SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Object-wise depth map loss. According to Eq. 1, the estimated object depth  $\hat{z}$  and coarse projected 3D center  $(\hat{u},\hat{v})$  jointly determine the location of the object 3D center  $(\hat{X},\hat{Y},\hat{Z})$ . However, many existing methods [20, 21, 23, 27] train the depth loss on the local region exactly on the GT center  $(u,v)$ , which can lead to significant errors in object depth estimation even with slightly inaccurate object center  $(\hat{u},\hat{v})$  during inference (i.e., center shifting in the image plane by a few pixels) [23].
To mitigate this issue, we train our network with an additional auxiliary head for object-wise depth estimation using the structure of [20]. Rather than solely providing depth supervision to an object center  $(u,v)$ , the auxiliary head is trained with depth supervision over the entire bounding box of the object. We create the foreground object-wise depth map  $\mathfrak{D} \in \mathbb{R}^{H' \times W'}$  following the policy of [40]. Then, we define the object-wise depth loss by adopting the Laplacian aleatoric uncertainty loss [9], the same as the object depth estimation task, only for the foreground regions:
$$
\mathcal {L} _ {o b j} = \frac {1}{| \mathfrak {D} |} \sum_ {z ^ {p} \in \mathfrak {D}} \frac {\sqrt {2}}{\hat {\sigma} ^ {p}} \left| z ^ {p} - \hat {z} ^ {p} \right| + \log (\hat {\sigma} ^ {p}), \tag {7}
$$
where  $z^p, \hat{z}^p$  and  $\hat{\sigma}^p$  are the corresponding GT object depth, prediction, and uncertainty of each pixel, respectively. Note that the additional task head is removed after training, so the proposed method does not increase the inference time.
Total loss. The total loss  $\mathcal{L}_{total}$  combines the loss used in 3D object detection baselines  $\mathcal{L}_{baseline}$ , the quasi-isometric loss  $\mathcal{L}_{qi}$ , and the object-wise depth map loss  $\mathcal{L}_{obj}$  as follows:
$$
\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {b a s e l i n e}} + \lambda_ {q i} \cdot \mathcal {L} _ {q i} + \lambda_ {o b j} \cdot \mathcal {L} _ {o b j}, \tag {8}
$$
where  $\lambda_{qi}$  and  $\lambda_{obj}$  represent the balancing weights for the quasi-isometric loss and the object-wise depth map loss, respectively. These values are set to 0.5 and 1. This total loss is applied to each baseline in our experiments, which we denote as "[Baseline] + Ours", as discussed further in Sec. 5.
Table 1: Evaluation results on KITTI validation set. (Blue/Red (positive/negative): relative performance from baseline.)
Extra data	Method	Car, AP3D|R40↑	Car, APBEV|R40↑
Easy	Moderate	Hard	Easy	Moderate	Hard
LiDAR	DID-M3D [27]	23.93	16.22	13.98	32.65	23.15	19.48
DID-M3D + Ours	24.77 (+3.5%)	17.12 (+5.5%)	14.30 (+2.3%)	34.32 (+5.1%)	23.45 (+1.3%)	20.64 (+6.0%)
None	MonoDLE [23]	17.32	14.35	12.22	24.62	20.25	17.75
MonoDLE + Ours	21.31 (+23.0%)	16.53 (+15.2%)	13.93 (+14.0%)	29.34 (+19.2%)	22.27 (+10.0%)	19.20 (+8.2%)
GUP-Net [22]	22.10	16.17	14.18	30.67	22.79	19.64
GUP-Net + Ours	24.21 (+9.5%)	17.82 (+10.2%)	15.01 (+5.9%)	32.38 (+5.6%)	23.61 (+3.6%)	21.20 (+7.9%)
MonoCon [20]	23.03	17.84	15.37	32.97	24.13	20.90
MonoCon + Ours	27.90 (+21.1%)	19.43 (+8.9%)	16.92 (+10.3%)	36.15 (+9.8%)	26.15 (+8.6%)	22.56 (+7.9%)
Additionally, to simplify the implementation of our proposed  $(K,B,\epsilon)$ -quasi-isometric loss, we detail the loss function in Alg.1.
