InterpIoU: IoU Loss for Bounding Box Regression

• InterpIoU is a bounding box regression loss that uses an interpolation method to compute an additional IoU term, ensuring useful gradients even when boxes do not overlap.
• It avoids reliance on auxiliary geometric penalties like center distance or aspect ratio, thus preventing misaligned optimization and box enlargement.
• Dynamic InterpIoU further adapts the interpolation coefficient based on current IoU, enhancing performance particularly for small objects and varied scales.

Searching arXiv for InterpIoU and closely related IoU-based object detection losses and verification work. arXiv search: "InterpIoU bounding box regression interpolation IoU optimization" InterpIoU is a bounding box regression loss for object detection that replaces handcrafted geometric penalties with an interpolation-based IoU term. Given a predicted box $B_{\text{pred}}$ and a ground-truth box $B_{\text{gt}}$, it augments the ordinary IoU loss with the IoU between the target and an interpolated box lying between prediction and target in box-parameter space. The method is motivated by two limitations of prior IoU-based losses: plain IoU has uninformative or vanishing gradients when boxes do not overlap, and auxiliary penalties such as center distance, aspect ratio, enclosing-box size, or angle can misalign optimization with the IoU objective and induce undesirable behaviors such as box enlargement. InterpIoU is presented as a way to preserve IoU as the regression target while restoring useful gradients in non-overlapping cases; the same work also introduces Dynamic InterpIoU, which adapts the interpolation coefficient according to the current IoU value (Liu et al., 16 Jul 2025).

1. Formal definition

InterpIoU is defined from three boxes: the prediction $B_{\text{pred}}$, the ground truth $B_{\text{gt}}$, and an interpolated box

$$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$

The loss is

$$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$

with standard IoU loss notation

$$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$

Equivalently,

$$\mathcal{L}_{\text{InterpIoU}}= \bigl(1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}})\bigr)+ \bigl(1-\mathrm{IoU}(B_{\text{int}},B_{\text{gt}})\bigr).$$

The variables are given explicitly as $B_{\text{pred}}=[x,y,w,h]$, $B_{\text{gt}}=[x,y,w,h]$, $B_{\text{gt}}$0, and $B_{\text{gt}}$1, with

$B_{\text{gt}}$2

In the paper’s formulation, the interpolated box is not an auxiliary geometric penalty in the style of GIoU, DIoU, CIoU, EIoU, SIoU, or PIoU; it is a second IoU term evaluated on a proxy box that is closer to the target (Liu et al., 16 Jul 2025).

This construction places the method squarely within IoU-based localization losses while altering the route by which gradients are obtained. A plausible implication is that the method attempts to remain metric-aligned at the loss level rather than correcting IoU’s optimization path by adding non-IoU objectives.

2. Motivation and relation to prior IoU-based losses

The stated motivation is that many existing IoU-based losses address the zero-gradient problem of non-overlapping boxes by adding handcrafted geometric penalties. The paper lists representative cases: GIoU uses an enclosing-box penalty, DIoU uses a center-distance penalty, CIoU uses center distance plus aspect-ratio penalty, EIoU uses width and height penalties, SIoU uses angle, distance, and shape penalties, and PIoU uses a target-adaptive penalty (Liu et al., 16 Jul 2025).

According to the paper, these penalties help optimization but are not directly IoU. They are described as sensitive to object shape, size, and distribution, and as potentially suboptimal for small objects. The paper further argues that such penalties can create misaligned objectives, with GIoU singled out as an example that can encourage the predicted box to expand in order to reduce the auxiliary term even when expansion is not the best way to increase true overlap. The same source characterizes this as a box enlargement issue (Liu et al., 16 Jul 2025).

The counter-position taken by InterpIoU is that IoU itself is the ideal regression target because IoU is the direct evaluation criterion for localization quality. In the broader literature, IoU is indeed the standard overlap measure for detection and tracking accuracy, even though its role there is evaluative rather than optimizational (Bottger et al., 2017). InterpIoU therefore does not redefine the target metric; it attempts to optimize that metric more faithfully under the non-overlap regime.

The paper’s simulation study is used to support this claim. It reports that traditional geometric penalties were often redundant, that InterpIoU consistently achieved superior optimization behavior, and that the loss landscape and convergence behavior suggested that the pure IoU-based interpolation signal was sufficient. This suggests a reinterpretation of IoU-loss design: the main deficiency may lie less in IoU as an objective than in the absence of a usable gradient path when overlap is zero (Liu et al., 16 Jul 2025).

3. Interpolation mechanism and gradient behavior

The central intuition is that the interpolated box

$B_{\text{gt}}$3

lies on the line segment between prediction and ground truth in box-parameter space. If $B_{\text{gt}}$4 is sufficiently large, $B_{\text{gt}}$5 becomes close enough to $B_{\text{gt}}$6 to overlap it even when $B_{\text{gt}}$7 does not. The paper identifies two consequences: non-overlapping cases receive gradients, and those gradients still push the prediction toward the target because backpropagation passes through a convex combination of prediction and ground truth (Liu et al., 16 Jul 2025).

