3.1.

Overview

We compare three windowing techniques: (i) No windowing is performed (i.e., features are extracted from all wavelengths $400–725\mathrm{nm}$ ), (ii) fixed window size of $20\mathrm{nm}$ (window size adopted from Ref. 5), and (iii) adaptive spectral window sizes determined by the novel algorithm presented in this paper.

3.2.

Data Collection

This analysis used a diffuse reflectance spectroscopy data set obtained in a previous investigation of oblique polarization reflectance spectroscopy (OPRS) of oral mucosa lesions.³⁰ In this study, we used only the diffuse reflectance subset of the OPRS data set because diffuse reflectance spectroscopy is more commonly used than polarized reflectance spectroscopy. Moreover, the purpose of this study is to demonstrate the effectiveness of a new windowing method that can be used with a broad range of spectral data; thus, the selection of a particular spectroscopy technique as an exemplar is arbitrary.

The data set was acquired at The University of Texas M. D. Anderson Cancer Center (UT MDACC) in Houston, Texas, as previously described.³⁰ Briefly, 27 patients over the age of $18\text{years}$ were referred to the Head and Neck Clinic at UT MDACC with oral mucosa lesions suspicious for dysplasia or carcinoma.³⁰ Spectroscopic measurements were typically performed on one or two visually abnormal sites and one visually normal site. Biopsies were taken of all measured tissue sites. We measured a total of 57 sites, of which 22 were visually and histopathologically normal (Normal), 13 sites were visually abnormal but histopathologically normal (Benign), 12 were visually abnormal sites that proved to be mild dysplasia (MD) on histopathology, and 10 were visually abnormal sites that proved to be severe high-grade dysplasia or carcinoma (SD) on histopathology. The resulting data set consisted of 618 diffuse reflectance spectroscopy data points within the wavelength range of $400–725\mathrm{nm}$ for 57 measurements. The resolution of the spectra is one data point every $0.5\mathrm{nm}$ .

We used MATLAB R14 (The MathWorks, Natick, Massachusetts) and its statistics toolbox for data analysis in this study.

3.3.

Preprocessing

The spectra were normalized to remove interpatient variation. As suggested by our previous study.³⁰ the spectra were dark subtracted, then divided by the diffuse reflectance from a white standard (Labsphere, SRS-99) to correct for the spectral response of the system and spectral profile of the source. Then each spectrum was normalized by dividing each intensity value by the intensity at $420\mathrm{nm}$ . No downsampling was performed in this study because detailed data were needed for piecewise linear regression.

3.4.

Piecewise Linear Regression for Spectral Feature Extraction

We developed an algorithm to adaptively adjust the spectral window size for feature extraction from optical spectra. The spectral window sizes are maximized given defined acceptable linear fits on the spectrum.

Our method first sets an initial spectral window size of $5\mathrm{nm}$ . The choice of this initial spectral window size is based on the spectral resolution $\left(5\mathrm{nm}\right)$ of our measured spectra. The spectral window size is iteratively increased by $5\mathrm{nm}$ , and simple linear regression is performed within the spectral window. A stopping criterion of $R^2$ of 0.8 is applied to ensure the goodness of each fit. The spectral window size and stopping criterion can be adjusted if other spectroscopy data, such as fluorescence spectra, are used. This algorithm ensures each piece of the linear model achieves the largest possible window size to fit a linear regression line with an $R^2$ value of 0.8 or greater. Once the $R^2$ value falls below 0.8, the iteration ends and the starting position of the next window is set at the ending position of the current window. The new window size is re-initialized to $5\mathrm{nm}$ , and the calculation for $R^2$ is repeated.

3.5.

Feature Extraction

Within every spectral window, eight spectral features are extracted to describe the signal (Table 1). Six features have been used previously by others to capture diagnostic information to detect cancer. These features were used to detect cancer in various organ sites: oral,^{12, 19} breast,⁵ and bladder,³¹ with diffuse reflectance spectroscopy,^{5, 12, 19, 31} and fluorescence spectroscopy.¹²

Features 1, 2, 4, 7, and 8 are adopted from literature.^{5, 12, 19, 31} In these previous studies, the features were extracted and fed to classifiers or clustering methods as inputs. In our study, we try to leverage this by extracting similar features to investigate the effect of adaptive windowing.

We contribute novel features 3, 5, and 6, which are the minimum, median, and maximum intensities, respectively, within each spectral window. These extreme points (maximum, minimum) provide additional information not represented by other summary features, such as the slope and intercept.

