Clustering Cum Polar Coordinate Feature Transformation (C-PCFT) Approach to Identify Pores in Carbonate Rocks

Most of the world’s oil reserves and natural gas are stored within carbonate rock’s pores and fractures. Pores and fractures are quite popular for predicting the amount of petroleum under an adequate trap condition. Hence, their petrophysical properties, such as shape and size, are paramount for accurately predicting the reservoir’s state and condition. Current modelling techniques are mostly based on manual and expert judgement which are time-consuming and cost-intensive. In this study, we devised a robust and scalable image processing framework that uses the combination of pixel-based clustering approach with a polar coordinate feature transformation technique to intelligently identify the pores of carbonate rock samples. We reported that such a method can be effective in detecting pores of different shapes and sizes in an automated fashion. We rigorously tested the proposed method on the computed tomography-scanned micro-images of a carbonate rock sample, and the results demonstrate improved identification accuracy of the proposed method than the current deep learning counterparts. Another key advantage compared to deep learning methods, the proposed method does not require extensive training on data, which saves time and effort without being computationally too expensive.


I. INTRODUCTION
The identification and modelling of pores and fractures statistics are becoming essential services within the oil and gas industry for predicting the state and condition of the reservoir.These advancements possess significant potential for various applications, such as facilitating contaminant transport analysis, enabling secure storage of radioactive waste in The associate editor coordinating the review of this manuscript and approving it for publication was Gang Mei .bedrock formations, and optimizing CO2 storage solutions.[1].These analyses primarily rely on an accurate estimation of the petrophysical characteristics of rocks, primarily as the porosity within reservoir rocks is accepted as key parameters for reservoir characterization.To our knowledge, most hydrocarbon reservoirs are composed of sedimentary rocks, in which porosity values vary between 10% and 40% for sandstones and between 5% and 25% for carbonates [2].Further we found that the sizes and shapes of the pores and fractures are extremely irregular.However, in some studies, the pore shapes are nearly-spherical, whereas the fracture shapes are mostly linear [3].To date, there are few methods that use classification techniques (such as linear support vector machines (SVMs)) to automatically detect pores and fractures using fractal dimensions on resistive image logs, in addition, some researchers have tried to experiment with deep learning systems.However, in such cases, the accuracy achieved may not translate good enough in low quantity and low-quality (poor resolution and chrominance) images.Such limitations are quite abundant in a variety of practical situations.Hence, in the current work, we attempted to propose a strategy that is robust even with low-quality image samples and with a comparatively smaller dataset (<300 images), as an input.The proposed method uses a combination of the pixel-based clustering approach combined with polar coordinate feature transformation and an image processing technique to efficiently identify the pores of carbonate rock samples.

