SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Evaluation metrics in our experiments are the mean and median error between the predicted and ground truth over all samples for both localization and orientation separately in meters and in degrees. Following [35], for the KITTI dataset, we additionally include the recall under a certain threshold for longitudinal (driving direction) and lateral localization error, and orientation estimation error. Our thresholds are set to 1m and 5m for localization and to  $1^{\circ}$  and  $5^{\circ}$  for orientation estimation.
Our network uses EfficientNet-B0 [29] with pretrained weights onImagenet [5] as both the ground and aerial feature extractors, with non-shared weights. The satellite image and bird's-eye-view (BEV) transformed from the ground image both have a size of  $512 \times 512$  on both the VIGOR [42] and KITTI [8] datasets. PyTorch is used for network implementation, and training is done using the AdamW [17] optimizer with a maximum learning rate of  $3.5 \times 10^{-4}$ . The network is trained with a batch size of 16 and a training iteration of 180000. We set the search radius of the correlation updater  $r = 4$  and set  $\alpha_{1} = 0.1$ ,  $\alpha_{2} = 10$ ,  $\alpha_{3} = 1.0$ ,  $\tau = 4$  in the loss function. The total iteration  $K = 6/10$  and the feature map size  $D \times H \times W = 320 \times 16 \times 16 / 320 \times 16 \times 16$  on VIGOR/KITTI.
VIGOR dataset On the VIGOR dataset, we compare our method against several state-of-the-art methods: CVR [42], SliceMatch [12], Boosting [26] and CCVPE [35]. For the performance evaluations of CVR [42], SliceMatch [12], and Boosting [26] with known orientation priors, we directly utilize the results provided by CCVPE or the corresponding paper. Regarding most state-of-the-art CCVPE [35], we executed its official code to obtain results not presented in the original
Table 1: Location and orientation estimation error on VIGOR [42] dataset. Best in bold. Different levels of noise are added to the orientation prior. " * " indicates methods using our corrected labels.
Noise	Method	Same-Area	Cross-Area
↓Localization(m)	↓Orientation(°)	↓Localization(m)	↓Orientation(°)
mean	median	mean	median	mean	median	mean	median
0°	CVR [42]	8.82	7.68	-	-	9.45	8.33	-	-
SliceMatch [12]	5.18	2.58	-	-	5.53	2.55	-	-
Boosting [26]	4.12	1.34	-	-	5.16	1.40	-	-
CCVPE [35]	3.60	1.36	10.59	5.43	4.97	1.68	27.78	14.11
CCVPE* [35]	3.37	1.33	9.47	5.23	4.96	1.69	26.30	14.21
HC-Net(ours)*	2.65	1.17	1.92	1.04	3.35	1.59	2.58	1.35
±20°	CCVPE* [35]	3.48	1.39	9.80	5.49	5.16	1.78	26.36	14.87
HC-Net(ours)*	2.65	1.17	1.93	1.02	3.36	1.59	2.62	1.34
±45°	CCVPE* [35]	3.50	1.39	10.56	5.95	5.16	1.78	26.77	15.29
HC-Net(ours)*	2.70	1.18	2.12	1.04	3.46	1.60	3.00	1.35
(a) example 1
(b) example 2
(c) example 3
Figure 3: Visualization of our method on the VIGOR [42] dataset with localization confidence probability indicated in the legend. (d) demonstrates a failure case of our method when dealing with scenes that are radially monotonous and repetitive-uninformative.
(d) example 4
paper and used the published results where applicable. We also re-train and evaluate CCVPE with our corrected labels to ensure a fairer comparison. We conduct a comprehensive evaluation with different levels of noise in the orientation prior to ground images, including  $0^{\circ}$ ,  $\pm 20^{\circ}$ ,  $\pm 45^{\circ}$ .
As shown in Table 1, our method outperforms all previous methods in both same-area and cross-area settings in terms of the mean localization error metric, which suggests that our method can cope with more challenging scenarios, as shown in Figure 3(a). The mean localization error is reduced by  $21.3\%$  and  $32.4\%$  respectively. The previously best-performing method [35] shows a notable performance gap between same-area and cross-area settings, while our method significantly narrows this gap. This suggests that our method possesses superior generalization capabilities. We further demonstrate our superior generalization ability across datasets in Section 4.4.
