1. Introduction

Although the use of hyperspectral imaging was restricted to the field of remote sensing over most of the past decade, it has recently come to be accepted as one of the most powerful nondestructive imaging technologies in a variety of fields, such as environmental monitoring, quality assessment in food science and agriculture, mineral exploration, and surveillance [1–3]. This technology is called “spectral imagery” because imaging measurements are made in a series of narrow and contiguous wavelength bands, using a particular spectrograph and an imaging sensor. The acquired hyperspectral images contain rich information in both the spectral and spatial domains. Therefore, it is possible to derive a continuous spectrum for each image pixel, making it possible to detect, identify, and quantify imaged objects in more detail than is possible using the ordinary three bands in “color” images (R, G, and B). The main advantage of spectral imaging is the fine resolution in the wavelength domain. However, this can be a disadvantage for image acquisition and processing, in that acquiring and analyzing a hyperspectral image is more time-consuming and expensive than by using the ordinary “three-band” image processing method. Therefore, the efficient processing of spectral data to extract particular information while maintaining a desirable level of accuracy or performance is important in spectral imaging technology. Several relevant methods have been proposed that reduce dimensionality by extracting low-rank structures within the spectral data, e.g., chemometric techniques for chemical imaging, such as principal component analysis (PCA) or partial least-squares (PLS). Although these methods are useful for reducing the number of parameters required to process the acquired spectral data, a spectral acquisition process that costs time and money is still required. In addition, the light is split over many detectors, and therefore, high illumination intensity is required to obtain a sufficient signal-to-noise ratio (SNR) in hyperspectral imaging.

As an alternative, wavelength selection is commonly used in many practical applications to reduce the cost of spectral acquisition [4–7] and multi-spectral imaging [8,9]. Images are acquired using a set of specific wavelengths for a given objective, and the acquired gray-level images are used in a calculation process to produce output images in which certain features are highlighted. The measured wavelengths are typically selected by spectral analysis of the target’s absorption wavelengths. To reduce costs, the number of wavelengths used is typically much lower than the number used in an ordinary hyperspectral imaging system. Thus, this kind of multi-spectral imaging system simply consists of an image sensor (i.e., a CCD camera) and a set of optical filters (i.e., interference filters) situated in front of the camera (or light source) or sequential illumination, with different LEDs emitting different wavelength bands. Interference filters transmit (or the LEDs emit) only a specific wavelength, or, to be precise, a very narrow wavelength band. Therefore, the camera used in a multi-spectral imaging system is typically required to be highly sensitive because the energy of the transmitted light tends to be weak. In a hyperspectral imaging system, this light energy problem becomes more serious because of the narrower wavelength bands, as mentioned above.

To reduce the dimensionality of the spectral data it is not necessary to select a set of individual wavelengths by using narrow-band filters. Instead, any type of transmittance function of the filter can be selected to reduce the dimensionality of the data, provided one can find a better low-rank structure within the given data set. Furthermore, it is easier to obtain a sufficient SNR by using filters with broad, rather than narrow, wavelength bands. The present study considers a data-driven optimization problem to formulate a method for selecting the optimal set of band-pass filters that can detect and quantify the target in a multi-spectral image. Herein, we report the theoretical design and realization of an optimal set of real optical band-pass filters for ice detection on roads, by focusing on the spectral difference between ice and water.

The detection of ice is important for a number of purposes. For example, ice formed on road pavements or on car windows causes serious traffic accidents [10]. In addition, layers of ice on aircraft wings can make flights unsafe [11–13], and ice accretion on leaves can cause damage to plants [14]. At present, the determination of ice presence is performed visually by technicians under most types of circumstances, followed by tactile inspections. However, visual inspection is difficult and time consuming, and can only produce rough results. Tactile inspection determines the presence of ice through changes in the acoustic or electrical properties of the material as detected by a sensor [10–14]. However, the drawback is that this often requires physical contact between the sensing device and the target, and the reliability of readings is low due to the narrow (usually a point or a line) sensing areas, which offer localized and sparse coverage. Alternatively, there is an automatic and non-tactile method to inspect ice using reflectance spectroscopy [15]. Ice contamination on aircraft wings and snow layers on road pavements have been successfully inspected in this way, but black ice, a slick, transparent form of roadway ice, cannot be successfully identified using this method because of its low light reflectance. These methods focus on the difference in the absorption wavelength of ice and water, and they use narrow-bands filters for ice detection, which leads to a low SNR. The present study selected the optimum broad wavelength bands to distinguish ice from water based on a measured radiance spectrum, achieving a higher level of accuracy of discrimination than that achieved by the previously discussed methods.