A. FEW CONTEMPORARY APPROACHES
Globally, scientists and researchers are trying to address the above-mentioned challenges by employing various intelligent pattern recognition techniques and deep learning approaches.In a study, Lucia et al. [4] devised a carbonate rock porosity classification strategy based on data derived from visual inspection of rock samples in the field or laboratory.However, the proposed strategy [4] does not implement any intelligent image analysis mechanism in an automated fashion; rather, it relies on the skills of concerned geologists for manual porosity classification.This might often suffer from inevitable human bias and error.In an early pioneering study, Ehrlich et al. [5] proposed a method that uses colourful (RGB) image samples as an input to identify their micropores and fractures.We suspect that this method may be relevant for low-quality grayscale images.Similar remarks can be made for the work of Funk et al. [6], who described the pore size distributions of rock samples by analysing the petrophysical properties of rocks extracted from their RGB image samples.Van den Berg et al. [7] presented a sophisticated image analysis method for porosity classification that required high-quality images (resolution and chrominance) as inputs.The resolutions of the input image samples used in [7] were of high quality, and the sizes of pores and fractures were big enough, as they were visually identifiable (seen from the presented image samples).Perring et al. [8] considered high-resolution images as input samples for a semi-automated rock texture identification technique based on simple image processing technique.However, this strategy may suffer in the absence of high-quality input samples.Moreover, only a few contemporary studies [9], [10], [11], [12], [13] can be found in the area of porosity classification that employs simple image analysis techniques.Most of them use high-resolution images; however, the above proposals might suffer in the absence of high-quality image samples, which is a problem that we are trying to solve.Moreover, the above illustrations were lacking modern machine-learning approaches and potential for scalability on a commercial basis.
A few approaches employing machine learning techniques, such as SVM classification, deep learning etc. [14] have also achieved porosity classifications of different reservoir rocks in contemporary literature, and noteworthy works are presented below.To identify natural open fractures, Leal et al. [15] employed the fractal dimension of images as well as gamma rays and resistivity logs as the input dataset to an SVM classification model.However, the detection performance accuracy was subpar and might have been hampered by the low image quality.In another interesting study, Abedini et al. [16] employed a deep learning strategy and an autoencoder to identify pores and fractures in rock image samples.However, the images used [16] are easily visually distinguishable and identifying pores and fractures in the image samples is unarguably effortless and does not require deep learning to address the problem.Above all, this approach will unquestionably fail if there are fewer training image specimens with a higher degree of similarity in the pore and fracture patterns.As shown above, current classification and identification techniques need to be improved in order to increase their accuracy specifically when dealing with small training sets of low-quality image samples, which poses a real challenge in many real-world scenarios and may reduce the accuracy of best-in-class models.As a result, in the current study, we made an attempt to come up with a method that is effective with low-quality grayscale image samples using a small dataset (<300 images).The proposed method uses a combination of a pixel-based clustering approach with polar coordinate feature transformation and image processing techniques to intelligently identify the pores of carbonate rock samples.Furthermore, it can be effective in detecting pores of different shapes and sizes, such as lines, circles or other parametric curves, by choosing the correct parameter space for transformation by fine-tuning.

B. NOVELTY/CONTRIBUTION OF THE CURRENT WORK
Based on the above reviews and illustrations, machine learning and deep learning-based image-processing algorithms face significant problems in the absence of large and high-quality training data.Furthermore, such algorithms need to be heavily tuned as per the quantity and quality of the input images used.In this study, we are presented with computerised tomography (CT) images of a carbonate rock slab, the quality of which is extremely poor owing to the low-light condition.
Moreover, owing to the irregular shapes of pores, deep convolutional neural networks (CNNs) can often fail to identify pores and fractures in low-quality grayscale image samples.Furthermore, it can be computationally expensive to train networks for each type of shape and require large training datasets.Hence, in this current strategy, we devised a pixel-based clustering mechanism cum polar coordinate transformation, which is robust even in the absence of large and high-quality image datasets.It can also be effective in detecting pores of different shapes and sizes, such as lines, circles, and other parametric curves, by selecting appropriate parameter space for polar coordinate transformation.The proposed approach can be combined with unsupervised machine learning algorithms to cater to irregular shape detection without being computationally expensive and without having the appetite for a considerable amount of datasets for training the model.
To summarize, our proposed algorithm uses a pipeline of image processing techniques, Clustering, and Hough transformation for detecting dark pores which is much ahead of the best-in-class deep-CNN algorithms in terms of performance and accuracy.We also did research if similar techniques have been used to improve the detection of pores in the petrophysics domain provided with low training data quantity and quality.We couldn't find any such comparable references.Hence, we are confident to state that our proposed algorithm demonstrates a novel approach for detecting the Pores given highly unfavourable inputs and constraints.

