# Contrastive learning and physics oriented evaluation for advanced segmentation in electron tomography

Cyril Li, Christophe Ducottet, Maxime Moreaud, Sylvain Desroziers, Valentina Girelli Consolaro, Virgile Rouchon, Ovidiu Ersen

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Cyril Li, Christophe Ducottet, Maxime Moreaud, Sylvain Desroziers, Valentina Girelli Consolaro, et al.. Contrastive learning and physics oriented evaluation for advanced segmentation in electron tomography. Expert Systems with Applications, 2026, 298 (A), pp.129606. 10.1016/j.eswa.2025.129606. ujm-05267710

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# Contrastive learning and physics oriented evaluation for advanced segmentation in electron tomography

Cyril Li<sup>a, *</sup>, Christophe Ducottet<sup>a</sup>, Maxime Moreaud<sup>b, c</sup>, Sylvain Desroziers<sup>c</sup>, Valentina Girelli Consolaro<sup>b, d</sup>, Virgile Rouchon<sup>b</sup>, Ovidiu Ersen<sup>d</sup>

## ARTICLE INFO

### Keywords:

Neural network

Electron tomography

Weakly annotated data

U-net

Contrastive learning

Semi-supervised training

## ABSTRACT

Deep learning methods are now achieving strong results for segmentation tasks, and the standard metric for evaluating methods is the Intersection over Union (IOU). However, we show in this paper that IOU is not efficient in evaluating the quality of segmentation for electron tomography (ET) images of zeolites. We perform a physics-oriented evaluation to ensure that the segmentation results yield coherent physical measures. We also formalize Mixed Supervised / Self-Supervised Contrastive Learning Segmentation (M3S-CLS), a semi-supervised approach using a contrastive learning approach that uses expert annotations to train the neural network model. A detailed comparison of this method with a standard cross-entropy-based model is provided. In addition, we publish a database of five fully segmented ET volumes along with corresponding baseline results. The code and the database is available at [http://gitlab.univ-st-etienne.fr/labhc-iscv/M3S-CLS](http://gitlab.univ-st-etienne.fr/labhc-iscv/M3S-CLS).

## 1. Introduction

Electron Tomography (ET) is a robust 3D imaging technique that can reconstruct material microstructure at the nanometer scale. With a Transmission Electron Microscope (TEM), a set of images is acquired from different angles, allowing a 3D reconstruction of the inner structure. However, this technique leads to reconstruction artifacts because of the limited range of angles, making these images notoriously hard to segment with classical segmentation techniques (Fig. 1). As a result, the preferred method used for ET segmentation in practical applications remains the manual expert-based method (Flores et al., 2019; Gay et al., 2016; Medeiros-Costa et al., 2019).

Deep learning (DL) based approaches recently made significant progress in 2D and 3D semantic segmentation. However, classical DL models require a huge amount of labeled data, a scarce and costly resource for ET. Recent work showed that DL models trained with specific learning paradigms, such as contrastive learning, require few annotated data for good segmentation results (Chen et al., 2020; Khosla et al., 2020). In particular, the promising method proposed in Li et al. (2023)

leverages both self-supervised and contrastive learning to segment a 3D volume using annotated data reduced as a few segmented pixels in a single slice of the volume. This method will be referred in the following to as M3S-CLS for Mixed Supervised / Self-Supervised Contrastive Learning Segmentation. This new method opens the door to semi-supervised settings where an expert interacts with the algorithm in a refinement loop. Typically, the expert starts to segment small regions in a 2D slice of the volume which is given as input to learn a segmentation model. The model then processes the whole volume to obtain a full segmentation.

However, the semi-supervised segmentation setting proposed in Li et al. (2023) has not been fully evaluated in the context of a realistic physical scenario. In this work, we consider the case of zeolites which are porous materials used for catalyst reactions. Porosity characteristics and the inner structure of the volume are crucial information to evaluate the chemical performance of the catalyst (Davis, 2002; de Jong et al., 2010). The Jaccard index or Intersection over Union (IOU), which is the standard evaluation metric used in semantic segmentation, is not adapted to characterize material porosity because it does not take the morphological texture of the material into account carefully. The

<footnote><sup>a</sup> Université Jean Monnet Saint-Etienne, CNRS, Institut d'Optique Graduate School, Laboratoire Hubert Curien UMR 5516, Saint-Etienne, F-42023, France</footnote>

<footnote><sup>b</sup> IFP Energies nouvelles, Rond-point de l'échangeur de Solaize BP 3, Solaize, 69360, France</footnote>

<footnote><sup>c</sup> Manufacture Française des Pneumatiques Michelin, 23 Place des Carmes Déchaux, Clermont-Ferrand, 63000, France</footnote>

<footnote><sup>d</sup> Institut de Physique et Chimie des Materiaux de Strasbourg (IPCMS), UMR 7504, IPCMS, CNRS - Université de Strasbourg, 23 rue du Loess, BP 43, Strasbourg, 67034, France</footnote>

<footnote>* Corresponding author.</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. A slice of a zeolite volume with a resolution of 1 nm per voxel. The slices from the volume exhibit noise and artifacts, which complicate segmentation tasks.</figcaption></figure>

particular assessment of porosity characteristics could only be done through physical metrics such as porosity, specific surface and tortuosity. These physical metrics are evaluated based on the segmented volume, and they strongly depend on the quality of the segmentation.

