Abstract

I. INTRODUCTION

Previous research has attempted to categorize vegetation by predicting these constituents and the physical characteristics of canopies using spectral measurements [9, 10, 11, 1]. Near Infrared Spectrometry (NIRS) protocols have been developed for determining biochemical composition from dried ground agricultural forage using stepwise multiple linear regression (SMLR) [12, 13]. These methods have been extended to remote sensing studies for detection of biochemistry [14, 15, 8, 16, 17, 18, 19]. However, inconsistencies in predictions among these studies have raised several concerns. The main criticism is their assumption of linear relationships between leaf biochemistry and leaf reflectance, ignoring the fact that spectral identification and spectral detectability depend strongly on the context. That is, on anatomy and chemical composition of the leaf and on the conditions under which the measurements are made. At the image level, spectral contrast varies with the image context and affects detectability [20]. The SMLR might be adjusted to reduce undesired and interfering signatures for good local predictions but still produce poor performance for general purposes. Grossman et al. [21], and Curran [9, 10] have reported that the SMLR is subject to several types of errors. From investigations of SMLRs which had good r² statistics but poor predictions they showed that the technique is sensitive to numerical errors. They also report that wavebands selected by SMLR in these studies vary with data sets or conditions. In some cases, other biochemical constituents partially covarying with the chemistry of interest have confounded interpretations because of absorptions in the same wavebands [1]. Also if the range of chemistry concentrations exceeds the SMLR calibration range, then the predictions will be in error [1]. In still other cases, the concentration of the biochemicals are small (e.g., nitrogen or starch) or express little variation among samples (e.g., total carbon content) [22]. Furthermore, Grossman et al. [21] and Jacquemoud et al. [22] observed that the relationships were strongly dependent on the mode of expressing reflectance and whether the chemistry was expressed on a weight (g/g) or area basis (g/m²).

The plant leaf or canopy is composed of a range of biochemical constituents, which are similar in composition due to a shared primary metabolism but varying in their proportions; the spectra of leaves or canopies represent that mixture. The main statistical tests to identify targets and classify (spectrally) mixed pixels have been provided by performing various multivariate linear comparisons. First, it is customary to use principal component analysis (PCA) to describe the channel-to-channel variance in multispectral data. However, with hundreds of channels (e.g., with hyperspectral sensors), PCA may have several potential numerical problems due to the relatively high dimensionality that may result in singular PCA systems [23] and also they are difficult to interpret because the orthogonal axes of statistical variance do not have consistent and simple equivalence to field and laboratory observations. It is desirable, instead, to classify images within the conventional frame of reference of field and laboratory observations with methods that avoid intrinsic singular problems. In this respect, spectral mixture analysis (SMA) has become a well-established procedure for analyzing imaging spectrometry data [24, 25, 26, 27, 28].

SMA is a structured and integrated framework that simultaneously addresses the mixed-pixel problem, calibration, and variations in lighting geometry and displays the results in terms of proportions of endmembers that can be related easily to standard ecological observational units (e.g., cover). The general form of the SMA equation for each band is expressed as:
 

(1)

where R_b is the radiance at band b, F_em is the fraction coefficient of each endmember R_em weighting their radiance at band b, and E_b is an error term accounting for the unmodeled radiance in band b. Endmembers are chosen to explain the spectrally distinct materials that form the convex hull of the spectral volume. This approach works best when describing a few spectral types that, in various mixtures, can account for most of the variance in an image data set. It does not mean, however, that it is possible to identify any specific material. SMA works less well when the spectral features of interest are minor components of the total variance. In fact, SMA has the disadvantage, at least for this application, of approximating linearly the natural (non-linear) complexity of materials represented by the mixture of endmembers. This produces a non-unique mixing model to identify and quantify materials that occur at the sub-pixel scale [20]. Moreover, mixtures of materials may mask the absorption bands needed for unique identification of particular biochemicals. In summary, the technique is relatively insensitive to subtle absorption features, and produces significant quantification errors due to endmember variability from linear and nonlinear mixtures (e.g. from scattering, and lighting geometry) in a pixel. Therefore, minor sources of spectral variation (e.g., discriminating stressed from unstressed vegetation and variations in canopy chemistry) can hardly be detected by SMA [29]. Aber et al. [30] explicitly evaluated the use of SMA and endmember spectra to estimate biochemical composition of foliage samples from northeastern U.S. forest species and concluded that this methodology was inappropriate for obtaining quantitative biochemical estimates. This conclusion was reaffirmed in the ACCP report [1]. A methodology that could minimize this undesired variation by projecting it onto a plane where it clusters into a common point normal to the plane of view is desired.

