A spectrum-domain instance segmentation model for casting defects

Abstract

Accurate segmentation of casting defects plays a positive role in the quality control of casting products, and is of great significance for accurate extraction of the mechanical properties of defects in the casting solidification process. However, as the shape of casting defects is complex and irregular, it is challenging to segment casting defects by existing segmentation methods. To address this, a spectrum domain instance segmentation model (SISN) is proposed for segmenting five types of casting defects with complex shapes accurately. The five defects are inclusion, shrinkage, hot tearing, cold tearing and micro pore. The proposed model consists of three sub-models: the spectrum domain region proposal model (SRPN), spectrum domain region of interest alignment model (SRoIAlign) and spectrum domain instance generation model (SIGN). SRPN uses a multi-scale anchoring mechanism to detect defects of various sizes, where the SSReLU and SCPool functions are used to solve the spectrum domain gradient explosion problem and the spectrum domain over-fitting problem. SRoIAlign uses the floating-point quantization operation and the tri-linear interpolation method to quantize the 3D proposals to the feature values in an accurate manner. SIGN is a full-spectrum domain neural network applied to 3D proposals, generating a segmentation instance of defects in a point-wise manner. In the experiments, we test the effectiveness of the proposed model from three aspects: segmentation accuracy, time performance and mechanical property extraction accuracy.

Keywords

Casting defect mechanical properties spectrum domain instance segmentation

1. Introduction

Segmenting casting defects is an essential and practical problem in the field of casting CAE and plays an important role in accurately extracting the mechanical properties of defects. Because accurately segmenting casting defects prepares the ground for the exact extraction of the mechanical properties of defects, it is essential to research defect segmentation methods that are applied to improve the extraction accuracy of the mechanical properties of casting defects. Most of the existing methods can only detect defects or extract simple feature parameters of defects. Even the most advanced method can only segment the casting defect region; that is, the segmentation range is limited to the region around the defect. This coarse-grained segmentation method will lead to inaccurate extraction of the mechanical properties of casting defects. In addition, because the shape of the casting defect is complex and irregular, it is difficult to segment casting defects by using the existing methods. In this paper, a spectrum domain instance segmentation model (SISN) is proposed for segmenting five types of casting defects with complex shapes accurately. The five defects are inclusion, shrinkage, hot tearing, cold tearing and micro pore. SISN extracts the three-dimensional topological characteristics from the input volumes and the micro-structure features of the defects from metallographic images and outputs the segmentation instance of the defect. First, the SRPN sub-model is proposed to extract three-dimensional topological characteristics from the input volumes and output multi-scale three-dimensional proposals of the defects. The three critical parts of the SRPN are the multi-scale anchoring mechanism, the spectrum domain activation function (SSReLU) and the spectrum domain pooling operation (SCPool). Second, the SRoIAlign sub-model uses the floating-point quantization operation at the boundary of the feature volume to transform the multi-scale 3D proposals into fixed-size feature volumes and uses the tri-linear interpolation method to calculate the feature values of the featurse bins, which are aggregated by the following SCPool layer. Finally, the SIGN sub-model is proposed to learn characteristics from the three-dimensional proposals of the defects and generate the segmentation instance for each defect. The segmentation instance of the defect is used to calculate the mechanical properties of the defect in the experiment section. The structure of this paper is as follows: the SISN model is outlined in Section 3. The SRPN, SRoIAlign and SIGN sub-models are introduced in Sections 4–6, respectively. Finally, the experimental results are introduced in Section 7. The innovative aspects of this paper include:

1.
A spectrum domain instance segmentation model is proposed to segment casting defects accurately.
2.
A spectrum domain region proposal network (SRPN) is proposed to detect defects of multiple sizes.
3.
A spectrum domain RoI alignment model (SRoIAlign) is proposed to quantize the multi-size 3D proposals to the feature values in point-wise manner.
4.
A spectrum domain instance generation network (SIGN) is proposed to generate the segmentation instance of the defect.

2. Related work

The segmentation and detection of casting defects have become a popular issue in the field of casting simulation. A large number of scholars and research institutions have carried out in-depth research on segmenting casting defects and achieved many results. In this section, based on the summary of existing articles, combined with some of the latest research results of artificial intelligence methods in the field of casting defect recognition, the related work of this paper is summarized and analyzed as follows.

Since 1922, the United States established the world’s first industrial X-ray laboratory, making X-rays to detect the quality of casting products a reality. The X-ray detection method uses the X-ray absorption principle of casting defects to identify defects through defect ray images with specific grey distributions generated by the absorption of photosensitive materials. The X-ray detection method is still widely used because of its simple, intuitive, and convenient operation [1, 2, 3, 4, 5, 6]. However, X-ray imaging mostly reflects the casting defect morphology above the centimetre level, and manual operation is needed to complete the detection work. Due to the rise of artificial intelligence methods, this manual defect detection method shows disadvantages.

The use of neural networks for detecting casting defects appeared approximately 10 years ago. Because large-scale datasets obtained from foundries can be used to train neural networks, the casting defect recognition method based on an artificial neural network can be realized. Lewis et al. constructed a simple neural network model to predict the influence of casting processing parameters on casting defects [7] and established the relationship between the casting temperature parameters and casting defects. This method can predict whether there are defects in casting products and the possible causes but cannot predict the key geometric information, such as the location, shape, and quantity of defects. Zheng et al. constructed a double-layer BP neural network to predict the relationship between casting defects and casting processing parameters [8]. However, the neural network structure is too simple, and the depth of layers is too shallow, which limits the accuracy of the network, and the network is not used to detect casting defects. To address this, researchers have proposed a variety of improvement strategies to improve the accuracy of neural networks for defect detection. In 2012, Masci et al. used multi-scale pyramid pooling technique to build a neural network to detect defects in steel castings [9, 10]. Liu et al. constructed a deep confidence network model to extract the defect information in high-dimensional feature space and introduced a parameter transfer mechanism to alleviate the over-fitting phenomenon in a neural network [11]. However, these methods are not designed for the segmentation of defects, which makes it impossible for them to extract the mechanical properties of defects. Additionally, a variety of artificial intelligence methods for defect detection have emerged. Hanzaei used the RIMLV operator and the close morphological operator to detect ceramic defects [12], which can extract the geometric features of the defects automatically. Dounias used machine learning method to detect the mechanical defect of washing machine in the production line [13, 14, 15, 16]. Yan et al. [17]. proposed an AASI-GTP model to detect low-contrast defects on micro3D textured surfaces. However, these methods are designed specifically for ceramic defects and low-contrast defects, which are not applicable to the detection or segmentation of casting defects. Therefore, using an artificial neural network to detect defects entered a bottleneck period until the appearance of convolutional neural networks. In 2017, Wang et al. [18]. designed a deep convolutional neural network (CNN) based on the classical convolution neural network benchmark framework [19, 20, 21], which can automatically extract the geometric characteristics from the defect regions. Fang et al. [21]. constructed a deep convolutional neural network based on the traditional Faster R-CNN [23, 24] framework to detect crack defects. Zhang et al. [25]. constructed a category-aware object detection network to classify and detect surface defects. These CNN-based methods make it possible to detect surface defects accurately. However, the traditional CNN has a lower recognition accuracy for inner defects, which limits the extraction accuracy of the characteristic parameters of the inner defects. To address this, in 2018, Lin et al. proposed an improved deep convolution neural network architecture to detect casting inner defects more accurately, and the inner defects included shrinkage, porosity and hot tearing [26]. However, this method still cannot segment casting defects or extract the mechanical properties of castings. In 2019, reference [27] proposed a spatial domain segmentation network to segment the casting defect region. However, this method was proposed to segment defects in a coarse-grained manner, and the segmentation is limited to the area around the defect. Therefore, for defects with complex shapes, this method cannot segment them accurately, and inaccurate segmentation instances will also lead to a decline in the extraction accuracy of the mechanical properties. To improve the segmentation accuracy of casting defects with complex shapes and maintain the time performance of the network at a better level, a spectrum domain instance segmentation network (SISN) is proposed in this paper. SISN is a full spectrum domain training neural network model, from which the mechanical properties can be extracted. The model can improve casting process simulation accuracy and improve the rationality of casting process selection to improve the quality of castings.