The gradient identity given in the paper is

$B_{\text{gt}}$8

and since

$B_{\text{gt}}$9

this becomes

$B_{\text{pred}}$0

Hence, even if the first term is weak or zero because the boxes do not overlap, the second term remains informative as long as $B_{\text{pred}}$1 overlaps the target (Liu et al., 16 Jul 2025).

The paper also derives a sufficient condition for overlap between the interpolated box and the ground truth. For axis $B_{\text{pred}}$2, with center and side length variables $B_{\text{pred}}$3 for the ground-truth box and $B_{\text{pred}}$4 for the predicted box, the interpolated box has

$B_{\text{pred}}$5

Defining the edge-to-edge gap as

$B_{\text{pred}}$6

a sufficient condition for non-zero overlap is

$B_{\text{pred}}$7

The same paper notes that in common assignment settings a practical choice like $B_{\text{pred}}$8 is often enough, while static InterpIoU often uses $B_{\text{pred}}$9 in experiments (Liu et al., 16 Jul 2025).

This mechanism differs from adding a distance-like regularizer. The extra signal is still an IoU derivative, but evaluated at a box that is more favorably positioned relative to the target. That distinction is the paper’s basis for calling the method purely IoU-based.

4. IoU as regression target and Dynamic InterpIoU

A major argumentative component of the method is the claim that IoU itself already contains the correct optimization behavior. Writing

$B_{\text{gt}}$0

where $B_{\text{gt}}$1 is intersection area, $B_{\text{gt}}$2 is union area, $B_{\text{gt}}$3 is predicted box area, and $B_{\text{gt}}$4 is ground-truth area, the paper gives

$B_{\text{gt}}$5

$B_{\text{gt}}$6

The interpretation stated in the paper is that the $B_{\text{gt}}$7 gradients push the box toward the target to increase intersection, while the $B_{\text{gt}}$8 gradients balance overlap increase against unnecessary area growth (Liu et al., 16 Jul 2025).

Dynamic InterpIoU is introduced because a fixed $B_{\text{gt}}$9 can become too rigid. If the prediction is already good, a large constant $$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$0 places the interpolated box too close to the target, weakening the additional training signal. The dynamic variant defines

$$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$1

The intended behavior is explicit: low IoU leads to large $$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$2, keeping $$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$3 close to the ground truth and making overlap easier; as IoU improves, $$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$4 decreases, moving the interpolated box away from the target and creating what the paper calls a gradient boost zone (Liu et al., 16 Jul 2025).

The reported clamp ranges are dataset-dependent, including $$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$5, $$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$6, and $$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$7. Static InterpIoU typically uses $$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$8. The paper’s interpretation is that Dynamic InterpIoU enhances adaptability to scenarios with diverse object distributions, although it also notes that on highly varied COCO distributions the dynamic mechanism may be less stable than a well-chosen fixed $$B_{\text{int}} = (1-\alpha)B_{\text{pred}} + \alpha B_{\text{gt}}, \qquad 0<\alpha<1.$$9 (Liu et al., 16 Jul 2025).

5. Simulation evidence and empirical performance

The simulation study uses 7 ground-truth boxes with area $$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$0 and aspect ratios

$$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$1

It samples 5000 anchor points in a circular region and combines 7 scales with 7 aspect ratios, producing

$$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$2

simulated regression cases. Optimization uses Adam with learning rate 0.01. In a controlled regression experiment with non-overlapping boxes, the paper reports that GIoU, CIoU, DIoU, EIoU, and SIoU often produced substantial box enlargement, whereas WIoU, PIoU, and InterpIoU kept area changes much smaller. Enlargement is quantified by

$$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$3

where large positive $$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$4 indicates the predicted box becomes much larger than the ground truth (Liu et al., 16 Jul 2025).

The empirical evaluation spans PASCAL VOC 2007, VisDrone, and MS COCO, using SSD, YOLOv8-m, and DINO-4scale. Training details are reported as follows: DINO uses the official training scheme with batch size 8 for 12 epochs; YOLOv8 uses 500 epochs with batch size 4 on COCO and 150 epochs with batch size 16 on VOC and VisDrone; SSD uses batch size 32 for 140k iterations (Liu et al., 16 Jul 2025).

On PASCAL VOC, the paper reports that InterpIoU and D-InterpIoU consistently outperform prior IoU losses with SSD and YOLOv8. For SSD, InterpIoU attains AP $$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$5, AP50 $$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$6, and D-InterpIoU reaches AP $$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$7 and AP75 $$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$8. For YOLOv8, D-InterpIoU gives AP50 $$\mathcal{L}_{\text{InterpIoU}}(B_{\text{pred}}, B_{\text{gt}})=\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})+\mathcal{L}_{\text{IoU}}(B_{\text{int}}, B_{\text{gt}}),$$9, AP75 $$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$0, and AP $$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$1, which the paper identifies as the best in that table. Small-object classes highlighted on VOC are bottle, pottedplant, and person (Liu et al., 16 Jul 2025).

