Impact Statement
A dramatic reduction in cost of computing and storage has led to a huge increase in volume of images acquired and intensity of image analysis. In this contribution, we focus on “digital rocks”, 2D and 3D images scanned from subsurface porous rock samples that are used for oil and gas reservoirs evaluation, in mining, or in planning of subsurface storage sites for CO2 and hydrogen. Finding rock images similar to a newly acquired sample image with the proposed Double Siamese Neural Network can be done in seconds, compared to weeks or months required by experimentation on rock samples. This orders-of-magnitude acceleration equates to millions of dollars in monetary terms for each sample, and in higher values for missed industry opportunities.
1. Introduction
Content-based image retrieval (CBIR) is an established application in computer vision (Gudivada and Raghavan, Reference Gudivada and Raghavan1995; Devareddi and Srikrishna, Reference Devareddi and Srikrishna2022; Kulkarni and Manu, Reference Kulkarni and Manu2022). The objective is to retrieve images similar in content to a query image from a large image database. Following the success of deep learning (DL) in other computer vision applications (Voulodimos et al., Reference Voulodimos, Doulamis, Doulamis and Protopapadakis2018), DL has almost entirely replaced targeted feature engineering and greedy-based matching approaches (see Alcantarilla et al., Reference Alcantarilla, Bartoli and Davison2012 for a particularly sophisticated incarnation) as the dominant technical approach for CBIR. Multiple comprehensive and comparative reviews demonstrate this in a variety of applications (Zagoruyko and Komodakis, Reference Zagoruyko and Komodakis2015; Zheng et al., Reference Zheng, Yang and Tian2018; Rian et al., Reference Rian, Christanti and Hendryli2019; Chen et al., Reference Chen, Liu, Wang, Bakker, Georgiou, Fieguth, Liu and Lew2022; Simran et al., Reference Simran, Kumar and Bachu2021; Dubey, Reference Dubey2022). The majority of CBIR studies use common photographic images, and new approaches are almost always developed and calibrated in this rich public domain.
Providing efficient CBIR in an industrial image domain allows experts to quickly infer properties about the query image without the need for expensive or time-consuming physical experimentation. Industrial images provide different challenges, and have different requirements and properties. While state-of-the-art DL models on photographic images can be trained on datasets in the millions, industrial image databases are usually smaller because the images are more costly to collect and curate. This lends itself to DL architectures of smaller capacity. Some prior work addresses the use of databases and convolutional neural networks (CNNs) for analyzing images of common industrial materials (e.g. segmentation and classification in Bell et al., Reference Bell, Upchurch, Snavely and Bala2015; Ahuja et al., Reference Ahuja, Gupta and Rapole2022), and the medical community is routinely using DL-based CBIR (Pradhan et al., Reference Pradhan, Pal, Banka and Elloumi2021). However, CBIR applications in the industrial and materials domain are scarce.
In this article, we consider a database of “digital rock” images acquired in 2D via light microscopy or projected from 3D microcomputer tomography to 2D (Saxena et al., Reference Saxena, Hows, Hofmann, Alpak, Dietderich, Appel, Freeman and De Jong2019b). They are of fundamental importance in the Earth science domain and have led to a revolution in the understanding of deep-Earth phenomena, such as fluid transport through porous media (Blunt et al., Reference Blunt, Jackson, Piri and Valvatne2002; Andrä et al., Reference Andrä, Combaret, Dvorkin, Glatt, Han, Kabel, Keehm, Krzikalla, Lee, Madonna and Marsh2013; Al-Marzouqi, Reference Al-Marzouqi2018; Saxena et al., Reference Saxena, Hows, Hofmann, Freeman and Appel2019a, Reference Saxena, Hows, Hofmann, Alpak, Dietderich, Appel, Freeman and De Jong2019b). Computer simulation and DL-based image analysis have already greatly advanced the field of rock and fluid science (Araya-Polo et al., Reference Araya-Polo, Alpak, Hunter, Hofmann and Saxena2018; Saxena et al., Reference Saxena, Day-Stirrat, Hows and Hofmann2021; Ahuja et al., Reference Ahuja, Gupta and Rapole2022). Here, we focus on the specific CBIR challenges presented by the materials imaging domain with digital rock as a high-value example. We focus on resolving the following critical issues: (a) data paucity, (b) “expert knowledge” subjectivity, and (c) uncommon statistical image properties.