3. The architecture of SISN

In casting process simulation, the segmentation accuracy of casting defects directly affects the extraction accuracy of the mechanical properties of defects. In this paper, a spectrum domain instance segmentation model (abbreviated as SISN) is proposed for segmenting defects accurately. SISN consists of three sub-models: the spectrum domain region proposal network, abbreviated as SRPN; the spectrum domain region of interest alignment network, abbreviated as SRoIAlign; and the spectrum domain instance generation network, abbreviated as SIGN. The SRPN sub-model extracts three-dimensional topological characteristics from the input volumes and output multi-scale three-dimensional defect proposals. In addition, a residual network (ResNet-50-C4) [28] is used to extract the micro-structure features of the defects from metallographic images. SISN uses the ORN training framework [29] to train the SRPN and ResNet-50-C4 jointly and projects the extracted micro-structure features to the three-dimensional defect proposals. SRoIAlign uses a floating-point quantization operation to subdivide the multi-scale three-dimensional proposals of the defects into fixed-size feature volumes and uses the tri-linear interpolation method to calculate the feature values of the feature bins. SIGN learns characteristics from the three-dimensional proposals of the defects and generates the segmentation instance of the defect. The segmentation instance of the defect is a three-dimensional geometric layout of the defect with micro-structure features. The SRPN, SRoIAlign and SIGN sub-models are completely trained and tested in the spectrum domain. The pipeline of the proposed method is shown in Fig. 1. The proposed method is introduced in detail in the following sections.

Figure 1.

The pipeline of the proposed method.

Figure 2.

The SRPN pipeline. The SRPN is a full-spectrum domain neural network model which includes the Sconv layers, SSReLU layers and SCPool layers. The Sconv layer is a convolution layer in spectrum domain, which uses dot product operation in each convolutional layer instead of the traditional convolutional operation.

4. The SRPN sub-model

The precondition for accurately segmenting casting defects is how to detect defects of various sizes accurately. Therefore, a spectrum domain region proposal network (SRPN) is designed for accurately detecting multi-size defects. The SRPN pipeline is shown in Fig. 2. On the one hand, the SRPN uses the multi-scale anchoring mechanism to detect defects of various sizes. On the other hand, the spectrum domain architecture of SRPN ensures that the SRPN quickly detects defects from hundreds of millions of input volumes. In the rest of this section, we explain the SRPN structure in detail from these two aspects.

First, the SRPN uses a multi-scale anchoring mechanism for detecting defects of various sizes. The essence of the anchor mechanism is to use a sliding window to detect defects in a multi-scale manner [30]. We use a feature pyramid mechanism to generate the multi-size volumes for each defect and use a sliding window to scroll on the multi-size volume to generate a set of candidate regions; that is, we assign a set of rectangular boxes (anchors) with different scales and ratios to each spatial point. The location of these anchors is the coordinates of the central spatial point contained in the rectangular box. Then, the SRPN model proposed in this section is used to classify the defect feature information in the candidate regions. The parameter settings of the anchor mechanism used in this paper are explained as follows.

Figure 3.

The calculation process of the SSReLU function and the SCPool operation. We assume that the size of the output volume is $6*6*2$ . $y_{qp^{\prime}}$ denotes the $p^{\prime}$ -th output feature volume of the $q$ -th layer. $Y_{qp^{\prime}}$ represents the FFT value of the output feature volume.

The camera coordinates of the input volume refer to the position coordinates of the input volume overlapping with the gravity direction. The effective scanning range of our 3D casting scanner is [ $-$ 200, 200] millimetres in the horizontal direction, [ $-$ 150, 100] vertical direction and [10, 90] in depth. The encoded 3D casting is a volume block combination with a size of $400*300*100$ (the volume block size is $2*2*2$ millimetres), which is used as the SRPN input.

The number of anchor boxes is defined as 20 within a sliding window. An anchor box corresponds to a 3D proposal, as shown in Fig. 2. When the anchor aspect ratio is different, we define two anchors with the same volume but different angles. For example, the aspect ratio of an anchor is 1:2:1, and the aspect ratio of another anchor is defined as 2:1:1, which are two independent and different anchors.

The aspect ratio of the anchor boxes ranges from 0.5 millimetres (gas porosity defect) to 3 millimetres (crack defect). Because the shape of the defects is complex and diverse, if we use single-scale candidate regions to predict the positions of all anchor boxes, there will be a large number of invalid mappings. Therefore, we use the multi-scale SRPN to detect defects of various sizes.

We define the multi-task loss function on each anchor box as $L=L_{\text{cls}}+\lambda p*L_{\text{box}}+L_{\text{ins}}$ . $L_{\text{cls}}$ is the classification loss, and $L_{\text{box}}$ is the bounding box loss. $L_{\text{ins}}$ is the instance segmentation loss, which is introduced in Section 6. $\lambda$ is an equilibrium parameter. $p*$ is the true value of an anchor being a defect.

Second, the SRPN sub-model is a whole-spectrum domain training model. The SRPN model uses the SSReLU function to eliminate the gradient explosion phenomenon in the spectrum domain back-propagation pass and uses the spectrum domain down sampling operation (SCPool) to solve the spectrum domain over-fitting problem [31]. Inspired by the spatial domain activation function [32, 33, 34], we design the SSReLU function as follows:

$\displaystyle A_{qp^{\prime}}(U)=S{\rm R}^{+}(Y_{qp^{\prime}}(U))+S{\rm R}^{-}% (\hat{Y}_{qp^{\prime}}(U))=\frac{1}{2}(Y_{qp^{\prime}}(U)+\|{Y_{qp^{\prime}}(U% )}\|+\hat{Y}_{qp^{\prime}}(U)-\|{\hat{Y}_{qp^{\prime}}(U)}\|)$ (1)

where $S{\rm R}^{+}(\cdot)$ is the forward stochastic deviation function and $S{\rm R}^{-}(\cdot)$ is the backward stochastic deviation function. $Y_{qp^{\prime}}(\cdot)$ is the FFT [35] value of the output feature volume for $S{\rm R}^{+}(\cdot)$ . $\hat{Y}_{qp^{\prime}}(\cdot)$ is the FFT value of the output feature volume for $S{\rm R}^{-}(\cdot)$ . $\|\cdot\|$ is an operator that takes the positive value item by item. The detailed calculation process of the SSReLU function is presented in Fig. 3(a). The SSReLU function adds the backward stochastic deviation term to the output feature volume. Under the effect of the forward and backward stochastic deviation term, the SSReLU function can characterize the stochastic state of the feature volumes, guarantee the effective transmission of the feature volumes in the spectrum domain propagation pass, and maintain a low arithmetic complexity (the time performance is evaluated in Section 7.2). In addition, the SCPool function is designed to extract the multiple eigenvalues of the feature volumes in the spectrum domain. These multiple eigenvalues can characterize the condition in which the same kind of characteristics appear many times and the location information of these eigenvalues can also be maintained. Inspired by the spatial domain pooling operation [36, 37, 38, 39], we design the spectrum domain chunk max-pooling operation (SCPool) as follows:

$\displaystyle Y_{qp^{\prime}k^{\prime}}(U)=\text{Sdown}(A_{qp^{\prime}k^{% \prime}}(\cap))=\sum_{n=\beta}\max(y_{qp^{\prime}k^{\prime}}(n))\cdot e^{-j% \frac{2\pi}{i_{qp^{\prime}k^{\prime}}}n\cap}$ (2)

where $\text{Sdown}(\cdot)$ is the spectrum domain down-sampling function. $\beta$ indexes the spectrum items with non-negative eigenvalues in each feature bin. $\cap$ indexes the maximum spectrum items in each feature bin. The feature bin is size of $((l_{qp^{\prime}k^{\prime}1}-k_{qp^{\prime\prime}1})/\text{str}_{1}+1)\times((% l_{qp^{\prime}k^{\prime}2}-k_{qp^{\prime\prime}2})/\text{str}_{2}+1)$ , and $U\in[0,(l_{qp^{\prime}k^{\prime}}-k_{qp^{\prime\prime}})/\text{str}]$ . The detailed calculation process of the SCPool function is presented in Fig. 3(b).

