The Py-Net architecture is a simple yet powerful encoder–decoder network (Figure 1) comprising three main components: the image encoder, seg decoder, and vote decoder. This design enables the generation of accurate landmark segmentation maps and voting maps for 2D-to-3D correspondence modeling. During training, Py-Net uses a landmark segmentation method based on patch labeling to generate segmentation coordinates for all pixels that correspond to patches surrounding the landmarks. Additionally, each pixel within the landmark patch predicts a 2D directional vector pointing toward the landmark, thereby enabling reliable and precise coordinate voting.

The main encoder comprises stacked Py-layers of different sizes. The Py-layer is a coding layer composed of 1 × 1 and pyramid convolutions. 1 × 1 convolution kernels perform dimensionality reduction and expansion, whereas pyramid convolutions use convolutions of multiple sizes to process the input image. It contains multiple levels of feature extraction layers, each with convolutions of different sizes and depths. Thus, the ability of the network to extract scene information enhances. Coordinate attention is added at the end of the Py-layer for feature correction. The output of the encoder is then fed into atrous spatial pyramid pooling, where features are sampled in parallel using dilated convolutions with different sampling rates. Finally, seg decoder and vote decoder branches produce landmark segmentation and voting maps, respectively.

In both decoder branches, depth over-parameterized convolution (DOConv) module was used as the convolution module. This design over-parameterizes the decoder, increasing the number of learnable parameters while maintaining the original computational complexity and thus enhancing the quality of scene image features.

Compared with standard convolution, pyramidal convolution offers a more robust capability to process input images. It uses multiple convolution sizes containing multiple levels of feature extraction layers, each with convolutions of different sizes and depths; thus, it captures rich details of the scene. Standard convolution contains only one convolution size, and the convolutional depth us equal to the depth of the feature map. In contrast, the convolution size increases and depth decreases in pyramidal convolution with increasing feature extraction levels. In outdoor scenes, the occlusion of building parts may occur, making it difficult for a single type of convolution to effectively capture details in the scene image. However, pyramidal convolution can use convolution kernels with different receptive fields to capture fine features, thereby improving the camera relocalization accuracy.

Figure 2 shows that the number of residual blocks in the backbone network increases via pyramidal convolution, enabling the network to process images using multiple convolution sizes and multiple feature extraction levels. Thus, the ability of the network to extract scene information is enhanced. When optimizing the encoder of the voting segmentation network, the encoding function is mainly completed by 3 × 3 convolution kernels because 1 × 1 convolution kernels in each module serve to reduce and increase the dimensionality. Therefore, 3 × 3 convolutions are first improved in each module of the backbone encoder. Using pyramidal convolution, the network model can process input images using multiple sizes of convolution containing multiple levels of feature extraction layers, each with convolutions of different sizes and depths. Thus, the ability of the network model to extract scene information is slightly enhanced without introducing additional parameters and computational overhead.

For the input feature map F M i ∈ R C i × H × W , the n-layer pyramidal convolution has n convolutional kernel sizes K 1 2 , K 2 2 , K 3 2 , … , K n 2 and n convolutional kernel depths F M i , F M i K 2 2 K 1 2 , F M i K 3 2 K 1 2 , … , F M i K n 2 K 1 2 ; Output Features The F M o ∈ R C o × H × W is stitched together from the outputs of n convolutional kernels, where the size of C o is given by Equation 1: C o = F M o 1 + F M o 2 + F M o 3 + … F M o n (1)

Here F M o i , i ∈ 1 … n is the output feature dimension corresponding to n convolutional kernels.

The number of parameters of the pyramidal convolution is given by Equation 2: p a r a m e t e r s = K n 2 ∙ F M i K n 2 K 1 2 ∙ F M 0 n + K 3 2 ∙ F M i K 3 2 K 1 2 ∙ F M 03 + K 2 2 ∙ F M i K 2 2 K 1 2 ∙ F M 02 + K 1 2 ∙ F M i ∙ F M 01 (2)

Therefore, if the number of output feature maps is equal for each layer of the pyramidal convolution, then its parameters and computational costs will be evenly distributed across each level of the pyramid.

As shown in Figure 3, standard convolution can only have a single size convolution kernel, with the same kernel depth as the input features. Moreover, the number of computational parameters generated by a single convolution kernel is given by Equation 3: p a r a m e t e r s = K 1 2 ∙ F M i ∙ F M 01 . (3)

When multiple standard convolution kernels of different sizes are used to process input features, the as-generated computational and parameter quantities are greater than those generated via pyramidal convolution. This is because the input channels of these kernels must match the input features.