A. APPROACH FOR CLUSTERING CUM POLAR COORDINATE FEATURE TRANSFORMATION (C-PCFT)
C-PCFT primarily comprises of 2 major components i.e.Clustering and Hough Transformation and is further enhanced using image processing techniques.Both these concepts are tried and tested for their usefulness and superiority in their respective domains.Next, we would like to dig deeper into both components for understanding the conceptual framework of C-PCFT.This will be followed by summary of C-PCFT and Algorithm

1) FUNDAMENTAL OF HOUGH TRANSFORMATION
The Hough technique is used to detect noise-laden images, with object such as circles, straight lines, parabolas, and other curves, that can be captured within a limited set of characteristics, such as radius, angle, and orthogonality.
If we try to understand this as a concept [17], [18], the group of all lines in the image translates into a 2 parameter combination.Any line can be defined by a point in the parameter space if we can constrain this parameter.As illustrated in Figure 1, a line is characterised by the angle θ of its normal and its algebric distance ρ from its origin.The XY -coordinate plane has 2 coordinate axes, the X and Y -axis.Assume that now we have a set 2, and we want to find a set of straight lines that fit them.We transform the points (x i , y i ) into the sinusoidal curves in the θ-ρ plane, as illustrated in Figure 3, defined by following of the line in terms of ρ and θ can be written as follows: Furthermore, it can be rewritten as follows: If θ can be constrained to a range of [0, π], with this constraint, every line in the x-y plane corresponds  to a unique point in the θ-ρ plane.We can show that the curves of the collinear points under the scope passes through a common point in the transformed θ-ρ plane.This point in the θ-ρ plane, that is, (θ 0 , ρ 0 ), as illustrated in Figure 3, defines the line passing through the collinear points.Therefore, the problem of detecting collinear points can be translated to the objective of identifying concurrent curves.
The transformation is implemented by quantifying the Hough parameter space as in Figure 4. Collinear points as evident in Figure 4(a) and the accumulator cells as in Figure 4(b) [18].Details of these accumulators are captured in Figure 5 as an Algorithm.As the algorithm progresses, each x i and y i is transformed into a (θ, ρ) plane/curve, and the accumulator cells that lie along this curve are incremented.The resulting peaks in the accumulator array represent strong evidence that a corresponding straight line exists in the image, as shown in Figure 4(b).
This procedure can be used to detect other features.For instance, in the case of circles, the parametric equation is where (a, b) is the centre of the circle in 2-dimension space and r is the radius.The parameter space would be 3-dimensional, that is, (a, b, r).Here, the computational complexity of the algorithm increases as we now have a 3D parameter space and 3D accumulator.In general, the computation and size of an accumulator array increase in the polynomial mode with the number of parameters.For ellipses, a 5D parameter space is constructed.

2) FUNDAMENTALS OF CLUSTERING
Clustering is an iterative algorithm that attempts to partition a dataset (pixels here) into K distinct clusters (chosen or optimised using some techniques).It makes decisions to assign data points to a particular cluster to minimise the sum of the distance between data points and a cluster's chosen centroid.
The basic idea behind testing each pixel is to determine whether it is similar to pixels already clustered to be in the same cluster or to a new cluster.The common criterion of similarity is based on the pixel intensity similarity.A simple method to compare whether a point and centroid of the cluster are similar is by measuring the Euclidean distance between their colour vectors (pixel).We need to check whether this distance is above or below a certain threshold.Initial centroids are grown as per an adaptation that does the similarity between a pixel and centroid as their distance.
The K -means algorithm, where k is the parameter, divides n samples into k clusters, such that the intra-clusters distance is low and the distance between inter-clusters is high.In this study, the black pixels in the binarised image are used as data inputs for the cluster.The steps are as follows: 1) Given n datapoint X: Partition each pixel to the cluster C j closest to the center, i.e., 4) Update each category center to the mean value of all sample pixels belonging to the category: x i (5) 5) Repeat the above till the change in the category center is less than a chosen threshold.

3) PROPOSED METHODOLOGY: C-PCFT
This section briefly summarises the proposed mechanism of C-PCFT for pore detection in an image.The proposed system comprises three stages as follows: 1.
Stage 1 Image pre-processing: Pre-processing on the images to convert petrographic assets (rock images) to conveniently highlight key petrographic properties and reduce noise, as shown in Figure 6.