To solve the data paucity issue, we use DL models that are pretrained on common photographic images in conjunction with “few-shot” transfer learning (Wang et al., Reference Wang, Yao, Kwok and Ni2020). A model pretrained on several related tasks is fine-tuned with limited new data which helps the model generalize to unseen samples. Rather than solely relying on expert subjectivity in image similarity labeling, we use the metadata from physical measurements as additional labels for training. We present a novel custom DL architecture that combines Siamese neural networks where two neural networks share the same weights and produce comparable output vectors for similarity learning (Bromley et al., Reference Bromley, Guyon, LeCun, Säckinger and Shah1993; Wiggers et al., Reference Wiggers, Britto, Heutte, Koerich and Oliveira2019; Chicco, Reference Chicco and Cartwright2021; Ahuja et al., Reference Ahuja, Gupta and Rapole2022) and the efficient ResNet architecture (He et al., Reference He, Zhang, Ren and Sun2016). We train this Double Siamese Neural Network (DSNN) with a loss function that integrates classification and regression terms to train with both image and metadata similarity (Thiele et al., Reference Thiele, Saxena and Hohl2020) in a form of “physics-informed learning” (Kim et al., Reference Kim, Kim, Lee and Lee2021). During inference, we use a computationally efficient search for the closest matches in the database using a highly compressed image feature representation. We describe an implementation in Pytorch (Paszke et al., Reference Paszke, Gross, Massa, Lerer, Bradbury, Chanan, Killeen, Lin, Gimelshein, Antiga, Desmaison, Wallach, Larochelle, Beygelzimer, d’Alché-Buc, Fox and Garnett2019) on an Nvidia V100 multi-GPU system.
The remainder of this article is organized as follows: Section 2 describes the novel Double Siamese DL architecture used in this study along with the training process, loss function, and inference process. Section 3 contains details of the experiments that were conducted, describes the computational platform and performance, and quantifies the approximation and prediction error. We analyze the DSNN performance in detail and compare it with alternative architectures. In Section 4, we present comparative visuals of the image retrieval performance on several representative “real-world” examples. We discuss the three main factors behind the superior performance of our novel architecture. Section 5 presents our conclusions and lists future work.
2. Network Architecture
The first CBIR DL architectures were based on convolutional neural networks (Krizhevsky et al., Reference Krizhevsky, Sutskever and Hinton2012) and autoencoders (Ballard, Reference Ballard1987). While convolution and autoencoders provide a natural mechanism for image dimensionality reduction, single autoencoder networks proved difficult and time-consuming to train, and were limited in image retrieval performance. Wiggers et al. (Reference Wiggers, Britto, Heutte, Koerich and Oliveira2019) have shown that Siamese networks perform better than simple CNNs and autoencoders, and we build upon this work. We propose to leverage both physics and “expert knowledge” in a novel “Double Siamese Neural Network” architecture composed of four networks, each using ResNet-18 and a regression module for incorporating physics-informed properties.
In contrast to a conventional Siamese network where only one backbone network is used for image similarity learning, we employ an additional network to leverage both metadata and image similarity. To do the latter, an additional regression term is added to the objective function, similar to multi-task learning (MTL). The purpose of MTL is to train one or multiple shared networks with different tasks for improved prediction performance over networks trained individually with each task. MTL has been used in other CBIR applications, such as in Pisov et al. (Reference Pisov, Makarchuk, Kostjuchenko, Dalechina, Golanov and Belyaev2018), where a ResNet-like network is trained with multiple heads for classification and segmentation.
Instead, we opted to follow the approach taken by Standley et al. (Reference Standley, Zamir, Chen, Guibas, Malik and Savarese2020), by designing a model that can improve our main image retrieval performance with another, possibly conflicting, task of estimating metadata (e.g., porosity and permeability for digital rocks). In this vein, we add another Siamese network for training image similarity along with metadata similarity. Our DSNN’s loss function is thus composed of a classification and a regression component (see Figure 1). The classification term determines whether a pair of images is similar and the regression term quantifies the mismatch in the metadata space of the image pair.
Figure 1. Proposed Double Siamese Neural Network (DSNN) architecture. The blue Siamese part of the network is trained with image similarity and the gray Siamese part is trained with image and metadata similarity in classification. The dark gray part is trained with metadata for regression. “x,” “+,” “-“designate multiplication, addition, and subtraction. The green and yellow images are pairs of images, with binary classification label “similar” or “different.” After the training phase the networks are used in the inference phase for image retrieval and metadata regression as shown in Figure 3. See text for the mathematical details and symbols.
2.1. Training and the loss function
To avoid the issue of overfitting while still maintaining high-capacity CNNs, we use transfer learning (Tan et al., Reference Tan, Sun, Kong, Zhang, Yang and Liu2018; Wiggers et al., Reference Wiggers, Britto, Heutte, Koerich and Oliveira2019) by incorporating two pretrained backbone ResNet-18 networks (He et al., Reference He, Zhang, Ren and Sun2016) that were trained with the ImageNet database (Russakovsky et al., Reference Russakovsky, Deng and Su2015). This warm-start of the model parameters helps the model transition from images in the natural domain to digital rock images taken from microscopy or other sensors. We also considered freezing layers of our network, but note superior empirical CBIR recovery using the warm starting approach.
Using similarity labels generated by human experts for a range of image pairs is too costly and work-intensive in our application. We have instead chosen to codify the process by which human experts assess similarity: Visual perception combined with the matching of physical properties is how Earth scientists perform such assessment, and we have mapped this process into our loss function.
