Our method predicts for each input image X_i an instance mask Y ^ i ∈ [ 0 , n i ] H × W , matching the ground truth segmentation Y_i as accurately as possible.

As discussed in the introduction, conventional computer vision approaches for instance segmentation tend to struggle with the specific challenges of ring segmentation. We therefore build on recent work by Gillert et al. (2023) and adopt a tailored approach that addresses the task iteratively, i.e., ring by ring. Our model processes the input image radially from pith to bark and regresses the distance to the next ring from the previous ring.

We present an overview of our method, named INBD-R, as pseudo-code in Algorithm 1 . The main components are as follows. A trainable semantic segmentation model processes the downscaled input image and is followed by the iterative model. The semantic segmentation model returns the location of the pith from which the iterative process starts, as well as a first estimation of the ring width. Next, starting from the position of the pith, the position of the next ring is iteratively predicted. Each iteration includes the following steps:

The iterative process, visualized in Figure 2 , ends when the bark or the edge/outline of the xylem tissue is reached, which is detected via the missing ring width estimate. After the iterative process, the ring positions are converted back to instance masks by drawing their polygons from outside to inside. To gain prediction at full resolution, the continuous polygon points are upscaled. We first train the semantic segmentation model on the data. In the second step, we use the trained model to preprocess the input data to prepare it for the training of the radial regression model. Third, we train the radial regression model using our fast iterative unrolling training procedure. We describe each part of INBD-R in further detail in the following sections.

Task: The semantic segmentation model aims to find the position of the pith and produce a first estimate of each ring’s width, as visualized in Figure 3 . Following Gillert et al. (2023), we achieve this through a semantic segmentation step, where each pixel is classified within the three classes:

The boundary class itself contains most of the target information. However, the boundary prediction as a final result is not a viable option since a one-pixel-wide boundary prediction cannot be easily converted to instance masks. This is because of the insufficient robustness to wrong predictions, struggles with hard-to-detect boundaries on small scales, and the possibility of connecting wedging rings.

We use a convolutional neural network with sigmoid activation in the final layer to predict one binary mask for each of the four classes. To reduce the computational load and to increase the spatial context, the semantic segmentation model operates on a ×4 downscaled version of the input image.

Architecture and training: We employ a UNet (Ronneberger et al., 2015) with a Res2Net (Gao et al., 2021) backbone and train it with a binary dice loss (Sudre et al., 2017) defined in Equations 1, 2. This loss is applied to each mask individually. Therefore, y represents the ground truth binary mask and y ^ the predicted binary mask.

The binary dice losses are weighted with α ₀ = 0.1, α ₁ = 0.01, and α ₂ = 1. Binary segmentation per class is superior to multi-class cross-entropy training because it prevents the model from exclusively deciding between the background and the pith since they can look similar if the pre-processing of the xylem damaged the pith tissue, as seen in Figure 4 .

Initial boundary position: We convert the predicted binary mask of the pith into the initial boundary position. This position is defined by the center point of the binary mask and equally spaced outer edge points of the mask of shape ℝ^{2× M} . M represents the adaptive angular resolution, which grows proportionally to the distance to the center, ensuring similar spacing between boundary points across ring positions. The center point is reused for all boundaries. The actual value M for the next ring is set to 2π times the average distance between the center and the current ring boundary.

Polar transformation: The downsampled input image is converted into an image in polar space (polar image) where the upper row of pixels corresponds to the current boundary. The current boundary equals the initial boundary in the first iteration and is updated with the next predicted boundary in each iteration step. The rough estimate for the next ring width P ∈ ℝ is computed based on the current ring and the binary mask for the boundary class of the semantic segmentation model. P is obtained by taking 1.5 × the 95th percentile of the distances from the current ring to the next boundary pixel in a radial direction evaluated at each boundary point within the current ring. This overcomes outliers and ensures that the next ring boundary is within the polar image. Further details on the angular resolution M and the rough ring width estimate P can be found in the work by Gillert et al. (2023). The polar image is constructed by interpolating the downsampled image on a polar grid of shape N × M, with N = 256 the number of points in a radial direction. These points start at the current boundary and fan out in a radial direction with a distance of P/N between the points. We generate the polar image of shape ℝ^{6× N × M} by interpolating the down-sampled image as well as the background and boundary predictions without applying the activation function of the semantic segmentation model. For the 6th channel, we calculate the distance from each interpolation point to the center and normalize its values from 0 to 1. This helps the model to understand jumps in the previous boundary and gives information on the global scale.