Boardman [31], used a geometric approach based on the convex hull of the spectra projected into the mixing space to find a solution that minimized spectral variation for some features while accentuating others. His technique is still a SMA approach that automatically derives the number of endmembers and estimates their pure spectral composition [31], but it is suboptimal in the presence of multiple mixing. More recently, Harsanyi and Chang [32] developed a mixture technique that rejects undesired interference by performing an orthogonal subspace projection (OSP). This technique simultaneously reduces data volume and emphasizes the presence of a signature of interest. Bolster et al. [33] seeking the same goals, instead use the first difference partial least squares regression (PLS) that is based on a singular value decomposition (SVD) of the whole spectrum data set. SVD reduces noise-related interference, common in a first difference analysis, and reduces the analysis into a smaller set of independent variables. Both, OSP and PLS, achieve good performance in detecting material abundances at low levels for a particular scenario by incorporating the variability of the material abundance into the more important independent variables (factors) but they are unable to extend the application to other scenarios. Existing approaches present a common problem in lack of robustness. In this respect, we lack systematic means for quantifying vegetation from spectral measurements (see ACCP [1] and Peterson and Hubbard [11] for references).

In order to develop a directed search methodology to locate the desired robustness (analytic) property, Smith et al. [29] proposed a revised SMA technique, that they termed Foreground/Background Analysis (FBA). Harsanyi's approach shares the properties of orthogonal space projection and a similar rationale with the FBA technique. In this technique, spectral measurements are divided in two groups of foreground and background spectra that comprise a selected subset of spectra which emphasizes the presence of a signature of interest. In defining both groups they do not include intermediate mixtures between foreground and background. In that way, FBA vectors should be sensitive to minor sources of foreground spectral variation and insensitive to background spectral variation. The goal of FBA is to project spectral variation along the most relevant axis of variance that maximizes the spectral differences between the foreground and background, while minimizing spectral variation within each group. Their FBA approach defines a weighting vector w = (w_1, w_2, ..., w_Nb), with components w_b at each channel b = 1, ..., Nb, such that all foreground spectral vectors, R_f = (R_f,1, R_f,2, ..., R_f,Nb), are projected to 1 while background spectral vectors, R_b, to 0.This property is defined by the FBA system of equations:
 

	(2)

where T provides a translation that is typically required to optimize the FBA system. As stated FBA is in essence another linear classifier of the spectra that can be applied to identify low and high material abundances. Pinzon et al. [34, 35] modified the FBA linear system to project a subset of spectra into relevant axis of continuous chemical variation. In this case, the system of equations is given by
 

(3)

The FBA system to be solved is composed only of the training (calibration) samples that reflect the distribution of the feature to be detected in the whole data set. When singular value decomposition is used to solve the FBA system, PLS used by Bolster et al. and FBA are equivalent. Pinzon et al. [35] found that the method presents good predictions and good r² statistics, however, it was not robust in this form. In particular, the formal analysis breaks down when we use samples with different types of tissues or with concentrations beyond the range of variation in the training data set (sensitivity to calibration). This sensitivity depends on the method used, and moreover, it varies with each biochemical and the ranges of reliable values, which are typically different for species from functionally different ecosystems. While water and pigment concentrations are measured in a straightforward way, nitrogen, carbon, and cellulose are calculated by indirect methods that increase the level of uncertainty [30, 16, 36, 37]. Furthermore, chemistry concentrations do not vary independently and for example, Jacquemoud et al. [22] found for the JRC fresh leaf data set (FL) that nitrogen concentrations were significantly positive correlated with water content and negatively correlated with cellulose concentration.

In order to address these concerns, we have examined an alternative multivariate hierarchical FBA procedure (HFBA) that derives sequentially a series of FBA vectors extracting simultaneously important general (anatomical) features and discriminates samples at different levels of chemical detection. In this case, the right hand side of Equation 3 is changed to p_i, where p_i, i = 1, ..., n, represents either anatomical features or a quantized range of chemical concentration. Each FBA subsystem differs from Bolster's approach in the use of quantized chemistry ranges and reflectance itself in the equation, thereby reducing calibration dependencies and extraneous noise interference. By solving each FBA subsystem with a SVD, yields vectors that pack efficiently the spectral information that highlights those desirable features, and can potentially be extended to other sites.