In the following text, we describe the mathematical formulation used for selecting the filter’s properties. We then compare the detection accuracy with the accuracy obtained using the ordinary wavelength selection method for ice detection, as applied to the simple cases of two narrow-band filters (hereafter referred to as “two-wavelength filters”) and two broad wavelength bands. Finally, the design method is extended and generalized for larger numbers of filters and applied to the same data set to evaluate the generalization of the method.

2. Basic idea

Figure 1 outlines the concept we have used to reduce the dimensionality of a given set of spectral data, e.g., measurements of three properties of objects are performed by selecting a limited number of wavelengths and wavelength bands. Two wavelengths are selected by placing two interference filters in front of the detector or image sensor, which allow the wavelengths λ₁ and λ₂ to pass. The outputs from these devices span a two-dimensional space, which is a subspace of the original spectral data space. The objective of the selection is to find the optimum subspace in which the projected data contains as much of the desired information as possible (i.e., the properties of the measured objects). Conventionally, a certain specific wavelength that depends on the category or condition (λ₁ in Fig. 1), and another wavelength, which does not depend on them and is used as a reference (λ₂), are selected. To do so, the correlation or another similar index is evaluated for each wavelength as the entire range of wavelengths is scanned, and the best wavelengths are selected.

 

Fig. 1 Basic concept used to reduce the dimensionality of a given spectral data set by selecting wavelength bands with band-pass filtering.

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Our proposed method focuses on selecting wavelength bands instead of individual wavelengths. The transmittance properties of an ideal BPF are described by two parameters: the lower cut-off and higher cut-off wavelengths. The problem of finding an optimum set of BPFs can be formulated as the need to find a set of parameters that maximize a pre-defined evaluation function describing the level of target detection or categorization achieved from the low-dimensional data.

Previous studies have proposed methods that focus on the selection of wavelength bands, and these have been applied to skin parameter recovery [16, 17], target detection for remote-sensing data [18], fruit sorting [19], and target detection in hyperspectral images using a matched filter [20]. Although this study shares the basic premise of these works, our formulation of the evaluation function for filter selection is different. In typical applications, the evaluation function is designed to recover the original signal or important parameters from the filtered signals with a minimal error [16–18]. In another study, the target detection accuracy of combinations of commercially available filters [19] or the contrast between target and background for various filters [20] was evaluated. In our own previous studies [21,22], a spectral filter with an arbitrary transmittance function, instead of a BPF, was designed to optimally modulate the spectral sensitivity of an RGB camera to detect and quantify cosmetic foundation on skin. The present study describes the evaluation function from a data-driven, statistical point of view, based on linear discriminant analysis of the data projected onto the low-dimensional feature space, as described in Section 3. Furthermore, the extension of this method to the selection of an arbitrary number of filters in a realistic time-frame is described in Section 4.

3. Theory

Consider a system consisting of a camera and a set of optical filters placed in front of the camera lens. Suppose that the output of the camera with the i-th filter $(i=1,\cdots ,N)$ is simply described as

The camera output is expressed as a vector in N-dimensional feature space as O = (O₁, O₂, …, O_N)^t and is defined by a set of filters. Suppose that the set of camera outputs for certain objects have to be categorized into the known classes A:$\left\{O^A\right\}$ and B: $\left\{O^B\right\}$. The optimization problem here is defined as the need “to find an optimal set of the filter parameters $\{{\lambda }_{\text{L}}^{\text{i}},{\lambda }_{\text{H}}^{\text{i}}\}$, $(i=1,\cdots ,N)$, to categorize the camera outputs $O$ into class A or B.”