Regression: Different from previous work (Gillert et al., 2023), we frame the prediction of the next boundary prediction as a regression task. To this end, we employ a second deep net, the radial regression model, visualized in Figure 5 . It predicts a real-valued distance to the next boundary for each of the M angular positions of the polar grid.

Addressing this problem as a regression task has several decisive advantages. First, it completely alleviates ambiguous predictions occurring with a segmentation approach such as predicting multiple edges within a pixel column, described by Gillert et al. (2023). In other words, our approach enforces by-design that only one boundary is predicted at each step. A second advantage is that the architecture implicitly enforces a smoothness constraint on the ring width because it uses bilinear upsampling in the segmentation head. Third, our design also makes the model predictions directly interpretable as they correspond to angular ring widths. In particular, this enables us to additionally predict an uncertainty value for the ring position, as described in Section 2.2.5.

Architecture and training: We use the DeeplabV3+ (Chen et al., 2018) architecture with a fast MobileNet (Howard et al., 2017) backbone as a basis and modified it. First, we reduce the number of convolution filters from 256 to 64 in the altrous spatial pyramid pooling module of the DeeplabV3+. Second, we remove the normalization layer after the atrous spatial pyramid pooling. Third, we convert the convolutions into circular convolutions proposed by Peng et al. (2020). These convolutions wrap around in angular direction, which accounts for the circular polar space. Finally, we replace the batch norm layers (Ioffe, 2015) with instance norms (Ulyanov et al., 2016).

The change to instance norms is necessary since the circular convolutions in combination with the adaptive angular resolution M prevent conventional batching of input samples by concatenating them along the batch dimension due to shape mismatches. The lack of batching prevents the use of batch norm layers. To emulate batch training, we use gradient accumulation, collecting gradients of multiple forward passes before the weights update. The normalization layer after the atrous spatial pyramid pooling had to be removed because of the necessary change from batch to instance norm.

We add a bypass for the features from the semantic segmentation model by concatenating them to the output of the previously described modified DeeplabV3+, as visualized in Figure 5 . This allows the further use of the already processed features from the semantic segmentation model. Since concatenations does not change the spatial shape, we still have the same shape as the input N × M polar image. We reduce the height dimension to one with a radial average pooling with a kernel of shape N × 1. Finally, we apply a 1 × 1 convolution to reduce the channel dimension to one, and output the predicted distances to the next ring at each angular position Δ = [δ ₁,···,δ_M ] ∈ ℝ ^M .

We train the radial regression model with L1 loss. We find that the optimization is more stable if the target distances is first normalized to values in [−1,1] as follows:

Hence our radial regression model is trained by minimizing (Equation 4)

with D = [d ₁,···,d_M ] ∈ ℝ ^M the ground truth distances to the next ring. To encounter every ring with a similar frequency in training, we initialize the iterative process with every ground truth ring boundary as a starting point and limit the number of iterations to K = 3.

In the previous work of Gillert et al. (2023), the prediction of the next boundary is cast as a pixelwise segmentation problem of the polar image. In our work, instead of pixelwise class scores, we regress a distance for each angular position m. Hence, the predictions returned by our model are straightforward to understand. This also enables us to train our model to predict an uncertainty value that is directly expressed in terms of ring width, instead of a more abstract uncertainty on each pixel’s class prediction. For this, we modify the final one-by-one convolution of our radial regression model to predict two parameters instead of one: in addition to the distance Δ, the model also outputs uncertainty values for each radial position: B = [b ₁,···,b_M ] ∈ ℝ ^M . We train these predictions using a Negative Log Likelihood (NLL) with Laplacian distribution, as recommended by Yeo et al. (2021), meaning that δ and b are interpreted as the mean and scaling parameter of a Laplace distribution (Equation 5):

and both are supervised using the following loss function (Equation 6):