In this paper, we apply this approach to the prediction of leaf chemical constituents specifically: carbon, nitrogen, cellulose concentrations and water content. In the next sections of the paper, we describe the data sets under study (section 2), introduce the underpinning concepts behind the HFBA system (section 3), show the results of the experimental tests and their application to other sites (section 4) and finally (section 5), we give some concluding remarks.

II. DATA SET AND METHODS

The second primary data set used in this study was obtained from the LOPEx (Leaf Optical Properties Experiment) done by the Joint Research Center (JRC), Ispra, Italy. The data set was collected in the vicinity of the center (45^o 58' N and 8^o 38.6' E) during the summer of 1993 and is described more fully in [37]. Data included 63 fresh leaf foliar samples (LOPEx-JRC) from a diverse range of 37 species of herbaceous and woody dicots and 9 species of monocots, both cultivated and native. Species include maple, alder, birch, laurel, walnut, chestnut, corn, alfalfa, figs, beets, lettuce, tomato, iris, and bamboo. Biochemical analyses were performed at the Centre de Recherches Agronomiques, Libramont, Belgium, as described in Jacquemoud et al. 1994 [37].

A third, similar but smaller secondary data set, was obtained from the Jasper Ridge Biological Preserve (JRBP) at Stanford University, California (37^o 24' N and 122^o 13' 30” W). It included 17 fresh leaf samples representing 14 native herbaceous and wood dicot species and one monocot, which were collected in the summer of 1992 and spring 1993 from several coastal California plant communities at the JR. Species include maple, buckeye, toyon, and oaks. Biochemical analyses were performed by the U. New Hampshire following ACCP protocols, and spectral measurements by U. California, Davis.

Lastly, another secondary fresh leaf data set that was obtained from the Santa Monica Mountains (SMM) in southern California (34^o 2' N and 118^o 30.5' W) was used to test model robustness and predictions [38]. Species at this site consisted of several chaparral shrubs and represent even more sclerophytic foliar conditions than the JR data set. These data consist of 42 foliar samples from 8 species of common dicot shrubs. Species include manzanita, chamise, sage, laurel, and California lilic. Only water content, which was measured as the difference between fresh and dry leaf weights is reported here.

The ACCP studies used dry ground leaves which were measured in a NIRS Model 6500 (NIRSystems Inc., Silver Spring, MD) Near Infrared Spectrophotometer that provides a wavelength range of 400-2500nm. Chemistry for nitrogen, cellulose, and total carbon were analyzed following methods described in [16, 36, 39]. LOPEx reflectance spectra were made on individual fresh leaves for the whole data set and 48 dry ground leaves, which were measured in a Perkin Elmer Lambda 19 spectrophotometer (Norwalk, CT), equipped with an integrating sphere. Spectral resolution ranged from 1-2 nm in the visible and NIR to 4-5 nm in the middle or SWIR obtaining a wavelength range of 400-2500nm. Jasper Ridge spectra were obtained on fresh individual leaves which were measured in a model NIRS Model 6500 (NIRSystems Inc., Silver Spring, MD). The NIRS system gives a 2 nm wavelength interval and a full width-half maximum slit width of 10 nm between 400 and 2490 nm. Measurement characteristics are described in more detail in [40]. The SMM data were measured on a Varian Cary 5E spectrophotometer (Sunnyvale, CA), with an integrating sphere (Labsphere, North Sutton, NH), which has variable wavelength resolution ranging from less than 1 nm in the visible to about 4-5 nm at 2500 nm. According to their size and shape, two to five leaves were cut and combined along their edges to ensure that all the light interacted the leaves [41]. The spectra were convolved to get 10nm wavelength intervals for future application of the HFBA vectors to Advance Visible Infrared Imaging Spectrometer (AVIRIS) images for a total of 211 wavebands in the range of 400-2500nm.

Tables 1 and 2 show a summary of chemical concentrations for fresh and dry ground leaves, respectively, for both training and testing data sets. We provide mean, standard deviation, and ranges of measured values, expressed on a percent dry weight basis (g/g). Same statistics are presented for water content.