On VisDrone, which the paper emphasizes because the average normalized object area is about $$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$2 of the image, D-InterpIoU achieves AP50 $$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$3, AP75 $$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$4, and AP $$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$5, while InterpIoU attains AP50 $$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$6 and AP $$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$7. Category-wise improvements are reported for people, car, van, and truck (Liu et al., 16 Jul 2025).

On MS COCO, the paper reports for YOLOv8 that InterpIoU obtains mAP $$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$8, AP50 $$\mathcal{L}_{\text{IoU}}(B_{\text{pred}}, B_{\text{gt}})=1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}}).$$9, and APs $$\mathcal{L}_{\text{InterpIoU}}= \bigl(1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}})\bigr)+ \bigl(1-\mathrm{IoU}(B_{\text{int}},B_{\text{gt}})\bigr).$$0, while D-InterpIoU attains AP75 $$\mathcal{L}_{\text{InterpIoU}}= \bigl(1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}})\bigr)+ \bigl(1-\mathrm{IoU}(B_{\text{int}},B_{\text{gt}})\bigr).$$1 and APs $$\mathcal{L}_{\text{InterpIoU}}= \bigl(1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}})\bigr)+ \bigl(1-\mathrm{IoU}(B_{\text{int}},B_{\text{gt}})\bigr).$$2. For DINO, InterpIoU reaches mAP $$\mathcal{L}_{\text{InterpIoU}}= \bigl(1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}})\bigr)+ \bigl(1-\mathrm{IoU}(B_{\text{int}},B_{\text{gt}})\bigr).$$3, AP50 $$\mathcal{L}_{\text{InterpIoU}}= \bigl(1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}})\bigr)+ \bigl(1-\mathrm{IoU}(B_{\text{int}},B_{\text{gt}})\bigr).$$4, and APs $$\mathcal{L}_{\text{InterpIoU}}= \bigl(1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}})\bigr)+ \bigl(1-\mathrm{IoU}(B_{\text{int}},B_{\text{gt}})\bigr).$$5, all identified there as best. The reported overall takeaway is that InterpIoU is usually the safest and most consistent improvement, whereas D-InterpIoU can be better on datasets that are more small-object-heavy or have more consistent scale patterns, such as VisDrone and VOC (Liu et al., 16 Jul 2025).

These results are used to support the paper’s stronger claim that handcrafted geometric penalties are both unnecessary and suboptimal. A cautious reading is that the evidence is strongest for the evaluated detectors and datasets, especially in regimes with dense or very small objects.

6. Scope, limitations, and adjacent IoU literature

InterpIoU operates at the level of bounding box regression loss. It is therefore distinct from IoU as an evaluation metric, from normalized IoU-style measures for dataset-dependent evaluation, and from formal verification frameworks that certify worst-case IoU under perturbations. For example, relative Intersection over Union (rIoU) normalizes box-over-segmentation IoU by the best achievable box for a segmentation in order to make scores comparable across object shapes on densely segmented benchmarks such as DAVIS and VOT2016 (Bottger et al., 2017). By contrast, IBP IoU and IoUCert study formal robustness verification: they bound or certify IoU under perturbations for single-object detection settings, rather than designing a training loss (Cohen et al., 2024, Brückner et al., 3 Mar 2026).

This distinction addresses a common ambiguity around the name. InterpIoU is neither a replacement evaluation metric nor a certification procedure. It is a BBR loss intended to improve optimization by using interpolated boxes inside an IoU objective (Liu et al., 16 Jul 2025).

The method’s reported strengths are particularly tied to non-overlapping predictions, small objects, and avoidance of box enlargement. Its practical caveat is the need to choose or schedule $$\mathcal{L}_{\text{InterpIoU}}= \bigl(1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}})\bigr)+ \bigl(1-\mathrm{IoU}(B_{\text{int}},B_{\text{gt}})\bigr).$$6. The paper explicitly presents static $$\mathcal{L}_{\text{InterpIoU}}= \bigl(1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}})\bigr)+ \bigl(1-\mathrm{IoU}(B_{\text{int}},B_{\text{gt}})\bigr).$$7 as potentially too rigid and Dynamic InterpIoU as an attempt to compensate; it also notes that the dynamic mechanism may be less stable on highly varied COCO distributions than a fixed $$\mathcal{L}_{\text{InterpIoU}}= \bigl(1-\mathrm{IoU}(B_{\text{pred}},B_{\text{gt}})\bigr)+ \bigl(1-\mathrm{IoU}(B_{\text{int}},B_{\text{gt}})\bigr).$$8 (Liu et al., 16 Jul 2025). This suggests that the interpolation coefficient functions as a central inductive bias of the method.

Within the broader IoU literature, InterpIoU can be situated as a proposal to keep optimization aligned with the evaluation target while restoring informative gradients through reparameterized supervision rather than auxiliary geometry. That positioning is consistent with adjacent work showing that IoU plays multiple roles in detection research: as an evaluation criterion for detectors and trackers (Bottger et al., 2017), as a certification target in robustness verification (Cohen et al., 2024, Brückner et al., 3 Mar 2026), and in InterpIoU’s case, as the sole semantic target of a regression loss (Liu et al., 16 Jul 2025).