Optimal LRP (oLRP) Metric

• Optimal LRP (oLRP) is a unified object detection error metric that integrates localization error, false positives, and false negatives to determine an optimal confidence threshold.
• It is computed by exhaustively searching over a discretized grid of thresholds to minimize the LRP error, yielding distinct error components for localization, false positives, and false negatives.
• oLRP provides enhanced interpretability and discrimination over Average Precision, enabling practical performance evaluation and tailored threshold selection in modern detectors.

Optimal LRP (oLRP) is a metric-theoretic object detection error that integrates localization error, false positives, and false negatives with respect to an optimally chosen confidence score threshold. Acting as a unified scalar error metric in $[0,1]$, oLRP provides both a quantitative summary of detection performance and guidance on threshold selection, addressing limitations of metrics such as Average Precision (AP) by offering enhanced discrimination power and interpretability for modern visual detectors (Oksuz et al., 2018).

1. Foundations: LRP Error Definition

Localization Recall Precision (LRP) error evaluates object detection quality by combining three core error sources: bounding-box localization, false positives, and false negatives. For a given object class, denote the set of ground-truth boxes as $X$ and the set of detections (with confidence scores) as $Y$. Given Intersection-over-Union (IoU) threshold $\tau$ and confidence threshold $s$, detections with scores below $s$ are discarded. The usual IoU-based matching assigns detections to ground truths, yielding counts $N_{TP}$ (true positives), $N_{FP}$ (false positives), and $N_{FN}$ (false negatives).

The LRP error at threshold $s$ is computed via three normalized terms:

• Localization error:

$X$0

where $X$1 denotes the matched ground-truth and detection pairs.

• False-positive rate:

$X$2

• False-negative rate:

$X$3

The combined LRP error is given by:

$X$4

This formulation ensures LRP $X$5, with lower values indicating better performance. All terms admit a clear operational interpretation, and the metric remains well-defined whenever genuine detections or ground truths are present.

2. Definition and Computation of Optimal LRP

Optimal LRP (oLRP) for a given class searches for the threshold $X$6 minimizing the LRP error:

$X$7

The optimization is typically performed over a discretized grid $X$8 by exhaustive search. For each $X$9, the error components $Y$0, $Y$1, and $Y$2 are evaluated. The threshold $Y$3 achieving

$Y$4

is selected, and its auxiliary components $Y$5, $Y$6, and $Y$7 quantify the optimal breakdown of localization and classification errors.

A mean over classes (moLRP) is defined analogously to mean Average Precision (mAP):

$Y$8

where $Y$9 is the set of evaluated classes.

3. Algorithmic Structure and Practical Computation

Core computation proceeds via the following for each class and IoU threshold:

1. For each $\tau$0:
   • Select detections $\tau$1.
   • Assign detections to ground-truths using one-to-one IoU matching ($\tau$2).
   • Compute $\tau$3, $\tau$4, $\tau$5.
   • Derive $\tau$6, $\tau$7, $\tau$8 as above.
   • Calculate $\tau$9.
   • If $s$0 is the minimum so far, store $s$1 and corresponding error components.
2. Report $s$2 and the values at $s$3.

This process applies independently per class. The exhaustive thresholding, performed at a grid resolution of 0.01, is computationally modest given contemporary resources.

4. Comparison with AP and Other Evaluation Metrics

Average Precision (AP) computes the area under the precision-recall (PR) curve by integrating performance over all confidence thresholds. AP, however, is insensitive to PR-curve shapes that yield identical areas and disregards bounding-box localization accuracy, except for the IoU binary qualification of a match. oLRP, in contrast, reports the single best operating point on the PR curve, directly characterizes the operational trade-off at deployment, and incorporates localization error as a primary component.

Distinctive theoretical properties include:

• oLRP reports separate components for localization, false positive, and false negative errors at $s$4.
• LRP satisfies the triangle inequality and is a bounded metric in $s$5.
• oLRP remains valid even in degenerate cases with no true positives, so long as ground-truth or detected objects exist.

Unlike Panoptic Quality (PQ)—which combines recognition and segmentation quality but is specialized for panoptic segmentation—oLRP generalizes to all detection tasks and offers a principled means of threshold selection.

5. Empirical Findings and Deployment Implications

Empirical analysis on detection benchmarks (e.g., MS COCO-val2017) encompassing diverse contemporary detectors (e.g., SSD-300, SSD-512, Faster R-CNN + FPN, RetinaNet + FPN with ResNet-50/101) demonstrates:

• moLRP correlates closely with mAP rankings but often discriminates detectors with identical mAP by revealing differences in PR-curve sharpness or localization fidelity.
• Class-specific optimal thresholds $s$6 exhibit high variability (from $s$7 to $s$8), indicating the suboptimality of a global threshold.
• In video object detection, switching from a general default threshold ($s$9) to class-specific $s$0 improved oLRP and mAP, especially in classes with very low/high optimal thresholds.

6. Limitations and Considerations

Potential limitations of oLRP include:

• Threshold discretization: Optimization over $s$1 is performed on a discretized grid, introducing small approximation error.
• Single IoU threshold: oLRP is computed at a fixed $s$2 (e.g., $s$3), though evaluation at multiple thresholds is straightforward.
• Summary at one operating point: Collapsing performance onto $s$4 may obscure detector behavior elsewhere on the PR curve.

Nevertheless, oLRP delivers a practical, interpretable single-value metric that unifies localization and classification error, provides actionable deployment thresholds, and supports nuanced comparative analysis of detection systems (Oksuz et al., 2018).