Datasets. The KITTI dataset [12] consists of 7,481 training images and 7,518 test images for official KITTI 3D object detection evaluation and contains three categories: Car, Pedestrian, and Cyclist. For additional experiments, we follow [23], which splits the training images into 3,712 and 3,769 images for training and validation sets, respectively. The Waymo dataset [32] is a recently released dataset comprising 798 training sequences and 202 validation sequences, with four categories: Vehicles, Pedestrians, Cyclists, and Signs. We use the split reported in [28], including 52,386 training and 39,848 validation images, to evaluate performance on the Waymo dataset.
Evaluation Metrics. We adhere to the protocol reported in [12] and [28] for KITTI and Waymo datasets, respectively. The KITTI 3D object detection performance is evaluated by the average precision of 3D bounding boxes  $(AP_{3D|R40})$  with IoU thresholds of 0.7 for Car and 0.5 for Pedestrian and Cyclist. The evaluation is split into three levels of difficulty: Easy, Moderate, and Hard, based on the 2D bounding box height, occlusion level, and truncation. In contrast, the evaluation metrics of the Waymo dataset are based on 3D IoU with mean average precision (3D mAP) and mean average precision weighted by heading (3D mAPH). Each object is divided into one or two levels and evaluated at three distances:  $[0, 30)$ ,  $[30, 50)$ , and  $[50, \infty)$  meters.
Implementation details. First, we incorporate our method into four different baselines, including CenterNet-based frameworks [20, 22, 23, 27] for the KITTI dataset. For the Waymo dataset, we choose [20], as it performs best on the KITTI dataset among our baselines. We primarily follow the experimental setup details (i.e., epochs, optimizer) of each paper [20, 22, 23, 27] for a fair comparison. We denote the plugged versions of baselines as "[Baseline] + Ours". As an exception, we set all batch sizes to 16 for the KITTI and Waymo benchmarks. We retrain all baselines on the KITTI/Waymo validation set with "Car" and "Vehicle" classes, respectively. For the  $(K,B,\epsilon)$ -quasi-isometric loss, we use  $K = 1.5$ ,  $B = 0.5$ ,  $\epsilon = 10.0$ ,  $d_{1}(\cdot ,\cdot) = |\cdot ,|$ ,  $d_{2}(\cdot ,\cdot) = \| \cdot ,\|_{2}$  for all experiments. We provide more detailed experimental setups for each baseline in the supplementary materials.
5.1 Evaluation Results on KITTI and Waymo Datasets
KITTI dataset. We demonstrate the effectiveness of our method on both the KITTI validation and test datasets. As shown in Tab. 1, our method can be widely applied to various baselines and significantly enhances performance by a considerable margin. Particularly, the improvement is more pronounced in models that do not utilize extra data, surpassing models that do (13.1% vs. 3.8%). We surmise that the model trained with additional LiDAR data learns more discriminative features regarding depths than the method without extra data. Notably, the performance of "MonoCon + Ours" exceeds that of models trained with additional LiDAR data. This result illustrates that our proposed metric learning method empowers MonoCon to extract depth-discriminative features without the need for extra data. Similar trends are observable in the evaluation with the KITTI test datasets, as shown in Tab. 2. The experiments demonstrate that our proposed method enhances depth discrimination not
Table 2: Evaluation results on KITTI test set.
Extra data	Method	Car, AP3D|R40↑	Ped., AP3D|R40↑	Cyc., AP3D|R40↑
Easy	Mod.	Hard	Easy	Mod.	Hard	Easy	Mod.	Hard
LiDAR	DID-M3D [27]	24.40	16.29	13.75	-	-	-	-	-	-
DID-M3D + Ours	27.04	16.42	13.37	14.41	9.05	8.05	4.86	3.11	2.97
Improvement	(+10.8%)	(+0.8%)	(-2.8%)	-	-	-	-	-	-
None	MonoDLE [23]	17.23	12.26	10.29	9.64	6.55	5.44	4.59	2.66	2.45
MonoDLE + Ours	22.11	15.30	12.72	11.75	7.80	6.29	6.02	4.12	3.42
Improvement	(+28.3%)	(+24.8%)	(+23.6%)	(+21.9%)	(+19.1%)	(+15.6%)	(+31.2%)	(+54.9%)	(+39.6%)
GUPNet [22]	20.11	14.20	11.77	14.72	9.53	7.87	4.18	2.56	2.09
GUPNet + Ours	23.19	15.78	13.02	14.23	9.03	8.06	5.68	3.61	3.13
Improvement	(+15.3%)	(+11.1%)	(+10.6%)	(-3.3%)	(-5.2%)	(+2.4%)	(+35.9%)	(+41.0%)	(+49.8%)
MonoCon [20]	22.50	16.46	13.95	13.10	8.41	6.94	2.80	1.92	1.55
MonoCon + Ours	23.31	16.36	13.73	14.90	10.28	8.70	5.38	2.89	2.83
Improvement	(+3.6%)	(-0.6%)	(-1.6%)	(+13.7%)	(+22.2%)	(+25.4%)	(+92.1%)	(+50.5%)	(+82.6%)
Table 3: Evaluation results on Waymo validation set.