In order to solve this problem, a measure of the separability between the camera outputs categorized into classes A:$\left\{O^A\right\}$and B:$\left\{O^B\right\}$must be defined. We seek to obtain a scalar value by projecting the camera outputs $O$ onto a line defined by a projection matrix w as$f(O)=w^{t}O$. We here define the separability of a given set of filters as the separability of the scores $f(O)$, which is maximized by selecting the best line from all possible lines. According to Fisher’s linear discriminant [23], the separability of the scores is explicitly expressed as the ratio of the variance between the classes to the variance within the classes as

Linear discriminant analysis (LDA) yields an equivalent result. Assume that the conditional probability functions, $p(O|class=A)$ and $p(O|class=B)$ are both normally distributed with means of ${\mu }_A$ and ${\mu }_B$, respectively, with the same covariance matrix for both classes, i.e., ${\Sigma }_A={\Sigma }_B=\Sigma$. The discriminant function is then expressed as

The first term in Eq. (6) corresponds to the projection of the camera outputs, $O$, onto a line defined by $w={\Sigma }^{-1}({\mu }_A-{\mu }_B)$, which maximizes the separability of the scores, as mentioned above. The second term is a scalar that is used to determine the category boundary. When$f(O)>0$, then $O$is categorized into class A. When$f(O)<0$, then $O$is categorized into class B, with the best separability between $f(O^A)$ and $f(O^B)$ under the given filter parameters$\{{\lambda }_{\text{L}}^{\text{i}},{\lambda }_{\text{H}}^{\text{i}}\}$.

The filters can be optimized by choosing parameters from all the possible sets of parameters that maximize the separability of the discriminant score $f(O^A)$ and $f(O^B)$, as determined by the camera outputs, depending on the set of filters. Separability as a function of the filter parameters can be expressed as

4. Materials and methods

4.1 Spectral data set

Spectra of ice and water were measured using a spectroradiometer (Field-Spec3, ASD Inc.). A layer of ice approximately 1-cm thick was generated on a small block of asphalt repairing material, as shown in Fig. 2 (height: 6 cm; width: 11 cm; depth: 1.5 cm); this was used as an ice sample for the measurement. Once melted, the ice sample was used as a water sample. Details of the measurement conditions are shown in Table 1 . The spectral radiance over the range of 900 - 2500 nm was measured at 10 nm intervals. Each sample was illuminated from both sides by two sets of halogen lamps (500 W), and a spectroradiometer probe was set perpendicular to the light sources. Both of the light sources and the probe were set at an angle of 45° with respect to the sample to exclude specular reflection, as shown in Fig. 2. A total of 54 radiance spectra of ice and water were obtained from 9 points on each of six different samples.

 

Fig. 2 Schematic diagram of experiment setup (left) and measured samples (right). The sample in the near side of the image is the water sample and the one behind it is the ice sample. After the ice sample melted, it was used as a water sample.

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Table 1. Measurement conditions

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Figure 3 shows each of the 54 individually measured radiance spectra, as well as their averages, for ice and water. The original spectra are quite noisy and overlap one another because of irregularities in the water layer and the black asphalt. There are a number of spectra with a peak around 1700 nm for some water samples (blue lines); these were probably caused by oil floating on water that originated from the asphalt.

 

Fig. 3 Radiance spectra of ice (solid red line) and water (dashed blue line) on asphalt. The thin lines show the original spectra and the bold lines show the average spectra for ice and water.

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4.2 Method for filter selection

As described in Section 3, the selection of the optimal filter set for target detection is considered as a problem of non-linear optimization of the filter parameters$\{{\lambda }_{\text{L}}^{\text{i}},{\lambda }_{\text{H}}^{\text{i}}\}$. In this study, the set of optical filter parameters was designed using two methods: (1) searching all possible combinations of filter parameters, and (2) a multiple selection method for arbitrary numbers of filters. Searching all possibilities is a simple approach that guarantees one will find the best filter set if the result is obtained within a realistic time. By comparison, the multiple selection method can obtain a pseudo-optimal result within a realistic time even when a large number of filters are designed. As described in the next section, we applied the all-possible-combinations search method to find two BPFs to confirm the best performance of the wavelength-band method, and we applied the multiple selection method to find more filters to investigate the effects of the number of filters on performance. Next, we describe the algorithm used for selecting an arbitrary number of filters.

