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Abstract
The surface morphology of electrospun fibers largely determines their application scenarios. Conventional scanning electron microscopy is usually used to observe the microstructure of polymer electrospun fibers, which is time consuming and will cause damage to the samples. In this paper, we use backscattering Mueller polarimetry to classify the microstructural features of materials by statistical learning methods. Before feeding the Mueller matrix (MM) data into the classifier, we use a two-stage feature extraction method to find out representative polarization parameters. First, we filter out the irrelevant MM elements according to their characteristic powers measured by mutual information. Then we use Correlation Explanation (CorEx) method to group interdependent elements and extract parameters that represent their relationships in each group. The extracted parameters are evaluated by the random forest classifier in a wrapper forward feature selection way and the results show the effectiveness in classification performance, which also shows the possibility to detect nonporous electrospun fibers automatically in real time.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electrospun fibers have been used in diverse applications ranging from materials science and technology to life sciences and biomedicine [1,2]. The properties of the fibers mainly depend on the morphologies. for example, nonporous fibers are used as scaffolds or vascular repair materials to imitate extracellular matrix for their regular surfaces and fine diameters [3]. Various factors, e.g. parameters on electrospinning, solutions and surrounding environments, can affect the surface features of the fibers in the production process [4], which increases the difficulty of getting the morphological information just from electrospinning parameters in advance. Researchers usually rely on scanning electron microscopy (SEM) to monitor the morphology of samples. However, SEM is time consuming and may damage the fibers. Therefore, it is necessary to develop non-destructive, fast and effective techniques to characterize different morphologies of fibers.
Polarimetric imaging approach can provide much richer information than intensity images and has demonstrated its potential in material characterization [5,6], atmospheric remote sensing [7] and biomedicine [8,9]. It is non-invasive and sensitive to the morphological changes of materials. Previously, we used a transmission MM microscope to measure the electrospun fibers and distinguish different morphologies with depolarizaion parameter MMD-$\Delta$ [10]. In this paper, we use a backscattering Mueller polarimetric imaging system which has lower resolution and is more adaptive to thicker samples.
There have been consistent efforts to derive new polarization parameters from Mueller matrix (MM) for better description of the samples. MM polar decomposition (MMPD) [11] and MM transformation (MMT) [12,13] have been used to derive new sets of parameters which are explicitly related to the optical and structural properties of the samples. He et al. [14] analyzed the frequency distribution histograms of MM elements and used central moments of them to distinguish different microstructures of tissues. Since the complexity of many materials, it is difficult to distinguish variant structures with certain optical or polarimetric properties. Therefore, many efforts, e.g. [15–19], have tried to utilize statistical learning methods to analyze MM data. Others [15,16] built a decision-theoretic framework consisting of several classifiers to identity precancerous uterine cervix tissues. Vaughn et al. [17] applied non-linear support vector machines to classify 47 different materials. Cameron et al. [18] used principal component analysis (PCA) and feature selection method to analyze the scattering coefficient contributions in turbid media. Hoover et al. [19] used PCA and nonlinear components analysis to distinguish among electromagnetic-wave scattering characteristics of materials.
As we know, irrelevant or redundant parameters will cause a negative impact on the classification performance. In this work, we used feature selection and extraction to explore representative polarization parameters before applying the classifier on MM data. In the feature extraction procedure, we pay attention to the correlative information among MM elements. We analyzed the discriminative power of each element considering the dependencies with the target microstructures and the redundancies and complementarities [20,21] among them. Then, we used Correlation Explanation (CorEx) method [22,23] to explore the interdependence among elements. There are three major contributions in this paper.
- - A low-resolution polarimetric imaging method that distinguishes among backscattering characteristics of electrospun fibers was built, which potentially provides a real-time and dynamic way of monitoring nonporous surface.
- - We utilized information-theoretic methods to quantitatively evaluate the importance of MM elements in distinguishing different microstructures of electrospun fibers.