2.
The following steps are performed: a.
Resize the sample images.b.
Denoise the resized image.Apply circular transformation on the generated image.c.The Hough's transformation utilises a voting mechanism, such that an 'accumulator' is voted up [19] for finding the interest structure, as shown in Figure 4. d.
Iterate through the entire recreated image and vote the accumulator using the Hough transformation to detect circular edges [19].Find the maximum of the accumulators to detect the edges.The detailed experimental verifications are given in Section III and in Table 2.

4) ALGORITHM
Algorithm as shown in Figure 5 is a stepwise pipeline of algorithms placed one after the other to leverage each other's interim outputs.Starting with the information related to the images with an average pixel intensity of 101/image, all of the photos are shown in grayscale.Each pixel in the grayscale image represents only the amount of light or the intensity information.Grayscale images are a black-and-white or grey monochrome that are exclusively composed of shades of grey.In general, the grey colour contains R, G and B of equal intensity.Specifically, we consider a pixel as dark grey if R = G = B = 50, medium grey if R = G = B = 120 and light grey if R = G = B = 200.We measured the intensity (L ( i, j)), the measure of brightness/darkness of any pixel P (i,j) ∈ G n ∈∈ M as L (i,j) = R (i,j) × 0.299 + G (i,j) × 0.587 + B (i,j) 0.114, where R (i,j) , G (i,j) and B (i,j) are the red, green and blue values respectively for P (i,j) identified by the system, respectively.For clustering purpose, we classify a pixel as dark or bright if L (i,j ∈) ∈ [0, I ] and L (i,j) ∈ (I , 255], where I is the threshold in the range 0 < I < 255.This is referenced in Algorithm as shown in Figure 5.
To summarize, we can observe the whole algorithm is divided into 7 steps.Steps 1 to 4 perform basic image processing, step 5 is related to clustering and finally, step 6 onwards undergoes Hough transformation which produces the final output of pores detection.

B. DATASET SPECIFICATIONS
To better comprehend the suggested algorithm's utility, we first go over the parameters of the rock picture samples used in this study.We used CT-scan images of a carbonate rock slab with a length of 256 mm length, 1 mm breadth and 1 mm width obtained at every 1 mm height in all xy, yz and zx planes (Figure 6).We call every image as a micro-image.
A total of 256 X 3 micro-images with varying pixel counts make up a single image dataset.Average image size is 0.17 MP and median image ratio is 416 × 416 which is stretched further to 640 × 640 to 1024 × 1024 image orientation based on algorithm needs.Pores are small in size and occupies less than 10% of the total image size.In general, each image sample has multiple pores, the total 205 image samples have 585 pores.Each pixel area is approximately limited to 4.4 µm 2 , and one micro-image creates a field view of 1 × 1 mm.Images with an average pixel intensity of 101/image, all of the photos are shown in grayscale.Each pixel in the grayscale image represents only the amount of light or the intensity information.Grayscale images are a black-and-white or grey monochrome that are exclusively composed of shades of grey.In general, the grey colour contains R, G and B of equal intensity.Specifically, we consider a pixel as dark grey if R = G = B = 50, medium grey if R = G = B = 120 and light grey if R = G = B = 200.A median filter was used to lessen the spectral noise in the CT-scan image samples that were gathered, and a linear contrast stretch was used to improve the contrast between the features.Contiguous dark areas on the edges of each image that are neither pores nor fractures were excluded before commencing the computations.Thus, we manually marked the dark regions' borders in red (R = 255, G = 0, B = 0) in order to remove them from the computation (Figure 6) during data pre-processing.Then, the areas enclosed by the red boundaries are referred to as ''effective regions.''The suggested image processing method analyses the brightness of the bordering pixels based on their L-values to comprehend the red boundary.
For validating the improvement, we maintained the data distribution splits as 60:20:20 i.e. size of training, validation and test dataset for the deep-CNN methods.In addition, KFold validation strategy was also deployed with varying K from 5 to 10, to ensure any bias due to sampling a small dataset is addressed (Table2).But for our proposed method, we did not follow any data distribution because the method does not call for any training.