A flowchart of the proposed algorithm of the multiple selection method is shown in Fig. 4 . The optimization consists of two processes: a global search process (shown in the left column) and a local optimization process (in the right column). First, the global search process is performed to select the filters, one-by-one, that can make the strongest contribution to target detection. Next, a local optimization process is performed to adjust the filter parameters to obtain the local maximum of the Fisher criterion. The basic idea of the optimization is similar to that of the conventional variable selection method (i.e., the stepwise forward selection method).

 

Fig. 4 Flowchart of the multiple selection method, which consists of a global search process (left column) and a local optimization process (right column).

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First, all parameters of the algorithm are initialized. PF is the set of filter parameters that are already optimized and fixed (to begin with, this is an empty set, i.e., no parameters have been optimized), Cnt(${\lambda }_{\text{L}}^{},{\lambda }_{\text{H}}^{}$) represents the number of times the filter with the set of parameters $\{{\lambda }_{\text{L}}^{},{\lambda }_{\text{H}}^{}\}$ is selected (0 to begin with), and Ev(${\lambda }_{\text{L}}^{},{\lambda }_{\text{H}}^{}$) represents the evaluation index value of these parameter sets (which are also 0, to begin with). The evaluation index is the average value of the Fisher criterion that is obtained with the filters of parameters $\{{\lambda }_{\text{L}}^{},{\lambda }_{\text{H}}^{}\}$ when the k-1 first filters are fixed and the others filters are randomly chosen. k is the index number of the filter whose parameters are being optimized (set at 1 to begin with). Next, the parameters of the filters that are not fixed (all of the parameters, to begin with) are randomly selected. P_V is the set of parameters of the filters that are varied and N is the total number of filters to be optimized.

For these randomly selected filters, the projection matrix w* as defined by Eq. (5) is first determined by the calibration data set, then the evaluation index value and therefore the Fisher criterion $J(\{{\lambda }_L^{},{\lambda }_H^{}\})$ is calculated and stored as Ev(${\lambda }_{\text{L}}^{},{\lambda }_{\text{H}}^{}$), using the validation data set (two-fold cross-validation). The number of selections for the parameters $\{{\lambda }_{\text{L}}^{},{\lambda }_{\text{H}}^{}\}$ is also stored as Cnt(${\lambda }_{\text{L}}^{},{\lambda }_{\text{H}}^{}$). After repeating this process, the optimal parameters $\{{\lambda }_{\text{L}}^{*},{\lambda }_{\text{H}}^{*}\}$, which maximize the evaluation index, are selected and stored in P_F. This process is repeated until all the filter parameters are fixed. Note that in the global optimization process, from k = 1 to k = N - 1, parameters are randomly selected, but when k = N, all possible parameter values for the last Nth filter are sequentially evaluated, because the number of parameters is small sufficient for all of them to be searched sequentially.

After the global optimization process, the local optimization process is performed. First, one filter (with index number k) is selected. Then, by fixing the parameters of the other N-1 filters, the parameters $\{{\lambda }_{\text{L}}^k,{\lambda }_{\text{H}}^k\}$ are optimized such that $J(\{{\lambda }_L^{},{\lambda }_H^{}\})$ is maximized by searching all possible parameter values. If the optimized parameter$\{{\lambda }_{\text{L}}^{k*},{\lambda }_{\text{H}}^{k*}\}$ is different from the one obtained by the global optimization process, then the filter parameter is updated. This process is repeated until no filters are updated by this local optimization procedure.

4.3 Optimization conditions

Filters were selected using spectral data measured in the 900–2500 nm range, as shown in Fig. 3. Here, the spectral sensitivity of the camera, $S(\lambda )$, was assumed to be $S(\lambda )$ = 1 over the entire wavelength range, even though there were no such imaging sensors. However, it would not have a great effect upon the wavelength band selection if the sensor were somehow sensitive in the wavelength range to be analyzed. As described in Section 5.4, after the band-pass filters are selected, it is possible to compensate for the discrepancies between reality and theory in the actual camera sensors, as well as in the transmittance of the filters, by using the actual camera outputs with the real filters to re-calibrate the projection matrix defined by Eq. (5) and the discriminant function defined by Eq. (6).