The loss function used for training the DSNN has classification and regression terms. As will be described in the next sections in detail, the classification term has an image similarity component based on the images’ reduced-dimension feature representation but also a component related to the mismatch of physical properties with additional regression terms in the loss function. Some examples of the approximately 10,000 images used in this study (a small subset of those in our rock image database) are shown in Figure 2. They are acquired through specialized borehole drilling operations and come from various deep subsurface porous rock formations (“reservoirs”) across the globe. Millimeter-size pieces of rock (“rock samples”) are then cut into slices and subjected to micrography (“thin slices”) or scanned and digitized using x-rays (“micro-CT”). We use the rock sample identifier as the label: If a pair of images come from the same sample, they have identical labels. This is because different rock samples from the same reservoir can still exhibit different appearance and properties due to subsurface heterogeneity. The reservoir identifier is used here only during testing, not during the training phase.
Figure 2. Examples of 2D micro-CT and thin-section rock images. Micro-CT images are gray scale, while thin-section images are RGB.
2.1.1. Binary classification
Consider two backbone network models
$ {f}_1 $
and
$ {f}_2 $
for feature extraction. The networks
$ {f}_1 $
and
$ {f}_2 $
have the same network architecture but the weights are not shared. As noted earlier, we select ImageNet-pretrained ResNet-18s as the initial networks. The ImageNet dataset includes a large number of annotated photographs intended for computer vision research. A pair of images
$ \overrightarrow{A} $
and
$ \overrightarrow{B} $
, which are randomly chosen with 50 % probability from the same or different samples, are passed on to the networks
$ {f}_1 $
and
$ {f}_2 $
for feature extraction.
$$ {\overrightarrow{h}}_1={f}_1\left(\overrightarrow{A}\right),{\overrightarrow{h}}_2={f}_1\left(\overrightarrow{B}\right),\hskip2em {\overrightarrow{h}}_3={f}_2\left(\overrightarrow{A}\right),{\overrightarrow{h}}_4={f}_2\left(\overrightarrow{B}\right). $$
After extracting the features from the images (here: a vector of length 512), the difference in latent space is calculated with element-wise absolute difference as follows.
$$ {\overrightarrow{w}}_1=\mid {\overrightarrow{h}}_1-{\overrightarrow{h}}_2\mid, \hskip2em {\overrightarrow{w}}_2=\mid {\overrightarrow{h}}_3-{\overrightarrow{h}}_4\mid $$
$ {\overrightarrow{w}}_1 $
and
$ {\overrightarrow{w}}_2 $
are then concatenated and the resulting vector
$ \overrightarrow{w} $
passed to a fully connected (FC) layer for gating the outputs,
$ {\overrightarrow{w}}_1^{\prime } $
and
$ {\overrightarrow{w}}_2^{\prime } $
, from each model (see Patel et al., Reference Patel, Choromanska, Krishnamurthy and Khorrami2017 and Figure 1) for late fusion. The gated outputs are fused with an element-wise mean operation
$ \left({\overrightarrow{w}}_1^{\prime }+{\overrightarrow{w}}_2^{\prime}\right)/2 $
and passed into an output layer, producing a scalar output
$ z $
as FC(
$ \overrightarrow{w} $
), which is processed through a sigmoid function
$ \sigma $
to quantify whether a pair of images
$ \overrightarrow{A} $
and
$ \overrightarrow{B} $
is similar or not. Lastly, each output (
$ {\overrightarrow{w}}_1 $
and
$ {\overrightarrow{w}}_2 $
) from the models
$ {f}_1 $
and
$ {f}_2 $
is individually passed into a fully connected output layer with a sigmoid nonlinearity and used in constructing the loss function over
$ N $
image pair samples,
$ \sigma \left({z}_1\right) $
and
$ \sigma \left({z}_2\right) $
, where
$ {z}_1 $
is FC(
$ {\overrightarrow{w}}_1 $
), and
$ {z}_2 $
is FC(
$ {\overrightarrow{w}}_2 $
).
The loss function in the classification training process employs the standard binary cross entropy (BCE) for image similarity in a Siamese neural network architecture,
$$ BCE\left(y,\hat{y}\right)=-\frac{1}{N}\sum \limits_{i=0}^N\left({y}_i\hskip0.2em \log \left({\hat{y}}_i\right)+\left(1-{y}_i\right)\log \left(1-{\hat{y}}_i\right)\right) $$
where
$ \hat{y}=\sigma (z) $
, a given target label
$ y $
(0 and 1 representing “not similar” and “similar,” respectively), and
$ N $
is the number of samples in the dataset.
We now define two sets of classification labels to compose the overall classification loss. If during training a pair of images is selected from the same rock sample, we define
$ y=1 $
, otherwise
$ y=0 $
. For metadata similarity labels, if a pair of images
$ \overrightarrow{A} $
and
$ \overrightarrow{B} $
have similar metadata, where the relative difference of the metadata is less than a threshold of 10%, then we define
$ {y}^{\prime }=1 $
, otherwise
$ {y}^{\prime }=0 $
. Based on these two labeling methods, the classification loss
$ {\mathrm{\mathcal{L}}}_{\mathrm{clf}} $
is defined as follows:
$$ {\mathrm{\mathcal{L}}}_{\mathrm{clf}}= BCE\left(y\cdot y^{\prime },\sigma (z)\right)+ BCE\left(y,\sigma \left({z}_1\right)\right)+ BCE\left(y^{\prime },\sigma \left({z}_2\right)\right) $$
(the blue, orange, and gray output sections of the network in Figure 1, respectively). The cross term
$ y\cdot y^{\prime } $
ensures that images are only counted as “similar” during training when both branches of the double Siamese network classify them as similar.