Figure 4.

The pipeline of SRoIAlign sub-model. The floating-point quantization operation is used to quantify the feature volumes, and the tri-linear interpolation method is used to calculate the feature values of the feature bins.

The SRPN sub-model extracts three-dimensional topological characteristics from the input volumes and output multi-scale three-dimensional proposals of the defects. In addition, the residual network (ResNet-50-C4) is used to extract the micro-structure features of the defects from metallographic images. SISN uses the ORN training framework to train the SRPN and ResNet-50-C4 jointly and maps the extracted micro-structure features to the three-dimensional proposals of the defects.

5. The SRoIAlign sub-model

Because the pooling layer only accepts the fixed-size feature volume, it is necessary to transform the multi-scale 3D proposals into fixed-size feature volumes. The traditional RoIPooling method uses truncated rounding quantization at the boundary of the feature volume to transform the multi-scale 3D proposals into fixed-size feature volumes [40, 41]. This quantization method leads to a mismatch between the 3D RoI and the sfeature values. To address this, the SRoIAlign sub-model is proposed to quantize the 3D proposals to the feature values accurately. The SRoIAlign sub-model completes the quantization task in three steps. We describe the three tasks in detail as follows.

Step 1:
The floating-point quantization operation is used to quantize the 3D proposal to the feature volume. We assume that the size of a 3D proposal output by SRPN is $125125125$ , and the stride size is 8, as shown in Fig. 4. Then, we use the floating-point quantization operation to quantize this proposal; that is, the size of the quantized proposal is $15.6215.6215.62$ , which is computed by 125/8. Unlike the traditional RoIPool method, which computes the size of the proposal by $[125/8]$ (the size of the quantized proposal is $151515$ ), our quantization method preserves the floating-point values of the boundaries of the proposals. This floating-point quantization operation reduces the misalignment phenomenon caused by quantization and improves the quantization accuracy.
Step 2:
The feature volume is subdivided into fixed-size feature bins. We assume that the size of the input feature volume of the SCPool is $777$ , as shown in Fig. 4. Then, we still use the floating-point quantization operation to quantize the feature volume; that is, the size of the quantized feature volume is $2.232.232.23$ , which is computed by 15.62/7. Therefore, the feature volume is subdivided into $777$ fixed-size feature bins, and the size of each feature box is $2.232.232.23$ . In the following step, there is no need to quantize these feature bins again, the feature values are only extracted from each feature bin.
Step 3:
The tri-linear interpolation method [42] is used to calculate the feature values of the feature bins. The tri-linear interpolation method implements the linear interpolation operation on the tensor product grid of eight discrete sampling points in the feature bin. We perform a tri-linear interpolation on each feature bin to obtain $777$ feature values. Then, the feature values are aggregated by the following SCPool layer. The tri-linear interpolation method is as follows: first, we use the left-handed coordinate system to ensure that the $z$ -axis is upward. Then, we interpolate the four edges in the $x$ -axis direction to obtain the four values; that is, we perform bilinear interpolation on each edge to obtain a single feature value. Therefore, a total of four feature values are generated, which are represented by C00, C01, C10 and C11, as shown in Fig. 4. Then, the four feature values (C00, C01, C10, C11) are interpolated in the $y$ -axis direction to obtain the two feature values (C0, C1). Finally, we interpolate the two feature values (C0, C1) in the $y$ -axis direction to obtain the final feature value.

Figure 5.
The pipeline of SIGN sub-model. The head of SIGN is built on the fifth layer of the ResNet50 backbone structure and the feature pyramid network structure. The numbers under the name of each layer indicate the volume resolution and the number of spectrum-domain channels. In addition, the porosity parameters of each defect instance in the solidification process are characterized by the hierarchical visualization method.

6. The SIGN sub-model

An important factor affecting the extraction accuracy of the mechanical properties of casting defects is whether an accurate segmentation instance of the defect can be obtained. However, it is a challenging task to obtain accurate segmentation instances of defects, especially for defects with complex shapes. To address this, a spectrum domain instance generation network (SIGN) is proposed to generate segmentation instances of defects with complex shapes accurately. The SIGN sub-model is a full-spectrum domain convolutional neural network applied to 3D proposal, generating a segmentation instance in a point-to-point way. In this section, we introduce the SIGN sub-model in detail from four aspects: instance representation, backbone framework, head framework, and loss function.

(1)
Instance representation. The segmentation instance encodes the geometric layout of the input volume. Therefore, it is different from the classification regression and bounding box regression that are projected into small output volumes by full-connection operations. SIGN extracts the geometric structure of the defect by point-to-point alignment and spectrum domain convolution operations [43]. The spectrum domain pipeline of the SIGN is shown in Fig. 5. SIGN uses a full-spectrum domain convolutional neural network pipeline to generate a segmentation instance from a 3D proposal. This pipeline structure allows a 3D proposal to pass through the pipeline in a complete geometric layout without projecting into a small volume representation that lacks geometric information. It is different from the previous networks that use the full spatial domain convolutional layers to predict the segmentation instances, SIGN uses fewer feature parameters to represent the segmentation instances and is more accurate than the spatial network, which is verified in the experiments in Section 7.
(2)
Backbone framework. The backbone framework is used to extract features from the input volumes. We use three typical networks as our backbone frameworks: AlexNet7, VGGNet19 and ResNet50. The number after the network name indicates the depth of the network. In addition, the feature pyramid network is used to extract the multi-scale features of the 3D proposal from the multi-level feature pyramid. This feature pyramid network is also used in the SRPN sub-network to generate the multi-size feature volumes for each defect. Therefore, the two models can share the same training process, which is helpful to improve the training speed of the network.
(3)
Head framework. The head framework is used to generate the classification label, the bounding box and the segmentation instance. The segmentation instance is applied exclusively to 3D proposals that are output by the SRPN sub-model. The head framework pipeline is shown in Fig. 5. The head of SIGN is built on the fifth layer of the ResNet50 backbone structure and the feature pyramid network structure. The size of the convolutional filter is $333$ , and the size of the transposed convolution filter is $22$ . The SSReLU function and SCPool operation (see Section 4) are used in hidden layers.
(4)
Loss function. We use the multi-task loss function to measure the regression loss of the instance segmentation results. The multi-task loss contains three parts: the classification loss denoted as $L_{\text{cls}}$ , the bounding box loss denoted as $L_{\text{box}}$ and the instance segmentation loss denoted as $L_{\text{ins}}$ . We use an $Nmmm$ dimension feature matrix to represent N different types of segmentation instances. The resolution of the segmentation instance is $mmm$ , and each segmentation instance corresponds to a feature matrix. For each feature matrix, we use the SIGMOID function to calculate the relative entropy loss. $L_{\text{ins}}$ is the average value of the relative entropy loss. The average relative entropy loss is defined as $L_{\text{ins}}(i,o)=-(i\times\log(o)+(1-i)\times\log(1-o))$ , where $i$ refers to the input volume and $o$ refers to the output volume (segmentation instance). For each 3D proposal, if it is detected that the 3D proposal belongs to a certain classification, only the relative entropy loss of this classification is used to measure the regression degree of the 3D proposal. For example, there are three types of defects in casting defect classification: inclusions, shrinkage and cracks. The current 3D proposal belongs to the category of “crack”, so the $L_{\text{ins}}$ of the “crack” is the relative entropy loss of the category of “crack”. Finally, the binary segmentation instance can be output by comparison with the threshold value of 0.5. In this way, competition between classes is avoided, and classification is performed by the SRPN sub-model.

7. Results and discussion

To verify that the proposed model plays a positive role in the segmentation of casting defects, we test and analyse the advantages of the SISN model from three aspects: segmentation accuracy, time performance and extraction accuracy. Before giving the experimental data, we first explain the pre-set parameters of the experiment.