All possible combinations of filter parameters were searched for the case of selecting two filters. To compare this approach with the conventional method, two optimal wavelengths were also selected using the same procedure by fixing the filter’s bandwidth at 10 nm. The multiple selection method was performed for the case of more than two and as many as eight filters (N = 2, 3, …, 8). The maximum number of iterations in the random selection process (MaxItr in Fig. 4) was set at 100,000. In these experiments, a logarithmic transformation was applied to the camera outputs as it is in the transformation from reflectance to absorbance used in conventional spectral analysis.

5. Results

5.1 Al-possible-combinations search

The BPFs and the optimal two individual wavelengths selected by the all-possible- combinations search are shown in Fig. 5 . The two selected BPFs have transmittance bands of 1260 to 1700 nm and 1300 to 1550 nm. The two optimal wavelengths were found to be 1260 nm and 1400 nm. In both cases, a value near the absorption wavelength of water (approximately 1450 nm) was selected as one band (1300–1550 nm) or wavelength (1400 nm). Other selected wavelength and a band, on the other hand, were different in two cases (1260nm and 1260–1700 nm), which are expected to work as a reference independent of the ice/water absorption. This may reflect the difference between their respective performances. This point will be addressed in Section 6.

 

Fig. 5 Optimization results: the two optimal designed bands and the two optimal wavelengths; superimposed mean and individual spectra shown in Fig. 3. The Fisher criteria J in both cases are displayed at the top of the figure.

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The Fisher criteria J for both cases, which were computed by leave-one-out cross-validation, were 74.57 for the selected BPFs and 18.08 for the two individual wavelengths, as shown in Fig. 6 . For comparison, the Fisher criterion obtained using conventional spectral analytical techniques in which multiplicative scattering correction (MSC), principal component analysis (PCA) and linear-discriminant analysis (LDA) was applied to the entire wavelength region (900–2500 nm), was 61.74. The number of principal components was chosen to maximize the Fisher criterion J. It should be noted that for the multivariate case, the Fisher criterion varied depending on the wavelength region (17.29 for 900–2000 nm, and 129.55 for 900–2370 nm, which showed the best performance of this method found by searching the wavelength range), as shown in Fig. 6. The proposed two BPFs (two bands) method is superior to the two-wavelength method and even to traditional spectral analysis that uses all the spectral information (900–2500 nm). The reason for this is probably that the data at a wavelength range longer than approximately 2000 nm were noisy, which had a large effect upon the accuracy of the method that uses the entire wavelength range. Further, the BPFs method itself includes the wavelength range selection, and thus, all data longer than approximately 2000 nm were automatically removed from the analyzed wavelength range by maximizing the Fisher criterion.

 

Fig. 6 Comparison of the Fisher criteria J as computed by conventional spectral analysis techniques (MCS + PCA + LDA) for various wavelength regions, for two wavelengths (Two WL), and for the proposed two BPFs (Two bands).

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Based upon Fig. 7 , which shows the distribution of discriminant scores expressed by Eq. (6), it is appropriate to assume that the camera outputs obtained by the BPFs method were normally distributed, whereas there were outliers in the two WL and MSC + PCA + LDA for the 900–2500 nm cases, and hence, this distribution appears non-Gaussian. This implies that LDA of signals obtained by the BPFs method works better than these two cases.

 

Fig. 7 Distributions of the discriminant score defined by Eq. (6) for the two wavelengths (Two WL), the proposed two BPFs (Two bands) and the multivariate cases (MSC + PCA + LDA) for 900–2500 and 900–2370 nm.

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5.2 Filters selected by multiple selection method

The Fisher criteria J for BPFs selected using the multiple selection method are shown in Fig. 8 . Note that the Fisher criteria were computed using leave-one-out cross-validation in the same manner as in the all-possible-combinations search shown in Fig. 6. In general, the Fisher criterion increases with the number of filters. For the case of two filters, the Fisher criterion was 36.75, which is smaller than it was in the all-possible-combinations search cases, which indicates that the all-possible-combinations search method offers a significant advantage. It is also worth mentioning that the Fisher criterion J is almost saturated when more than five filters are used, which indicates that there are approximately five spectral bands that mainly contribute to the class separation.

 

Fig. 8 The Fisher criteria J of the number of filters selected using the multiple selection method.

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5.3 Filter realization

A trade-off exists between the performance and the cost for selecting optimized filters. Because the result of the all-possible-combinations search, J = 74.57, was considered sufficiently accurate for the application of ice detection on asphalt, we produced and tested real optical filters corresponding to the two BPFs selected by the all-possible-combinations search method.