Carbon concentration is the most homogeneous between the sites. However, JRC carbon data has a standard deviation of almost 1:5 times the standard deviation of the ACCP and JR data sets. The higher means and standard deviations of nitrogen concentrations in the JRC data set were due to the large number of cultivated plants, which typically have values near or greater than 3%. Significantly different ranges of chemical values are noticeable among data sets for cellulose and nitrogen. However, ranges of nitrogen concentration in the calibration data set (JRC and BH sites) essentially encompass those in the validation data set (JR and HF-HO sites). That is not the case for cellulose concentration: we notice significantly higher values in the JR dicot samples with respect to JRC samples, possibly due to the larger number of cultivated plants in JRC since cellulose and nitrogen were found significantly negative correlated [22]. Also cellulose had low values in HF-HO with respect to BH, this was mainly due to the high number of conifers, especially hemlock found in HO which are absent in the BH data set.

We exploit the species specific properties of the leaf spectra in order to find HFBA vectors (to be defined below) that discriminate the spectra into different groups and ranges of chemical variation. Chemical values are quantized into discrete ranges (low, intermediate and high concentrations) depending on their variance in order to reduce the sensitivity to calibration that all regression techniques have. We have also normalized each spectra in order to reduce dependencies on the conditions under which the measurements are made. The normalization is done by dividing each spectra by its respective Euclidean norm, that is, the normalized reflectance at channel i, which is given by

This normalization transforms each spectra to a common framework.

In the next section we analyze the properties of HFBA vectors.

III.  THE ANALYSIS

In the HFBA system, at the first level, we spectrally discriminate samples in the study into different categories given by a species classification or relevant chemistry variation: low, intermediate and high concentrations. Random selection of samples for the training subset usually have good results, however, robustness is not guaranteed. By selecting samples in the HFBA system to preserve the distribution observed in the property of interest for the whole data set, we can emphasize their general features. In that way, any sample that does not meet the first level of classification could be considered as outliers and rejected from application of the subsequent levels; they should be beyond the expected range of biochemical variation. In subsequent levels, we concentrate the FBA vectors that best relates leaf reflectance to a specific quantized range of chemical concentration (based on the variance in the chemistry data). We sequentially go down in scale narrowing the range, to obtain a closer approximation. This methodology allows us to refine searching of (subtle) spectral features that are related to biochemical differences in low, medium, and high quantized ranges. Also, we can group samples having similar anatomical properties and chemical concentrations to evaluate the reliability of the FBA vectors. The hierarchical detection tree constructed in this way, piecewise linearizes the spectral non-linearities observed in the overall chemical variation. In each step, the variance associated with each biochemical is considered to be the required level of detection necessary to make a valid interpretation. We quantized the chemical variation into few discrete values (no more than five) at each level. There are two basic reasons for seeking such quantization. First, following this methodology, the variance of each chemical constituent can be optimized in sequence by obtaining the FBA w vectors from Equation 3, and we can also determine when we cannot extend the analysis to a further level. Ability to determine the noise floor is an element of supreme importance in applying this analysis to remotely sensed images. Second, for classification purposes the quantized chemical concentrations contain discriminant features that can be related to observed spectral variation. Therefore, by quantizing chemical concentrations, biochemical detection by HFBA is, in essence, a classification problem at each level: to discriminate samples by associating their projected spectra (via HFBA vectors) into discrete ranges of biochemical variation.

At each level the HFBA equation (Equation 3) is solved by using a singular value decomposition (SVD) of the reflectance matrix R_fb. SVD becomes attractive and relevant to questions involving the behavior of the reflectance matrix R_fb itself, or its pseudo-inverse because it packs the spectral information into a few relevant axes of variation [42]. The power of the HFBA method becomes apparent as we begin to catalogue more precisely the performance of the SVD in information packing and avoidance of overfitting problems. For robustness, it means that we are training HFBA vectors with general but discriminating spectral properties. For more details about SVD see [42].