- - We analyzed the interdependent information among MM elements and extracted new parameters representing these relationships. The parameters were evaluated by a random forest classifier in a wrapper forward feature selection way.
2. Materials and polarimetric imaging acquisition system
In the electrospinning procedure, different proportions of Poly-(L-lactic acid) (PLLA) and Poly($\varepsilon$-caprolactone) (PCL) were mixed and dissolved in methylene chloride. The experimental preparation of fibers is detailed in [10]. Figure 1 shows the SEM photographs of fibers with variant morphologies.
Fig. 1. SEM photographs of electrospun fibers with different surface morphologies. (a) Nonporous surface, PLLA/PCL (25:75), Spinning solution (8wt%), N,N-Dimethylformamide (DMF, 6g); (b) Nonporous surface, PLLA/PCL (100:0), Spinning solution (8wt%), DMF (6g); (c) Meshed surface PLLA/PCL (100:0), Spinning solution (8wt%), DMF (4.6g); (d) Porous microspheres (25:75), Spinning solution (4wt%), DMF (4.8g).
We adopted a typical backscattering MM measurements with dual rotating retarder configuration [24–26] as illustrated in Fig. 2. The polarimeter consists of one light-emitting diode (Source, 633nm, 3w), two lens (L1 and L2, Thorlabs, USA), two rotating quater-wave plates (R1 and R2, Daheng, Optic, China), two polarizers (P1 and P2, extinction ratio > 1000:1) and a CCD camera (Qimaging 32-0122A, 12bit, Canada). The exposure time used for the CCD camera is 200ms. To reduce the influence of surface reflection, we set an oblique angle ($\theta =30^\circ$) between the illumination path and the axis of detection. In each measurement, 30 images with specific incident and output polarization states are obtained by the rotation of two retarders with the angular speed of $\theta _{2}=5\theta _{1}$. The acquisition time of the data for each MM measurement is 90 seconds and the subsequent prediction time is within 1 second on one desk computer. The MM is calculated by using the Fourier coefficients $a_{n}$ and $b_{n}$ as shown in Eq. (1) [27].
Fig. 2. Experimental setup. (a) Intensity distribution from the CCD where the center region (red box) is cropped and analyzed. (b)Mueller matrix backscattering imaging system with two active polarimetric elements: a Polarization State Generator (PSG) and a Polarization State Analyzer (PSA). (c) Photograph of an electrospun fiber sample.
The MM images of electrospun fibers with different morphologies are shown in Fig. 3. The non-diagonal matrices indicate that the materials are anisotropic. The variations of three diagonal elements ($M_{22},M_{33}$ and $M_{44}$) show the differences of depolarization abilities. The values of $M_{24},M_{34}, M_{42}$ and $M_{43}$ mean that the samples are birefringent.
Fig. 3. 2D images of MMs of electrospun fibers. (a) Nonporous surface; (b) Meshed surface; (c) Porous microspheres. The color bar is from −0.5 to 0.5 for diagonal elements and from −0.1 to 0.1 for other elements.
The individual matrix elements were cropped to $300\times 300$ pixels (approximate $0.6cm\times 0.6cm$), with the center located at the sample region. In addition, all elements were normalized to the first element $M_{11}$ that was excluded in the next data analyzing procedure. To reduce the impact of noise, we filtered all remained 15 images by a $5\times 5$ slide window (sampling with a 5-pixel step) and computed the average of these 25 pixels. We handle data in the pixel level. For a filtered pixel in MM, 15 elements would be stacked as a $15\times 1$ vector.
3. Feature selection and extraction
Mueller matrix (MM) has qualitative information of the microstructures and analyzing the discriminative parameters is useful for revealing the polarization characteristics about samples. In this section, we resort to information measures to select and extract the most representative parameters.