Therefore, in this paper, we propose a comprehensive evaluation of M3S-CLS for semi-supervised segmentation in the particular context of the assessment of zeolite catalytic properties. Our main contributions are:

* A physics-oriented evaluation of M3S-CLS with a study of the influence of the annotation ratio, i.e. the surface of the slice annotated by the expert, in the semi-supervised setting.
* A full formalization of both M3S-CLS method and the semi-supervised setting that models the partial annotation process provided by the expert.
* A comparison of M3S-CLS with the standard Cross-Entropy-based Learning (CEL) in the semi-supervised setting.
* A dataset of five fully segmented ET volumes with the corresponding baseline results for both M3S-CLS and CEL.

## 2. Related works

**Electron tomography segmentation** Segmentation of ET volumes is a notoriously difficult task. Manual segmentation with visualization tools remains the preferred method (Fernandez, 2012; He et al., 2008) often combined with image processing methods such as watershed transform (Volkmann, 2010). With the remarkable results from DL methods in standard semantic segmentation tasks (Chen et al., 2018; Çiçek et al., 2016; Milletari et al., 2016; Ronneberger et al., 2015; Sun et al., 2019), DL approaches have been naturally adapted to segmentation tasks for 2D electron microscopy (Akers et al., 2021; Khadangi et al., 2021). One requirement to train the DL model efficiently is access to a large amount of labeled training data. The availability of such datasets is limited for 2D ET segmentation tasks, and new techniques that require only a scarce amount of annotated data (Khadangi et al., 2021) such as semi-supervised few-shot were developed (Akers et al., 2021). Some works also investigate 3D segmentation tasks. In Genc et al. (2022), only 30 slices are annotated and used to train a standard U-Net model to segment γ-alumina/Pt catalytic material. In our work, we will consider a much more challenging scenario where only few pixels of one slice are annotated.

**Self-Supervised Contrastive Learning** With such a limited pool of annotated data for ET segmentation, extracting the maximum amount of information with the available data is crucial. Contrastive learning and Siamese networks (Zhao et al., 2021) aim to build a better representation of the latent space without using labels by constraining

objects from the training data to be close or far away in that latent space depending on the similarity of the objects with a contrastive loss function. The contrastive loss function computes the distance in the latent space between the two objects. Pairs of images are constructed. The pairs are either positive pairs or negative pairs. For positive pairs, the contrastive loss function minimizes the distance between the members of the pair. For negative pairs, the distance between the two objects of the pair is maximized. This method has shown great performance for image classification and does not require annotation to build pairs (Chaitanya et al., 2020; Chen et al., 2020; Chen & He, 2021; Grill et al., 2020; Khosla et al., 2020). A positive pair is built with one image and a transformed version of this image. A negative pair is built with two different images from the database. In those works, an encoder is trained with a contrastive learning loss on a large database that does not require labels to learn the visual representation of the image and a discriminator is trained on a smaller but labeled dataset to make the final prediction. This method introduces a bias on negative pairs when the number of classes in the database is limited (Khosla et al., 2020; Li et al., 2023). When building a negative pair, a random image is chosen in the database. If an image with the same class is unfortunately chosen for a negative pair, the contrastive loss tries to maximize the distance between objects that resemble each other. This is negligible for image classification tasks where there are many different classes in the database. However, for binary segmentation tasks, the occurrence of a wrong categorization for negative pairs happens often.

**Semi-supervised methods in semantic segmentation** Contrastive learning is also used in supervised settings. Positive and negative pairs are built with the knowledge of labels to avoid negative pairs with two images of the same class (Khosla et al., 2020). In Li et al. (2023), a novel semi-supervised setting is used. This work proposes to segment an entire volume with the annotation of only a portion of one slice. Both labeled and unlabeled pixels are used to train a standard U-Net, with a contrastive learning paradigm in a supervised setting for labeled pixels and in a self-supervised setting for unlabeled pixels. The trained DL model is then used to make a segmentation prediction on the remaining volume. This method was applied to 3D ET volumes and is perfectly adapted to contexts where annotation data is scarce. This method produces a full-volume segmentation, but only a segmentation evaluation has been carried out. No real evaluation to physically characterize the volume was considered. We propose to evaluate the method with physics measurement.

## 3. Physics-oriented evaluation

In this section, the evaluation criteria are detailed. This approach differs from previous work (Li et al., 2023), in which only a segmentation metric, the Intersection over Union (IOU), is considered. However, despite the excellent results with standard evaluation of the segmentation, the method provides segmentation results where the segmentation metrics and the error of the physical measures might not be correlated. This is due to the evaluation metric used in most segmentation tasks. The IOU metric computes the ratio between the area of overlap between the ground truth and the prediction and their union. The score is close to 1 when the prediction is close to the ground truth and 0 otherwise. The chemical performance of ET volumes is greatly correlated to its porous network, and near-perfect segmentation results and finer details on the inside of the ET volumes are needed to compute reliable, relevant metrics properly. As the IOU computes a global resemblance, there are cases where prediction gets a high IOU score result but completely fails to segment the porous network correctly. In other words, the generic IOU provides poor information about the internal morphology of the volume. To compute a finer evaluation for precise volumic characterization, we propose relevant metrics for a physics-oriented evaluation of porous materials.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Segmentation mask of a slice.</figcaption></figure>

## 3.1. Physics oriented criteria

Three morpho-structural parameters are chosen to describe the physical properties of a porous zeolite. These parameters are extracted from the segmented volumes.