(1)
Datasets. The in-house EA88 dataset is constructed for training and testing the spectrum domain networks in our experiments. The EA88 dataset consists of two parts: the three-dimensional volume models of EA88 and the metallographic images. The EA88 dataset contains 12,000 pieces of metallographic images, each metallographic image contains 400 attribute labels, and each attribute label contains 8 micro-structure feature parameters; that is, the EA88 dataset contains $12{,}0004008$ micro-structure feature parameters of the defects. A Prisma E scanning electron microscope was used to scan the defect samples to obtain metallographic images of the defects. We use the metallographic images and the three-dimensional volume models of the EA88 cylinder as the input dataset to train and test the spatial domain and spectrum domain models in this paper. We divided the input dataset into three sub-sets: a training dataset (70%), a validation dataset (10%) and a testing dataset (20%). In our experiment, we use three classical convolutional neural network models as the benchmark framework for training and testing the spatial domain network. The three benchmark frameworks are AlexNet7 [44], VGG19 [45] and ResNet50 [28]. We use the radix-2 FFT method to transform the input data set from the spatial domain to the spectrum domain. The spatial domain reference models trained and tested in this section include LCNN and DCNN; the spectrum domain models include koCNN [46], fbFFT [47], FPN [48], SRPN, SRoIAlign and SIGN. The spatial variant of the SIGN sub-model is denoted as IGN, which can be used as the segmentation sub-networks of the LCNN [49] and DCNN [26] models. The GPU specifications used in our experiment include 1920 NVIDIA CUDA cores, 1,683 MHz core frequency, and 8.92 TFLOPS peak speed per second.
(2)
Casting defect samples. The casting defect samples used in this paper are all from an EA88 cylinder. The EA88 cylinder is made by a sand casting process. The process parameters of EA88 cylinder casting are as follows: 1. the pouring temperature is 1,390 ${{}^{\circ}}$ C; 2. the pouring time is 20 seconds; 3. the casting materials are grey iron 250 and AlSi7Cu3; 4. the sand material is silicon; 5. the sand temperature is 20 ${{}^{\circ}}$ C; and 6. the filter material is foam ceramic. The EA88 cylinder block is divided into four test bars with defects, and each test bar is divided into 30 defect samples of the same size; that is, 120 defect samples are included in our tests. The size of the defect sample is as follows: the top width of both sides is 30 mm; the top length of both sides is 40 mm; the middle width is 15 mm; and the semicircle angle is 22 radians. We used a Prism E electron microscope to scan the 120 defect samples and collected 100 pieces of metallographic images from each defect sample. The sampling process is shown in Fig. 6. In addition, we use the bar tensile test and the ball stress test to collect the mechanical properties from each defect sample. The collected data are the mechanical properties truth values, which are used to calculate the porosity error, tensile strength error and hardness error in our experiment. The EA88 cylinder block used in this paper is provided by FAW Foundry Co., Ltd.

Table 1
Detection and segmentation performance of DCNN, koCNN and SISN

Input/backbone AP ${}_{\text{I}}$ AP ${}_{\text{S}}$ AP ${}_{\text{HT}}$ AP ${}_{\text{CT}}$ AP ${}_{\text{P}}$ AP ${}_{\text{I$+$S}}$ AP ${}_{\text{I$+$HT}}$ AP ${}_{\text{S$+$CT}}$ AP ${}_{\text{HT$+$P}}$ AP ${}_{\text{I$+$S$+$HT$+$P}}$

DCNN TSDF/ResNet50 44.9 44.1 41.5 41.9 44.8 41.1 38.5 40.9 39.0 35.1

koCNN volume/fbFFT 42.8 42.5 38.8 38.9 42.4 39.2 35.8 37.5 36.1 34.5

DCNN $+$ IGN (DCNN-I) TSDF/IGN 31.9 32.0 29.8 28.9 31.5 28.5 24.9 26.5 25.9 23.9

koCNN $+$ SIGN (koCNN-I) volume/SIGN 34.5 34.0 30.2 30.4 33.7 31.0 28.2 30.4 29.1 24.5

SISN volume/SRPN $+$ ResNet50 46.2 46.1 44.6 44.9 46.8 44.5 43.7 43.8 43.9 40.1

SRoIAlign $+$ SIGN 41.8 41.6 40.1 40.5 41.9 39.9 39.2 39.2 39.5 36.6

2DMI $+$ volume/SRPN $+$ SRoIAlign $+$ SIGN 45.6 45.5 45.0 45.3 45.6 43.6 43.1 43.0 43.4 40.5

Figure 6.
The experiment streamline. The casting defect samples used in this paper are all from an EA88 cylinder which is made by a sand casting process. A Prisma E scanning electron microscope was used to scan the defect samples to obtain metallographic images of the defects.

7.1 Segmentation performance

	Input/backbone	AP ${}_{\text{I}}$	AP ${}_{\text{S}}$	AP ${}_{\text{HT}}$	AP ${}_{\text{CT}}$	AP ${}_{\text{P}}$	AP ${}_{\text{I$+$S}}$	AP ${}_{\text{I$+$HT}}$	AP ${}_{\text{S$+$CT}}$	AP ${}_{\text{HT$+$P}}$	AP ${}_{\text{I$+$S$+$HT$+$P}}$
DCNN	TSDF/ResNet50	44.9	44.1	41.5	41.9	44.8	41.1	38.5	40.9	39.0	35.1
koCNN	volume/fbFFT	42.8	42.5	38.8	38.9	42.4	39.2	35.8	37.5	36.1	34.5
DCNN $+$ IGN (DCNN-I)	TSDF/IGN	31.9	32.0	29.8	28.9	31.5	28.5	24.9	26.5	25.9	23.9
koCNN $+$ SIGN (koCNN-I)	volume/SIGN	34.5	34.0	30.2	30.4	33.7	31.0	28.2	30.4	29.1	24.5
SISN	volume/SRPN $+$ ResNet50	46.2	46.1	44.6	44.9	46.8	44.5	43.7	43.8	43.9	40.1
	SRoIAlign $+$ SIGN	41.8	41.6	40.1	40.5	41.9	39.9	39.2	39.2	39.5	36.6
	2DMI $+$ volume/SRPN $+$ SRoIAlign $+$ SIGN	45.6	45.5	45.0	45.3	45.6	43.6	43.1	43.0	43.4	40.5

In this experiment, we select four typical neural network models to test their detection accuracy and segmentation accuracy for casting defects and then compare the testing results with our models. We use the parameter AP to evaluate the defect detection accuracy and segmentation accuracy of these models. AP is the abbreviation of average precision, and the AP error threshold is set to 0.25 and 0.15. Due to the different microstructures and formation mechanisms of casting defects, we further subdivide the AP parameters into five grades, which are represented by five AP parameters with subscripts: $\text{AP}_{\text{I}}$ , $\text{AP}_{\text{S}}$ , $\text{AP}_{\text{HT}}$ , $\text{AP}_{\text{CT}}$ and $\text{AP}_{\text{P}}$ . $\text{AP}_{\text{I}}$ is the average precision value for detecting and segmenting inclusion defects. $\text{AP}_{\text{S}}$ is the average precision value for detecting and segmenting shrinkage defects. $\text{AP}_{\text{HT}}$ is the average precision value for micro hot tearing defects. $\text{AP}_{\text{CT}}$ is for micro cold tearing defects. $\text{AP}_{\text{P}}$ is for micro pore defects. In addition, we recombine these five kinds of defects and construct five kinds of defect combinations. The detection (segmentation) accuracy is expressed by $\text{AP}_{\text{I$+$S}}$ , $\text{AP}_{\text{I$+$HT}}$ , $\text{AP}_{\text{S$+$CT}}$ , $\text{AP}_{\text{HT$+$P}}$ and $\text{AP}_{\text{I$+$S$+$HT$+$P}}$ . First, we use the spatial domain model (DCNN) to detect five kinds of micro defects. The $\text{AP}_{\text{I}}$ value of DCNN is 44.9, the $\text{AP}_{\text{S}}$ value of DCNN is 44.1, and the $\text{AP}_{\text{P}}$ value of DCNN is 44.8, as shown in Table 1. The detection accuracy of DCNN for these three kinds of defects is similar. However, the AP values of hot tearing and cold tearing are 41.5 $\text{AP}_{\text{HT}}$ and 41.9 $\text{AP}_{\text{CT}}$ , the average accuracy is 3.0 points lower than that of the first three kinds of defects. This indicates that the DCNN is more suitable for detecting defects with simple shapes. In addition, the detection accuracy of the combination of inclusion and shrinkage defects is 41.1 $\text{AP}_{\text{I$+$S}}$ , which is the maximum accuracy of all other defect combinations. However, this value is still 3.0 points lower than the detection accuracy of any single defect, which shows that the complexity of defects has a great impact on the detection accuracy of the DCNN model.