3.6.

Performance Evaluation

In this study, and generally in spectroscopy data processing, performance is evaluated based on individual windows. In other words, performances of wavelength bands are evaluated separately. This step is necessary for two reasons. First, the number of features is very large if all of the wavelengths are involved in the analysis, especially with fixed-size windows. Second, this method can provide important insights for instrument development. Analyzing different wavelength bands enables measurements only at certain wavelengths, making it inherently suitable for filter-based imaging instrumentation design.

The wavelength-based performance analysis uses all eight features extracted from a window to predict to which of the two diagnostic categories each spectrum belongs. We consider only two of the possible diagnostic tasks in order to simplify the analyses, as follows:

In our analyses, three windowing techniques are evaluated, as described earlier in Sec. 4.1, (i) no windowing, (ii) fixed-size windowing, and (iii) adaptive windowing. For each window throughout the spectrum, the eight features listed in Table 1 are extracted. A two-class linear discriminant analysis (LDA) classifier is used to combine the eight features. A leave-one-out cross-validation strategy is employed, and the area under the receiver-operating-characteristic (ROC) curve (AUC) is used as an evaluation metric for the diagnostic power of each wavelength. The features are extracted from each window, but the AUCs are evaluated per wavelength because the window definitions vary across spectra.

One AUC value is reported for each window. Therefore, in the adaptive and fixed-size windowing methods, the wavelength space is divided into fixed-size or adaptive size windows; thus, each window results in one AUC value. In comparison, in the no-windowing method, only one set of features is extracted; thus, there is only one AUC value for the entire wavelength space.

3.7.

Classification and ROC Analysis

The classification task in this study is carried out using LDA, which is a linear classifier that searches for a hypersurface in the feature space that has an orientation to effectively discriminate between the two classes of data. The choice of this classifier is based on both its high computational efficiency and the previous success of LDA with this type of data set. In our previous study, we used LDA exhaustively to search for the most discriminatory features and identified that LDA can achieve high AUC on this task.³⁰ The purpose of this study is to compare the different windowing techniques in feature extraction; thus, the same classifier is used throughout this paper.

ROC analysis is used to quantify performance for two-class classification tasks.³³ Sensitivity and specificity indicate the ability of the diagnostic method to distinguish between two groups (e.g., healthy and diseased). By varying the threshold, an ROC curve of sensitivity versus (1-specificity) is generated. However, the AUC is often used as a metric to quantitatively summarize the performance of a clinical decision support system. Therefore, we use AUC as the metric to evaluate performance. The higher the AUC metric is, the better the performance is. An AUC value of 1 represents perfect discrimination, while an AUC value of 0.5 represents performance expected by chance alone.

3.8.

Denoising

Noisy data can interfere with classifier training. We used two procedures for denoising: (i) Thresholding on classifier outputs and (ii) outlier removal. These two procedures are described as follows.

Thresholding on classifier outputs uses the classifier output values to identify situations where the classifier has not been properly trained. Specifically, if all of the normalized classifier outputs fall in the range 0.4 to 0.6 (0 being negative cases and 1 being positive cases), the classifier has not been effectively trained to distinguish between the target groups. Likewise, if the mean classifier output for the positive cases is smaller than the mean classifier output for negative cases, then the classifier has not been effectively trained to distinguish between the two classes. If either of these two criteria is met, then we consider the predictive LDA model to be too noisy to make a proper prediction and we remove this LDA and its corresponding AUC from our analysis.

Unlike thresholding of classifier outputs, outlier removal is performed to identify a particular spectrum as distinct from all other spectra. Outlier removal is performed in the feature space and is based on the Mahalanobis distance (MDist) measure.³⁴ The MDist is a multivariate measure (in square units) of the separation of an unknown data set from a known set (with mean $\mu$ and covariance matrix $S$ ) in space. The Mdist of a data set, when applied to itself, can be used to find outliers. It has been shown that for a large sample of multivariate normal data, the MDist follows approximately a ${\chi }^2$ distribution with the degrees of freedom being the number of the variables.³⁴ We consider a spectrum to be an outlier if its MDist is larger than the critical point (significance level $\alpha =0.001$ ) of the ${\chi }^2$ distribution with the degrees of freedom being the number of variables participating in the MDist.

The denoising of the data is different from the preprocessing step described in Sec. 4.3. The denoising of the data is necessary because it removes unwanted information from the feature space, while the data preprocessing rescales the data from the spectroscopic space.