First of all, it is worth mentioning that two types of voxels contribute to the 3D representation of a porous object in a segmented volume. The first category consists of the voxels assigned to the zeolite matrix (white voxels in Fig. 2), while the second corresponds to the matrix-free voxels of the zeolite (black voxels in Fig. 2), which are still contained in the zeolite volume. The latter class is here identified with the pores of the zeolite. The definition of porosity corresponds to the percentage of pores within the total volume of the zeolite. In practice, it is obtained from the percentage of voxels assigned to the pores compared to the sum of all voxels involved in the 3D representation.

The surface of a porous object includes both the outer and inner surfaces. The outer surface separates the zeolite matrix from the background, while the inner surface separates it from the pores. The surface area of a zeolite is estimated as the sum of the area contribution of all its contour voxels (Lindblad, 2005). Therefore, the specific surface area, which represents the accessible area per unit volume, gives the ratio of the surface area to the volume of the zeolite.

The tortuosity quantifies the complexity of the pore arrangement inside a porous object. To obtain a global representation of the tortuosity, the M-tortuosity (Chaniot et al., 2019) performs a calculation based on a stochastic consideration of the pore ends that are present in the entire porosity. From a physical point of view, while geometric tortuosity focuses on the shortest path to traverse a pore, M-tortuosity selects the most accessible one of the entire pore network. In particular, tortuosity becomes an informative parameter when the porous object represents a truly interconnected system of pores, also called a porous network, as in zeolites.

These parameters are voxel-wise sensitive, so a high-quality segmented volume must be considered for accurate analysis. To show that a good score of IOU is not equivalent to a small error in those physics metrics, a Person Correlation Coefficient (PCC) was computed on the IOU and the physics metrics error measured between a segmented volume and the ground truth for the 5 volumes. The PCC is a correlation coefficient indicating whether two variables are linearly correlated. If the PCC is close to 1 or -1, the variables are correlated; if it is close to 0, it is not. The result for the PCC between the IOU and the physical metrics errors are reported in the Table 1 for 300 different results. The IOU does not correlate with the physics-oriented metrics, meaning that the score of the IOU only is not enough to categorize the ET volume.

<table><caption>**Table 1**
PCC values between the segmentation metrics and the physical metrics for 5 different volumes segmented. The IOU does not correlate with the physics-oriented metrics.</caption><thead><tr><td></td><td>Porosity</td><td>Specific surface</td><td>M-tortuosity</td></tr></thead><tbody><tr><td>IOU</td><td>-0.568</td><td>-0.564</td><td>-0.184</td></tr></tbody></table>

<table><caption>**Table 2**
PCC values between the segmentation metrics and the physical metrics for 5 segmented volumes. The Focused IOU correlates better with the IOU for the porosity and specific surface. Both metrics are not correlated with the M-Tortuosity.</caption><thead><tr><td></td><td>Porosity</td><td>Specific surface</td><td>M-tortuosity</td></tr></thead><tbody><tr><td>IOU</td><td>-0.568</td><td>-0.564</td><td><b>-0.184</b></td></tr><tr><td>f-IOU</td><td><b>-0.856</b></td><td><b>-0.862</b></td><td>-0.107</td></tr></tbody></table>

## 3.2. Focused IOU

The standard IOU, as defined as Eq. (1), compares the whole volume prediction $\hat{Y}_s$ to the ground truth $Y_s$.

$$ IOU(V) = \frac{\sum_{s=1}^{S} \hat{Y}_s \cap Y_s}{\sum_{s=1}^{S} \hat{Y}_s \cup Y_s} \qquad (1) $$

As there are way more voxels of the object than those representing the porous network, slight variations of the finer details of the porous network do not significantly impact the IOU score despite drastically affecting physical measures. We propose a modified version of the IOU, called focused IOU (f-IOU), to better focus on the core of the object, making predicting correctly the porous network of the volume necessary to obtain a good score. We use a mask $M_s = fill_holes(Y_s)$ to focus the IOU on the object's core. The mask is built with mathematical morphology operations to fill holes in the ground truth.

$$ f\text{-}IOU(V) = \frac{\sum_{s=1}^{S} (M_s * \hat{Y}_s) \cap (M_s * Y_s)}{\sum_{s=1}^{S} (M_s * \hat{Y}_s) \cup (M_s * Y_s)} \qquad (2) $$

A Person Correlation Coefficient (PCC) was computed on the f-IOU and the physical metrics error. The result for the PCC between the f-IOU and the physical metrics errors are reported in Table 2. By construction, f-IOU shows a better correlation with the specific surface and the porosity. As expected, the quality of the segmentation metric is improved by paying more attention to the internal structure. The poor correlation results with M-tortuosity show that M-tortuosity is very sensitive to the segmentation of porosities, since the metric computes values related to porosity connectivity. If a few small holes are not properly segmented, it can have a significant impact on the computed porous network as the path connecting the pores will no longer exist. This results in a different M-tortuosity score.