III. RESULTS
This section briefly presents the results of applying the proposed algorithm to the above-mentioned dataset described 98490 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.above.Stage 1 performs the binarisation of the input images shown in Figure 7a, and the sample output is shown in Figure 7b, where the image processing clearly deals with the low-light condition robustly.The images were also considerably denoised.Note that the additional background was segmented from the image.This image can be used for the next stage, that is, Stage 2 (clustering) Figure 7c.Clustering can group pixels based on their intensity levels, and it performs fairly well in clustering pores into less-darker groups.
These clustered groups will then be used to detect edges using the Sobel convolution filter for edge detection and circular lines using the Hough transformation, and then highlight the visualised circles, as shown in Figure 8.We have showcased several pictures from different shape types and backgrounds.

A. COMPARISON WITH OTHER SIMILAR METHODS
There were several trials performed by the research so as to address similar problems.Alexander Sheshkus, et al. [20] subjected the raw image to morphological contrast, followed by a Hough Transform, before being fed into a few convolutional filters of the CNN network.Then they used 2 datasets 98492 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.namely object classification using CIFAR-10 and printed character recognition that included symbols from Russian passports.This approach is something similar to our approach in terms of combining Hough transformation with another well-known technique like CNN in this case.Improvement in the loss metric expected does not seem to be significant as opposed to the author's claim like 47.5% to 43.4% for the light model and with a heavier model from 35.7% to 34.1% on VOLUME 11, 2023 98493 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE 2.
Comparing the accuracy of the proposed approach with that of the supervised deep-CNN accuracy measure (mAP IOU 0.5-0.95).CIFAR-10 data set.Whereas for custom datasets from 7.2% to 6.9% and 6.1% to 6.0% on light and heavier models respectively.Further, there is a piece of evidence that the approach was trading off accuracy with speed.In comparison to the above approach, our proposed approach is faster compared to deep learning models like YOLO [21] and RCNN [22] as per our computation complexity which makes it a good candidate for on-edge implementation.
In order to identify curves in a binary image, Zhang and Liu [23] used a novel clustering method on the Hough space.They try to address the issue of predicting the optimal no. of prototypes (line in an image) beforehand so that 98494 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Hough transformation can detect most of the straight lines.To achieve the above they followed ''one prototype take one cluster'' approach instead of the conventional approach of ''one prototype take multi-clusters''.The results were presented in an unconventional manner using a single example which is difficult to compare.
Liu et al. [24] used a 3d point cloud of building reconstruction over urban areas.They used innovative Hough transformation to detect lines with noise and then used clustering to find missing lines and allocate to segments of the building.Post this, the above data was used as training data in the neural network for building the 3-D construction.
Lin and Yongze [25] used a combination of manual parameters, hough transformation for lane detection and finally calibration using clustering methods.The approach was quite robust in detecting the lanes (straight and curved) and during day and night.
In comparison to the above approaches, our proposed approach is lighter and faster like deep learning models such as YOLO and RCNN as per our computation complexity which makes it a good candidate for on-edge implementation.Further, our approach has a small effort of tuning the model which is focused on hyperparameters only which saves on the training of the model.Further to add that our proposed approach is novel and has not been tried before as quoted under this section [20], [21], [22], [23], [24], [25].