Attention mechanism is widely used in computer vision to enable network models to focus on relevant information. Some operations in convolutional neural networks, such as convolution, pooling, and fully connected layers, only consider nonself-desired clues. Contrarily, attention mechanism is purposeful and can explicitly model cues that align with its own intentions. As convolutions are conducted within local windows in the main encoder, their representation capability for global features is relatively weak. To address the issue of weak antiinterference capability in outdoor scenes with SCR networks, an attention mechanism is embedded in the encoder. This enables the network model to completely leverage global and local information. Consequently, the model can prioritize the areas of interest in an image, increase the corresponding weights of those areas, and ultimately highlight useful features while suppressing or ignoring irrelevant ones. These factors further enhance the localization accuracy.

Coordinate attention is a lightweight attention mechanism that considers channels and spaces in parallel (Figure 4). With the input of a feature X, X ∈ R H × W × C , the coordinated attention mechanism first pools along the height and width of the feature using two pooling kernels of size H × 1 and 1 × W, respectively. The output of height h and channel c can be expressed as follows (Equation 4): z c h h = 1 W ∑ 0 ≤ i ≤ W x c h , i (4)

The output of width w and the cth channel can be expressed as Equation 5: z c w w = 1 H ∑ 0 ≤ j ≤ H x c j , w (5) where z c h ∈ R C × H × 1 and z c w ∈ R C × 1 × W .

After the two pooling operations, the two embedding features z c h and z c w can be stitched together. Then, the stitched features are reduced by 1 × 1 convolution and input to the sigmoid function (Equation 6): f = δ F 1 z h , z w (6) where f ∈ R C r × 1 × H + W , F 1 denotes the convolution, and δ denotes the sigmoid activation function. Then, the feature f is split in the spatial dimension using two 1 × 1 convolutions to obtain two feature maps. After transforming these maps, the attention vector is obtained as follows (Equations 7–8): g h = σ F h f h (7) g w = σ F w f w (8) where   F w and F h denote the 1 × 1 convolutional transformation and f w and f h denote the split features f w ∈ R C r × w and f h ∈ R C r × H , respectively. Eventually, the input features can be corrected using the two attention vectors (Equation 9):

y c i , j = x c i , j × g c h i × g c w j (9)

CA has higher computational efficiency and less computational overhead in the spatial dimension than other attention modules.

In standard convolution, all kernels in a convolutional layer are convolved with the input image. Standard convolution emphasizes the spatial relation of pixels and considers it as channels but at the expense of relatively high computational complexity. In contrast, depthwise convolution assigns each kernel to a specific input channel. Each channel of the input image is convolved with a dedicated kernel. Depthwise convolution focuses more on the features represented in the depth of an image and has relatively lower computational complexity.

Deep over-parameterized convolution (Cao et al., 2022) is an extension of standard convolution, in which an additional depthwise convolution is incorporated. This additional convolutional structure increases the number of trainable model parameters and enhances the learning capacity of the model. The quality of scene features extracted by this model is also consequently improved.

The feature quality in the SCR network deteriorates due to the influence of repetitive structures and low-texture objects in scene images. To address this issue, standard convolutions must be substituted with deep over-parameterized convolutions. This replacement will enhance the quality of features extracted by the network and improve the overall nonlinear capability of the network, making it more robust. Figure 5 shows the structure of deep over-parameterized convolution.

Deep over-parameterized convolution comprises standard and depthwise convolutions. First, the convolutional kernel D from the standard convolution and the depthwise convolution kernel W are used to compute a new convolutional kernel W . Then, W is convolved with the feature P in the same manner as the standard convolutional kernel (Equation 10): O = D T ○ W × p (10)

As shown in Figure 6, W represents the convolutional kernels of standard and depthwise convolutions and P represents the feature. W in standard convolution is a 3D tensor, w ∈ R c o u t × M × N × C i n . P is a 2D tensor, p ∈ R M × N × C i n . where C o u t and C i n denote the output and input channel dimensions of the feature and M × N denotes the spatial dimensions of the feature. Each convolutional kernel on the Cout dimension of W performs a dot product operation with P , yielding the output O (Equation 11): O C o u t = ∑ i M × N × C i n w C o u t i p i . (11)

The dimensions of O correspond to the output channel dimension of the convolutional kernel.