## 4. Semi-supervised contrastive segmentation for volume images

In this paper, we have chosen to evaluate M3S-CLS (Li et al., 2023) which uses contrastive learning in a mixed supervised and self-supervised manner. This method reduces the annotation needed, to several small regions in a slice of the volume, allowing the entire volume to be segmented in a semi-supervised setting. In this setting, as summarized in Fig. 3, in step 1 (partial annotation) the annotator is tasked to label a small chunk of one slice highlighted in blue and green. In step 2 (train), the partially annotated slice is then used to train the DL model through contrastive learning. Every other slice of the volume is then passed through the trained model to achieve a complete volume segmentation during step 3 (inference). We then evaluate the computed segmentation. This means that for each volume, a small part of the volume has to be annotated, and a training step is performed.
<figure><img src="image_4.png" /><figcaption>Fig. 3. Overview of the M3L-CLS's setup. This method requires the annotation of an expert of a small portion of one slice of the volume. The partial annotation is used to train our model with a contrastive learning approach. To obtain the complete volume segmentation, each slice of the volume is passed through the trained model.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Overview of the M3L-CLS's architecture. The encoder-decoder *f* projects the image in the latent space and the linear classifier *h* makes the final prediction.</figcaption></figure>

## 4.1. Formalization

Although some full 3D convolution models exist for 3D segmentation, Kern et al. (2021) shows that processing each slice of the volume to provide the full segmentation mask of the 3D volume is more memory efficient with no degradation in the results. Contrastive learning in a semi-supervised setting has proven to be an incredibly efficient paradigm when the available annotated data is extremely limited, as both annotated and unannotated data can provide critical information for the segmentation task (Li et al., 2023).

The architecture comprises an encoder-decoder *f* trained with the contrastive loss. The supervised path is applied for the labeled data, and the unlabeled data passes through the self-supervised path. A classifier layer *h* trained with a classification loss is concatenated at the end of *f*. A weighted cross entropy loss function is used. Fig. 4 shows the complete architecture.

## 4.2. Formalization

The input volume *V* is decomposed in *S* slices {$X_s$}$_{s=1,...,S}$ where each slice $X_s \in \mathbb{R}^{W \times H \times I}$ with *W* the width of the slice, *H* the height of the slice. $x_{s,i}$ is the grayscale value of the pixel $i \in I$ in the slice *s* with *I* being the image spatial support. Each slice is passed through the architecture independently $\hat{Y}_s = h(f(X_s))$. The output of the encoder-decoder *f* of $X_s$ is $Z_s = (z_{s,i})_{i \in I} = f(X_s) \in \mathbb{R}^{W \times H \times D}$ where *D* is the dimension of the latent space. Finally, $\hat{Y}_s = h(Z_s)$ is the pixel-wise class prediction of the classification layer *h* with $\hat{Y}_s \in \mathbb{R}^{W \times H \times C}$, where *C* is the number of classes.

The pixel-wise class label map $Y_s$ of the slice *s* has to consider both labeled and unlabeled data. Let assign $\emptyset$ to unlabeled pixels, then $Y_s = (y_{s,i})_{i \in I}$ with $y_{s,i} \in \{\emptyset, 1, 2, ..., C\}$ where 1, 2, ..., *C* are the labeled classes.

We note $\mathcal{L}_s$ the set of labeled pixels and $\mathcal{V}_s$ the set of unlabeled pixels:

$$ \mathcal{L}_s = \{i \in I, y_{s,i} \neq \emptyset\}, \quad \mathcal{V}_s = \{i \in I, y_{s,i} = \emptyset\} \quad (3) $$

## 4.3. Supervised contrastive loss

We compute the contrastive loss (Khosla et al., 2020) with the output of the encoder-decoder *f* to learn a representation space where pixels with the same label are close to each other in the representation space, while pixels from different labels are far from each other in the representation space. For labeled pixels, we build positive pairs $\mathcal{P}_{s,i}^+$ with pixels of the same class and negative pairs $\mathcal{P}_{s,i}^-$ with pixels of different classes (Fig. 5):

$$ \mathcal{P}_{s,i}^{+} = \{j \in \mathcal{L}_s, i \neq j, y_{s,j} = y_{s,i}\} \quad (4) $$

$$ \mathcal{P}_{s,i}^{-} = \{j \in \mathcal{L}_s, i \neq j, y_{s,i} \neq y_{s,j}\} \quad (5) $$

The list of pairs formed for each pixel $x_{s,i}$ is therefore $\mathcal{P}_{s,i} = \mathcal{P}_{s,i}^{+} \cup \mathcal{P}_{s,i}^{-}$ with $N_p$ the number of pairs.