The two selected BPFs were realized as real optical filters, which we then tested for visualization of the detection result by placing them in front of a near-infrared camera. The filters were developed using a multilayer thin film coating technology. The spectral transmittances of the realized optical filters are shown in Fig. 9 alongside the theoretically designed transmittances. Here, filter 1, with cut-off edges at 1260 nm and 1700 nm, was developed as a high pass filter because the image sensor of the near-infrared camera used in this experiment had no sensitivity for wavelengths longer than 1700 nm. The Fisher criterion J calculated for the set of real filters was 67.20. The obtained performance was high, despite the mismatches between the theoretical and the realized transmittance functions.

 

Fig. 9 Theoretical and realized spectral transmittances of tested filters. The computed Fisher criterion is displayed at the top of the figure.

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5.4 Ice detection by real optical filters

Figure 10 shows the experimental setup for ice and water discrimination and visualization on asphalt using a camera with the realized filters. A near-infrared camera XSVA XS FPA-1.7–320 (InGaAs sensors with 320 × 256 pixels; Xenics) was used in this experiment. Filters were placed in front of the camera by a rotating filter wheel (Newport Japan) controlled by a PC through a USB interface. The integration time was approximately 10 ms for all conditions. The samples were the same as those used for the previously described spectral measurements shown in Fig. 2. Two sets of halogen lamps (300 W each) illuminated the samples from both sides. The camera was set just above the samples. A white reference (Spectralon) was used for the camera calibration. Images of the ice and water samples were taken at 1-minute intervals for 14 minutes to monitor the ice melting over time. The background area surrounding the asphalt sample was masked manually for visualization purposes.

 

Fig. 10 Measurement setup: a near-infrared camera XSVA XS FPA-1.7–320 (Xenics) with installed developed filters was used to take the spectral images. Two sets of halogen lamps (300 W each) illuminated the samples from both sides. The camera was set just above the sample.

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The visualization result at 0 minutes is shown in Fig. 11 . The left image shows an image taken with an ordinary digital camera and the right image shows the detection result. The top sample has an ice layer and the bottom sample has only a water layer. The classification accuracy, which is defined by the pixel count of the ice and water regions, was found to be 90.28%. The accuracy of ice detection obtained using the proposed method is sufficiently high for real applications, and with two BPFs, the cost for implementation is relatively low. This high accuracy and low cost allowed us to use the proposed method to monitor ice over a period of time. The results, which are shown in Fig. 12 , clearly show the detected ice region decreasing with time due to the ice melting. We also confirmed that the same filter worked well for other backgrounds, for example, for detection of ice on leaves (frost detection: 93.56%) and on glass (ice on car window: 98.99%), as shown in Fig. 13 . The detection of ice on asphalt was more difficult than it was on these backgrounds because of the low reflectance of asphalt.

 

Fig. 11 Visualization results: the left image shows the RGB color image taken with a digital camera, and the right image shows the detection result computed from the images taken with the NIR camera with the installed optimized filters. The top sample had a 1-cm-thick ice layer and the bottom one had a water layer. The pixels classified as ice were colored red and the areas classified as water were colored blue.

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Fig. 12 Visualization results over time, which clearly show the area classified as ice (the sample in the top row) decreasing with time.

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Fig. 13 Visualization results for other materials: ice on leaf (93.56%) and glass (98.99%). Colors are shown in the same manner as in Figs. 11 and 12.

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6. Discussion

It is also worth noting that the two selected BPFs shown in Fig. 5, and the BPFs selected by the multiple selection method are ‘nested’. Furthermore, when the proposed method was applied to spectral data sets for purposes other than ice detection, nested filters were also obtained in most cases. In the following, we discuss why nested filters provide a good result.

A graphical image of the evaluation index, Ev, as calculated using the multiple selections method (N = 2), and the obtained transmittance function of the filters, is shown in Fig. 14 . The horizontal axis shows the bandwidth and the vertical axis shows the center wavelength of the searching BPFs (originally represented by${\lambda }_{\text{L}}^{}\text{,}{\lambda }_{\text{H}}^{}$, as shown in Fig. 4, but shown here as the bandwidth and center wavelength, for purposes of explanation). The gray level shows the average Fisher criterion J. The left and right panels respectively show the evaluation index obtained when the first and the second filters were being optimized.