Difficulty	Method	Vehicle, AP3D ↑	Overall Vehicle, APH3D ↑
Overall	0-30 m	30-50m	50m-∞	0-30 m	30-50 m	50 m-∞
LEVEL_1(IOU = 0.7)	MonoCon [20]	2.30	6.66	0.67	0.02	2.29	6.62	0.66	0.02
MonoCon + Ours	2.50	7.62	0.72	0.02	2.48	7.57	0.72	0.02
Improvement	(+8.7%)	(+14.4%)	(+7.5%)	(+0.0%)	(+8.3%)	(+14.4%)	(+9.1%)	(+0.0%)
LEVEL_2(IOU = 0.7)	MonoCon	2.16	6.64	0.64	0.02	2.15	6.59	0.64	0.02
MonoCon + Ours	2.34	7.59	0.70	0.02	2.33	7.54	0.69	0.02
Improvement	(+8.3%)	(+14.3%)	(+9.4%)	(+0.0%)	(+8.4%)	(+14.4%)	(+7.8%)	(+0.0%)
LEVEL_1(IOU = 0.5)	MonoCon	10.07	27.47	3.84	0.16	9.99	27.26	3.81	0.16
MonoCon + Ours	10.14	28.51	3.99	0.17	10.06	28.28	3.96	0.17
Improvement	(+0.7%)	(+3.8%)	(+3.9%)	(+6.3%)	(+0.7%)	(+3.7%)	(+3.9%)	(+6.3%)
LEVEL_2(IOU = 0.5)	MonoCon	9.44	27.37	3.71	0.14	9.37	27.17	3.68	0.14
MonoCon + Ours	9.50	28.40	3.85	0.15	9.43	28.18	3.82	0.15
Improvement	(+0.6%)	(+3.8%)	(+3.8%)	(+7.1%)	(+0.6%)	(+3.7%)	(+3.8%)	(+7.1%)
only for Car but also for Pedestrian and Cyclist. The average performance increases for these object categories are  $10.3\%$ ,  $12.4\%$ , and  $53.1\%$ , respectively.
Waymo dataset. We extend the evaluations to the Waymo dataset to further demonstrate the generality of our method, as shown in Tab. 3. The results indicate that our proposed method consistently enhances performance by an average of  $4.6\%$  and  $4.5\%$  in terms of mAP and mAPH metrics across all levels, IoU thresholds, and distance ranges. Significantly, the proposed loss boosts the depth discrimination not only within specific depth ranges but across all depth ranges. This underscores the effectiveness of our method not just on the KITTI dataset, but on the Waymo dataset as well.
5.2 Additional Experiments
Ablation studies. To demonstrate the effectiveness of our proposed method, we conduct the ablation studies of two components of our method:  $(K,B,\epsilon)$ -quasi-isometric loss  $\mathcal{L}_{qi}$  and object-wise depth map loss  $\mathcal{L}_{obj}$ . We employ DID-M3D [27] and MonoCon [20] as baselines of a method with and without extra data. The results in Tab. 4 indicate that any method incorporating either the quasi-isometric or object-wise depth losses sees a performance improvement. Notably, the  $(K,B,\epsilon)$ -quasi-isometric loss contributes to a larger performance gain than the object-wise depth loss ( $+2.3\%$  vs.  $+1.9\%$  for DID-M3D and  $+8.7\%$  vs.  $+5.0\%$  for MonoCon). The models trained with both of the proposed losses significantly surpass the performance of the baselines.