Therefore, the supervised contrastive loss is defined as:

$$ L_1(Z_s) = \frac{-1}{N_{\mathcal{L}_s}} \sum_{i \in \mathcal{L}_s} \frac{1}{N_p} \sum_{j \in \mathcal{P}_{s,i}^{+}} \log \frac{\exp(\mathrm{sim}(z_{s,i}, z_{s,j}))}{\sum_{k \in \mathcal{P}_{s,i}^{-}} \exp(\mathrm{sim}(z_{s,i}, z_{s,k}))} \quad (6) $$

with $N_{\mathcal{L}_s}$ the number of pixels in $\mathcal{L}_s$, $N_p$ the number of pairs for each pixel and sim is the cosine similarity, defined as:

$$ \mathrm{sim}(u, v) = \frac{u \cdot v}{\|u\| \cdot \|v\|} \quad (7) $$

## 4.4. Self-supervised contrastive loss

A self-supervised method is applied for both labeled and unlabeled pixels. To build a set of positive and negative pairs, a random transformation $\mathcal{T}$ is applied to the output of *f*. The pixels of the resulting
SELF-SUPERVISED CONTRASTIVE LOSS

<figure><img src="image_6.png" /><figcaption>SUPERVISED CONTRASTIVE LOSS</figcaption></figure>

**Fig. 5.** The contrastive learning setup can be applied to both labeled and unlabeled pixels. Pairs of pixels are constructed. To construct a pair, an anchor pixel is selected (blue). For labeled pixels (yellow rectangle), positive pairs (green) are constructed with pixels of the same class. For negative pairs (red), pixels of different classes are chosen. For unlabeled pixels (purple rectangle), the slice is transformed. A positive pair is built with the pixel in the same location as the anchor pixel in the transformed image (green). For a negative pair, a random pixel in the transformed image is chosen (red).

slice $\tilde{Z}_s = f(\mathcal{T}(X_s)) = (\tilde{z}_{s,i})_{i \in I}$ are paired with the corresponding pixels of the original slice $Z_s$ to form positive pairs. Negative pairs are built with one pixel in the original slice and one pixel in another location in the transformed slice.

The self-supervised contrastive loss $X_s$ is defined as:

$$L_2(Z_s) = \frac{-1}{N_I} \sum_{i \in I} \log \frac{\exp(\operatorname{sim}(z_{s,i}, \tilde{z}_{s,i}))}{\sum_{j \in I, i \neq j} \exp(\operatorname{sim}(z_{s,i}, \tilde{z}_{s,j}))} \quad (8)$$

where $N_I$ is the number of pixels in $I$.

The final loss is a combination of the loss of both paths:

$$L(Z_s) = L_1(Z_s) + L_2(Z_s) \quad (9)$$

## 4.5. Biased patch selection

To evaluate the performance of M3S-CLS in a realistic semi-supervised scenario, we need to simulate the annotation made by the user for different ratio of annotated pixels. For that purpose, we consider a fully annotated slice and we generate random partially annotated data trying to mimic the partial annotation made by the user. To take into account that a user will rather annotate pixels from the zeolite matrix than from the background, we prioritize the inner object of the training slice. Formally, we denote $r$ as the labeling rate, i.e. the ratio of the slice that is annotated. The annotated training slice denoted $Y_s$ is divided into a grid of $N_c \times N_c$ fragments, where $N_c$ is a parameter of the simulation process. Each fragment of size $W/N_c \times H/N_c$, is denoted $A^k$ with $k \in \{1, 2, 3, ..., N_c^2\}$. The $N_c^2$ fragments are then separated into two sets, one representing object fragments, denoted $\mathcal{O}$, and one representing the background fragments, denoted $\mathcal{B}$ (Fig. 6). A fragment is considered as an object fragment if the ratio of object pixels in this fragment is greater than a threshold $\alpha$. If $\alpha = 0.5$, we then consider that if a fragment has more object pixels than background pixels, then it is considered as an object fragment. When $\alpha$ is small, background pixels can be included in object fragments thus object edges are mostly located inside these fragments (see Fig. 6). In practice, edge information is valuable for accurate segmentation during training, so we choose a low alpha value (see Section 5.2).

$$\mathcal{O} = \left\{ A^k; \frac{\sum_{i \in A^k} y_{s,i}}{N_c^2} \geq \alpha \right\} \quad (10)$$

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Strategy of the generation of partial annotations. The object pixels $\mathcal{O}$ are in green and the background pixels $\mathcal{B}$ are in red with $\alpha = 0.01$.</figcaption></figure>

$$B = \left\{ A^k; \frac{\sum_{i \in A^k} y_{s,i}}{N_c^2} < \alpha \right\} \quad (11)$$

To simulate a user annotation with a labeling rate $r$, we randomly pick fragments from $\mathcal{O}$ until the desired labeling rate is satisfied. If all elements of $\mathcal{O}$ are picked, the fragments from $\mathcal{B}$ are randomly chosen to complete the set. This way of picking inside the object gives more training examples containing object pixels. This is a more realistic scenario which, in addition, contributes to a better segmentation of the inner structure of the material.

## 5. Experiments and results

In this section, we evaluate the method on five volumes of zeolite materials. To assess the competitiveness of M3S-CLS, we compare it with a standard cross-entropy-based learning, considering only the annotated pixels and training the same UNet-based network architecture. The IOU and the f-IOU are computed, as well as the physical measures. Then, the corresponding results are discussed.
<caption>**Table 3**</caption>

The architecture of our method M3S-CLS. This table features the layers of out U-Net like architecture. Convolution #8 projects each pixel in the latent space of dimension D=16 that is trained with the contrastive loss and Convolution #9 corresponds to the classification h that is trained with the cross-entropy loss.

