Optimal combination of data modes in inverse problems: maximum compatibility estimate

Abstract: We present an optimal strategy for the relative weighting of different data modes in inverse problems, and derive the maximum compatibility estimate (MCE) that corresponds to the maximum likelihood or maximum a posteriori estimates in the case of a single data mode. MCE is not explicitly dependent on the noise levels, scale factors or numbers of data points of the complementary data modes, and can be determined without the mode weight parameters. As a case study, we consider the problem of reconstructing the shape of a body in $\R^3$ from the boundary curves (profiles) and volumes (brightness values) of its generalized projections.

• The paper introduces the maximum compatibility estimate (MCE) as a robust, scale-invariant approach for integrating heterogeneous data modes in inverse problems.
• It presents a χ²-based likelihood framework with adaptive weighting that eliminates reliance on noise level specifications and arbitrary scaling.
• Numerical demonstrations on asteroid shape reconstruction validate the method's stability, objectivity, and broad applicability.

Optimal Combination of Data Modes in Inverse Problems: Maximum Compatibility Estimate

Background and Problem Setting

The paper "Optimal combination of data modes in inverse problems: maximum compatibility estimate" (1001.2634) addresses the challenge of combining heterogeneous data modes within the context of inverse problems, with a focus on asteroid shape reconstruction. In many practical inverse scenarios, practitioners are presented with complementary datasets—for example, boundary curves extracted from projection images and disk-integrated brightness values—each with distinctive characteristics such as noise levels, scaling, and the number of data points. Crafting an objective function that optimally integrates such modes is nontrivial since heuristic or ad hoc weight selection introduces bias and instability into the recovery process.

The manuscript frames a general methodology for optimally weighting and fusing multiple data sources by introducing the maximum compatibility estimate (MCE), a scale-invariant point estimate for the model parameters. The proposed approach is independent of explicit noise level specifications, data scaling, and data mode cardinalities.

Methodological Contributions

The methodology formalizes the joint $\chi^2$-based likelihood of multiple data sources with adaptive weights and seeks the optimal weighting with respect to relative model-compatibility, not just noise-based considerations. For the case of two data modes, the total cost function is:

$$\chi^2_{\text{tot}}(P, D) = \chi^2_1(P, D_1) + \lambda \chi^2_2(P, D_2)$$

where $P$ is the parameter vector, $D_i$ is data from mode $i$, and $\lambda$ is the weighting parameter.

Key to the analysis is the introduction of the "compatibility curve" $\mathcal{S}(\lambda)$ in $\big(\log \chi^2_1, \log \chi^2_2\big)$ space, analogous but not restricted to the Tikhonov L-curve. Points on this curve are obtained by varying $\lambda$ and minimizing $\chi^2_{\text{tot}}$ for each value. The invariance of $\mathcal{S}$ under scale and unit changes derives from the logarithmic transformation in each mode.

The main result is that optimal combination is achieved at the point (model parameters) which minimizes the quadratic distance in log-$\chi^2$ space to the “ideal” point (the vector of single-mode optima). Formally, for the two-mode case,

$$P_0 = \operatorname{arg\,min}_P \left( [\log \chi^2_1(P) - \hat{x}_0]^2 + [\log \chi^2_2(P) - \hat{y}_0]^2 \right)$$

where $\hat{x}_0$ and $\hat{y}_0$ are the single-mode minima.

This extends naturally to $n$ data modes, with a simple sum over squared log-ratios to the minima. Importantly, the number of data points and scaling in each mode have no effect on the optimal solution, and the approach is robust to the shape of the compatibility region.

The MCE corresponds to the traditional maximum likelihood estimate (MLE) for the single-mode case and maximum a posteriori (MAP) when regularization is present.

Case Study: Shape Reconstruction from Generalized Projections

A central case explored is the joint inversion of asteroid shapes from boundary profiles extracted from images and total brightness measurements, both as functions of viewing geometry. The methodology is demonstrated by representing the shape as a 3D polytope or using truncated spherical harmonics with exponential representation for ensuring positivity:

$$r(\theta, \varphi) = \exp\Bigg[\sum_{l,m} c_{lm} Y^m_l(\theta, \varphi)\Bigg]$$

where $c_{lm}$ are the parameters to be estimated.

Model fit is evaluated using $\chi^2$ for brightness and profile data individually, and the approach determines the optimal compromise in parameter space. A key observation is that using only brightness data results in ambiguity with respect to non-convexities, while profile data is more informative about deviations from convexity. The MCE approach allows for an optimal, rigorous integration of both.

Numerical Results

Demonstrations include asteroid 2 Pallas and 41 Daphne. In these, the compatibility curve is evaluated, and the region nearest the lower left (minimal joint misfit) is unambiguously identified, validating the stability and objectivity of the MCE-based procedure. The method is robust to computational noise, and the optimal weighting is insensitive to small numerical errors in estimating data mode minima.

Theoretical and Practical Implications

The MCE approach provides a principled and general framework for multi-modal inverse problems unaffected by arbitrary normalization, scale, or data cardinality choices. This addresses a significant shortcoming in prior practice, where subjective choices in weight settings could undermine the statistical validity of the recovered model. Because MCE avoids assumptions about the curvature or monotonicity of the multi-fit region (unlike the L-curve), it is broadly applicable, especially when data modes are not commensurate.

On the practical side, the methodology is directly extendable to problems involving more than two data sources and can incorporate regularization terms using the same framework. The explicit demonstration with asteroid data shows its applicability to fields such as computational geometry, astronomical imaging, and other domains where inverse problems with multimodal data are prevalent.

Future Directions

Potential avenues for further development include extending the approach to highly non-regular inverse problems where model sufficiency fails or the mapping from model parameters to data is noncontinuous. Advanced methods such as adaptive Monte Carlo could be deployed for visualizing compatibility regions in high dimensional settings, although computational cost is a limiting factor. The method's generality suggests applicability to multi-view 3D reconstruction, multimodal medical imaging, and beyond, especially as the complexity and diversity of measurement systems continue to expand.

Conclusion

The paper provides a rigorous, principled methodology for the optimal combination of heterogeneous data modes in inverse problems via the maximum compatibility estimate. The MCE framework eliminates dependence on arbitrary weightings, scale factors, and data mode cardinalities. The method is shown to yield stable, reliable reconstructions in the context of asteroid modeling but stands as a general solution for multi-modal inverse problems, offering significant theoretical and practical advantages in any setting demanding objective combination of complementary datasets.