Determining how many abstract factors are needed in order to retain important information and avoid overfitting is a key step in any factor-based technique. The trick is to keep only factors that contain analytical information and discard factors that contain redundancies and noise. While keeping too many factors creates a dangerous tendency to overfit the data and adds undesired noise to the discriminating vectors, too few factors generate a poor set of discriminating vectors. Several indicator functions are available to aid in identifying the optimum number of factors (rank or dimensionality). We use Malinowski's methods to estimate the number of factors [43], and to determine the rank of the spectral system, although we use singular values instead of eigenvalues in our definitions. We must be careful when using these methods. First, not all linear problems are subject to such indicator functions, they are also limited to problems in which errors are not systematic or erratic. In our cases, spectral measurements contain relatively uniform errors throughout. Secondly, the dimensionality can not be replicated for all nodes of the hierarchical tree (levels of detection). At each node, we define a system of equations using the training set to define and highlight the desired properties. Therefore, Malinowski's indicators can be applied at each level in the analysis to determine the noise level. By adding to the solution system only singular vectors having the highest singular values and provide a satisfactory reproduction of the original data (within experimental error), the HFBA vector efficiently focuses the relevant spectral information that fits the variation of the desired property (species, chemical discrimination or other) at that level. The hierarchical procedure using this classification tree may account for the effects of non-linear chemistry dependencies on geometry and on anatomy. By the SVD, we also maximize the distinguishing characteristics of each site and relate directly the most common and relevant ranges of chemical concentrations to the spectral information.

In summary, HFBA methodology focuses in general but discriminating characteristics in the spectra that allow us to relate them to foliar chemistry variation at different levels of detection. Our principal assumption is that we can not detect all possible ranges of canopy or foliar biochemistry in one step. The need to break the chemistry variation into different levels of detection is due to the complex and nonlinear dependencies in the composition of a mixture. We do not see a mixture as a simple linear operator, but rather as a material that is composed of mixtures at different scales. The HFBA methodology is based on this concept of mixture: detecting accurate discriminating characteristics at each level allows us to reduce the range of variability in the next step. Poor discrimination at one step limits the level of detection that is possible with the HFBA methodology, which is bounded by the instrumental uncertainty, by nonlinear dependencies on geometry, and on the conditions under which the experiment was done.

For the fresh leaf data set, we have two distinctive features to use at the first level. One is the discrimination between dicot and monocot leaf samples due to their respective anatomical differences, and the other is a quantized water content due to the strong absorption features of water that could mask minor sources of spectral variation. To represent the dicot-monocot features we have selected 15 dicots and 5 monocots from JRC data set, which had 48 dicots and 15 monocots in total. To discriminate fresh leaf samples according to their water content (quantized into 2 main values: low-high water content), we trained the HFBA system at this level using 21 samples (17 low, 4 high) from JRC data set with a total of 53 and 10 (low-high, respectively) quantized water contents. The number of factors used for each classification was 8 for dicots-monocots and 7 for water quantization.

We considered the spectral discrimination of conifers and deciduous species as the first step of spectral biochemical detection in ACCP samples due to their anatomical and biochemical differences. Their differences in spectral properties were characterized using 13 conifers (from white and red pine samples) and 47 deciduous leaf samples taken from the BH data set which represented 40 conifer and 142 deciduous samples. In total, ACCP data set had 202 conifers and 356 deciduous samples. The number of factors derived from Malinowski's approach was 8. Here, a high level of information packing (compression) was obtained by SVD. It is worth noticing that hemlock and spruce species were present in HO and HF but not in BH.

After the primary classification step, we applied a second HFBA level that trained two vectors from the samples classified at the first level. As before, samples in the training set are selected to reflect the distribution of the quantized chemical concentration in the classified subset. Tables 3 and 4 present the five-centered distribution of quantized chemical values used to train the two HFBA vectors. Each column shows the number of samples in each site and the training set for each range. Dimension, shown in the top corner cell of each sub-table, represents the number of factors used after each SVD. Observe that the distribution of the full chemical data set is retained in the training set. In general, we used about 1/3 of the data set to train the vectors. The aforementioned out-of-range cellulose concentration for JR and HO-HF samples has been accounted by including 2 samples from JR and 40 samples from HO-HF (28-HO, 12-HF) in the training of cellulose HFBA vectors. Otherwise, the high (JR) or low (HO-HF) cellulose values, will be projected erroneously into intermediate ranges, R4 for JR and R2 for HO-HF. Nevertheless, the main features across low-medium-high ranges are still captured.