Table 2
Classification performance for SISN, koCNN and DCNN

Type of defect	The first level			The second level			The third level
(unit: mAP)	SISN-L1	koCNN	DCNN	SISN-L2	koCNN-I	DCNN-I	SISN-L3	koCNN-I	DCNN-I
Type-A	90.0	65.8	78.5	94.5	71.2	84.2	95.2	71.2	84.2
Type-B	89.2	60.2	77.5	92.1	67.5	74.5	94.4	67.5	74.5
Type-C	86.6	62.5	74.3	88.2	68.1	80.6	90.5	68.1	80.6
Type-D	88.4	65.1	75.5	89.5	70.9	81.9	91.5	70.9	81.9
Type-E	85.5	59.6	71.2	88.0	67.1	77.8	89.1	67.1	77.8
Mean average precision	87.9	62.6	75.4	90.4	68.9	79.8	92.1	68.9	79.8

Then, we use the spectrum domain model koCNN to detect five kinds of defects. For the five defects, the average detection accuracy of koCNN is 41.08 AP and that of DCNN is 43.44 AP, which is 2.36 points higher than the former. This is because koCNN uses sinc interpolation and Hermitian symmetry to speed up the detection. These two functions use boundary interception to reduce the training time, which also reduces the detection accuracy. Similar results are obtained for the five defect combinations. However, we integrate the proposed SIGN sub-model into the koCNN and the IGN sub-model into the DCNN model (denoted as koCNN-I and DCNN-I) so that koCNN-I and DCNN-I can segment the defects. For the five defects, the average detection accuracy of koCNN-I is 30.82 AP and that of DCNN-I is 32.56 AP, which is 1.74 points higher than the former. Furthermore, for the five defect combinations, the average detection accuracy of koCNN-I is 25.94AP and that of DCNN-I is 28.64AP, which is 2.7 points higher than the former. It can be seen that the complexity of the defect increases, but the segmentation accuracy of koCNN-I and DCNN-I does not decrease, and is even nearly one point higher than that of koCNN and DCNN. This shows that the proposed SIGN framework has good compatibility and high accuracy. It can be integrated into other detection frameworks to segment complex defects accurately.

Finally, we evaluate the detection and segmentation performance of our model in three levels. In the first level, we combine the SRPN sub-model and ResNet50 network to generate the first-level detection framework (denoted as SISN-L1) and calculate the detection accuracy of the first-level framework for five kinds of defects and five kinds of defect combinations. In the second level, we combine the SRoIAlign and SIGN sub-model to generate the second-level segmentation framework (denoted as SISN-L2) and calculate the segmentation accuracy of the second-level framework. In the third level, we combine the SRPN, SRoIAlign and SIGN sub-models to generate the last level segmentation framework (denoted as SISN-L3) and calculate the segmentation accuracy of the SISN for five defects and five defect combinations. As shown in Table 1, the average detection accuracy of SISN-L1 is 45.72 AP, which is 2.28 points higher than that of the DCNN model and 4.64 points higher than that of the koCNN model. This is due to the use of the proposed SRPN sub-model, in which the SSReLU and the SCPool are employed to train the network completely in the spectrum domain. The average segmentation accuracy of SISN-L2 is 41.18 AP, which is 8.62 points higher than that of the koCNN-I model and 10.36 points higher than that of the DCNN-I model. This is due to the proposed SRoIAlign sub-model, in which the floating-point quantization operation is used to quantize the 3D proposal to the feature volume. However, the integrated models (koCNN-I and DCNN-I) use truncated rounding quantization at the boundary of the feature volume, which reduces the segmentation accuracy. In addition, SRoIAlign uses the tri-linear interpolation method to calculate the feature value of the centre point of the feature bin, which also improves the segmentation accuracy. For the five single defects, the average segmentation accuracy of SISN-L3 is 45.40 AP, which is 4.22 points higher than that of SISN-L2. For the five defect combinations, the average segmentation accuracy of SISN-L3 is 42.72 AP, which is 3.84 points higher than that of SISN-L2. The difference in accuracy increment between the two is only 0.38 points. This is attributed to the SIGN sub-model which generates the segmentation instance of the defect in a point-to-point way. The results show that regardless of whether the geometric complexity of defects is consistent, the combination of SRPN, SRoIAlign and SIGN greatly improves the defect segmentation accuracy, and the three sub-models are compatible with each other.

To evaluate the classification accuracy of the proposed model for casting defects, we present another set of experimental data. We select the seven models in Table 1 as the test models and calculate the classification values of the seven models for five types of defects(Type-A, Type-B, Type-C, Type-D and Type-E). The calculation results are shown in Table 2. For Type-A defect, the classification value of the SISN-L1 model and koCNN model are 90.0 AP and 65.8 AP respectively. The mean Average Precision of SISN-L1 is 87.9 mAP and that of koCNN is 62.6 mAP. This shows that the SISN-L1 model has better classification performance than koCNN. This is due to the SSReLU and SCPool functions, which can better maintain the feature extraction precision of the whole-spectrum domain framework. Although koCNN also uses the whole-spectrum domain framework, the sinc interpolation and Hermitian symmetry method used by koCNN increases the feature extraction error of defects. Furthermore, the mean Average Precision of DCNN is 75.4 mAP, which is 12.5 mAP lower than that of SISN-L1 and 12.8 mAP higher than that of koCNN. This indicates that the classification precision of koCNN is not better than that of DCNN without integrating the proposed SIGN. For Type-E defect, the classification values of the SISN-L2 model and koCNN-I model are 88.0 AP and 67.1 AP respectively. The mean Average Precision of SISN-L2 is 90.4 mAP and that of koCNN-I is 68.9 mAP, with a difference of nearly 21.5 mAP. This is due to the adoption of the SRoIAlign sub-model in SISN-L2. The SRoIAlign sub-model implements the floating-point quantization operation on feature volumes, which eliminates the decline in feature extraction accuracy caused by the traditional quantization method. Furthermore, The mean Average Precision of DCNN-I is 79.8 mAP, which is 10.9 mAP higher than that of koCNN-I. This shows that the SIGN sub-model has better compatibility with koCNN than the DCNN. In addition, the classification error of DCNN-I is 4.4 mAP lower than that of DCNN, and the classification error of koCNN-I is 6.3 mAP lower than that of koCNN. This shows that SIGN performs well on the whole spectrum-domain framework, and it also plays a positive role in improving the classification performance of the spatial domain framework. Finally, the mean Average Precision values of SISN-L3 and koCNN-I are 92.1 mAP and 68.9 mAP, with a difference of 23.2 mAP. This is due to the SRPN and SRoIAlign sub-models which are used in SISN-L3. The SRPN sub-model can detect defects with complex shapes and accurately generate multi-scale 3D proposals. Even if SRPN gives rise to classification errors in the forward propagation pass, SRoIAlign can also correct the errors to a certain extent. Note that when choosing different backbone frameworks, the classification performance of koCNN and DCNN will be greatly affected. When the accuracy of backbone frameworks is low, the classification error of koCNN and DCNN will also significantly increase. However, because the proposed SISN adopts the spectrum domain training mechanism, the training accuracy of the SISN model is less affected by the backbone frameworks, which indicates that SISN is robust.