For a given instance set comprising $N$ observations $\{m^{n}\}$, where $m^{n}\in \mathcal {M}$ and $n=1,\ldots ,N$, together with corresponding morphology labels (or target classes) $\{c^{n}\}$ where $c^n\in \mathcal {C}$. We denote $M_{ij}$ as the element (we use "element" $M_{ij}$ rather than standard notation "feature" $X$ according to the context of this work) in MM with $i^{th}$ row and $j^{th}$ colunm index, where $i,j\in \{1,2,3,4\}$ and denote $C$ as the label variable. $c_i$ is considered as the $i^{th}$ label where $i=1,\ldots ,|\mathcal {C}|$ (in this work, $|\mathcal {C}|=2$). The Shannon’s entropy [28] function $H(\cdot )$ and $H(\cdot |\cdot )$ are used to measure the uncertainty of variables with and without condition respectively. For the prior class variable $C$, the information gain, also known as mutual information (MI), from one element $M_{ij}$ is the reduction in the prior uncertainty due to the knowledge of this element as shown in Eq. (2).
This gives a way to quantify how much information an element or an element set can provide with respect to the class variable $C$. $I(M_{ij},C)$ is non-negative and is zero if $M_{ij}$ and $C$ are statistically independent. The maximum value happens when $C$ is completely determined by $M_{ij}$, i.e. $H(C|M_{ij})=0$.MI has been widely used in the feature selection literature. We consider $S=\{M_{ij}\}$ as the selected subset with $k$ items ($|S|=k$). Since the hardness of the estimation for joint probability density $p(S)$ and $p(S,C)$, [29] selects the top $k$ features with high MI score individually, i.e., $\max {\sum _{M_{ij}\in {S}}{I(M_{ij},C)}}$, where $k$ is predefined. However, for highly interdependent elements, the discriminative power would not change much if one of them were removed. To get a nested subset, besides the MI score with respect to the class variable, [30–32] consider the small similarity or the low dependency between elements, i.e., $\min {\sum _{M_{ij},M_{i'j'}\in {S}}I(M_{ij},M_{i'j'})}$. However, [20,33] pointed out that dependency does not simply imply redundancy and the interaction between elements may provide new information about $C$, measured as conditional dependency: $\sum _{M_{ij},M_{i'j'}\in {S}}I(M_{ij},M_{i'j'}|C)$. Brown et al. [20] summarized the selection criteria from the literature as a linear combination of these three components in Eq. (3).
Some feature extraction methods [38,39] based on information theory have been proposed. In this work, we mainly focus on the interdependence among elements. In the selection procedure, we filter out irrelevant elements only considering the first term in Eq. (3). To represent the correlative information among elements, we utilize an unsupervised information-theoretic method, Correlation Explanation (CorEx) [22,23], that groups the highly interdependent elements together and extracts the representative parameters from them, which will also preserve the interaction information in each group. Terminologically, we are trying to find representative variables $Y_{1},Y_{2},\ldots ,Y_{l} \in G$ that can explain the relationships in $S$.
Total correlation (TC) or multivariate mutual information [34,40] is a general measure of non-independence among elements from a variable set $S$ shown as Eq. (4). For $|S|=2$, $TC(S)$ is the expression of mutual information.
The extent to which $G$ explains the relationships in $S$ can be measured by looking as how much the total correlation is reduced after conditioning on $G$ [22,23] shown as Eq. (5) where $TC(S|G)=\sum _{M_{ij}\in {S}}{H(M_{ij}|G)-H(S|G)}$. The maximum is achieved when $TC(S|G)=0$, in which case that variables in S are independent conditioned on $G$. This indicates that variables $Y_{j}$s in $G$ explain all the relationships in $S$. CorEx divides the set $S$ into several subgroups and simultaneously searches the representative $Y_{j}$ in each group by optimizing the information-theoretic objective function shown as Eq. (6) where $S_{j}$ is denoted as a subgroup and $l$ is the total group number. $Y_{j}$, the probabilistic function of $S_{j}$, is a discrete random variable and $r$ is the number of the possible values of it. The search strategy and implementations are discussed in [22]. Before feeding the data into CorEx, we need to determine the subgroup number $l$. Here we both consider the interdependence among elements and their polarization properties of MM elements.- Step1: Rank the mutual information between each element and the target class variable $C$. For a given threshold $T_{0}$ denoted as the amount of information, select the top $k$ ones where $k$ is the minimum value that $\frac {\sum _{k}{I(M_{ij},C)}}{\sum _{15}{I(M_{ij},C)}} \geq T_{0}$.