B. COMPARING WITH THE BEST IN CLASS DEEP-CNN METHODS
The basic supervised deep-CNN network diagram shown in Figure 9 was then further customised for YOLO or region-based CNN (RCNN) models.We compared the identification accuracy of the proposed approach based on the detected pores (from every individual image sample of the dataset) a priori by exploiting our expert judgement, and we found that the mean average precision (mAP) was approximately 79 %.On the contrary, deep-CNN methods show a much inferior performance (mAP 59% for RCNN detectron2, 47.3% for YOLOv5 and Faster Masked R-CNN 45.6%) (see Table 2, Figure 10-13), than the proposed approach.
To compare the identification accuracy of the proposed algorithm compared to those of the deep-CNN methods, we manually classified each pore in every image sample after analysing its visual properties (area and shape) and corroborating with its geophysical properties in the original sample carbonate rock slab.Next, we classified each individual pore using the proposed algorithm and deep-CNN methods and compared their results with those of manual classification.The subsequent results, listed in Table 2, demonstrate that the accuracy of our proposed algorithm is considerably better than that of deep CNN methods.The few observed deviations in the geometrical forms of the pores from the identification logic employed in the suggested approach were blamed for this error rate.Contrarily, the deep CNN methods' error rate was linked to their inability to learn many shape-related  mrcnn mask loss = mask binary cross-entropy loss for the mask head.The loss metrics as train prefix are training losses, and the ones with the val prefix are the validation losses.The combined loss metric given is the aggregation of the above 5 losses as defined by the Mask R-CNN's authors [22].13 demonstrates the Loss and mAP (mean Average Precision) trend.mAP computes the average precision value for recall value over the range of 0 to 1 [21].The horizontal axis is the representation of epochs number and the vertical is the loss/mAP metric.
Here, we reported the accuracy of the proposed approach in comparison to best-in-class YOLO and Mask R-CNN deep-CNN methods.The proposed method clearly outperforms the deep-CNN methods as shown in Table 2.

C. COMPUTING THE COMPUTATIONAL COMPLEXITY
In addition to the above comparison, we have also highlight the computational complexity along with time complexity of the proposed algorithm.We assume that the computer takes the same time for each operation and n is the input size.In addition same computing platform like Google colab is assumed.The image length = n/2 and breadth = n/2, assuming a square image for the convenience of calculation.n 1 is the size of L (bag of pixels) ∀ n 1 < n; n 1 ⊂ n.As can be seen, we primarily have a few blocks of codes, presented as follows: 1) COMPUTING THE COMPUTATIONAL COMPLEXITY OF THE PROPOSED APPROACH REFERRING TO THE FIGURE 5 1.
Ingest pixels for process # runs 3n times, step 1 to 4.
The outer ''for'' loop to iterate on the length of the image, runs n/2 times.b.
The inner ''for'' loop to iterate on the height of the image, runs n/2 times.2.

3.
Similarly, the next loops for the Hough transformation, runs for n 2 1 times, step 6.The combined execution time is 3n + 2X (n 2 /4) + n 2  1 .Now, we ignore the lower-order terms because the lower-order terms are relatively insignificant impact for large inputs as compared to the highest-order term.So considering only the highest-order terms (without constant), and n 1 < n.So, the calculation complexity Big(0) will be O(n 2 ).
2) COMPUTING THE COMPUTATIONAL COMPLEXITY OF YOLO AND RCNN

1.
In general, an MLP (Mult-Layer Perceptron) with n inputs and m hidden layers, where the i-th hidden layer contains m i hidden neurons and k output neurons, will perform the following multiplications (excluding activation functions): which in a big-O notation can be written as where is the lower bound and big-O notation is the upper bound.

3.
By putting them together, we have By generalising the above-mentioned steps for the RCNN and YOLO, the computation complexity for both will be (n 4 ).The comparison of both shows that the RCNN and YOLO tend to be more complex than the proposed approach In general, we would also like to provide readers with the computation complexity with respect to computation runtime Table 2.The issue we found here is that our deep CNN models are trained and inferring using GPU CUDA.Whereas our proposed model (C-PCFT) uses CPU or at best can say that even though it runs in GPU environment, it is not configured to use GPU processing threads and hence not able to use GPU efficiency.Nevertheless, we are providing you with comparable timing for all the methods mentioned here on the same Google COLAB platform.Proposed algorithm C-PCFT, clocked 0.001063 sec per image which is equivalent to 10.63 ms per image.Faster Mask R-CNN is able to clock 0.2705 s/iteration.R-CNN detectron2 clocked 8.84 ms.YoloV8 clocked 2.0 ms inference Google COLAB.Common configuration for Google COLAB is T4 GPU with a standard of 4 device configuration with cu118 CUDA and GPU as Tesla T4, VRAM 16 GB.
So we observe, as expected YoloV8 is the fastest in time complexity and MASK R-CNN seems the slowest.Proposed Algorithm C-PCFT is clocking 10.63 ms which is without any CUDA support whereas remaining deep CNN algorithm are able to use CUDA in the backend.So we can expect up to 5 times the speed when we are able to configure the C-PCFT algorithm with GPU CUDA to support it which makes it comparable to YoloV8 algorithm in terms of time complexity.As concluded before computational complexity evidences that proposed approach (O(n 2 )) is several times efficient than Deep-CNN approaches O(n 4 ).

