in multidimensional shape analysis.

Bias-free mean plane measurement refers to rigorous methodologies and statistical frameworks aimed at estimating the average (“mean”) geometric plane within a dataset (or population), such that the resulting estimator is robust to various sources of systematic error (“bias”). Bias in this context typically arises from analytic choices (e.g., metric or coordinate definition), data selection, measurement procedure, or inherent properties of the shape space or physical acquisition system. The goal is to ensure that the computed mean plane is representative and statistically reliable, not artificially displaced due to sample concentration, selection procedures, coordinate system artifacts, or modeling heuristic errors. The development of bias-free mean plane measurement methodologies spans multiple domains including shape analysis, geometric inference, sensor data processing, and statistical correction theory.

1. Foundational Concepts: Mean Plane in Shape Spaces

The mean plane concept generalizes traditional averaging from Euclidean settings to non-Euclidean shape spaces formed as quotients of Riemannian manifolds by isometric group actions. Notably, mean plane is a special case of the mean shape problem in spaces such as Kendall’s 3D shape space, where the measurement space may contain non-manifold or singular regions. Three principal definitions are recognized:

• Intrinsic mean: Defined by minimizing the Fréchet function with respect to the intrinsic (geodesic) metric. This corresponds to the center-of-gravity within the manifold portion of the shape space.
• Ziezold (extrinsic) mean: Formed via isometric embedding of the pre-shape manifold into an ambient Euclidean space; computed using the Euclidean metric and projected back to the shape space. The resulting “manifold mean” becomes the Ziezold mean in the quotient.
• Residual/Procrustes mean: Computed by projecting the difference between configurations and candidate mean onto the tangent space. For Kendall’s spaces, such means inherit from the residual metric and are termed full Procrustes means.

The explicit mathematical relationships between these means for concentrated data are central, most notably the result that, for high concentration, the Ziezold mean divides the geodesic segment between intrinsic and Procrustes means in a $1:3$ ratio.

2. Manifold Stability and Statistical Testing

Bias-free mean plane estimation depends critically on manifold stability. A mean is said to possess manifold stability if, whenever the random shape lies within the manifold part of the space with positive probability, the computed mean also resides within the manifold part. The intrinsic and Ziezold means satisfy this criterion under broad conditions (the distribution has at most countably many point masses on singular strata).

Manifold stability is a precondition for the statistical validity of inference based on means in non-manifold spaces. If stability holds, the central limit theorem applies locally and both one-sample and two-sample hypothesis tests are justified. Conversely, Procrustes means may lose manifold stability in sparsely concentrated or multimodal configurations, yielding means that fall into singular (non-regular) strata and thus producing systematically biased measurements. The selection of intrinsic or Ziezold means is therefore fundamental to preserving statistical rigor in bias-free mean plane analysis.

3. Detailed Mathematical Relationships and the 1:3 Ratio

For highly concentrated data on spheres or Kendall’s shape spaces, the relationship between the three canonical means can be stated to first order (up to $O((1-\|\mathbb{E}(X)\|)^2)$ error terms):

• Residual mean: $$x^{(r_0)} \approx (\|\mathbb{E}(X)\|/\lambda^{(r)})[x^{(e)} - (1/\|\mathbb{E}(X)\|) \mathbb{E}(\langle X - x^{(e)}, X\rangle X) + \ldots]$$
• Intrinsic mean: $$x^{(i)} \approx (\|\mathbb{E}(X)\|/\lambda^{(i)})[x^{(e)} + (1/3)(1/\|\mathbb{E}(X)\|) \mathbb{E}(\langle X - x^{(e)}, X\rangle X) + \ldots]$$

Here, $x^{(e)}$ is the extrinsic/Ziezold mean, $x^{(r_0)}$ the Procrustean mean, and $x^{(i)}$ the intrinsic mean. The tangent separating $x^{(e)}$ and $x^{(i)}$ is approximately one-third the length (and direction) as that from $x^{(r_0)}$ to $x^{(e)}$. This “1:3 property” allows practitioners to verify the internal consistency and stability of their mean plane estimators in practical applications.

4. Practical Examples, Simulations, and Bias Manifestations

Implementations on empirical and simulated datasets—for instance, planar quadrangles (“poplar leaves”), landmark-based digits, 3D macaque skulls, and tetrahedral brooches—confirm the theoretical findings. For data with tight concentration, intrinsic and Ziezold means closely coincide and exhibit the expected geometric division with Procrustes means.

Simulations further reveal that using intrinsic or Ziezold means yields manifold-stable, bias-free estimates suitable for standard two-sample testing. In contrast, Procrustes means introduce bias under low-concentration or multimodal settings, sometimes ignoring dispersed data points. Extrinsic Schoenberg means, based on embedding into matrix spaces, may inadvertently elevate the dimension of the mean via jumping between strata.

The choice of tangent space coordinates also impacts statistical power: residual tangent coordinates typically enhance the power of two-sample tests over intrinsic coordinates, with the Ziezold/Frechet framework providing computational efficiency and higher finite-sample power.

5. Selection Function and Bias Correction Paradigms

Bias due to selection functions arises in contexts where the procedure for including data in analysis couples with the underlying measurement quantity, as in weak lensing shear estimation. The design of shape-independent selection criteria—for example, using the total flux measured at $k=0$ in Fourier space ($\nu_F = |\tilde{f}^s(0)|/(\sqrt{N}\cdot\sigma)$)—minimizes such bias. These principles generalize to mean plane measurement: selection must be independent of shape or geometric configuration to guarantee bias-free results, as shown by extrapolation from shear measurement pipelines.

Bias in roughness and plane parameters due to aggressive background removal or finite scan areas can be corrected using self-consistent formulations of the autocorrelation function (ACF). Here, the expectation of the measured ACF is expressed recursively in terms of the true ACF, and either bias-modified analytical models (with explicit terms in $\alpha = T/L$) or direct inversion of the relation provides bias-free parameter estimation.

6. Implications, Computational Trade-offs, and Limitations

Bias-free mean plane measurement is essential for robust morphometric and geometric inference in various disciplines (shape analysis, sensor calibration, robotics, astronomy). Intrinsic and Ziezold means are statistically justified, stable on the manifold part, and avoid “blindness” or pathological behaviors seen in residual or naive extrinsic means. The choice of coordinate system, computational scheme (Ziezold means offer speed, Procrustes may require iterative optimization), and sample concentration all play roles in balancing bias, variance, and statistical test power.

Residual coordinates, Ziezold means, and bias-corrected selection or filtering techniques are preferable for finite samples and under realistic measurement conditions. However, all correction schemes increase estimator variance as bias is reduced. The application of these methods necessitates careful consideration of each dataset’s geometric and statistical properties, scan dimensions, and analytic framework.

7. Synthesis and Outlook

Ensuring bias-free mean plane measurement requires adopting definitions (intrinsic or Ziezold means) and inferential procedures that preserve manifold stability and statistical validity. Mathematical results such as the 1:3 geodesic ratio, as well as practical corrections for background subtraction and selection function design, facilitate unbiased estimation of mean shape (or mean plane) in both classical shape spaces and contemporary imaging datasets.

This approach integrates geometric analysis via the Slice Theorem, horizontal lifting, and orbit types, with rigorous statistical methods (central limit theory, hypothesis testing, variance correction), to establish a robust protocol for mean plane measurement. The resulting bias-free estimators are central to accurate morphometric analysis, geometric reconstruction, and hypothesis testing in multidimensional shape spaces and sensor-acquired data.