2.1.2. Metadata regression
As described earlier, we augment the loss function with a metadata regression term to preferentially retrieve images with similar properties in the inference phase. For our digital rock application, we choose porosity and permeability (see Saxena et al., Reference Saxena, Hows, Hofmann, Freeman and Appel2019a). Model
$ {f}_2 $
is regularized with a regression loss
$ {\mathrm{\mathcal{L}}}_{\mathrm{reg}} $
. A regression module composed of a fully connected layer and an output layer is added in
$ {f}_2 $
, as colored with dark gray in Figure 1. Our approach does not rely on segmentation but predicts rock properties directly from input images via the regression module trained with lab measurements as labels. We use an L1 loss for the metadata regression task
$ {\mathrm{\mathcal{L}}}_{\mathrm{reg}} $
because it provides the best performance.
The total loss function is a combination of the classification and regression loss:
$$ \mathrm{\mathcal{L}}={\mathrm{\mathcal{L}}}_{\mathrm{clf}}+\alpha {\mathrm{\mathcal{L}}}_{\mathrm{reg}} $$
Note that the regression term in (2.5) is multiplied with an adjustable scaling factor
$ \alpha $
to balance the two loss terms so that the magnitude of the classification loss approximately equals the regression loss. Here, we selected a scaling factor
$ \alpha =10 $
. Please see pseudo code of training DSNN in supplementary material section below for details.
2.2. Inference
After training our DSNN, the backbone networks
$ {f}_1 $
and
$ {f}_2 $
are used for image retrieval and metadata prediction as shown in Figure 3. Before inference, we first precompute the encoded features from the two trained backbone networks
$ {f}_1 $
and
$ {f}_2 $
for all samples in the existing database, including partially labeled samples that were not used for training. When a new query image is presented, its encoded features
$ {\overrightarrow{h}}_1 $
and
$ {\overrightarrow{h}}_3 $
are computed through the two trained backbone networks
$ {f}_1 $
and
$ {f}_2 $
, respectively. A similarity measure containing both image and metadata terms can be computed for all existing samples in our data base using these encoded features. First, the outputs of both networks,
$ {f}_1 $
and
$ {f}_2 $
, are fused with an element-wise mean operation as
$ \overrightarrow{z}=\left({\overrightarrow{h}}_1+{\overrightarrow{h}}_3\right)/2 $
. Then, the Euclidean distance is used to compare a query image with the images in the data base (see Figure 3) and retrieve the closest matches. In addition, the output of model
$ {f}_2 $
is connected with the trained regression module to separately predict metadata for the query image (here: the physical properties porosity and permeability). The DSNN then predicts physical properties in addition to the main CBIR task.
Figure 3. Image retrieval system with the trained DSNN. Note that the feature encoding for the images in the data base (top) is computed offline and not during inference for a new query image (bottom).
3. Results and Discussion
3.1. Experimental setup
We now demonstrate the performance of our physics-informed DSNN architecture on the two datasets of micro-CT and thin-section images, as shown in Figure 2. All calculations were run on a workstation with a 2.50Ghz Intel Xeon® Gold 6248 v80 CPU with 4 Nvidia Tesla® V100-SXM2 GPUs. Our DSNN takes 8 hours for training using
$ 7740 $
images of 2D micro-CT with
$ 1000 $
epochs and 12 hours for training with
$ 12240 $
images of the thin-section dataset with
$ 600 $
epochs. The most time-consuming part of the training process in every epoch is convolution. The DSNN is implemented with Pytorch 1.7.1 (Paszke et al., Reference Paszke, Gross, Massa, Lerer, Bradbury, Chanan, Killeen, Lin, Gimelshein, Antiga, Desmaison, Wallach, Larochelle, Beygelzimer, d’Alché-Buc, Fox and Garnett2019). The (pretrained) ResNet-18 (He et al., Reference He, Zhang, Ren and Sun2016) is used as a backbone network for DSNN, the last two ResNet layers were removed for feature extraction. The number of trainable parameters is 22.4 million with 26 GB memory usage on each GPU.
The DSNN is trained with a balanced set of selected pairs of images (i.e. half of the pairs are from the same rock samples, see above). We down-sampled all images to
$ 224\times 224 $
, randomly flipped images horizontally and vertically with 50% probability for data augmentation, and then normalized with z-scaling. For hyperparameters, we select 1000 epochs for micro-CT and 600 epochs for the thin-section dataset. We set the batch size to 1024 and the learning rate to 0.00001 and 0.0001 for the micro-CT and thin-section dataset, respectively. We use the Adam optimizer to minimize the objective function. To compare with other approaches, we use an ImageNet-pretrained ResNet-34 ("Vanilla ResNet") as the baseline for image retrieval performance. It matches the number of trainable weight parameters of our Double Siamese ResNet-18 network to allow for a fair comparison.