When prior information exhibits varying degrees of noise, our method still outperforms CCVPE [35] and provides a more accurate estimation of the orientation. Moreover, our network generates confidence probabilities for localization results, allowing us to filter out potential misestimations. As shown in Figure 3 (c), when the scene contains a symmetric layout like zebra crossings and noise exceeds  $90^{\circ}$ , the homography estimation may incorrectly align the ground image with the satellite map. However, by rotating the BEV image four times and selecting the prediction with the highest confidence probability, we can obtain the correct result as in Figure 3 (b). Despite the mislocalized point having a similar appearance to the ground observation, its probability output is significantly smaller than the localization confidence probability from the correct location. This property is crucial for safety-critical applications like autonomous driving.
Table 2: Location and orientation estimation error on KITTI [8] dataset. Best in bold. Long. and Orien. are abbreviations for Longitudinal and Orientation, respectively.
Method	Area	↓Location(m)	↑Lateral(%)	↑Long.(%)	↓Orien.(°)	↑Orien.(%)
Mean	Median	R@1m	R@5m	R@1m	R@5m	Mean	Median	R@1°	R@5°
LM [24]	Same	12.08	11.42	35.54	80.36	5.22	26.13	3.72	2.83	19.64	71.72
SliceMatch [12]	Same	7.96	4.39	49.09	98.52	15.19	57.35	4.12	3.65	13.41	64.17
Boosting [26]	Same	-	-	76.44	98.89	23.54	62.18	-	-	99.10	100.00
CCVPE [35]	Same	1.22	0.62	97.35	99.71	77.13	97.16	0.67	0.54	77.39	99.95
HC-Net(Ours)	Same	0.80	0.50	99.01	99.73	92.20	99.25	0.45	0.33	91.35	99.84
LM [24]	Cross	12.58	12.11	27.82	72.89	5.75	26.48	3.95	3.03	18.42	71.00
SliceMatch [12]	Cross	13.50	9.77	32.43	76.44	8.30	35.57	4.20	6.61	46.82	46.82
Boosting [26]	Cross	-	-	57.72	91.16	14.15	45.00	-	-	98.98	100.00
CCVPE [35]	Cross	9.16	3.33	44.06	90.23	23.08	64.31	1.55	0.84	57.72	96.19
HC-Net(Ours)	Cross	8.47	4.57	75.00	97.76	58.93	76.46	3.22	1.63	33.58	83.78
(a) Projection of KITTI
Figure 4: (a) depicts the projection of the frontal perspective into a bird's-eye view in the KITTI dataset [8], accompanied by the corresponding satellite imagery. (b) visualizes our method's localization results on KITTI [8]: the top row shows aligned ground and satellite images using our estimated homography matrix; the bottom row presents original satellite images for reference.
(b) Localization Results
KITTI dataset On the KITTI dataset, we compare our method with the state-of-the-art LM [24], SliceMatch [12], Boosting [26] and CCVPE[35]. Similar to the evaluation on VIGOR, we directly use the results provided by CCVPE [35] or the corresponding paper. We utilize an increased number of iteration steps in response to the limited coverage of ground images from KITTI [8] within satellite images. The results are shown in Table 2. When a  $\pm 10^{\circ}$  orientation prior is considered in both Training and Test1, our method has a  $34.4\%$  lower mean error for localization than CCVPE [35]. Figure 4 presents the visualization results on the KITTI dataset. Boosting [26] achieves exceptional accuracy in estimating orientation. This may be attributed to their utilization of a two-stage approach, where the first stage is dedicated to orientation estimation.
4.4 Cross-Dataset Generalization
The CVUSA dataset [39] is commonly used for cross-view localization, but it only provides retrieval labels and cannot be used for training precise localization models. Despite the panoramas in CVUSA being cropped and having a non-standard aspect ratio, our spherical transform method successfully projects them to a bird's-eye view by completing it to the correct proportion, as shown in Figure 5. We use images from CVUSA in different cities to test our model's generalization ability, which is trained on VIGOR. The alignment results in Figure 5 demonstrate that our model has strong potential and can align significantly different BEV images with their corresponding satellite images for precise localization, even in unfamiliar cities without retraining.
Table 3: Ablation Study on Pseudo-Siamese Backbone and Homography Estimation Module.
Experiment	Mean Error	Median Error
Non-shared weights	2.65 m	1.17 m
Shared weights	3.36 m	1.36 m
SuperGlue [22]	13.06 m	3.76 m
LoFTR [28]	28.85 m	4.89 m
Table 4: Comparison of computational efficiency between our method and CCVPE [35].
CCVPE [35]	HC-Net(ours)
Parameters	57.40 M	11.21 M
Memory Usage	4730 MiB	1900 MiB
Inference Time	33 ms	31 ms
Mean Error	3.37 m	2.65 m
Figure 5: Three examples in CVUSA[39]. The top of each example shows the completed panorama. The bottom-left shows the bird's-eye view obtained using the spherical transform. The bottom-right shows the alignment result of our method between the BEV and the satellite image.