3.9.

Statistical Comparison of AUC Values

A bootstrapping technique was used to estimate the significance of the observed difference in the AUC between LDA models.³⁵ $P$ values below the conventional threshold of 0.05 were regarded as statistically significant. In some comparisons in this study, a few cases were removed by the outlier identification algorithm for one windowing method but not the other. For example, case 19 is removed in adaptive windowing in Normal versus $\mathrm{MD}+\mathrm{SD}$ , but not removed in the fixed windows’ analysis. In order to compare the two windowing techniques, we remove all of the cases that are defined as outliers in either of the two methods under study.

3.J.

Studying Effects of Different Initialization Points

In principle, the initialization point of the first spectral window in piecewise linear regression may be a factor that affects performance. In other words, different starting points may cause the entire spectrum to be adaptively divided into different windows. To investigate this possibility, we experimented with multiple initialization points to determine if different initialization points result in different piecewise linear regression models.

Three different experiments were conducted:

4.1.

Piecewise Linear Regression Models

Sample results of our piecewise linear regression model are shown in Fig. 2 . These four examples show diffuse reflectance spectra measured on four sites with different histopathology statuses (solid curves). These measured spectra are fitted by our piecewise linear regression models to iteratively search for maximum window sizes with acceptable goodness of fit (dashed curves). These fitted piecewise linear models define different sizes of spectral windows for feature extraction. These fitted spectra are not intended to replace measured signals. On the contrary, features are extracted from the measured spectra.

Fig. 2

Sample diffuse reflectance spectra for (a) normal (b) benign (c) MD, and (d) SD patients. The solid curve indicates the measured diffuse reflectance spectrum, while the dashed curve indicate indicates the fitted spectrum based on our piecewise linear regression model.

A qualitative look at these piecewise linear regression models reveals that the regression models capture most of the variability within each spectrum; thus, most of the spectral information is retained while the number of windows needed in feature extraction is reduced relative to using a smaller fixed window size. For example, Fig. 2 has a window that starts at $425\mathrm{nm}$ and ends at $500\mathrm{nm}$ —a window size of $175\mathrm{nm}$ , which is about nine times the size of the $20\text{-}\mathrm{nm}$ fixed windows. This larger spectral window captures the relevant features necessary for classification that are also contained in several smaller windows, but using one large window has less redundancy. The features slope (features 2) and intercept (feature 3) are the same with large or small windows. The intensities features (features 4–7) may change for larger windows, but they are linearly proportional as shown in previous theoretical derivations (Sec. 2) such that the classifier will compensate for it. Therefore, this piecewise linear regression method decreases the redundancies in feature extraction relative to a fixed window size. An important finding of this study is that the adaptive windowing technique uses fewer windows to cover regions that behave similarly. In other words, several fixed-size windows may cover a wide range of the spectrum that could instead be represented equally effectively in a single window. Adaptive windowing, on the other hand, preserves diagnostic information while decreasing the redundancy.

4.2.

Feature Extraction

After defining the adaptive windows, eight features are extracted from each window. In this section, we evaluate the three windowing techniques (no windowing, fixed-size windowing, and adaptive windowing) by looking at the AUC performances and the feature redundancies of each windowing technique.

In Table 2 , we compare the predictive power of the three windowing techniques. For each window, all eight features were used to train a LDA classifier via leave-one-out cross validation to obtain the AUC. Because there are multiple windows involved in windowing techniques (fixed-size, adaptive windows), multiple AUCs are reported. In Table 2, the maximum and median AUCs of each windowing method are listed. The complete list of AUCs (not only maxium and median) are shown in Table 3 . The classifiers based on features extracted from adaptive or fixed-size windows outperform the classifier based on features extracted from the entire spectrum (no windowing). This same trend is apparent for both the normal versus $\mathrm{MD}+\mathrm{SD}$ and benign versus $\mathrm{MD}+\mathrm{SD}$ classification tasks. Specifically, the maximum AUC of classifiers trained on features extracted from adaptive windows (max $\mathrm{AUC}=0.73$ ) is statistically significantly $(p=0.04)$ larger than that of the classifier based on features extracted from the entire spectrum (no windowing, $\mathrm{AUC}=0.61$ ) for normal versus $\mathrm{MD}+\mathrm{SD}$ . Likewise, the maximum AUC of the classifiers trained based on features extracted from fixed windows (max $\mathrm{AUC}=0.73$ ) is statistically significantly larger $(p=0.04)$ than that of the classifier based on features extracted from the entire spectrum (no windowing, max $\mathrm{AUC}=0.61$ ) for normal versus $\mathrm{MD}+\mathrm{SD}$ and also for benign versus $\mathrm{MD}+\mathrm{SD}$ with adaptive (max AUC 0.79 versus AUC 0.68, $p=0.04$ ) and fixed windows (max AUC 0.83 versus AUC 0.68, $p=0.01$ ). From these statistical analyses, we found that the maximum AUCs by windowing techniques are all higher than no-windowing techniques, in both normal versus $\mathrm{MD}+\mathrm{SD}$ and benign versus $\mathrm{MD}+\mathrm{SD}$ , and these AUCs are significantly higher $(p<0.05)$ in all of four comparisons.