Comparison of our method with SupCR. To demonstrate the advantage of our quasi-isometric loss over existing metric learning schemes, we compare the task performance of our method with that of SupCR (Supervised Contrastive Regression) [39]. SupCR is the first regression-aware representation learning method that effectively applies metric learning to regression tasks using GT labels. Similar to our quasi-isometric loss  $\mathcal{L}_{qi}$ , SupCR selectively chooses the relative negative pair object features based on positive pair distance. In Tab. 5, we report the  $AP_{3D|R40}$  for Car, Moderate and errors between the GT and prediction of four key tasks that determine the location of the 3D bounding box:  $t\in z,(h,w,l),\gamma ,c,$  independently. The errors include the absolute difference  $E_z,E_{dim}$  for  $z,(h,w,l)$ , respectively, the mean angular distance  $\Delta \gamma$  [11] for  $\gamma$ , and the accuracy Acc.  $^c$  for object
Table 4: Ablation studies on KITTI validation set.
Components
Lqi Lobj	Baseline	Car, AP3D|R40 ↑	Overall
Easy	Moderate	Hard
DID-M3D [27]	23.93	16.22	13.98	0.0%
✓	24.61 (+2.8%)	16.57 (+2.2%)	14.23 (+1.8%)	+2.3%
✓	24.53 (+2.5%)	16.84 (+3.8%)	13.87 (-0.8%)	+1.8%
✓	✓	24.77 (+3.5%)	17.12 (+5.5%)	14.30 (+2.3%)	+3.8%
MonoCon [20]	23.03	17.84	15.37	0.0%
✓	27.00 (+17.2%)	18.68 (+4.7%)	16.02(+4.2%)	+8.7%
✓	24.90 (+8.1%)	18.65 (+4.5%)	15.73 (+2.3%)	+5.0%
✓	✓	27.90 (+21.1%)	19.43 (+8.9%)	16.92 (+10.3%)	+13.5%
Table 5: Comparison of our method with SupCR on KITTI validation set.
Baseline	AP3D|R40 ↑	Ez(m) ↓	Edim(m) ↓	Δγ(rad) ↓	Acc.c(%) ↑
MonoCon [20]	17.84	0.019	0.025	π/371.79	93.15
MonoCon + SupCR	17.55 (-1.6%)	0.014 (-26.3%)	0.031 (+24.0%)	π/312.78 (+18.9%)	91.11 (-2.2%)
MonoCon + Lqi	18.68 (+4.7%)	0.011 (-42.1%)	0.024 (-4.0%)	π/368.11 (+1.0%)	93.07 (-0.1%)
classification  $c$ . The results illustrate that our method enhances 3D object detection performance, whereas SupCR diminishes it. Interestingly, both methods reduce depth errors  $E_{z}$  as they train the encoder to extract more depth-discriminative features. However, SupCR significantly increases the errors in bounding box size and angle estimation  $E_{dim}$ ,  $\Delta \gamma$  by approximately  $24.0\%$  and  $18.9\%$ , respectively. It demonstrates that SupCR, which forcibly arranges the feature manifold, can negatively impact the performance of other tasks due to the complex shared representations across multiple sub-tasks.
Scalability of our quasi-isometric loss with the anchor-based method and BEV paradigm. In Tab. 1-4, we note that our method improves performance when applied to CenterNet-based baselines (anchor-free). We further demonstrate the effectiveness of the proposed metric learning incorporated into anchor-based methods and the bird-eye-view (BEV) paradigm: MonoDTR [15] and ImVoxelNet [30]. The object-wise depth map loss is not applied to these baselines, since it already uses an auxiliary depth loss or inherent nature of the BEV paradigm.
As shown in Tab. 6, although the performance enhancement compared to anchor-free methods is not as significant, our quasi-isometric loss still shows consistent improvements across all metrics for each baseline. These results suggest that the loss has broad applicability to the tasks that extract object features on image spatial coordinates given object depth labels.
Model performance with respect to the ratio of property-violated objects. The proposed quasi-isometric loss is designed to arrange the feature space in accordance with object depth labels, aiming to meet quasi-isometric properties. To demonstrate the correlation between 3D object detection performance and the ratio of features that satisfy the quasi-isometric condition, we conduct an additional experiment. In Fig. 2, we plot the ratio of object pairs violating the properties (where the Ratio  $= (|\mathbf{P}^{+}| + |\mathbf{P}^{-}|) / |\mathbf{P}|$ ,  $|\mathbf{P}| = \binom{L}{2}$ , and  $L$  is the total number of objects in the dataset.) and model performance in relation to various loss scales  $\lambda_{qi}$ . For this experi
iment, we use the baseline model [20] trained with different loss scales  $[10^{-5}, 10^{-3}, 10^{-2}, 10^{-1}, 0.5]$ . We measure the  $AP_{3D}$  with IoU thresholds of 0.7 for Car, Moderate on the KITTI validation dataset. The results indicate that an increase in the loss scale corresponds with a decrease in the ratio of object pairs violating the quasi-isometric properties, leading to an improvement in model performance. This
Figure 2: Loss scale (log-scaled) with respect to property-violated pairs ratio and performance.