Table 3

The multiplication arithmetic complexity of koCNN, DCNN, LCNN, fbFFT and SISN

	LDNN/DCNN	fbFFT/koCNN(FFT-based)	SISN
fprop/Ffprop	[l] $S\cdot f^{2}\cdot f^{\prime}(n_{1}\cdot n_{2})\cdot$
$(k_{1}\cdot k_{2})$	[l] $S\cdot f^{\prime}(n_{1}\cdot n_{2}+k_{1}\cdot k_{2}-1)\cdot$
$(1+3/2\cdot\log(n_{1}\cdot n_{2}+$
$k_{1}\cdot k_{2}-1))$	[l] $S_{l}\cdot f_{l}^{\prime}\cdot l_{qp^{\prime}}\cdot(\max(l_{qp^{\prime}k^{% \prime}1}l_{qp^{\prime}k^{\prime}2})+$
$k_{qp^{\prime\prime}1}k_{qp^{\prime\prime}2}-1)(1+$
$\log(\max(l_{qp^{\prime}k^{\prime}1}l_{qp^{\prime}k^{\prime}2})+$
$k_{qp^{\prime\prime}1}k_{qp^{\prime\prime}2}-1))$
bprop/Fbprop	[l] $S\cdot f\cdot{f^{\prime}}^{2}(n_{1}-k_{1}+1)\cdot$
$(n_{2}-k_{2}+1)(k_{1}\cdot k_{2})$	[l] $S\cdot f^{\prime}(m_{1}\cdot m_{2}+k_{1}\cdot k_{2}-1)\cdot$
$(1+3/2\cdot\log(m_{1}\cdot m_{2}+$
$k_{1}\cdot k_{2}-1))$	[l] $S_{l}\cdot f_{l}\cdot l_{qp^{\prime}}\cdot(\max(l_{qp^{\prime}k^{\prime}1}l_{% qp^{\prime}k^{\prime}2})+$
$k_{qp^{\prime\prime}1}k_{qp^{\prime\prime}2}-1)(1+$
$\log(\max(l_{qp^{\prime}k^{\prime}1}l_{qp^{\prime}k^{\prime}2})+$
$k_{qp^{\prime\prime}1}k_{qp^{\prime\prime}2}-1))$
Weight calc.	[l] $S\cdot f\cdot{f^{\prime}}(n_{1}-k_{1}+1)\cdot$
$(n_{2}-k_{2}+1)(n_{1}\cdot n_{2})$	[l] $f\cdot f^{\prime}(m_{1}\cdot m_{2}+n_{1}\cdot n_{2}-1)\cdot$
$(1+3/2\cdot\log(m_{1}\cdot m_{2}+$
$n_{1}\cdot n_{2}-1))$	[l] $f_{i}\cdot f_{i}^{\prime}\cdot\max(l_{q},l_{q}^{\prime})\cdot(\max(l_{qp^{% \prime}k^{\prime}1}l_{qp^{\prime}k^{\prime}2}))+$
$\max(l_{qpk1}l_{qpk2}-1)(1+$
$\log(\log(\max(l_{qp^{\prime}k^{\prime}1}l_{qp^{\prime}k^{\prime}2})+$
$\max(l_{qpk1}l_{qpk2})-1))$
memoryframes	LDNN/DCNN	fbFFT/koCNN(FFT-based)	SISN
Weight	$f\cdot f^{\prime}\cdot k_{1}\cdot k_{2}$	$f\cdot f^{\prime}\cdot k_{1}\cdot k_{2}$	$f\cdot f^{\prime}\cdot k_{qp^{\prime\prime}1}\cdot k_{qp^{\prime\prime}2}$
Feature	[l] $S\cdot f\cdot n_{1}\cdot n_{2}/S\cdot$
$f^{\prime}m_{1}\cdot m_{2}$	$S\cdot f\cdot n_{1}\cdot n_{2}/S\cdot f^{\prime}m_{1}\cdot m_{2}$	$S_{i}\cdot f_{i}\cdot n_{qp1}\cdot n_{qp2}/S_{i}\cdot f_{i}^{\prime}m_{qp^{% \prime}1}\cdot m_{qp^{\prime}2}$
Cache for FFT	–	$4\tilde{n}(\tilde{n}_{1})(S\cdot f+S\cdot f^{\prime}+f\cdot f^{\prime})$	$4\tilde{n}(\tilde{n}_{1})(S\cdot f+S\cdot f^{\prime}+f\cdot f^{\prime})$

7.2 Time performance

We select four typical models as the Ref Models to test the time performance of the SISN. The four Ref Models are koCNN, DCNN, LCNN and fbFFT. We design a complexity measurement unit to calculate the time performance of the SISN model. The complexity measurement unit is the average arithmetic complexity reduction unit (abbreviated as AACr). We use ResNet50, VGGnet19 and AlexNet7 as the baseline framework to calculate the AACr values of the SISN under the four Ref Models.

Figure 7.

The four AACr values of SISN with respect to the three baseline frameworks. The four AACr values of SISN with respect to the three baseline frameworks are computed.

AACr is a rate coefficient designed for estimating the training time of the model under a particular baseline framework. As we choose koCNN as the Ref Model, the AACr value of SISN is calculated by dividing the training time of koCNN by the training time of SISN. The AACr value of SISN calculated by using koCNN as the Ref Model is denoted as AACr ${}_{ko}$ . Similarly, the AACr values of SISN calculated by using DCNN, LCNN, and fbFFT as the Ref Models are denoted as AACr ${}_{\text{DC}}$ , AACr ${}_{cu}$ and AACr ${}_{fb}$ , respectively. The training time of the framework is the sum of the convolution operation time in each propagation pipeline. Therefore, the spatial domain model needs $S\cdot f^{2}\cdot{f}^{\prime}(n_{1}\cdot n_{2})(k_{1}\cdot k_{2})$ operations. The frequency domain model needs $S\cdot{f}^{\prime}(n_{1}n_{2}+k_{1}k_{2}-1)(1+3/2\cdot\log(n_{1}n_{2}+k_{1}k_{% 2}-1))$ convolution operations. The SISN needs $S_{i}\cdot f_{i}^{\prime}\cdot l_{qp^{\prime}}\cdot(\max(l_{qp^{\prime}k^{% \prime}1}l_{qp^{\prime}k^{\prime}2})+k_{qp^{\prime\prime}1}k_{qp^{\prime\prime% }2}-1)(1+\log(\max(l_{qp^{\prime}k^{\prime}1}l_{qp^{\prime}k^{\prime}2})+k_{qp% ^{\prime\prime}1}k_{qp^{\prime\prime}2}-1))$ convolution operations. Because the time of a convolution operation is the time of a complex multiplication, the training time of the model can be evaluated by the multiplication arithmetic complexity. The multiplication arithmetic complexity of koCNN, DCNN, LCNN, fbFFT and SISN are shown in Table 3. Therefore, the AACr value of SISN is computed by:

$\displaystyle AACr_{{ko,DC,LC,fb}}=$ (3) $\displaystyle\sum_{i=1}^{128}\frac{T_{ko,DC,cu,fb}}{\makecell[l]{S_{i}\cdot f^% {\prime}_{i}\cdot l_{qp^{\prime}}\cdot(\max(l_{qp^{\prime}k^{\prime}1}l_{qp^{% \prime}k^{\prime}2})+}}$