D. FACTORS BEHIND FAILURE OF CONVENTIONAL (DEEP-CNN) APPROACHES
This section basically explores the potential causes of the shortcomings of conventional techniques.

1) UNAVAILABILITY OF HIGH-QUALITY DATA
In the design of machine vision systems, image quality is generally regarded as a significant challenge that is not regarded as crucial in experiments.Although high-quality image datasets are frequently used for trained and tested for machine vision experiments for academic purposes, this is not necessarily the case in real-world applications [26].As a result, several modern machine vision algorithms have performed poorly in practical field settings.It is reasonable to predict that when provided with a grayscale CT-scan image, the detection quality of the deep-CNN approaches will drop.However, as we already mentioned, the proposed C-PCFT method is reliable as it uses thorough search methods to find rock pores and fractures in each and every image sample.As a result, in the current work, we provide thorough data to support the assertion that the suggested approach has a good level of recognition accuracy even with constraints mentioned above.

2) SAMPLES WITH LOW DIVERGENCE
Low divergence has been highlighted as a significant issue with current deep learning techniques.Most of the samples in the dataset available for this study have relatively little divergence.In other words, the majority of samples have fairly similar visual quality.This may impact the learning process and results in low accuracy due to low generalization of the model.

3) SMALL TRAINING SETS
The lack of sizable training samples is the primary issue preventing deep-CNN algorithms from being successful [27].So as to guarantee that deep-CNN algorithms produce the expected results, large datasets are frequently required [16], [28].Large and diverse training samples are frequently essential to the effectiveness of deep learning techniques.Due to a suboptimal dataset size in the current situation, the accuracy of the deep-CNN methods were less accurate compared to its performance in several computer vision applications.

4) DETECTING OBJECTS WITH IRREGULAR SHAPES
Deep-CNN methods extract more effective and problemspecific features for pore and fracture diagnosis with each layer.In the current situation, prior to classification, it is crucial to understand the type of features that deep-CNN algorithms have extracted.Therefore, it is also crucial to precisely specify their traits before classification in order to recognise items with irregular shapes, such as pores (as in the current situation).Due to the peculiar forms of the pores, this stage is particularly challenging, and as a result, there are more misclassification incidents.The percentage of misclassifications made with deep-CNN techniques [16] is higher with a small number of training samples.

IV. CONCLUSION
The benefits of the suggested algorithm are clear from the aforementioned sections, and they are as follows: 1.
The suggested method is simple to comprehend and use.Additionally, even with the current dataset's low quality, recognition accuracy is still rather good.

2.
Because the suggested technique does not require training for item identification, it may be used right out of the box with some model tuning.

3.
The suggested technique can be used to recognise objects that don't have set geometric proportions and shapes like pores and fractures.Contrarily, deep-CNN approaches are ineffective in this situation because they lack sufficient knowledge of the shape-related elements, such as polygons, that are necessary to accurately categorise pores or fractures.