IV. RESULTS

A. Species Classification: first level   Reflectance HFBA Vector

Table 5 presents the results of classification using these features. Ninety-nine percent of deciduous samples were correctly identified, however 39 of the conifers (20%) were misclassified, 9 of these were larch samples (deciduous-conifer) and 21 were hemlock species having low cellulose concentration. Considering only results for BH, a hundred percent of samples were correctly classified. The HFBA vector captures the spectral features that distinguish conifer from deciduous samples.

Similarly, Figure 2 shows the properties of the monocot-dicot and low-high classification vectors applied to 120 fresh leaf samples. Monocot and dicot samples are identified by their spectral features in the visible region, where monocots are brighter (Figure 2a). The highest HFBA coefficient is found around the red edge, which has been used as an estimator of chlorophyll content [44]. In fact, monocot JRC samples had higher means in chlorophyll a (39.3 g/cm²) and chlorophyll b (11.78 g/cm²) than dicot JRC samples (35.4 g/cm² and 11.5 g/cm², respectively) [37]. Low and high water contents are spectrally discriminated by the main water absorption features at 1400nm and 1900nm, and the way these features interact in the blue visible region (around 400nm), where spectra of low water content has lower reflectance than spectra of high water content.

Table 6 presents the results of monocots-dicots and low-high water content classification. Ninety-five percent of monocot and dicot samples and eighty-eight of low-high water content samples were correctly classified. In the classification of low-high water content we focused the HFBA vector on identifyng samples that belong in the range R1, given in Table 3. Samples in range R2, in the same table, were considered as having intermediate water content and 16 were correctly separated from R1 samples. However, the HFBA vector projected the remaining 11 misclassified R2 samples into the top of the range of R1 samples. The spectral properties of these 11 samples around 1400nm were very closed to the properties of R1 samples, e.g. higher reflectance or (Figure 2b).
 

B. Second level: prediction

Figure 3 presents the two vectors used after classification for predicting each biochemical in the ACCP data set. While vectors on the left-hand side correlate quantized chemical concentrations for samples classified as deciduous, vectors on the right hand side do so for conifer samples. Carbon vectors present similar features in deciduous and conifer samples with major peaks about 700nm and 1920nm, especially in conifer vectors. Vectors for cellulose show some differences: deciduous vector (left hand side) presents high coefficients starting about the red edge waveband, 700nm and about 2120nm, an O-H bond bending band features (see similar results in [9]). On the other hand, conifer vectors had broad chlorophyll features, another indication of a higher correlation in conifers between pigments and cellulose concentration. We also found for the conifer-cellulose vector a broader range of wavebands with high coefficients between 1700 and 1900nm, which Curran reported as strong absorption wavebands for cellulose between 1780-1820nm [9]. Nitrogen vectors present main peaks at different wavebands. Deciduous vector peaks start about 1400nm (a water absorption band) and the conifer vector near the red edge. In this respect, wavelengths needed for spectral nitrogen predictions depend on the species. A common secondary positive peak about 1900nm indicates a positive correlation between water and nitrogen, which probably dampens the weak nitrogen absorption in fresh leaf data.

Similarly, Figure 4 presents the two vectors for each biochemical in each classified group of fresh leaf samples (monocot-dicot vectors for carbon and cellulose and low-high vectors for nitrogen and water). Overall, wavebands around the red edge and the primary water absorption bands (1400nm, 1900nm) have the highest coefficients. However, in the monocot-cellulose vector (second row, left side) the red edge contributes the most, jointly with the near infrared region which is greatly influenced by cellular structure [45]. This vectors has only two more regions with high coefficients, about 1100nm and between 1300 and 1600nm with highest peak at 1380nm where a C-H stretching vibrations occur [11].

Although, nitrogen vectors are very similar in shape to those for water, their coefficients had moderate values at wavebands between 970nm and 1200nm (other water absorption bands). Instead the red edge waveband region is highlighted. Carbon vectors, as in ACCP samples, are qualitatively similar. Same wavebands are highlighted in both groups, indicating that carbon features are consistent between monocots and dicots. As expected, water vectors have high coefficients at major water absorption bands, 1400nm and 1900nm, especially when the vector is applied to the high water content group. The other water absorption bands around 970nm and 1200nm are detected only by the vector trained with high water content samples. The highest value in water vectors is found at 1900nm. This value is much higher for the high water content vector.
 