This loss enables the model to predict a higher uncertainty b for samples where the regression error is hard to minimize, and thus reduce their impact on the overall loss at the cost of increasing the second term. After convergence, the model learns to predict higher uncertainty b only for samples for which the error is more likely to be high. The predictive uncertainty b is the scaling factor of the Laplace distribution expressed in normalized units, we transform it to a standard deviation σ_m of the distance expressed in original units as follows (Equation 7):

Dataset splits: We follow the given training and testing splits, resulting in only 22, 24, and 22 training samples with average diameters of 3,700, 3,260, and 3,979 pixels for DO, EH, and VM. Example images and ground truth labels can be seen in Figure 1 .

Semantic Segmentation Model: We train this model for 1,000 epochs using a cosine annealing learning rate schedule with a starting learning rate of 1e − 3 and an end learning rate of 1e − 5. The samples are augmented with random scaling, rotation, flipping, and color jitter and cropped to a size of 512. We apply the standard ImageNet pixel normalization. These samples are stacked into batches of size 8. For the backbone, we use the default hyperparameters and pre-trained on ImageNet.

Radial Regression Model: We train our radial regression model for 500 epochs with a cosine annealing learning rate schedule starting at 1e − 3 and ending at 1e − 5. As augmentation, we use color jitter since the other augmentations we used for the semantic segmentation model do not work in polar space. We apply the standard ImageNet pixel normalisation. As described in section 2.2.4, we emulate the batch size of 8 with gradient accumulation.

In the original implementation by Gillert et al. (2023), a training step consists of running the semantic segmentation model, K = 3 consecutive iterative steps, which include polar grid construction and interpolation and running the regression model.

We propose a more efficient implementation that enables faster training. Our efficient implementation is based on 1) saving the predictions of the segmentation model to disk instead of re-running it at each epoch and 2) unrolling the iterative steps onto different epochs. Indeed, we identified the polar grid construction and interpolation as the main bottleneck in the training process.

The polar grid construction cannot be done in an offline pre-processing step since it depends on the predicted ring of the previous iterative step. However, we can offload the polar grid construction and interpolation to other threads, allowing for parallelizability during radial regression model training. To achieve this, we propose a new training process named iterative unrolling. We unroll the iterative steps over K epochs. In iterative unrolling, we only run one iterative step per training step. However, we require the main thread to save the predicted ring to disk in the current epoch. In the next epoch, this boundary will be read by the data loader, which calculates the polar grid and interpolates the next polar image. Therefore, this sample is effectively in the next iterative step for this epoch. Splitting the iterative process over multiple epochs avoids race conditions between saving and loading the boundary files. After 3 epochs, we start over with the ground truth ring, as is done in the original implementation.

In the original iterative implementation, the model sees each sample K times per epoch. To mimic this, we duplicate each sample K times and start the iterative process in a staggered manner where the i-th copy starts in the i-th epoch using the predicted values. This allows for the duplicate samples to be in different steps in the iterative process. Another advantage of iterative unrolling is the possibility to gain batches with polar images from many different input images, as visualized in Figure 6 . This is more difficult in the original implementation due to the K steps with the same input image. These more diverse batches result in more stable gradients, which is especially useful for multi-species training where images of different species display larger diversity. Besides the more stable gradients, we achieve a speedup of two to three times using four parallel threads.

One of the benefits of our efficient implementation of the iterative detection is that it enables multi-species training. In this section, we showcase how multi-species training leads to better generalization to species not seen in the training data.

To measure the better generalization, we compare our multi-species model to a species-specific model on images from unseen shrub and tree species. These samples belong to Salix polaris, Fagus sylvatica, Fraxinus excelsior, and Vaccinium vitis-idea. All samples are of markedly different quality to those in the training set as they were produced in different labs using different equipment and slightly different protocols. Due to the large visual differences, no method was able to detect the pith. Therefore, we provide the methods with the ground truth pith, which is an acceptable amount of user input if the subsequent automatic ring segmentation is of sufficient quality.