where $T_{{ko,DC,LC,fb}}$ is the training time of the Ref Models. $S$ is the float multiplication cell of the CUDA. In Fig. 7, we employ Eq. (3) to compute the four AACr values of SISN with respect to the three baseline frameworks. Because the training domain of the Ref Model is different, we divide the four AACr values into two groups: the first group contains the AACr values of SISN calculated by using DCNN and LCNN as the Ref Models, and the second group contains the AACr values of SISN calculated by using fbFFT and koDNN as the Ref Models. With respect to the first group, the AACr ${}_{\text{LC}}$ value of SISN is 8.5843 in batch 128, and the AACr ${}_{\text{LC}}$ value of SISN is 6.2952 in batch 1, in Fig. 7(a). The AACr ${}_{\text{LC}}$ value of SISN decreases by 2.2891 points, and the mean AACr ${}_{\text{LC}}$ value of SISN under the eight batches is 7.5112, which is 1.0731 less than that of SISN in batch 128. This shows that the AACr ${}_{\text{LC}}$ value of the SISN obtains the peak value in batch 128. This also indicates that the LCNN training time will keep increasing, while the SISN training time tends to increase gradually. The reason is because the SISN is trained in the spectrum domain; that is, the SISN employs the SSReLU and SCPool operations to avoid the convolution operations in the training period. In contrast, the LCNN performs the convolution operations in the whole training period, and the convolution arithmetic complexity increases proportionally. In addition, the AACr ${}_{\text{DC}}$ value of SISN is 3.2189 in batch 128, and the AACr ${}_{\text{DC}}$ value of SISN is 2.3605 in batch 1. The AACr ${}_{\text{DC}}$ value of SISN decreases by 0.8584 points, and the mean AACr ${}_{\text{DC}}$ value of SISN under the eight batches is 2.8162, which is 4.6947 less than that of AACr ${}_{\text{LC}}$ . This finding shows that the time performance of DCNN is better than that of LCNN. However, the time performance of SISN is still better than both. With respect to the second group, when the batch size decreases from 128 to 1, the AACr of SISN shows an upward trend. For instance, in Fig. 7(c), the AACr ${}_{fb}$ value of SISN is 6.1794 in batch 128, and the AACr ${}_{fb}$ value of SISN is 24.7979 in batch 1. The AACr ${}_{fb}$ value of SISN decreases by 18.6185 points, nearly four times that before. This result is because the SISN performs the fast Fourier transforms only in the initialization phase, but the fbFFT requires transforming the feature volumes from the spatial domain to the spectrum domain in the whole convolution period. In addition, the AACr ${}_{ko}$ value of the SISN decreases by 6.8957 points, and the mean AACr ${}_{ko}$ value of the SISN under the eight batches is 4.0920, which is 1.8033 more than that of the SISN in batch 128. This shows that the AACr ${}_{ko}$ value of the SISN obtains the minimum value in batch 128. This also indicates that the training time of the SISN will keep decreasing, while the training time of the koCNN tends to decrease gradually until it is lower than the mean AACr ${}_{ko}$ . However, the maximum training time of koCNN is twice that of the SISN. This is due to the spectrum bins that are used in the SISN architecture. In the forward and backward propagation passes of the SISN, the input and output feature volumes are divided into a number of spectrum bins. These bins are computed in parallel by the float multiplication cell of the CUDA, that decreases the arithmetic complexity of the SISN in parallel with the GPU bandwidth required for spectrum domain training.

7.3 The extraction accuracy

Accurate extraction of the mechanical properties of casting defects is the key factor in evaluating the quality of casting products. However, due to the low segmentation accuracy for casting defects, it is not easy to calculate the mechanical properties of casting defects accurately. To address this, the SISN model is proposed in this work. To measure the influence of the SISN on the mechanical property extraction accuracy, we present five typical casting defects segmented by the SISN in this section and use the stress finite element method [50] to calculate the mechanical properties of these defects. Then, the calculation results are compared with the two representative models: DCNN-I and koCNN-I.

Figure 8.

Casting defects segmented by the SISN model. We select five representative defect instances as test samples (denoted as Type-A, Type-B, Type-C, Type-D and Type-E), calculate their mechanical properties, and characterize the porosity parameters of each defect instance in the solidification process by the hierarchical visualization method.

Figure 9.

The mechanical properties of Type-B, Type-C, Type-D and Type-E defect instances. Note that the mechanical properties of Type-A defect is analyzed in the next experiment.

First, we use the in-house EA88 dataset to train the SISN model and select the EA88 cylinder block casting as the segmentation object. The segmentation results are shown in Fig. 8. The EA88 model contains 101 defect instances, and the maximum diameter of each defect instance is in the range of 0.02 to 1.5 cm. We select five representative defect instances as test samples (denoted as Type-A, Type-B, Type-C, Type-D and Type-E), calculate their mechanical properties, and characterize the porosity parameters of each defect instance in the solidification process by the hierarchical visualization method, as shown in Fig. 8. According to the solidification sequence of Type-A instances, the porosity value of Type-A fluctuates from 250 seconds to 400 seconds and tends to be stable after 400 seconds. The fluctuation process in the solidification sequence is characterized by the colour hybrid transition at the intersection of the segmentation lines. The porosity of Type-B defects fluctuates during the solidification process from 440 seconds to 700 seconds and tends to be stable after 700 seconds. The porosity of Type-C defects fluctuates from 490 seconds to 610 seconds and tends to be stable after 610 seconds. The porosity of Type-D defects fluctuates during the solidification process from 130 seconds to 250 seconds and tends to be stable after more than 250 seconds. The solidification sequence of the five types of defects is consistent with their porosity distribution curve (see Fig. 9). The results show that the proposed model can accurately segment the defects and characterize the solidification sequence of the segmented defects, which lays the foundation for the next step of the calculation of the mechanical properties of defects.

Second, we calculate the mechanical properties of the four defect instances (Type-B, Type-C, Type-D and Type-E; Note that the Type-A defect is analyzed in the next experiment), and select 15 sampling points on each type of defect instances to calculate the mechanical properties of each point in the solidification process, as shown in Fig. 9. We use the characters Pt1, Pt2, $\ldots$ , Pt15 to mark these 15 sampling points, and each sampling point is distributed in a different solidification sequence. By using these sampling points, we can evaluate the effect of the segmentation model on the calculation of mechanical properties more comprehensively. Next, we analyze the mechanical properties of 15 points of four type defect instances. (1) Type-B defects. At the 700th second of solidification process, the von mise stress of 15 points tend to increase steadily. This indicates that the liquid metal begins to solidify at 700 seconds. From the porosity curves of 15 points, it can be seen that when the solidification process reaches 700 seconds, the porosity curves of 15 points show a horizontal extension state. It shows that the von mise stress values of the 15 points are consistent with the porosity values of the defect instances segmented by the proposed model. In addition, the tensile strength of Type-B defect is about 5 MPa higher than that of Type-E defect. However, the shape of the Type-B defect is more complex than Type-E defects. The result shows that the shape of defects will affect their tensile strength and the segmentation accuracy of defects plays a key role in the calculation of mechanical properties. Note that the scatter diagrams of Type-B defects is similar to that of Type-E defects. (2) Type-C defects. The tensile strength and yield strength of Type-C are the highest among the four types. In the 500th second of solidification process, the tensile strength and yield strength of 15 points are over 30 MPa and 22 Mpa respectively. This is due to the fact that the volume of Type-C defect instances is smaller than that of other defects and their shapes are not too complex. (3) Type-D defects. When the minimum value of the von mise stress of 15 points reaches 2.3, the von mise stress is no longer proportional to the plasticstrain; that is, the plasticstrain does not fluctuate anymore. This shows that the defect instances will maintain permanent deformation after the 700th seconds of solidification process, which is consistent with the experimental data. (4) Type-E defects. During the solidification process, the von mise stress curve of 15 points began to separate and decrease at the 320th second, and ended at the 400th second. This indicates that the liquid metal enters into the liquid-solid state at 320 seconds, and begins to enter the solidification state at 400 seconds. When the liquid metal enters the solidification state, the defects will be produced and enter a stable state. From the porosity curves of 15 points, it can be seen that when the solidification process reaches 400 seconds, the porosity curves of 15 points show horizontal extension state. In addition, it can be seen that the tensile strength curve and yield strength curve of 15 points are basically fitted. This indicates that the strength distribution of Type-E defects is uniform. From the cold crack curves of 15 points, it can be seen that the cold crack parameters of Pt1 and Pt2 are more than 0.25, that indicates that cold crack defects are easy to appear at these two points. In addition, the scatter diagrams of porosity and tensile strength of Type-E defects are presented. When the porosity value is around 0 to 1, the tensile strength value is around 6.5 MPa to 8.0 MPa; when the porosity value is around 1 to 2.5, the tensile strength value is around 5.0 mpa to 6.5 MPa; finally, when the porosity value is around 3.5 to 5, the tensile strength value is around 1.5 MPa to 2.5 MPa; this shows that the porosity value of Type-E defect is inversely proportional to their tensile strength value, that is in good agreement with the experimental data. The results show that the proposed model plays a positive role in the accurate calculation of mechanical properties of defects. It can be seen that the shape and volume of the defect instances have an effect on the mechanical properties of the defects. Therefore, it is further proved that the segmentation accuracy of defects plays an important role in the calculation of mechanical properties.