C. Statistical results

The expected dampening of the weak nitrogen absorptions in the fresh leaf data by the strong water absorptions, is the main cause for the poor approximation obtained by using PLS. HFBA worked better because, we have grouped samples by water content, reducing this effect.

Figure 7 shows same statistics for the application of HFBA and PLS vectors to the available 120 fresh leaf samples (63-JRC, 17-JR, 40-SMM). A clear shift is noticed in PLS results, mainly in the samples of the testing data set (JR-SMM). Three main ranges are identified: low (less than 2.2 g/dm²), intermediate 2.2-3.6), and high (greater than 3.6). HFBA detects better such ranges, mainly by identifying the low water content, higher in number (see histograms in Figure 7). Again, the predicted distribution of nitrogen in the HFBA samples are closer to the measured values than was found for PLS method.

Tables 7 and 8 present statistical results of HFBA and PLS predictions of carbon, cellulose, and nitrogen concentrations for ACCP and FL data sets, and water content for Fl data set. For a reconstruction of the distribution, both tables provide five columns that indicate the number of samples projected to the ranges defined in tables 3 and 4, ACCP and FL respectively. Also, for a quick statistical comparison mean, standard deviation, minimum maximum and r² values of the prediction are given for each chemical. The tendency in PLS predictions is a clear shifting toward the ranges of the training set. This tendency is reduced in HFBA application. HFBA vectors, in this respect, provide more general characteristics, independent of the site, than PLS.

Overall, HFBA presented better results than PLS. For the whole ACCP data set the r²'s were improved from 0.31 to 0.49 in carbon prediction, 0.28 to 0.51 in cellulose prediction, and 0.63 to 0.69 in nitrogen. HFBA also approximates better the distribution (ranges), and therefore, mean, minimum and maximum of each data set when considered separately. While the r² statistics in cellulose indicates poor predictions, a more closed look to the ranges show a very good approximation of the distribution of the original data, compared with Table 3.

In the FL data set, HFBA also has better approximations than PLS. Coefficients of regression, r², are improved from 0.15 to 0.28 in carbon, 0.01 to 0.52 in cellulose, 0.13 to 0.71 in nitrogen, and from 0.43 to 0.75 in water. Although, carbon has a poor  r² for JRC data (0.22), the distribution of predicted and original data are very close, having R4 as the range with most samples. In general,  r² statistics reflect how well the prediction fits the ranges with highest number of samples. That means, in the case of carbon, that the prediction is spread across the whole range of values. Therefore, we got a poor  r².  Better results are obtained for cellulose. Mainly, because the vector for monocots weighs highly the near infrared region. The best results were obtained for nitrogen and water. The original data, as we mentioned in section 2, had high positive correlation between water and nitrogen. This aspect was captured by the nitrogen HFBA vectors (Figure 4), but not by the PLS vector. A clear shift for JR in nitrogen PLS predictions is noticed. This shift is most clear in the water content PLS predictions. Basically, shifting quantization ranges forces nitrogen shifting in high water content samples. HFBA controls better this relation and better preserves the original ranges.

V. DISCUSSION AND CONCLUSION

By the iterative hierarchical procedure we force the system to account for important non-linear dependencies directly related to spectral scaling. In that respect, one of the strong points of the proposed method is that we can group together samples with similar anatomical properties manifested spectrally. However, if the distribution of these properties is continuous, samples near the boundaries of the discriminant regions could be misclassified weakening the helpfulness of the classification step. In particular, as spatial variation of vegetation is high, the selection of a training set that explains the mixing presented at different spatial scales is critical. This process seems to be a key factor for understanding the good performance of HFBA dealing with sub-pixel scaling issues in this application, although HFBA was not properly equipped to deal directly with these spatial issues. There are more appropriate image analysis methodologies concerning spatial scaling problems such as wavelet transforms. The wavelet decomposition will give a better representation of spatial distribution (at different scales) of the data, and especially a better description of the properties of samples near to discriminant boundaries. Clearly, these points have to be further investigated to identify the relationship between spatial-spectral scales. As a conclusion, we consider that a combination of HFBA and wavelets or other spatial scaling transforms has significant potential and certainly deserves further investigation.
 

ACKNOWLEDGMENTS

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