The results in Table 2 show clear improvement for the multi-species method, surpassing the single-species methods by more than 10 pts for mAR, 10 pts for ARAND, and 20 pts for PQ. For MAE and MedAE, we observe high values in all cases, but the multi-species values are clearly smaller. This is especially impressive since the higher RQ value of 0.544 indicates that more rings are included for the MAE and MedAE calculations. These additional rings include rings that were too difficult to detect for the single species model. The displayed performance improvements are achieved by a larger and more diverse dataset. Nonetheless, the multi-species training still contains only 68 images, which explains the large performance drop for unseen species. Further diversifying and increasing the training set will reduce the number of completely unseen species and allow for better generalization of our model.

We display qualitative results on unseen species in Figure 9 . These segmentations vary widely in quality depending on the specific images, as seen in the first two rows, which display results for visually similar images. Therefore, validating the results for unseen species is even more important. This figure also displays plausible mistakes (b) and missing species-specific knowledge (b and c) from the multi-species model. However, it is the only model that provides acceptable ring estimates for unseen species.

We single out the design decisions for the radial regression model loss and show the step-wise improvements achieved by each component. All experiments are done with the same semantic segmentation model per species to exclude variances in the semantic prediction. We investigate the influence of the loss type and the target normalization, which maps the regression values from 0 to 255 to −1 to 1. Table 3 shows clear improvements for each step, supporting our model design choices. Switching to the L1 loss, which is more robust against outliers, significantly improves performance with an average gain of 7 pts for mAR and 6 pts for PQ. For the DO, the performance difference is drastic, changing the MAE from 26.6 to 16.1, which is a reduction of nearly 40%. Adding the target normalization, formally described in Equation 3, displays similar improvements for EH and DO. For VM, however, this step is necessary to gain proper segmentation results, improving the performance by 34.6 pts for mAR, 20 pts for ARAND, and 27.5 pts for PQ. This resulted in an improvement from 37.1 to 7.96 for MAE and 22.2 to 2.66 for MedAE, which represents error reductions of 78% for MAE and 88% for MedAE. These results show the need for the L1 loss in combination with target normalization, which stabilizes the training and, therefore, results in the best performance for each species.

We investigate how well uncertainties are calibrated and determine if the additional uncertainty estimation deteriorates the overall performance. This is done for each species individually. We evaluate the uncertainty calibration with the previously described ENCE metric, which gives a good intuition of the uncertainty quality.

We report the ring segmentation metrics in Table 4 . They show similar performance between the method with and without uncertainty calibration if we use the Laplace distribution for the NLL loss. We additionally investigate the performance with the Gaussian distribution that is commonly used by default for uncertainty estimation. Since NLL with a Gaussian distribution is related to an L2 loss, we can see some performance degradation, as shown in Table 4 . Additionally, we see a large difference between the Gaussian and Laplacian NLL for the ENCE metric. On average, the ENCE metric is 97% lower for the Laplacian NLL, clearly showing a better calibration of the uncertainty.

In this evaluation, we further investigate the results of our multi-species training and, specifically, the performance increase for EH and VM and the decrease for DO. By looking at the performance of the semantic segmentation model, we see a clear difference between the species, shown in Table 5 . In each species, the boundary segmentation improves by roughly 1pt, however, we can see a drop in performance for the pith and background only for DO. The DO species contains samples with piths that were torn off during the thin-sectioning process, giving it the same visual appearance as the background. Additionally, some samples have no rings on one side, so there is a direct connection between the pith and the background. Both these cases make differentiating between the pith and the background more difficult. An example for both cases can be seen in Figure 4 . This is especially true for the multi-species case, where these difficult cases are an even smaller percentage compared to the single-species case. These errors in pith prediction will propagate through multiple iterative steps, resulting in a decreased segmentation performance for DO. In the other species, the improved boundary prediction increases the overall performance even further.