<table><thead><tr><th colspan="4">Encoder</th><th colspan="4">Decoder</th></tr><tr><th>N</th><th>Layer</th><th>Kernels</th><th>Output</th><th>N</th><th>Layer</th><th>Kernels</th><th>Output</th></tr></thead><tbody><tr><td>#1</td><td>Convolution</td><td>3×3×1</td><td>W,H,16</td><td>#5</td><td>Up-convolution</td><td>3×3×128</td><td>W/4,H/4,64</td></tr><tr><td>#1</td><td>Convolution</td><td>3×3×16</td><td>W,H,16</td><td>#5</td><td>Concatenate</td><td>With #3</td><td>W/4,H/4,128</td></tr><tr><td>#1</td><td>MaxPooling</td><td></td><td>W/2,H/2,16</td><td>#5</td><td>Convolution</td><td>3×3×64</td><td>W/4,H/4,64</td></tr><tr><td></td><td>#5</td><td>Convolution</td><td>3×3×64</td><td></td><td></td><td></td><td></td></tr><tr><td>#2</td><td>Convolution</td><td>3×3×16</td><td>W/2,H/2,32</td><td>#6</td><td>Up-convolution</td><td>3×3×64</td><td>W/2,H/2,32</td></tr><tr><td>#2</td><td>Convolution</td><td>3×3×32</td><td>W/2,H/2,32</td><td>#6</td><td>Concatenate</td><td>With #2</td><td>W/2,H/2,64</td></tr><tr><td>#2</td><td>MaxPooling</td><td></td><td>W/4,H/4,32</td><td>#6</td><td>Convolution</td><td>3×3×32</td><td>W/2,H/2,32</td></tr><tr><td></td><td>#6</td><td>Convolution</td><td>3×3×32</td><td></td><td></td><td></td><td></td></tr><tr><td>#3</td><td>Convolution</td><td>3×3×32</td><td>W/4,H/4,64</td><td>#7</td><td>Up-convolution</td><td>3×3×32</td><td>W,H,16</td></tr><tr><td>#3</td><td>Convolution</td><td>3×3×64</td><td>W/4,H/4,64</td><td>#7</td><td>Concatenate</td><td>With #1</td><td>W,H,32</td></tr><tr><td>#3</td><td>MaxPooling</td><td></td><td>W/8,H/8,64</td><td>#7</td><td>Convolution</td><td>3×3×16</td><td>W,H,16</td></tr><tr><td></td><td>#7</td><td>Convolution</td><td>3×3×16</td><td></td><td></td><td></td><td></td></tr><tr><td>#4</td><td>Convolution</td><td>3×3×64</td><td>W/8,H/8,128</td><td>#8</td><td>Convolution</td><td>1×1×16</td><td>W,H,D</td></tr><tr><td></td><td>#9</td><td>Convolution</td><td>1×1×D</td><td></td><td></td><td></td><td></td></tr></tbody></table>

## 5.1. E.T. catalyst volumes

The database consists of five 500 nm mesoporous zeolites ET volumes reconstructed from 75 to 90 projections with an angular range of at least 120 degrees. For each volume, a data cube of 512 × 512 × 512 voxels containing the reconstructed object is created using the SIRT algorithm (Giibert, 1972). This data cube is adjusted to the size of the zeolite by removing the sample-free regions. A more detailed analysis of the data can be found in Girelli Consolaro et al. (2025).

## 5.2. Implementation details

The encoder-decoder *f* is a standard U-Net (Ronneberger et al., 2015) with the last layer modified to project in a *D* = 16 dimensions representation space. The encoder is composed of three layers, each consisting of two 3 × 3 convolution layers, a ReLU layer, and a Max Pooling layer, halving the dimension of the input and doubling the number of channels for each layer. The decoder is composed of three layers, with an up-convolution layer and an upscaling layer, doubling the input resolution and halving its channels. Skip connections are established between each corresponding encoder and decoder layer. Finally, a 1 × 1 convolution layer with 16 filters is appended to the decoder to project the output in the representation space. Table 3 gives more details on the architecture.

The classification layer *h* consists of a single 1 × 1 convolution layer. The classification layer's goal is to predict each pixel's class. The classification layer is trained with a weighted cross-entropy where the weight of unlabeled pixels is set to 0 (Çiçek et al., 2016).

For the generated partial annotation, we chose $N_c = 10$ and $\alpha = 0.1$.

<caption>**Table 4**</caption>

Random transformations used in the self-supervised path.

<table><thead><tr><th>Transformation</th><th>Probability</th><th>Parameters</th></tr></thead><tbody><tr><td>Gaussian noise</td><td>1</td><td>$\lambda'(0, [0.01; 0.06])$</td></tr><tr><td>Gray level shift</td><td>0.5</td><td>$[-0.01; 0.01]$</td></tr><tr><td>Gaussian blur</td><td>0.5</td><td>$\sigma = [0.5 - 1.5]$</td></tr></tbody></table>

We chose to pick ten positive pairs and ten negative pairs for each pixel to train with the contrastive loss. The set of random transformations used for the unsupervised contrastive loss is detailed in the Table 4.