Table 4

The porosity error, tensile strength error and hardness error of the three models at 15 sampling points

Model error points	DCNN-I			koCNN-I			SISN
	Ep	Et	Eh	Ep	Et	Eh	Ep	Et	Eh
Pt1	7.1030	11.1150	14.1560	6.2341	9.2461	9.2871	1.0010	1.0130	1.0290
Pt2	8.5654	12.5765	15.9531	7.4453	10.4564	10.833	1.2522	1.2633	1.3655
Pt3	8.5090	12.5411	15.9288	7.4456	10.4777	10.8654	1.1955	1.2276	1.3556
Pt4	8.5910	13.6865	17.1819	7.5131	10.6086	11.1040	1.2100	1.3055	1.3999
Pt5	7.4052	11.4863	14.9685	6.3652	9.4463	9.9285	1.1721	1.2532	1.4011
Pt6	8.408	13.3867	17.7509	7.3616	11.3403	11.8045	1.1785	1.2572	1.3855
Pt7	8.4164	13.3816	17.7124	7.3708	11.336	11.7668	1.1777	1.2429	1.3656
Pt8	9.4114	13.3826	16.7263	8.3633	11.3345	11.7782	1.1802	1.2514	1.3725
Pt9	9.4106	13.2804	16.5154	8.3629	11.2327	11.6677	1.1798	1.2496	1.3652
Pt10	8.381	12.2462	15.4568	7.3476	10.2128	10.6234	1.1655	1.2307	1.3454
Pt11	8.3844	12.2386	15.4953	7.3500	10.2042	10.6609	1.1665	1.2207	1.4025
Pt12	7.3846	11.0997	13.2272	6.3499	8.0650	8.5285	1.1668	1.2179	1.4124
Pt13	7.3710	10.0188	11.9210	6.3411	7.9889	8.2911	1.1620	1.2098	1.2544
Pt14	7.3656	10.0055	11.8775	6.3379	7.9778	8.2498	1.1598	1.1997	1.2321
Pt15	6.2974	8.3275	8.5577	5.2975	5.3276	5.5578	1.1320	1.1621	1.2001
error	8.0669	11.9182	14.8952	7.0323	9.6836	10.0631	1.1666	1.2203	1.3257

Table 5

List of validation parameters and functions

Coefficients	Description	Function
AACr	Average Arithmetic Complexity reduction	A rate coefficient designed for estimating the training time of the model under a particular baseline framework
TFLOPS	Tera Floating-point operations per second	One trillion ( $=10^{12}$ ) floating point operations per second
AP	Average Precision	Evaluating the defect detection accuracy and segmentation accuracy
mAP	Mean Absolute Precision	Quantifying the accuracy of the predicted data
$L_{\text{ins}}$	$L_{\text{ins}}(i,o)=-(i\times\log(o)+(1-i)\times\log(1-o))$	The average value of the relative entropy loss
$L_{\text{box}}$	Each anchor box	The bounding box loss
$p\ast$	$L=L_{\text{cls}}+\lambda p\ast L_{\text{box}}+L_{\text{ins}}$	The true value of an anchor being a defect
$T_{B}$	The time complexity of CUDA pipeline is $T_{B}\log_{2}N$	The calculation time of a butterfly operation
Sdown $(\cdot)$	$\sum_{n=\beta}{\max(y_{qp^{\prime}k^{\prime}}(n))\cdot e^{-j\frac{2\pi}{i_{qp^% {\prime}k^{\prime}}}n\cap}}$	The spectrum domain down-sampling function
$S{\rm R}^{+}(\cdot)$	$\frac{1}{2}(Y_{qp^{\prime}}(U)+\\|{Y_{qp^{\prime}}(U)}\\|)$	The forward stochastic deviation function

Figure 10.

Mechanical property extraction pipeline for a Type-A defect instance. Fifteen sampling points are selected from Type-A defect to calculate their mechanical properties during solidification process.

At last, to verify the superiority of the proposed model in accurate calculation of mechanical properties of defects, we select two representative models: DCNN-I and koCNN-I as the comparison model. We use the Type-A defect to test the extraction performance of DCNN-I, koCNN-I and SISN. We select 15 sampling points on the Type-A defect instance to calculate the mechanical properties of each point in the solidification process, as shown in Fig. 10. Then, we use the stress finite element method to calculate the porosity, tensile strength and hardness of the three models at 15 sampling points. Note: the sampling time of the solidification sequence is from 0 to 800 seconds. Finally, we calculate the average values of the porosity, tensile strength and hardness of the three models at 15 sampling points and calculate the difference between these average values and the truth values of the mechanical properties; that is, the difference values are the porosity error values, tensile strength error values and hardness error values of the three models at 15 sampling points, as shown in Table 4. We use Ep to represent the porosity error, Et to represent the tensile strength error and Eh to represent the hardness error. The Ep value of SISN model at Pt1 is equal to 1.1010. Compared with other sampling points, the Ep value of the SISN model at Pt1 is the smallest. The Ep values of DCNN and koCNN are 7.1030 and 6.2341 at Pt1, respectively. Therefore, the error values are six to seven times that of the SISN. This shows that the SISN model can extract the porosity of defects with complex shapes more accurately than the other two models. In addition, the maximum tensile strength error of the SISN model appears at Pt4, where Et is equal to 1.3055; the maximum tensile strength error of the DCNN model appears at Pt8, where Et is equal to 9.4114; and the maximum tensile strength error of the koCNN model appears at pt8, where Et is equal to 11.3345. The results show that in the worst case, the tensile strength error of the SISN model is still lower than that of the other two models, and the error value of SISN is only 1/11 that of the koCNN model. Furthermore, the average porosity error of the SISN model at all points is equal to 1.1666, the average tensile strength error is equal to 1.2203, and the average hardness error is equal to 1.3257. Compared with the DCNN model and koCNN model, the average error is at least 6 orders of magnitude lower. The results show that the mechanical properties of 15 sampling defects in the solidification process can be extracted more accurately by using SISN to segment casting defects. Note that we list the main parameters used in this paper in Table 5.

8. Conclusion and future work

Taking SRPN, SRoIAlign and SIGN as three crucial sub-models, a spectrum domain instance segmentation model is proposed for accurately segmenting casting defects. The recognition accuracy and segmentation accuracy of the proposed model for casting defects are at least 3.4AP and 11.1AP higher than those of the state-of-the-art models. However, in our spectrum domain training architecture, the spectrum domain activation function is limited to the stochastic rectified linear unit operation, which needs to be implemented in the high-performance GPU-CUDA unit. This limitation restricts the wide-spread use of spectrum domain models in the field of casting CAE. Furthermore, our model is designed for segmenting five types of casting defects: inclusion, shrinkage, hot tearing, cold tearing and micro pores exclusively. In the future, we intend to probe the potential segmentation ability of the spectrum domain model for various types of casting defects. We also try to extend our idea to 3D data simplification [51] and 3D data exchange [52] and other powerful supervised machine learning/classification algorithms [53, 54, 55]. The above generality exploration and recent references will be interesting to the readers of ICAE journal.

Footnotes

Acknowledgments

The authors would like to thank the casting metallographic Research Institute of FAW Foundry Co., Ltd. This work was supported in part by the National Natural Science Foundation of China under Grant 62006027, in part by the National High-tech Research and Development Program under Grant 2014AA7031010B, and in part by the Science and Technology Research Project of Jilin Provincial Department of Education (13th five year plan) under Grant JJKH20200681KJ, in part by the enterprise cooperation project under Grant 1834003XXXX.

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A spectrum-domain instance segmentation model for casting defects

Abstract

Keywords

1. Introduction

3. The architecture of SISN

Table 2 Classification performance for SISN, koCNN and DCNN

Footnotes

Acknowledgments

References

Table 2
Classification performance for SISN, koCNN and DCNN