Table 2

Statistics from three windowing techniques. The maximum and median AUCs for LDA classifiers trained using all eight features from each spectral window. Pairwise comparison of the windowing techniques found that the maximum AUC for both the adaptive and fixed window technique perform significantly better (p=0.04) than the no-windowing method for the classification task normal versus MD+SD . Both the adaptive and fixed window techniques also showed improved performance ( p=0.04 and 0.01) over the no-windowing technique for the classification task benign versus MD+SD . For both classification tasks, the median AUC did not show statistically better performance over the no-windowing technique. The results demonstrate the value of windowing, as adaptive/fixed windowing higher maximum AUCs than no windowing. It also suggests that the adaptive windowing technique yields classifiers as effective as the fixed windowing technique.

Table 3

Supplementary data. A list of all the AUCs generated using LDA classifiers trained using all eight features from each spectral window. The maximum and median values of these AUCs are shown in Table 2. The number of AUCs is higher for adaptive windows because they have more unique (nonrepeated) features. Repeated features are commonly seen in fixed windows; they are removed here. The no-windowing technique takes the entire spectrum to do feature extraction and classification; therefore, only one AUC is calculated.

Because the maximum AUCs represent the best diagnostic power of the spectrum across all wavelengths, we showed that windowing techniques provide better diagnostic accuracy than no windowing. Moreover, the classifiers based on adaptive windows are as good as those based on fixed windows in both normal versus $\mathrm{MD}+\mathrm{SD}$ and benign versus $\mathrm{MD}+\mathrm{SD}$ .

On the other hand, the median AUCs for both normal versus $\mathrm{MD}+\mathrm{SD}$ and benign versus $\mathrm{MD}+\mathrm{SD}$ did not show a statistically significant improvement over the no-windowing technique. The results tell us that the median AUCs of windowing and those of no windowing are similar. This is equivalent to comparing a group of mediocre performers to the average of all performers. Therefore, comparison of median performance does not predict which window has the most diagnostic power.

In addition to evaluating the windowing techniques based on the resulting classifier efficacy, we also investigated the efficiency of the classifiers. Specifically, we examined the number of unique features extracted, the average number of windows used per spectrum, and the total number of windows used (Table 4 ). The unique features are calculated by simply removing repeated features extracted in our program. These repeated features are commonly seen in fixed-size windows. For both the normal versus $\mathrm{MD}+\mathrm{SD}$ and benign versus $\mathrm{MD}+\mathrm{SD}$ diagnostic tasks, the adaptive windowing technique requires fewer windows (8 windows instead of 16) but produces more unique features (60 unique features instead of 17) relative to the fixed windowing method. This comparison demonstrates that adaptive windowing is able to maximize the information obtained in one window and consequently reduce the number of windows needed to maintain the diagnostic power. In other words, adaptive windowing avoids the use of redundant windows that are employed by the fixed-size windowing method. Likewise, while reducing the number of data points used for feature extraction, the number of unique features remains high in adaptive windows. The adaptive windowing technique is able to retain the variability of data while reducing the data dimensionality in feature space.

Table 4

Numbers of unique features extracted and of windows per spectrum are presented. It shows that adaptive windows require fewer windows on average, but produce most unique features. The no-windowing technique has 44 and 35 “total number of windows” because there are 44 and 35 spectra in each case, and each spectrum has one window.

This observation agrees with our theoretical assessment in Sec. 2 that adaptive windowing preserves information by using a larger adaptive window to cover the region where fixed-size windows behave similarly and extract similar and redundant features.

4.3.