## 5.3. Comparing contrastive and entropy-based learning

We use the same settings as Li et al. (2023). A random slice of the volume is selected and is partially hidden with a fixed labeling rate *r* as mentioned previously. This is the only data used in training. We evaluate the trained model on every frame of the volume. This process is repeated $N_F = 5$ times with a different training slice. This ensures that our method is more generalizable and is independent of the annotated slice. The IOU (Eq. (1)) and the f-IOU (Eq. (2)), as well as the mean relative absolute error (MRAE) of the porosity, the specific surface, and the M-tortuosity, are then calculated on the whole volume.

$$MRAE = \frac{1}{N_V} \frac{1}{N_F} \sum_{i=1}^{N_V} \sum_{j=1}^{N_F} \frac{|y_{i,j} - GT|}{GT} \quad (12)$$

Where $N_V$ is the number of volumes, $y_{i,j}$ is the prediction of the physical measure of the *i*th volume with the *j*th annotated slice and *GT*

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** IOU and f-IOU for different values of the labeling rate *r* for the contrastive method (blue) and U-Net (orange) for all tested volumes. The colored areas represent the standard variation. The values are in Table 5. Both methods reach very high scores, but M3S-CLS performs better and has smaller variations.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Mean Relative Absolute Errors (MRAE) for the physical metrics for different values of the labeling rate *r* for the contrastive method (blue) and U-Net (orange). The colored areas represent the standard variation. The values are in Table 5. Only in M3S-CLS does the MRAE score under one and perform significantly better than U-Net.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 9.** Visual comparison between the contrastive method and U-Net. The 3D reconstruction is computed with the contrastive method and a labeling rate of *r* = 0.25. Values of different metrics are in Table 6. A random slice is extracted to compare both methods visually. Green pixels are true positive, black pixels are true negative, purple pixels are false positives and red pixels are false negatives. Qualitatively, both methods produce good segmentation masks, but M3S-CLS shows fewer red and purple regions than the slices from U-Net, meaning better segmentation of the object's shape and the volumes' porosities.</figcaption></figure>

is the measured ground truth. The results are compared with the same
settings but with a modeled trained with a cross-entropy loss using only
the annotated pixels. This baseline method will be referred to as U-Net
because it is exactly the same as the standard U-Net architecture. This
process is repeated for different labeling rates *r* and ET volumes. The re-
sults are shown in Table 5 and in Figs. 7 and 8. The contrastive method
outperforms the standard version of U-Net in IOU, f-IOU and relative
errors of the metrics. This demonstrates that training with a contrastive
approach results in better segmentation, even with very few annotated
data. It is also worth noting that for low labeling rates, the contrastive

method gives more stable results than the U-Net approach, as the blue
area that represents the standard variation on all N<sub>f</sub> = 5 training on all 5
volumes for M3S-CLS is smaller than the U-Net's orange area. This shows
that M3S-CLS is more generalizable as it is less dependent on the training
slice than U-Net. Fig. 9 visually compares the volume segmentation for
both volumes and a 3D reconstruction. The models used for the illustra-
tions are trained with a labeling rate r = 0.25. Quantitatively, the models
trained with the contrastive approach gives better results for the vol-
ume's global shape and the volumes' cavities. The 3D reconstruction is
computed using the volume segmentation produced with the contrastive
<caption>**Table 5**</caption>

Values of IOU, Focused IOU, MRAE of porosity (P), specific surface (SpS), and M-tortuosity (M-t) for different values of the labeling rate *r* depending on the training loss for M3S-CLS and the standard U-Net models. For almost every metric, M3S-CLS outperforms the U-Net model trained with a cross-entropy loss.

<table><caption>**Table 6**</caption><thead><tr><th></th><th>0.02</th><th>0.06</th><th>0.12</th><th>0.25</th><th>0.5</th><th>1.0</th></tr></thead><tbody><tr><td>M3S-CLS - IOU</td><td><strong>0.779</strong></td><td><strong>0.810</strong></td><td><strong>0.865</strong></td><td><strong>0.887</strong></td><td><strong>0.898</strong></td><td><strong>0.848</strong></td></tr><tr><td>U-Net - IOU</td><td>0.723</td><td>0.751</td><td>0.793</td><td>0.811</td><td>0.784</td><td>0.806</td></tr><tr><td>M3S-CLS - f-IOU</td><td><strong>0.924</strong></td><td><strong>0.936</strong></td><td><strong>0.944</strong></td><td><strong>0.954</strong></td><td><strong>0.955</strong></td><td><strong>0.952</strong></td></tr><tr><td>U-Net - f-IOU</td><td>0.905</td><td>0.928</td><td>0.923</td><td>0.945</td><td>0.954</td><td>0.949</td></tr><tr><td>M3S-CLS - P MRAE</td><td><strong>0.796</strong></td><td><strong>0.576</strong></td><td><strong>0.432</strong></td><td><strong>0.267</strong></td><td><strong>0.260</strong></td><td><strong>0.354</strong></td></tr><tr><td>U-Net - P MRAE</td><td>1.397</td><td>0.833</td><td>0.963</td><td>0.429</td><td>0.592</td><td>1.123</td></tr><tr><td>M3S-CLS - SpS MRAE</td><td><strong>0.741</strong></td><td><strong>0.620</strong></td><td><strong>0.411</strong></td><td><strong>0.232</strong></td><td><strong>0.227</strong></td><td><strong>0.299</strong></td></tr><tr><td>U-Net - SpS MRAE</td><td>2.879</td><td>2.500</td><td>1.690</td><td>1.331</td><td>2.192</td><td>3.167</td></tr><tr><td>M3S-CLS - M-t MRAE</td><td>1.191</td><td><strong>0.315</strong></td><td>0.383</td><td><strong>0.277</strong></td><td><strong>0.325</strong></td><td><strong>0.265</strong></td></tr><tr><td>U-Net - M-t MRAE</td><td><strong>0.444</strong></td><td>0.566</td><td><strong>0.343</strong></td><td>1.275</td><td>1.677</td><td>0.943</td></tr></tbody></table>

Values of metrics for the volumes showcased in Fig. 9. The volumes are computed with M3S-CLS and a labeling rate *r* = 0.25. The ground truth values for the physics metrics are reported in parentheses.