Pseudo-Siamese Backbone We employ a Pseudo-Siamese backbone with non-shared weights for processing images from two different sources. To demonstrate the effectiveness of this structure, we train two models with shared and non-shared weights. The results in Table 3 show that applying the Pseudo-Siamese backbone leads to a mean localization error reduction of  $0.71\mathrm{m}$ .
Homography Estimation Module We also conduct ablation experiments to demonstrate the effectiveness of our homography estimation module. We replace the homography estimation module with feature-based methods SuperGlue [22] and LoFTR [28], utilizing RANSAC-based functions from OpenCV to compute homography matrix from matched feature points. The experimental results are presented in Table 3. The fact that the feature-based methods have low median localization errors demonstrates the effectiveness of transforming localization into 2D alignment. However, the significantly lower mean errors achieved by our proposed deep homography estimator indicate its superiority in achieving more accurate localization.
4.6 Computational Efficiency Analysis
We assess our method's computational efficiency against the state-of-the-art CCVPE [35]. Table 4 compares model parameters, inference memory, per-frame inference time, and mean localization error on the VIGOR dataset using a 12th Gen Intel(R) Core(TM) i5-12490F processor, 16GB memory, and an NVIDIA RTX 3050 GPU. Our method achieves higher accuracy, faster speed, and lower memory usage, demonstrating its computational efficiency.
In this study, we introduce HC-Net, an end-to-end network designed for fine-grained cross-view geo-localization. The network processes spherical-transformed ground images and GPS-tagged satellite images as inputs and generates the homography matrix between them, along with the precise GPS location and orientation of the ground camera. Compared to the previous state-of-the-art, HC-Net demonstrates a reduction in mean localization error by  $21.3\%$  and  $32.4\%$  on the same-area and cross-area splits of the VIGOR dataset, respectively.
This work was supported by NSFC 62088101 Autonomous Intelligent Unmanned Systems, in part by TI 2030-Major Projects 2021ZD0201403.
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Supplementary Material for Fine-Grained Cross-View Geo-Localization Using a Correlation-Aware Homography Estimator
In this supplementary material, we provide detailed information on projecting ground images to bird's-eye view for VIGOR [42] and KITTI [8] datasets in Section A, as well as the computation details for converting GPS labels to pixel coordinates used during training in Section B. We also discuss the potential for future research based on our method in Section C and broader societal and industry impact of our approach in Section D.
A Projection Details
A.1 Details for Spherical Transform
In the main paper, we provide an overview of the derivation process of the Spherical Transform and the final result in Equation 1. Here we will provide detailed information on the process for the projection. We will use the expressions defined in Section 3.1. To clarify the presentation, we also provide the schematic diagram of the projection process, as shown in Figure 6.
In the camera coordinates, the conversion formula between the Cartesian coordinates  $P = (x_{1},y_{1},z_{1})$  and the spherical coordinates  $(\phi ,\theta)$  is given by:
$$
\left\{ \begin{array}{l l} \phi = \arctan 2 \left(y _ {1}, x _ {1}\right) & \in [ - \pi , \pi ], \\ \theta = \arctan 2 \left(z _ {1}, \sqrt {x _ {1} ^ {2} + y _ {1} ^ {2}}\right) & \in [ - \pi / 2, \pi / 2 ]. \end{array} \right. \tag {6}
$$
The equirectangular projection is used to project spherical coordinates onto a plane, which is the display format for panoramic images. The conversion formula between the spherical coordinates  $(\phi, \theta)$  and the Normalised Equirectangular coordinates  $P' = (x_2, y_2)$  is given by:
$$
\left\{ \begin{array}{l l} x _ {2} = \frac {- \phi}{\pi} & \in [ - 1, 1 ], \\ y _ {2} = \frac {\theta}{\pi / 2} & \in [ - 1, 1 ]. \end{array} \right. \tag {7}
$$
The negative sign in the  $x_{2}$  expression is due to the fact that panoramic images display the scene as viewed from the camera optical center to the outside. Therefore, when  $\phi$  is positive, it corresponds to the negative half-plane of the Equirectangular plane in terms of  $x_{2}$ . The mapping between pixel coordinates  $(u_{p}, v_{p})$  and Normalised Equirectangular coordinates  $(x_{2}, y_{2})$  on the panorama is established as:
$$
\left\{ \begin{array}{l l} u _ {p} = (x _ {2} + 1) W _ {p} / 2 & \in [ 0, W _ {p} ], \\ v _ {p} = (- y _ {2} + 1) H _ {p} / 2 & \in [ 0, H _ {p} ]. \end{array} \right. \tag {8}
$$
To obtain the corresponding bird's-eye view of the panorama, we place a tangent plane at the south pole of the spherical imaging plane as a new imaging plane, as shown in Figure 6 (c). The focal length of the BEV in the imaging process is  $f = 0.5W_{b} / \tan (fov)$ . The camera coordinate system coordinates  $(x_{1},y_{1},z_{1})$  corresponding to a pixel  $(u_b,v_b)$  on the bird's-eye view (BEV) imaging plane are determined as:
$$
\left\{ \begin{array}{l} x _ {1} = - v _ {b} + H _ {b} / 2, \\ y _ {1} = - u _ {b} + W _ {b} / 2, \\ z _ {1} = - f. \end{array} \right. \tag {9}
$$
By substituting Equation 9 into Equation 6, and then substituting the result into Equation 7, we can obtain the Normalised Equirectangular coordinates  $(x_{2},y_{2})$  for a pixel on the bird's-eye view (BEV) image plane. Finally, by substituting the Normalised Equirectangular coordinates into Equation 8, we can obtain the pixel coordinates  $(u_p,v_p)$  on the panoramic image. This leads to the Spherical Transform mapping formula in Equation 1.
(a) Normalised Equirectangular coordinates
(b) Spherical coordinates
(c) BEV Project
(d) BEV coordinates
Figure 6: Illustration of panorama imaging model and the spherical transform mechanism for projecting pancreas to a bird's-eye view.
We test the effectiveness of the bird's-eye view projection with different field of view (fov) parameters, as shown in Figure 7. As seen in (d), when  $fov = 85^{\circ}$ , the field of view of the bird's-eye view image is similar to that of the corresponding satellite image, and there are few invalid parts in the image. Therefore, we use this parameter for all of our experiments.
The spherical transform projects the ground image to a bird's eye view, while the polar transform [25] or projective transform [24] projects the satellite image to the perspective of the ground image (aligning with a panoramic view). There is a growing recognition and research on the role of Bird's Eye View (BEV) representation in localization, navigation, and related tasks.
The advantage of projecting to a bird's eye view, as compared to projecting the satellite image to the perspective of the ground image, lies in its more intuitive nature for the localization task and the need for only one projection at the start of the localization process. On the other hand, the latter approach requires selecting projection points (initial pose or candidate poses) for projecting the satellite image, and if these points are not accurately chosen, the resulting projection may not align well with the ground image. Additionally, for more accurate localization, the latter approach requires multiple applications of the projection method from different positions, which could potentially make it less real-time compared to our method.
A.2 Details for Projecting Front View Images in KITTI
In the KITTI dataset [8], the ground is a front view image, so the process of projecting it onto a bird's-eye view is completely different from that of the panorama in VIGOR [42]. We illustrate the mechanism of this projection process in Figure 8, where  $AB$  represents the height of the frontal view image, and  $AC$  represents the desired height of the bird's-eye view image.
We define the size of the front view image as  $H_{f} \times W_{f}$ , and the target size of the bird's-eye view image as  $H_{b} \times W_{b}$ . Using a method similar to that described in Section 3.1, we imagine a bird's-eye projection plane parallel to the ground and adjacent to the front view image. Then we connect the camera optical center with each pixel on the bird's-eye view image to obtain the corresponding pixel coordinates on the front view image.
In Figure 8 (b),  $fov$  represents the field of view of the front view image ( $\angle ADC$ ),  $f$  represents the focal length  $DE$ ,  $h$  represents the distance from the camera optical center  $D$  to the bird's-eye
(a) Ground Image
(b) Satellite Image
(c)  $\mathrm{fov} = 75^{\circ}$
(d)  $\mathsf{fov} = 85^{\circ}$
(e)  $\mathsf{fov} = 88^{\circ}$
(a) Front-View and BEV Image
Figure 7: The examples of bird's-eye view images obtained through the Spherical Transform at different field of view (fov) parameters.