Effect of Different Initialization Points

As described in Sec. 3J, there are three experiments conducted. First, the adaptive windowing technique was applied with different initialization points: 400, 405, 407, 410, 415, and $420\mathrm{nm}$ (Fig. 3 ). In Fig. 3, we selected one spectrum of category SD to visualize the window definitions of different initialization points. It is visually apparent that the adaptive window definitions are only slightly shifted from each other. For a range of $20\text{-}\mathrm{nm}$ changes in initialization points, the window definitions are only shifted within $10\mathrm{nm}$ . Compared to the wavelength span of $325\mathrm{nm}$ (400–725) of the spectra, the shift is very small. Hence, from our first experiment, because the window definitions are similar, the features extracted from windows identified with different initialization points should be similar. We further investigated this in the third experiment.

Fig. 3

Adaptive window technique applied on a spectrum with variable initialization points within a small range. This is one example out of the 57 spectra in the data set. The patient belongs to SD pathology group. The solid black curve is the measured spectrum, and the dashed straight lines are the windows. This example shows that the variable initialization points do not affect the window definitions very much (i.e., the changes are within $10\mathrm{nm}$ , which is 1.4% of the entire spectrum).

In the second experiment, we investigated a larger range of the initialization points. Because the ranges are larger, we modified the algorithm such that we do not discard the spectral data to the left of the initialization point (as we did in the previous experiment). The modified version of the algorithm “grows” windows to the left or right side of the initialization point. Three initialization points were tested: 400 (the smallest wavelength), 562 (the median of the wavelengths), and 725 (the largest wavelength). The results are shown in Fig. 4 . Similar to Fig. 3, in Fig. 4, the same spectrum is used to visualize the window definitions of different initialization points. By visually assessing the window definitions, we can see that the initialization point of 400 and $562\mathrm{nm}$ have more similar window definitions; whereas the ones with $725\mathrm{nm}$ have larger variations from those of 400 and $562\mathrm{nm}$ .

Fig. 4

Adaptive window technique applied on a spectrum with variable initialization points within a large range. This is one example out of the 57 spectra in the data set. The patient belongs to SD pathology group. The solid black curve is the measured spectrum, and the dashed straight lines are the windows. In this experiment, the initialization points differ (in a larger range) from the previous experiment (Figure 3); Windows “grow” to either the left or right side of the initialization point—they grow to the right with $400\mathrm{nm}$ initialization, to both left and right with $562\mathrm{nm}$ , to left side with $725\mathrm{nm}$ . This example shows that the window definitions are visually similar for 400 and $562\mathrm{nm}$ , but the ones with $725\mathrm{nm}$ are visually distinguishable.

In the third experiment, we tested the performance of these different initialization points. We calculated the AUCs for the classifiers based on features extracted from the adaptive windows defined by these different initialization points. All eight features are used to train an LDA classifier using leave-one-out cross validation. The maximum AUC ranges from 0.65 to 0.82 (Table 5 ) based on the choice of initialization point. Similarly, the median AUC ranges from 0.56 to 0.66 based on the choice of the initialization point. Conversely, the AUCs for fixed windows do not change at all with different initialization points.

Table 5

AUCs calculated from different initialization points. It is shown that the AUCs of adaptive windows differ when the initialization point is varied, whereas the AUCs from fixed windows remain the same. Some of these AUC changes are statistically significant (0.82 compared to 0.65). This shows that adaptive windowing has a dependency on where the window definition starts, which gives the adaptive windows flexibility to achieve higher accuracy to predict the disease (AUC=0.82) . However, this also shows that fixed windows have better robustness because adaptive windows depend on their initialization points.

These results in Table 5 give us two interesting perspectives: first, the adaptive windowing technique has flexibility to achieve higher AUC (0.82) to outperform fixed windowing (0.73). However, this also raises the concern that the adaptive windowing has variability where it is not guaranteed to have the optimum performance and consistency in terms of the initialization points. In other words, the initialization point needs to be optimized to produce the best performance. Second, while the window definitions of initialization points 400 and $725\mathrm{nm}$ may seem very different, their AUCs are not that different. The AUCs for the 725 and $405\mathrm{nm}$ initialization points are the same $(\mathrm{AUC}=0.78)$ . Therefore, optimizing the AUC from adaptive window does not require running through all the possible wavelengths. Instead, running a number of initialization points in a small range and looking for the best AUC may be sufficient; in our example, we considered 6 points in $20\mathrm{nm}$ . This $20\text{-}\mathrm{nm}$ range is adjustable and needs to correspond with the starting window size of the adaptive window–defining algorithm.