In depthwise convolution, each input channel of P undergoes a dot product operation with D m u l channels of the convolutional kernel. The dimension of each input channel of P is transformed from M × N to D m u l , where D m u l is the depth multiplier. The final output O is obtained accordingly (Equation 12): O d m u l C i n = ∑ i M × N w i d m u l c i n p i c i n , (12) where the depthwise convolutional kernel W is a 3D tensor, w ∈ R M × N × D m u l × C i n .

DOConv and standard convolution have the same receptive field. For a feature input P , the output feature dimensions obtained via DOConv and output feature dimensions processed via standard convolution are identical. The linear transformation of standard convolution can be parameterized using C o u t × M × N × C i n Cout training weights. The linear transformation of DOConv can be parameterized using the training weights of two convolutional kernels. Only when D m u l ≥ M × N , the newly combined convolutional kernel W ʹ can exhibit the same linear transformation as the standard convolution kernel W when formed via combination. DOConv introduces over-parameterization to the network, thereby increasing the number of learnable parameters while maintaining the original computational complexity and enhancing the feature quality of scene images.

The scene is first reconstructed in 3D using Py-Net. The resulting 3D surface is divided into 3D image patches using the center point of each patch as a 3D landmark, q 1 , … , q n ∈ R 3 . Segmentation maps S ∈ Z H × W and voting maps d ∈ R H × W × 2 are obtained by projecting 3D image patches and 3D landmarks onto 2D images. To generate a segmentation map, values are assigned to each pixel point p i u i , v i by determining its projected 3D image patch coordinate on the 2D image. A value of 0 is assigned to the pixel point coordinate p i u i , v i if the associated area is not covered by the projected 3D surface, indicating that they did not influence the positioning. The landmark voting map is generated by first projecting the landmark q j on a 3D surface to a 2D plane to obtain 2D coordinates (Equation 13): I j = P q j , K , C ∈ R 2 , (13) where K is the camera internal reference matrix and C is the camera pose parameter. Each pixel belonging to an image patch containing the coordinate q j is responsible for projecting a 2D direction vector d i ∈ R 2 pointing to q j . By unitizing this coordinate wth p i , the unit vector is obtained (Equation 14): d i = I j − p i I j − p i 2 (14)

This vector is used to represent the orientation of the landmarks on the 2D plane. Supervised training of the voting segmentation network is possible by using the obtained segmentation map and voting map as training truths. This in turn establishes the relationship from 2D to 3D, thus enabling camera relocation.

The Vote Segmentation Network uses a prototype-based ternary loss function and a negative sample mining strategy to supervise the training of the network. Training the network in this way requires maintaining and updating a set of learnable class prototype embeddings P. Where each class prototype embedding represents a class and P j represents the class prototype embedding of the jth class. Therefore, the pixel embeddings belonging to the jth category should be as close as possible to, and as far away as possible from, the class prototype embeddings of other categories.

Pixel embeddings can be obtained by voting on the segmentation branches of the segmentation network and the class of prototype embeddings P. Pixel embeddings can form a pixel-by-pixel embedding graph E. The prototype-based ternary loss function first first L2 normalizes each pixel embedding in the embedding graph E, and then minimizes the error between each pixel embedding E i and the prototype embedding P i (Equation 15): L s e g = ∑ a l l   i max   0 , m + s i m E i , P i − − s i m E i , P i + (15) where sim a , b = a T b b ∙ b is used to measure the cosine similarity between the pixel embedding and prototype-like embedding. P i + denotes the true value of prototype-like embedding corresponding to pixel i and P i − denotes the true value of prototype-like embedding unrelated to pixel i . m denotes the boundary of the loss function.

The voting decoder of the voting segmentation network is then used to determine the landmark’s cast position in the 2D image.The voting decoder is used to determine the casting position of landmarks in a two-dimensional image. Inputting an image, the voting decoder outputs a voting map d. For each pixel i in the input, it generates a two-dimensional direction vector d i , which points to the two-dimensional position of the landmark. The voting decoder is supervised trained under the voting loss function Equation 16: L v o t e i = ∑ a l l   i 1 S i ≠ 0 d ^ i − d i (16)

Where one denotes the indicator function and d ^ i and d i denote the direction vectors predicted for pixel i and their direction vector truth values, respectively. The overall loss function of VS-Net is represented as in Equation 17: L o v e r a l l = L s e g i + λ L v o t e i (17) where λ denotes the weight of voting loss.

