Application-driven Validation of Posteriors in Inverse Problems

TL;DR

This work tackles the challenge of validating posterior-based solutions for ambiguous inverse problems by introducing an application-driven, mode-centric framework inspired by object detection. It formalizes a problem fingerprint to tailor metric choice and splits validation into distribution-based and object-detection-inspired approaches, with decision guides to navigate tradeoffs. The framework is instantiated using conditional Invertible Neural Networks (cINNs) across a toy root-finding problem and two medical vision use cases (pose estimation in intraoperative 2D/3D registration and functional tissue parameter quantification via photoacoustic imaging), demonstrating that mode-centric validation reveals practical advantages over traditional MAP-focused validation. The findings suggest that embracing mode-level evaluation yields more meaningful, application-relevant insights for clinical deployment and sets the stage for standardized validation in inverse problems. Overall, the approach enables more interpretable, decision-relevant assessment of posterior-based methods and encourages broader adoption in practice.

Abstract

Current deep learning-based solutions for image analysis tasks are commonly incapable of handling problems to which multiple different plausible solutions exist. In response, posterior-based methods such as conditional Diffusion Models and Invertible Neural Networks have emerged; however, their translation is hampered by a lack of research on adequate validation. In other words, the way progress is measured often does not reflect the needs of the driving practical application. Closing this gap in the literature, we present the first systematic framework for the application-driven validation of posterior-based methods in inverse problems. As a methodological novelty, it adopts key principles from the field of object detection validation, which has a long history of addressing the question of how to locate and match multiple object instances in an image. Treating modes as instances enables us to perform mode-centric validation, using well-interpretable metrics from the application perspective. We demonstrate the value of our framework through instantiations for a synthetic toy example and two medical vision use cases: pose estimation in surgery and imaging-based quantification of functional tissue parameters for diagnostics. Our framework offers key advantages over common approaches to posterior validation in all three examples and could thus revolutionize performance assessment in inverse problems.

Figures (9)

• Figure 1: Computational tasks in healthcare may be inherently ambiguous. When multiple substantially different plausible solutions exist, they can be encoded as modes in the output posterior. (a): In image-guided surgery, the task of determining the pose of a patient relative to an intraoperative X-ray system may be ill-posed as different patient poses (here: posterior-anterior and anterior-posterior) can yield almost identical 2D projection images for the same device pose. (b): Inverse kinematics in robotic surgery presents inherent ambiguities when determining optimal joint configurations to reach desired target positions, as different joint angles can achieve the same end-effector position. (c): When analyzing screw placement from a 2D frontal X-ray view, the screw angle in the lateral plane cannot be recovered due to the lack of depth information in the projection image. Knee X-ray adapted from fang_extra-articular_2021 (CC BY 4.0).
• Figure 2: Object detection validation methodology lends itself well to posterior validation. This validation is subdivided into the steps of instance localization, assignment, and computing of classification metrics. These steps have natural analogs in the posterior validation case. Used abbreviations: Average Precision (AP), True Positive (TP), False Positive (FP), False Negative (FN), Standard Deviation (STD).
• Figure 3: Overview of metric selection framework for posterior validation. Depending on the reference granularity (reference posterior with/without labeled modes, exhaustive or non-exhaustive list of reference modes), the user follows the correspondingly colored path in the decision tree. When a tree branches, the fingerprint items determine which exact path to take. Recommendations for distribution-based metrics (Subprocess S1) are provided in Fig. \ref{['fig:subprocess_1']}. The main novelty of the proposal relates to the selection of object detection-inspired metrics, which is presented in a separate Subprocess S2 (Fig. \ref{['fig:subprocess_2']}). The notation Metric1@Metric2 refers to providing the value for Metric1 for a specific target value (e.g. Recall = 0.95) of Metric 2.
• Figure 4: Subprocess S1 for selecting distribution-based metrics. Based on the exact representation of the predicted posterior and the dimensionality of the problem, different metrics become available.
• Figure 5: Subprocess S2 for selecting object detection-inspired metrics, comprising the steps of selecting the localization criterion, the assignment strategy, and the actual classification metric(s). The notation Metric1@Metric2 refers to providing the value for Metric1 for a specific target value (e.g. Recall = 0.95) of Metric 2. Decision guides for selecting a suitable option from a list of candidates are provided in section \ref{['sec. decision guide']}. Used abbreviations: Average Precision (AP), Free-response Receiver Operating Characteristic (FROC), False Positives Per Image (FPPI).