4.
Furthermore, the proposed method does not require high computational power such as GPUs/TPUs or servers, and can perform inference on edge devices, such as Raspberry Pi and Jetson Nano [29], [30].
Hence, the proposed method can easily be deployed in industrial machines and devices.
This study can be extended in several areas.The proposed system can be prepared as a module and attached with imaging devices for simultaneous identification of pores and fractures during operations like drilling.Furthermore, another classifier can be attached to process pores identified for further decisions on their petrophysical properties.It can be effective in detecting pores of different shapes and sizes, such as lines, circles or other parametric curves by selecting the right parameter space for transformation.The proposed approach can be combined with machine learning algorithms to cater to irregular shape detection without being computationally too expensive.

FIGURE 1 .
FIGURE 1. Parameter of a line.

FIGURE 2 .
FIGURE 2. Collinear points on a line.

FIGURE 3 .
FIGURE 3. Sinusoidal curves with intersection point as the parameters for line passing through these points.

3 . Stage 2 :
Pixel classification and dark cluster (region of interest) identification using a pixel-based region growing algorithm, as shown in Figure 7. 4. Stage 3: Boundary identification and extraction using Hough transformation to find circular/line objects in the image classification post-region growing algorithm as shown in Figure 8. Below, we briefly present the Hough's algorithm: a. Pre-process image as per stages 1 and 2. b.

VOLUME 11, 2023 98491
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

FIGURE 6 .
FIGURE 6.(a) Collection of raw CT-scan image samples of carbonate rock.The samples are grayscale images.(b) Sample raw CT-scan images.(c) Pre-processed CT-scan image samples with already marked boundary with red color.

FIGURE 7 .
FIGURE 7. Image pre-processing cum clustering step: (a) The original CT-scan image of the carbonate rock.(b) The actual binarization of the image.(c) clustering dark spots on image using region growing technique (clustering) (less darker spots).

FIGURE 8 .
FIGURE 8. Visual results with detected pores using C-PCFT algorithm.

FIGURE 9 .
FIGURE 9. Conceptual diagram of the supervised deep-CNN.

FIGURE 10 .
FIGURE 10.Sample results with detected pores using deep learning Mask R-CNN.
features, which are necessary to accurately identify pores in a limited number of training examples.Specifically referring to the deep-CNN method like Mask Figure 11 A and Figure 11 B show the trend of the loss metrics algorithm and each relates to Training and Validation steps respectively.The horizontal axis is the representation of epochs number and the vertical is the loss metric.As can be seen that training loss has been trending consistently downwards but the validation is not consistently downward.This is an indication of a overtraining and partially inability of deep CNN to learn and generalize from give dataset.Each of them is a combination of 5 loss types as below.1. rpn class loss = RPN anchor classifier loss.2. rpn bbox loss = RPN bounding box loss graph.3. mrcnn class loss = loss for the classifier head of Mask R-CNN.4. mrcnn bbox loss = loss for Mask R-CNN bounding box refinement. 5.

FIGURE 13 .
FIGURE 13.Loss trend and mAP trend performance by confidence using Yolo 5 Deep-CNN.

Figure 12
Figure12demonstrates visual detection of over-prediction and missed prediction of YOLO algorithm.Missed predictions are marked in yellow outline.Figure13demonstrates the Loss and mAP (mean Average Precision) trend.mAP computes the average precision value for recall value over the range of 0 to 1[21].The horizontal axis is the representation of epochs number and the vertical is the loss/mAP metric.Here, we reported the accuracy of the proposed approach in comparison to best-in-class YOLO and Mask R-CNN deep-CNN methods.The proposed method clearly outperforms the deep-CNN methods as shown in Table2.

Figure
Figure12demonstrates visual detection of over-prediction and missed prediction of YOLO algorithm.Missed predictions are marked in yellow outline.Figure13demonstrates the Loss and mAP (mean Average Precision) trend.mAP computes the average precision value for recall value over the range of 0 to 1[21].The horizontal axis is the representation of epochs number and the vertical is the loss/mAP metric.Here, we reported the accuracy of the proposed approach in comparison to best-in-class YOLO and Mask R-CNN deep-CNN methods.The proposed method clearly outperforms the deep-CNN methods as shown in Table2.