(b) Geometric Mechanism
Figure 8: Illustration of the projection mechanism for KITTI [8].
projection plane, and  $\theta$  represents the angle between the camera imaging plane  $AB$  and the vertical direction. To facilitate the subsequent derivation, we also define  $\delta$  to represent  $\angle FAD$ ,  $\varphi$  to represent  $\angle BAD$ ,  $\theta_{2}$  to represent  $\angle AGD$ ,  $\theta_{3}$  to represent  $\angle AHG$ ,  $f'$  to represent  $FA$ , and  $l_{0}$  to represent  $AD$ . According to [8],  $fov = 17.5^{\circ}$  in the KITTI dataset. These variables are calculated as follows:
$$
\left\{ \begin{array}{l} f = \frac {H _ {f}}{2} / \tan (f o v) \\ \varphi = \frac {\pi}{2} - f o v \\ \delta = \frac {\pi}{2} - (\varphi - \theta) \\ l _ {0} = \sqrt {f ^ {2} + \left(\frac {H _ {f}}{2}\right) ^ {2}} \\ h = l _ {0} \sin \delta \\ f ^ {\prime} = l _ {0} \cos \delta . \end{array} \right. \tag {10}
$$
We denote the pixel coordinates on the bird's-eye view imaging plane as  $(u_b, v_b)$ . The values of  $\theta_2$  and  $\theta_3$  can be calculated using the arctangent formula and the properties of exterior angle as follows:
$$
\left\{ \begin{array}{l} \theta_ {2} = \arctan \left(h / \left(f ^ {\prime} + H _ {b} - v _ {b}\right)\right) \\ \theta_ {3} = \frac {\pi}{2} + \theta - \theta_ {2}. \end{array} \right. \tag {11}
$$
Then, based on the sine and similarity triangle theorems, we can calculate the corresponding pixel coordinates  $(u_{f},v_{f})$  on the front view image for each pixel on the bird's-eye view image. The calculation is as follows:
$$
\left\{ \begin{array}{c} \frac {H _ {f} - v _ {f}}{H _ {b} - v _ {b}} = \frac {\sin \theta_ {2}}{\sin \theta_ {3}} \\ \frac {W _ {f} / 2 - u _ {f}}{W _ {b} / 2 - u _ {b}} = \frac {f ^ {\prime} + \left(H _ {f} - v _ {f}\right) \sin \theta}{f ^ {\prime} + H _ {b} - v _ {b}} \end{array} \right. \tag {12}
$$
(a) Front-View Image
(c)  $\theta = 0^{\circ}$
Figure 9: The examples of bird's-eye view images obtained from front view image at different  $\theta$  parameters.
(d)  $\theta = 0.8^{\circ}$
(e)  $\theta = 1.6^{\circ}$
(f)  $\theta = 0.8^{\circ}$  and Rotate
$$
\Rightarrow \left\{ \begin{array}{l} v _ {f} = H _ {f} - \frac {\sin \theta_ {2}}{\sin \theta_ {3}} \left(H _ {b} - v _ {b}\right) \\ u _ {f} = \frac {W _ {f}}{2} - \frac {f ^ {\prime} + \left(H _ {f} - v _ {f}\right) \sin \theta}{f ^ {\prime} + H _ {b} - v _ {b}} \left(\frac {W _ {b}}{2} - u _ {b}\right) \end{array} \right. \tag {13}
$$
We introduce the angle  $\theta$  because we found that in practical testing, the imaging plane of the ground camera may not be perfectly vertical, as shown in Figure 9. Based on experimental results, we use  $\theta = 0.8^{\circ}$  for all experiments. During the actual projection process, we set the target size of the bird's-eye view image to be  $H_{b} \times W_{b} = (6 * W_{f}) \times (6 * W_{f})$ , which corresponds to a field of view of approximately  $40m \times 40m$ . After the first projection calculation, we directly applied the homography matrix  $H$  obtained from the above projection process to obtain the BEV. Then, we used scaling and rotation homography matrices  $H_{1}$  and  $H_{2}$  to obtain the final BEV with the specified size.
$$
H _ {1} = \left[ \begin{array}{c c c} \operatorname {s c a l e} & 0 & 0 \\ 0 & \operatorname {s c a l e} & 0 \\ 0 & 0 & 1 \end{array} \right], H _ {2} = \left[ \begin{array}{c c c} \cos \gamma & - \sin \gamma & u _ {c} (1 - \cos \gamma) + v _ {c} \sin \gamma \\ \sin \gamma & \cos \gamma & v _ {c} (1 - \cos \gamma) - u _ {c} \sin \gamma \\ 0 & 0 & 1 \end{array} \right] \tag {14}
$$
Here, scale represents the ratio of the current  $H_{b} \times W_{b}$  to the desired input size of the network  $(512 \times 512)$ .  $\gamma$  represents the yaw angle of the ground camera with noise, and  $(u_{c}, v_{c})$  represent the pixel coordinates of the center of the final BEV image.
B Label Correction
B.1 Label Correction in VIGOR Dataset