3.1.1. 2D micro-CT image dataset
We have access to a proprietary database of 3D micro-CT images with image sizes
$ {1000}^3 $
or
$ {1300}^3 $
, where pixel size ranges from 1.01 μm to 2.12 μm. From these raw samples, 90 2D slices are sampled from each 3D image, which provides a total of 9000 2D slices. 7740 slices are used for training and 1260 slices used for testing the regression performance. To test the image retrieval performance, a separate test set of 1527 2D slices out of 509 3D images is used.
3.1.2. Thin-section Image dataset
The thin-section image dataset is composed of 264 16-bit RGB raw 2D images of image size
$ 5000\times 5000 $
with pixel size of 1.41 μm. Similar to the 2D micro-CT image dataset, 60 2D slices are extracted from each raw image, for a total of 15840 slices. 12240 images are used for training, and 3600 images for testing the regression performance. For testing image retrieval performance, 37 out-of-sample images are used, a random slice from each of the 37 samples. Porosity is used for regression in this dataset because the permeability is not broadly available.
3.2. Image retrieval performance
For the retrieval performance (estimation error) metric, we use mean average precision at k (mAP@k) (Christopher et al., Reference Christopher, Manning and Raghavan2008). It captures how many of the known similar images the network can retrieve on average when a query input image is presented. The total similarity between the query image and the retrievals in the testing phase is quantified with the known “reservoir” designation of the rock samples from which the images were captured (see Section 2). True to the practical application of our method, images are labeled “similar” in the test set if their origin is the same subsurface reservoir. Note that this is different from the DSNN training process where “rock sample” was used as label. Based on this metric, our DSNN outperforms the pretrained (“vanilla”) ResNet-34 by 9.8%, the Siamese ResNet-18 by 3.1% and the Siamese ResNet-34 by 1.4% in the 2D micro-CT dataset (see Table 1). In addition to the vanilla ResNet-34, a single Siamese ResNet-18 and a ResNet-34 network was post-trained with our rock images as described above for the DSNN. Comparing with these single Siamese networks, our Double Siamese ResNet-18 shows superior performance.
Table 1. Image retrieval performance comparison.
Note. The KAZE results (Thiele et al., Reference Thiele, Saxena and Hohl2020) are listed for reference.
On the thin-section dataset, our DSNN also outperforms the vanilla ResNet-34 by 22.5%, a Siamese ResNet-18 by 4.3% and a Siamese ResNet-34 by 4.0%. Even though ResNet-34 has more trainable parameters, it does not perform as well as a ResNet-18, which we attribute to overfitting. Generally speaking, we note that all algorithms and models do not perform as well on thin-section data as the micro-CT data due to the lower number of raw samples. Despite that, our DSNN again shows the best image retrieval performance due to the added rock properties.
3.3. Regression performance
As noted earlier, the regression portion of the DSNN network not only improves the image retrieval performance but also enables the prediction of physical properties for a query image. This can happen in two ways: First, the user retrieves the set of best-matching images from the database using the DSNN and “manually” looks up the physical properties of those. Second, and in a more automated fashion, the trained regression module (Figure 1) can be used directly to predict the physical properties of a query image. The performance of this process (approximation error and estimation error from out-of-sample testing) is shown in Figures 4 and 5. The coefficient of determination
$ {R}^2 $
is defined as
$ 1-{\sum}_i{\left({y}_i-{f}_i\right)}^2/{\sum}_i{\left({y}_i-\overline{y}\right)}^2 $
, where
$ {f}_i $
and
$ \overline{y} $
represent prediction and mean of prediction, respectively. Note that only our DSNN has a regression module that can be used in this way, the other neural networks used for comparison in this study cannot predict physical properties. The log of permeability is scaled with a min-max normalization for the training process and permeability itself is then calculated back for testing the regression performance. In terms of rock property prediction accuracy, the
$ {R}^2 $
for both porosity and permeability, respectively, is 0.8 and 0.9 with our DSNN on the test set after 1000 training epochs.
Figure 4. Regression coefficient of determination
$ {R}^2 $
for porosity and permeability in the 2D micro-CT dataset.
Figure 5. Regression coefficient of determination
$ {R}^2 $
for porosity in the thin-section dataset.
For the thin-section dataset, we only use porosity in the regression due to the small sample size of available permeability data. Figure 5 shows a final
$ {R}^2=0.7 $
for porosity prediction. This is still considered excellent predictive performance in rock physics and on par with orders-of-magnitude more time-consuming direct computer simulation (Saxena et al., Reference Saxena, Hofmann, Alpak, Berg, Dietderich, Agarwal, Tandon, Hunter, Freeman and Wilson2017). Our predictive DL approach thus provides a viable alternative to computationally expensive pore-scale flow simulations.