- Step2: Calculate the interdependence among the selected elements. Given a threshold $T_{1}$, for $M_{ij}$ and $M_{i'j'}$, group them together when $\frac {I(M_{ij},M_{i'j'}|C)}{H(M_{ij},M_{i'j'}|C)} \geq T_{1}$. Then count the group number $l$.
- Step3: Given the predefined group number $l$, use CorEx to search the representative parameters about correlative level. The parameter $r$ is determined through trail-and-error.
4. Experiment results
Backscattered Mueller imaging were performed on 19 electrospun fiber samples. After being filtered and stacked as described in Section 2, the dataset encompasses 68,400 averaged pixels as instances. We used 10-fold cross validation strategy and the test samples did not overlap with the samples used in the training set.
In this work, we used a feature ranking and selecting strategy as the first step and set $T_{0}=0.85$. Figure 4(a) shows the MIs between elements and the class label $C$, sorted by decreasing values. The curve indicates the percentage of cumulative MI. As we can see, the top seven elements include most of the characteristic information (nearly 86%) which represent the variants of different morphologies.
Fig. 4. Mutual information (MI) scores of individual elements. (a) MI between each element and the class variable $C$, i.e. $I(M_{ij},C)$. The curve represents the cumulative percentage of MI. (b) An example of multi-information between $M_{44}$ and other elements, i.e. $I(M_{44},M_{ij}|C)-I(M_{44},M_{ij})$. Here we set the interaction values with $M_{44}$ itself as zero since the value is the opposite of $I(M_{44},C)$ (non-positive).
For given MMs, besides the individual characteristic power of each element, the redundant or synergistic information among them should also be considered. Figure 4(b) shows the multi-information ($I(M_{44},M_{ij}|C)-I(M_{44},M_{ij})$) of $M_{44}$ interacting with other elements. The positive values mean that the kind of interaction information between them are complementary rather than redundancy, especially with other two diagonal elements $M_{22}$ and $M_{33}$. For the top seven elements, the three diagonal elements are related to the depolarization ability while the others are related to birefringence properties ($q_{L}=\sqrt {M_{42}^{2}+M_{43}^2}$ and $r_{L}=\sqrt {M_{24}^2+M_{34}^2}$ are rotation invariant parameters that represent the capability of transforming between linear and circular polarization [41]). We divided the selected elements into two groups and then extracted the parameters which could represent the correlative information among them in each group.
We set $l=2$ in Eq. (6) and fed the seven elements into CorEx model. Figure 5(a) shows the MI results between the extracted parameters, $Y_{1}$ and $Y_{2}$, from each group and the class variable $C$. The results converge as the cardinality $|Y_{j}|=r$ increases. Two dashed or dotted lines paralleling the horizontal axis are the MIs of $M_{22}$ and $M_{34}$ which are the highest individual scores in each group respectively. The greater MI values of extracted parameters may indicate that the correlative information among elements are characteristic. After $r=3$, the separation results were stable where the selected elements were divided into two groups: diagonal set $S_{1}=\{M_{22}, M_{33}, M_{44}\}$ and non-diagonal set $S_{2}=\{M_{24}, M_{34}, M_{42}, M_{43}\}$ as illustrated in Fig. 5(b), which satisfied the multi-information results in Fig. 4(b) to some extend. It should be noted that we only predefined the group number, and CorEx divided the element set according to the conditional total correlation maximization criterion without any other prior knowledge.