Table 6: Evaluation results of anchor-based and bird-eye-view paradigms on KITTI validation set.
Method	Car, AP3D|R40 ↑	Overall
Easy	Moderate	Hard
MonoDTR [15]	24.57	18.45	15.37	-
MonoDTR + Ours	26.17 (+6.5%)	19.07 (+3.4%)	15.93 (+3.6%)	+4.5%
ImVoxelNet [30]	24.54	17.80	15.87	-
ImVoxelNet + Ours	26.14 (+8.0%)	18.20 (+7.2%)	15.81 (+3.8%)	+6.3%
Table 7: Performance of KITTI validation with respect to  $K, B, \epsilon$ .
Components	Car, AP3D|R40 ↑	Overall
K	B	ε	Easy	Moderate	Hard
MonoCon (w/o Lobj)	23.03	17.84	15.37	+0.0%
1.0	0.5	10.0	24.67 (+7.1%)	18.09 (+1.4%)	15.34 (-0.2%)	+2.8%
1.5	0.5	10.0	27.00 (+17.2%)	18.68 (+4.7%)	16.02 (+4.2%)	+8.7%
2.0	0.5	10.0	23.34 (+1.3%)	17.44 (-2.2%)	14.80 (-3.7%)	-1.5%
1.5	0.0	10.0	24.54 (+6.6%)	18.01 (+1.0%)	15.37 (+0.0%)	+2.5%
1.5	1.0	10.0	25.49 (+10.7%)	18.70 (+4.8%)	15.82 (+2.9%)	+6.1%
1.5	5.0	10.0	23.70 (+2.9%)	18.00 (+0.9%)	15.12 (+-1.6%)	+0.9%
1.5	0.5	1.0	24.71 (+7.3%)	17.84 (+0.0%)	15.43 (+0.4%)	+2.6%
1.5	0.5	5.0	24.99 (+8.5%)	18.23 (+2.2%)	15.80 (+2.8%)	+4.5%
1.5	0.5	∞	22.85 (-0.8%)	16.91 (-5.2%)	14.14 (-8.0%)	-4.7%
suggests a potential relationship between the reduction in the ratio of property-violated object pairs and the enhancement of model performance.
Influence of hyperparameters  $(K, B, \epsilon)$  in quasi-isometric loss. The choice of the tunable hyperparameter  $\epsilon$  is crucial. An excessively small  $\epsilon$  would sample a few objects within a mini-batch, making representation learning infeasible. On the other hand, quasi-isometric properties with too large an  $\epsilon$  would harm the non-linearity of the feature manifold in  $\mathbf{P}$ -space. To investigate the influence of hyperparameters on our quasi-isometric loss, we evaluate the performance of "Monocon + Ours" using KITTI [12] validation set for various hyperparameters  $(K, B, \epsilon)$ , as presented in Tab. 7. Although most hyperparameter combinations improved performance, there are notable exceptions, especially when  $K = 2.0$  and  $\epsilon = \infty$ . We observe a decline in performance with excessively large values of  $K$  or  $\epsilon$ , which define the criteria for property violation and the non-linearity of the feature manifold.
In this paper, we address the challenge of monocular 3D object detection in RGB images by proposing a novel metric learning scheme. Our method, which does not rely on extra parameters, modules, or data, concentrates on extracting depth-discriminative features without increasing the inference time or model size. By employing a distance-preserving function and the  $(K,B,\epsilon)$ -quasi-isometric loss, we successfully arrange the feature space manifold in accordance with ground-truth object depth, while preserving the non-linearity of the natural feature manifold. Furthermore, by introducing an auxiliary head for object-wise depth estimation, we improve object depth quality without increasing inference time. Our experimental results on the KITTI and Waymo datasets illustrate consistent performance enhancements across different baselines, highlighting the effectiveness of our proposed method. As a potential avenue for future work, our method could feasibly be extended to multi-camera 3D object detection scenarios and other regression tasks that involve multiple sub-tasks.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2023-00210908).
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