 

Fig. 14 Evaluation index calculated using the multiple selection method and the transmittance functions of the selected filters (N = 2). (a), (b) when the first filter is selected (k = 2); and (c), (d) when the second filter is selected (k = 2). The horizontal axis shows the bandwidth and the vertical axis shows the center wavelength of BPFs. The gray level corresponds to the average Fisher criterion J of the given filter parameters. The white arrow in (a) points to the wavelength selected using the two-wavelength method, and the arrow in (c) points to the parameter of the second filter selected using the multiple selection method.

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When k = 1 and the first filter (Filter 1) is selected in the “Filter selection” process, the parameter set of Filter 1 at center wavelength = 1395 nm and bandwidth = 10 shows the highest value, as shown and highlighted in Fig. 14(a). On this map, the wavelength of 1400 nm with a narrow bandwidth that was selected using the two-wavelength method also shows a high value. In addition, 1260 nm with a narrow bandwidth, which was selected using the two-wavelength method as another “reference” wavelength, as mentioned in Section 5.1, also shows a high value (indicated by a white arrow). However, when the second filter (Filter 2) is selected after fixing the parameters of Filter 1, the 1260 nm region is no longer high, as shown in Fig. 14(c). Instead, the parameter region that forms the “nested” filter, which is indicated by an arrow, shows higher values. The two-wavelength method is restricted to select only the parameters around the bottom edge of Figs. 14(a) and (c), whereas the proposed method can exhibit significantly improved performance than the fixed narrow-band filter (two wavelength), when the second broadband filter is selected such that the first filter is nested in the second one. This implies that a nested filter is more effective for baseline correction than the reference wavelength, as used in the two-wavelength method.

7. Conclusions

In this paper, we proposed a method for reducing the dimensionality of a spectral data set by using optimal band-pass filters, which facilitates hyperspectral imaging. This method was then applied to ice-region detection on asphalt for roads. The general idea governing this method is the selection of an optimum set of wavelength bands, instead of the individual wavelengths that are conventionally used. The selected BPFs can be easily implemented as real optical interference filters that are mounted in front of a conventional camera. Furthermore, it is easier to obtain sufficient SNR by using filters that have broader wavelength bands than narrow-band filters. The set of band-pass filters can be formulated as the projection function from the spectral domain to the low-dimensional feature space. The filter can be optimized by finding the parameters that maximize the separability, as defined by the discriminant score (Fisher criterion J) among the possible sets of filter parameters.

Optimal wavelength bands were selected in two ways: an all-possible-combinations search and a multiple selection method. The two wavelength bands selected by the all-possible-combinations search demonstrated better performance than the conventional two-wavelength method, and even better than the conventional spectral analysis technique (MSC + PCA + LDA). The results of the multiple selection method indicated that a higher number of filters provides a better performance than the two bands selected by an all-possible-combinations search. Because the cost of implementing the system and capturing the spectral images with a large number of filters generally tends to be higher, a suitable number of filters should be chosen depending on the application.

Regarding future work, the dependence of the method’s performance on the thickness of ice need to be investigated. In our experiment setup, the thickness was 1.0 cm on average, and it is probable that the performance would diminish as the thickness of the water/ice becomes thinner. To improve the method’s performance, a more effective method for searching a large number of filters also should be investigated. In our multiple selection method, although only pseudo-optimal filters were obtained, higher performance obtained with a smaller number of filters is desirable. We compared the proposed method with two optimization algorithms: simulated annealing and the Nelder-Mead method (“fminsearch” of the MATLAB function). Although the method described in this paper proved superior to both of these methods, it would still be worthwhile to improve the search method. Additionally, other statistical classification methods, including a kernel method such as SVM, should be considered to obtain a more powerful classifier.

Obviously, the proposed method is not restricted to the problem of ice detection. It can be used for many other purposes such as quality assessment of food, mineral exploration, and medical imaging. In addition, our method can be extended from detection to quantification by replacing the target function (separability) with other indexes that are commonly used for the evaluation of quantification accuracy (e.g., the standard error of prediction or the determination coefficient).