Fig. 5. Grouping results from CorEx given the group number $l=2$. (a) The MI between extracted parameters from each group and class variable. The horizontal axis is the number of possible values $r$ in Eq. (6). (b) The schematic diagram of grouping results generating by CorEx. Elements in the bottom row represents the input. $Y_{1}$ and $Y_{2}$ in the higher layers are learned parameters which explain the relationships in each group.
In order to demonstrate the effectiveness of our method, we conducted a serious of experiments on two parameter sets: One is the original parameter set; Another is the enhanced set including $\{Y_{1}, Y_{2}\}$. We used the random forest (RF) classifier as an evaluator to do the feature selection in a wrapper forward way. Figure 6 shows the accuracy results with the increasing feature number. The overall performance on the element set with extracted parameters are better than that without them. The classifier picked $Y_{1}$ as the first feature whose individual prediction was at a higher significance level than others in the data. Although the MI score of $Y_{2}$ is much higher, it was selected further back. The multi-information between $Y_{1}$ and $Y_{2}$ was negative, which means that they were redundant and adding $Y_{2}$ would diminish the dependence between $Y_{1}$ and $C$. If we set the first feature as $Y_{2}$ (the second best individual feature according to MI score), $Y_{1}$ would be added in the fourth step and would improve the performance slightly, which might be caused since the dependency between $Y_{1}$ and $C$ is much more important that the feature redundancy.
Fig. 6. The RF classification accuracy with the wrapper forward selection. Three lines denote the results on: (1) All 15 MM elements and two extracted parameters; (2) All 15 MM elements and two extracted parameters with setting the first picked feature as $Y_{2}$; (3) All 15 MM elements.
After wrapper selection by RF, 8 parameters were selected from the enhanced set while 6 parameters remained from the original set. The maximum accuracies were 0.859 and 0.842 respectively. Table. 1 shows the overall performance on these two sets before and after wrapper section. The results show that the enhanced parameter set with $\{Y_{1}, Y_{2}\}$ lead to better performance than the original set. Furthermore, we visualized the receiver operating characteristic (ROC) curves for these experiments as illustrated in Fig. 7. The AUCs from the enhanced set are relatively higher.
Table 1. Performance comparison with and without $\{Y_{1}, Y_{2}\}$
Fig. 7. ROC curves for classification results with RF. (a) Parameter sets before wrapper selection. The AUCs on original and enhanced sets are 0.894 and 0.865 respectively. (b) Parameter sets after wrapper selection. The AUCs on original and enhanced sets are 0.940 and 0.908 respectively.
When the trees in RF were being trained, the splitting strategy on every node was based on how much each element or parameter contributes to decreasing the weighted impurity. By averaging the decrease in impurity over trees, we could get the feature importance as illustrated in Fig. 8. The extracted parameters $Y_{1}$ and $Y_{2}$ were in top 3.
5. Conclusion
In the past decades, many efforts have tried to analyze the Mueller matrix (MM) and have extracted many parameters to describe different polarization properties based on certain assumptions. In this paper, we analyze the correlative information among elements in a statistical view. We used backscattering Mueller polarimetry to detect nonporous morphologies of electro-spun fibers. We selected the most representative elements based on the results of mutual information with target morphology variable individually. Then, we used CorEx method to divide the selected elements into two groups and extracted new parameters which represented the relationships in each group.
The selected elements in two groups are related to the depolarization power and birefringence properties respectively. In the future work, we plan to use Monte Carlo simulations based on the sphere-cylinder scatterers model to analyze the underlying meaning of these parameters. Meanwhile, the correctness of the estimation of entropy from the data is the key to guide the feature selection and extraction procedure. More estimation methods, e.g. network-based methods [42], will be compared to determine the most suitable one for MM data.
Funding
National Natural Science Foundation of China (11974206, 61527826); Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20160818143050110, JCYJ20170412170814624).
Disclosures
The authors declare no conflicts of interest.
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