3.4. Quantitative image retrieval analysis
In this section, we assess the DSNN performance for rock property predictions. We compare query images with their best-matching three retrievals in all samples of a separate test set. This is of paramount importance for digital rocks because users often infer rock properties by studying “similar” rocks returned from the CBIR query.
For the 2D micro-CT dataset, porosity and permeability of a query image and the top three retrievals are compared in Figure 6, with a mean and a standard deviation of (0.21, 0.07) for the porosity and (1.52, 1.01) for permeability. For a high-performing network all top three retrievals should lie on the diagonal line. From Figure 6a it is apparent that the vanilla ResNet-34 does not perform as well as our DSNN. This is because it was not trained on rock images. The average perpendicular distance of the top-1,2,3 retrievals to the diagonal line is 0.0043 from the vanilla ResNet, and 0.002 from our DSNN, about a two times better retrieval quality in terms of similar porosity. Figure 6b shows that our DSNN retrieves most of the images in each sample correctly. The average perpendicular distance from the diagonal of the top-1,2,3 retrievals for permeability is 2.17 from the vanilla ResNet, and 0.42 from our DSNN, about five times better in terms of similar permeability. While some images are not retrieved from the same sample, the porosity differences between the query and the retrievals are small. Due to the rock property similarity-based labels and the regularization with the property regression module, the image retrieval quality goes up. In Figure 7, the improved retrieval performance of our algorithm is shown in histogram form where the horizontal axis represents metadata difference between query image and its top three retrievals. Our DSNN retrieves rock samples which have better matching porosity and permeability than the vanilla ResNet-34, as demonstrated by higher peaks near zero and lower peaks away from zero. The permeability recovery performance for DSNN is also shown in Figure 7.
Figure 6. Cross plot of porosity and permeability of query images and their top three matching retrieval images for the 2D micro-CT dataset. (a and b) are for porosity and (c and d) are for permeability. Our DSNN (right panels) reliably retrieves rock samples with similar rock properties; most of the retrievals are on the diagonal line. The top, second-, and third-best matches are colored blue, orange, and green, respectively.
Figure 7. (Porosity, permeability) of query and (porosity, permeability) of top three retrievals compared for the 2D micro-CT dataset. (a and b) are for porosity and (c and d) are for permeability. The error histogram for our DSNN shows a narrower spread around zero (perfect match) than the vanilla ResNet-34.
The thin-section image results are shown in Figure 8. Only porosity is used as physical quantity with (0.26, 0.06) as mean and a standard deviation. Figure 8a shows that the vanilla ResNet-34 is unable to consistently retrieve samples with similar porosity. More than half of the samples are far off the diagonal line. Our DSNN (Figure 8b) reliably retrieves rock samples with similar porosity. Figure 8c,d demonstrate that the DSNN retrieves more similar rock samples with a higher peak at zero compared with the vanilla ResNet-34. In terms of the average perpendicular distance metric, the vanilla ResNet shows 0.03, and our DSNN still shows improvements as 0.02.
Figure 8. Comparison of the porosity of a query image and the top three ranked images retrieved in the thin-section dataset. (a) and (b) are cross plots of porosity of query images and the top3 matching retrieval images. (c) and (d) are histograms to display the cross plots in terms of distributions. Similar to the results in the 2D micro-CT dataset, our DSNN in (d) has a higher peak at zero.
4. Practical Examples with Discussion
Figures 9–12 show examples of actual (both Micro-CT and thin-section) rock images retrieved from the Shell image database using both a vanilla ResNet-34 and our DSNN. A comparison demonstrates the superior performance of our DSNN architecture and training procedure. For the micro-CT dataset, in Figures 9 and 10, the first row (red box) contains the top three ranking images retrieved from the same rock sample as the query image. The ranking metric is the L2 distance of the feature vectors described in Section 2.2. The second row (green box) includes the top three ranked retrievals from the same geological subsurface rock formation (“reservoir”) as the query but not from the same physical rock sample. The third row (blue box) shows the top three ranked retrievals from images of rock samples that originate in different reservoirs than the query image. This similarity assessment is used in practice to identify geologically similar rocks already in the database that can serve as useful analogs with known properties. In these figures, porosity is abbreviated as por, permeability as perm, and absolute retrieval ranking as ABS ranking. Note that the rock properties of many images in the database were not measured, as indicated by NaN.
Figure 9. Qualitative retrieval results with reservoir E2 on 2D micro-CT dataset. Vanilla ResNet-34 (yellow) vs our DSNN (purple).
$ por $
and
$ perm $
are lab-measured porosity and permeability.
$ po{r}^{\prime } $
and
$ per{m}^{\prime } $
represent predicted porosity and permeability from DSNN.
Figure 10. Qualitative retrieval results with Reservoir K on 2D micro-CT dataset. Vanilla ResNet-34 (yellow) vs our DSNN (purple).