<table><caption>**Table 7**</caption><thead><tr><th></th><th>f-IOU</th><th>Porosity</th><th>Specific surface</th><th>M-tortuosity</th></tr></thead><tbody><tr><td>(1)</td><td>0.986</td><td>0.001 (0.001)</td><td>0.674 m²/g (0.529)</td><td>2.32 (3.34)</td></tr><tr><td>(2)</td><td>0.960</td><td>0.013 (0.011)</td><td>7.15 m²/g (6.59)</td><td>1.29 (2.54)</td></tr><tr><td>(3)</td><td>0.956</td><td>0.005 (0.007)</td><td>4.09 m²/g (3.97)</td><td>2.17 (2.24)</td></tr><tr><td>(4)</td><td>0.942</td><td>0.009 (0.012)</td><td>4.18 m²/g (5.53)</td><td>1.69 (1.43)</td></tr><tr><td>(5)</td><td>0.932</td><td>0.015 (0.017)</td><td>6.63 m²/g (7.05)</td><td>1.75 (1.49)</td></tr></tbody></table>

Time in minutes required to complete each training for both M3S-CLS and U-Net.

<table><thead><tr><th></th><th>M3S-CLS</th><th>U-Net</th></tr></thead><tbody><tr><td>(1)</td><td>22 minutes</td><td>5 minutes</td></tr><tr><td>(2)</td><td>18 minutes</td><td>3 minutes</td></tr><tr><td>(3)</td><td>19 minutes</td><td>3 minutes</td></tr><tr><td>(4)</td><td>13 minutes</td><td>2 minutes</td></tr><tr><td>(5)</td><td>19 minutes</td><td>4 minutes</td></tr></tbody></table>

method. The values of some metrics for these volumes are reported in Table 6. Both methods achieve visually high-quality segmentation, demonstrating the importance of computing physics-specific measures. Moreover, the predicted values for the metrics are close to the ground truth measured for each volume in parentheses in Table 5. For example, for volume 1, the porosity is perfectly measured, the specific surface is estimated at 0.67m²/g with 0.52m²/g for the ground truth which corresponds to a 20% error. For the M-tortuosity, 2.32 was measured on the segmented volume while 3.34 was reported for the ground truth volume. This represents a 30% error. This confirms that our method gives accurate segmentation results.

These results show that an accurate estimation of these two parameters, which are voxel-wise sensitive, is obtained with a contrastive learning approach. Since these criteria are obtained from the segmented volumes of the reconstruction, this leads to a better allocation of the voxels representing the zeolite, in particular those associated with the internal pores and those associated with the zeolite matrix. A realistic 3D representation and morphological quantification is achieved with M3S-CLS, which is consistent with the measurements derived by the manual, time-consuming method.

## 5.4. Computational efficiency

As M3S-CLS is a semi-supervised approach, computational efficiency is essential in determining its viability. The results of the previous ex-

periments were computed on a NVIDIA TITAN RTX. Table 7 shows the average training time for each volume and each method. There are two reasons why M3S-CLS takes longer to train a volume. First a heavier pre-processing step is required for M3S-CLS. For each pixel, it is necessary to find $N_c$ positive pairs and $N_c$ negative pairs. Secondly, M3S-CLS requires to train *f* first with the semi-supervised contrastive loss and then train *h* with a supervised cross-entropy. However, the time needed to train M3S-CLS is still reasonable, and we believe that the significant performance increase makes up for this.

## 6. Conclusion

In this paper, we carry out a comprehensive physics-oriented evaluation of M3S-CLS on a dataset of fully segmented ET volumes. A detailed comparison between a contrastive learning approach and a cross-entropy based learning is performed and shows that in a semi-supervised context with a very little annotated data, M3S-CLS achieves better segmentation. In future work, we can take advantage of methods coming from physics-informed machine learning to guide the training process of M3S-CLS with losses computed through the physics-oriented metrics highlighted in this paper. This could provide even more accurate results around the porous region of the volume.

### CRediT authorship contribution statement

Cyril Li: Conceptualization, Methodology, Investigation, Software, Writing; Christophe Ducottet: Conceptualization, Methodology, Reviewing & Editing, Supervision; Maxime Moreaud: Conceptualization, Methodology, Resources, Reviewing & Editing, Supervision; Sylvain Desroziers: Methodology, Reviewing; Valentina Girelli Consolaro: Resources, Data Curation, Reviewing & Editing; Virgile Rouchon: Resources, Reviewing; Ovidiu Ersen: Resources, Reviewing.

### Data availability

The code and the database is available at https://gitlab.univ-st-etienne.fr/labhc-iscv/M3S-CLS.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR).

### Supplementary material

Supplementary material associated with this article can be found in the online version at 10.1016/j.eswa.2025.129606

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