The improvement in the proposed model was validated using the Cambridge Landmarks (Kendall Alex et al., 2015) dataset containing five outdoor landmarks scenes: Great Court, Kings College, Old Hospital, Shop Facade, and St. Mary’s Church. The dataset was more complex than indoor scenes; therefore, it better demonstrated the robustness of the model, as exterior environments undergo drastic environmental changes. For instance, the outdoor camera moves faster than the indoor camera, which may result in blurry images.

As the camera relocalization error in outdoor environments is significant than that in indoor settings, using the a 1-cm distance error and 1° angle error as well as 2-cm distance error and 2° angle error are the evaluation metrics is not ideal. Therefore, the median distance error within 5 cm and median angle error of 5° were used as the evaluation metrics instead. We also report the percentage of high-precision localization points with a distance error within 5 cm and an angle error within 5°

The performance of Py-Net was compared with existing visual localization systems on the Cambridge Landmarks dataset. Detailed results are shown in Table 1, where the primary indicators for evaluating the accuracy of camera relocalization are compared: translation error (m) and rotation error (°). Experimental results indicate that Py-Net considerably outperforms the existing methods. The translation error decreased by 40% using the SCR method than DSAC. This indicates that the voting segmentation network can considerably reduce the number of outliers in the scene and improve the relocalization accuracy. Py-Net outperformed VS-Net, which also employs a voting segmentation architecture, in the majority of scenes, particularly in the Great Court scene, where the translation and rotation errors decreased by 14% and 50%, respectively. This suggests that using Py-layer to enhance the backbone encoder, the network can learn multiscale features. Thus, the ability of backbone encoder to extract scene information enhances, resulting in better performance.

Figure 7 compares the scene coordinate prediction between VS-Net and Py-Net. In outdoor scenes, the enlarged scene scale results in a substantial number of invalid scene coordinate points. This affects the prediction accuracy and limits the scene information available for camera relocation.

However, as shown in Figure 7A, Py-Net provides higher amounts of usable information within the predicted scene coordinate maps. In the Great Court scene, Py-Net generates a scene coordinate map with an increased level of scene information (Figure 7A). In the Kings College scene, the scene coordinate map predicted by Py-Net contains noticeably increased usable information (Figure 7B). The Shop Facade scene contains a substantial amount of usable scene information, with fewer background pixels representing the sky (Figure 7C). This indicates that incorporating depthwise separable convolutional modules into the scene coordinate decoder effectively enriches the amount of scene information, thereby enhancing the information representation capability of the network.

Figure 8 shows the positioning trajectories of the improved Py-Net on the Cambridge Landmark dataset. The figure compares the distance errors between VS-Net and Py-Net. Panels (a), (c), and (e) show the positioning trajectories of VS-Net, and panels (b), (d), and (f) show the positioning trajectories of the improved Py-Net. When zooming in on the positioning errors in the Great Court scene, a certain degree of reduction in distance errors can be observed. In the Kings College scene, Py-Net reduces the distance errors of marked points. As the test samples of the Shop Facade scene were limited, only a few positioning points with reduced errors are shown in the figure.

Figure 9 shows that in multiple scenes, Py-Net outperforms existing methods in the proportion of high-precision points with a distance error of less than 5 cm and an angle error of less than 5°. The increase in the proportion of high-precision localization points indicates that the number of invalid localization points in the scene has decreased, meaning that outliers have been filtered out.

The Py-layer and depthwise separable convolution were introduced to enhance the performances of the main encoder and decoder, respectively. As a result, the size of Py-Net considerably compared with that of VS-Net (Table 2). Specifically, the model size decreased from 236 to 170 MB, and the parameter count was only 68.15% of the original. This improvement considerably enhanced the applicability of the model on devices with limited storage space.

Ablation experiments were conducted to demonstrate the effectiveness of the improvements proposed herein. By introducing the Py-layer and coordinated attention mechanism, Py-Net can effectively extract outdoor scene information and filter out invalid localization points in the scene. Thus, the model performance is improved across various scenes. In the Great Court scene, the distance error decreased by 16%, whereas in the Shop Facade and St. Mary’s Church scenes, the angle error decreased by 33%. As shown in Table 3, after incorporating the depthwise separable convolution module, the angle error of the network considerably increased across multiple scenes. This indicates that Py-Net can retain more scene information, thus achieving more accurate estimation results.