In Figure 9, one slice of a rock sample from reservoir “E2” is used as query (top left in the first row). Ideally, the images retrieved from the same rock sample as the query image should be among the top-ranking images, and the query image itself should be the highest-ranked result. Note that the images retrieved by the vanilla ResNet-34 from the same rock sample are the same ones as those retrieved by our DSNN but have lower absolute ranking, an undesirable discrepancy. The secondand third-ranking images retrieved by the vanilla ResNet-34 are from a different rock sample; the two retrieved images are also visually dissimilar to the query image and belong to different rock types defined by distinct rock property trends. That is despite the fact that the database included multiple 2D slices from the same rock sample from which the query image was selected. It is apparent that a pretrained vanilla ResNet-34 does not function well for retrieving similar images in the materials domain.
On the other hand, our DSNN successfully retrieves the secondand third-ranked images from the same rock sample as the query image. Our DSNN also shows stable regression performance on porosity por’ and permeability perm’: A 10% error in porosity and log(permeability) is considered satisfactory in practice (Andrä et al., Reference Andrä, Combaret, Dvorkin, Glatt, Han, Kabel, Keehm, Krzikalla, Lee, Madonna and Marsh2013; Woods, Reference Woods2014; Saxena et al., Reference Saxena, Hofmann, Alpak, Berg, Dietderich, Agarwal, Tandon, Hunter, Freeman and Wilson2017). The blue box (third row; top three ranked retrievals from different reservoirs) is a tool to assess which other reservoir rocks can be used as analogs for the query rock sample. Rocks of similar geologic origin can be expected to have similar rock properties. Reservoirs E2 (query image), E1, and Z are from the same geological formation but were penetrated by different wells several kilometers away from reservoir E2. The three reservoirs are located at different depths. Reservoirs E2, E1, and Z have seen the same degree of rock compaction over geologic times, and are known to have similar rock properties and trends. Note how all images retrieved by our DSNN are visually similar to the query image. Our DSNN correctly predicts reservoir E1 as a good analog for reservoir E2. It also predicts reservoir Z as the next best analog for reservoir E2. The retrievals from the vanilla ResNet-34 are not satisfactory since the top retrieved image from a different reservoir (blue box) is from reservoir C which is from a very different geological formation. The next two predictions from the vanilla ResNet-34, in the blue box, are from reservoir E1 which is the same geological formation as reservoir E2 but is known to be less compacted and of higher porosity. Note also how multiple retrieved images are visually dissimilar to the query image.
Figure 10 shows the same results as Figure 9 but for a different query image obtained from a rock sample from reservoir “K”. The key difference between the results predicted by the two networks is in the prediction of similar images from different reservoirs (bottom blue box in Figure 10). Results from the vanilla ResNet-34 are not satisfactory for two important reasons: Firstly, for the query image from reservoir K, the retrieved images from reservoirs R and T have significantly higher concentration of “heavy” minerals (brighter pixels in the images) and are visually dissimilar to the query image. Secondly, reservoirs R and T have different geologic origins to reservoir K. Reservoirs R and T are aeolian deposits, whereas reservoir K has a turbitic origin. Our DSNN predicts reservoir Z as a good analog for reservoir K; both of these reservoirs are turbidite deposits and are correctly predicted to have similar porosity and permeability.
The thin-section dataset results are shown in Figures 11 and 12. Only one thin-section sample was analyzed for the purpose of our discussion here so that only two rows of images are shown. The first row (green box) represents the top three retrievals from the same reservoir. The bottom row (blue box) includes the top three retrievals from different reservoirs in our database. One slice of reservoir “U” is used as query (Figure 11). Based on the absolute ranking from the retrieval, both the vanilla ResNet-34 and our DSNN generally retrieve similar rock samples from the same reservoir. The key difference between the two networks is seen in the blue box (second row; top three retrievals from different reservoirs): The vanilla ResNet-34 predicts rocks from reservoir D as the top analogs, whereas our DSNN predicts reservoir W as the closest analog for reservoir rock U. Reservoir W (but not reservoir D) is known to have a similar degree of rock compaction and range of rock properties as reservoir U. This confirms that our DSNN is able to capture attributes that identify petrophysically similar rocks from different reservoir. The retrieval performance is improved by adding a second Siamese network with a regression module and by training with both image and metadata simultaneously.
Figure 11. Qualitative retrieval results with reservoir U on the 2D thin-section dataset. Vanilla ResNet-34 (yellow) vs our DSNN (purple).
$ por $
is a lab-measured porosity and
$ po{r}^{\prime } $
represents porosity predicted from DSNN.
Figure 12. Qualitative retrieval results for reservoir D on the 2D thin-section dataset. Vanilla ResNet-34 (yellow) vs our DSNN (purple).
Figure 12 shows the same analysis as in Figure 11 but for a query image from reservoir D. The vanilla ResNet-34 predicts reservoir U as one of the top three images retrieved from different reservoirs (blue box). However, the two rock images from reservoirs D and U are not only different in porosity but are also dissimilar in texture and mineralogy. The images retrieved from our DSNN are consistent with the known rock physics.
5. Conclusions
We propose a novel double Siamese network for content-based image retrieval of industrial materials images in the form of “digital rocks.” The performance-limiting challenge is the low number of available labeled images for supervised learning. To still maintain high-capacity networks without overfitting, we leverage pretrained ResNet-18 models to warm-start the training process. We also augment the training loss by incorporating prior knowledge in the form of physical parameters, that is, porosity and permeability. We demonstrate superior empirical performance for image similarity through the domain standard map@k metric. Our approach also allows for estimation of metadata from images, and we observe excellent predictive power in terms of
$ {R}^2 $
. We present a sample of qualitative search results, demonstrating the ability of our DSNN to provide relevant and impactful image retrieval results. Future work includes an extension to full 3D, that is, to using 3D images of rock samples and using 3D kernels in the neural network. Permeability prediction could thus potentially be improved because its volumetric nature cannot be captured completely with 2D kernels. Our novel approach has the potential to replace direct computer simulation of physical properties in digital industrial materials images with orders-of-magnitude faster direct prediction.
Pseudo code of our proposed DSNN on the training process is shown below.
Algorithm 1 Pseudocode for training the double Siamese Neural Network. See Figure 1 for the diagrammatic representation.
Input: Pairs of input images
$ \left({X}_A,{X}_B\right) $
, image similarity-based label y, metadata similarity-based label y’, sigmoid
$ \sigma $
, pairs of metadata label
$ \left({y}_{metaA},{y}_{metaB}\right) $
, loss balancing term
$ \alpha =10 $
, number of iterations i
1: for 1 to i do
2:
$ {\overrightarrow{h}}_1={f}_1\left({X}_A\right),{\overrightarrow{h}}_2={f}_1\left({X}_B\right) $
⊳ Feature extraction via backbone network
$ {f}_1 $
3:
$ {\overrightarrow{h}}_3={f}_2\left({X}_A\right),{\overrightarrow{h}}_4={f}_2\left({X}_B\right) $
⊳ Feature extraction via backbone network
$ {f}_2 $
4:
$ {\overrightarrow{w}}_1=\mid {\overrightarrow{h}}_1-{\overrightarrow{h}}_2\mid, {\overrightarrow{w}}_2=\mid {\overrightarrow{h}}_3-{\overrightarrow{h}}_4\mid $
⊳ Calculate feature absolute difference
5:
$ \overrightarrow{w}= softmax\left({\overrightarrow{w}}_1\Big\Vert {\overrightarrow{w}}_2\right) $
⊳ Concatenate two vectors to generate two scalars
$ {s}_1,{s}_2 $
6:
$ {\overrightarrow{w}}_1^{\prime }={s}_1\cdot {\overrightarrow{w}}_1,{\overrightarrow{w}}_2^{\prime }={s}_2\cdot {\overrightarrow{w}}_2 $
⊳ Gating feature vectors
7:
$ z= FC\left(\left({\overrightarrow{w}}_1^{\prime }+{\overrightarrow{w}}_2^{\prime}\right)/2\right) $
⊳ A final output with late fusion
8:
$ {z}_1= FC\left({\overrightarrow{w}}_1\right) $
⊳ A final output from model
$ {f}_1 $
9:
$ {z}_2= FC\left({\overrightarrow{w}}_2\right) $
⊳ A final output from model
$ {f}_2 $
10:
$ {\mathrm{\mathcal{L}}}_{\mathrm{clf}}= BCE\left(y,\sigma \left({z}_1\right)\right)+ BCE\left(y\cdot {y}^{\prime },\sigma (z)\right)+ BCE\left({y}^{\prime },\sigma \left({z}_2\right)\right) $
⊳ Apply BCE loss for classification
11:
$ {\mathrm{\mathcal{L}}}_{\mathrm{reg}}= SmoothL1\left({y}_{metaA},{\overrightarrow{h}}_3\right)+ SmoothL1\left({y}_{metaB},{\overrightarrow{h}}_4\right) $
⊳ Apply SmoothL1 loss for regression
12:
$ \mathrm{\mathcal{L}}={\mathrm{\mathcal{L}}}_{\mathrm{clf}}+\alpha {\mathrm{\mathcal{L}}}_{\mathrm{reg}} $
⊳ Total loss function for backpropagation
13: end for
Acknowledgments
We thank Ronny Hofmann, Dipanwita Nandy, Rishu Saxena, and Lalit Gupta for helpful discussions, and Shell for granting permission to publish this work.
Author contribution
Conceptualization and investigation: M.S.S. and C.T.; Visualization and writing original draft: M.S.S.; Methodology, software, and validation: M.S.S., J.V., N.S., and D.H.; Project administration, resources, and supervision: J.V., N.S., and D.H.; Reviewing and editing: M.S.S., C.T., J.V., N.S., and D.H.; All authors approved the final manuscript.
Competing interest
The authors declare none.
Data availability statement
Source codes are uploaded on this url: https://github.com/sede-open/RockIR.
Funding statement
This research was funded by Shell International Exploration and Production (US) Inc.
Open access
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