Spatial Validation of Spectral Unmixing Results: A Case Study of Venice City

Abstract

Since remote sensing images offer unique access to the distribution of land cover on earth, many countries are investing in this technique to monitor urban sprawl. For this purpose, the most widely used methodology is spectral unmixing which, after identifying the spectra of the mixed-pixel constituents, determines their fractional abundances in the pixel. However, the literature highlights shortcomings in spatial validation due to the lack of detailed ground truth knowledge and proposes five key requirements for accurate reference fractional abundance maps: they should cover most of the area, their spatial resolution should be higher than that of the results, they should be validated using other ground truth data, the full range of abundances should be validated, and errors in co-localization and spatial resampling should be minimized. However, most proposed reference maps met two or three requirements and none met all five. In situ and remote data acquired in Venice were exploited to meet all five requirements. Moreover, to obtain more information about the validation procedure, not only reference spectra, synthetic image, and fractional abundance models (FAMs) that met all the requirements, but also other data, that no previous work exploited, were employed: reference fractional abundance maps that met four out of five requirements, and fractional abundance maps retrieved from the synthetic image. Briefly summarizing the main results obtained from MIVIS data, the average of spectral accuracies in root mean square error was equal to 0.025; using FAMs, the average of spatial accuracies in mean absolute error (MAE_k-Totals) was equal to 1.32 and more than 78% of these values were related to sensor characteristics; using reference fractional abundance maps, the average MAE_k-Totals value increased to 1.97 because errors in co-localization and spatial-resampling affected about 29% of these values. In conclusion, meeting all requirements and the exploitation of different reference data increase the spatial accuracy, upgrade the validation procedure, and improve the knowledge of accuracy.

1. Introduction

The spatial resolution (i.e., the ability to resolve two adjacent points as separate in the image) is one of the features of remote sensing imaging systems [1]. Some researchers highlight that, no matter how high the spatial resolution might be, the instantaneous field of view of image sensors always includes several discrete elements and no image pixel is completely homogeneous in spectral characteristics [2,3,4]. Therefore, the spectrum of each pixel is caused by a mixture of spectral signals (mixed pixels) and image processing techniques that are based on spatial analysis are not applicable [2,3,4,5]. In any case, since 1979, there has been a combined research effort to retrieve mixed pixel information from remote sensing images [6]. This effort has made possible the development of a systematic methodology that was given several slightly different names: linear mixing, spectral mixing models, spectral mixture analysis, spectral mixture modeling, and spectral unmixing [7,8,9,10,11,12]. In this paper, the term spectral unmixing was chosen from these. This methodology can be decomposed in two consecutive procedures: the first compares the spectrum of each mixed pixel with reference spectra to decompose it into a collection of constituent spectra (called endmembers); the second determines the proportion of each endmember present in the pixel (called fractional abundance) [4,12,13]. Since the number of endmembers must be less than the number of bands [7], some authors associated the development and use of spectral unmixing only with hyperspectral data [3,12,13,14]. However, the literature shows that, for over 40 years, spectral unmixing has been the most commonly used procedure for analyzing not only hyperspectral, but also multispectral images (e.g., [15]). Additionally, it is used for more and more applications from monitoring terrestrial environments (e.g., [16]) to extraterrestrial ones (e.g., [7]). One among these applications is urban land cover mapping which is now, more than ever, essential to understanding the impact of growing urban areas on the environment [12,17,18,19,20]. Spectral unmixing was exploited to distinguish urban surface materials from each other and from the background using hyperspectral (e.g., [21]) and multispectral (e.g., [22]) images at high spatial resolution and using hyperspectral (e.g., [18]) and multispectral (e.g., [23]) images at moderate spatial resolution, whereas it was exploited to distinguish urban area from the background using multispectral images at coarse spatial resolution (e.g., [20]).

To separate mixed pixel spectra into a collection of endmembers (i.e., the first procedure of spectral unmixing), the literature highlights two main models: linear (LMM) and nonlinear mixture models (NLMM). They assume two different behaviors of radiation reflected from each constituent material of the mixed pixel [4,12,24]. The light is completely separated before being mixed within the sensor in LMM’s case, whereas it is multiply scattered in NLMM’s case [4]. The choice between LMM or NLMM is still disputed; some authors pointed out that it might depend on the scale of the endmembers and the accuracy of the fractional abundance required [4]. However, LMM was the most widely employed in many scenarios [4,24], whereas NLMM was mainly applied to retrieve fractional abundance of minerals in soils [4]. To solve the inverse method which estimates the fractional abundances of each endmember (i.e., the second procedure), many solutions were proposed (e.g., Gram–Schmidt orthogonalization [25], least square methods [26], minimum variance methods [4], singular value decomposition [27], and variable endmember methods [4]). The least squares methods employed in LMM are the most utilized approaches because they provide an acceptable approximation of the light scattering mechanisms by minimizing errors [4,12].

Since 1986, many spectral unmixing techniques have been introduced to upgrade the mathematical validity of the model (e.g., [14,28]), but spatial and spectral validation of the results with ground truth data is still difficult to achieve (e.g., [4,24]). The spectral accuracy of spectral unmixing results can be evaluated using spectral signatures that were acquired in situ and/or in the laboratory and/or obtained from images. The previous works employed the spectra as reference spectral library to determine the endmembers present in the pixel and validate the results measuring their spectral accuracy (e.g., [14,20]). The flowchart of this spectral validation method is shown in Figure 1a. In the flowchart, reference data utilized as input into a process is connected to it by a red arrow, whereas reference data used for the validation process are connected to it by a green arrow (Figure 1a). The selection of an accurate and complete reference spectral library is essential because the key to spectral unmixing is the determination of endmembers [4,24,29,30]. Some authors created a synthetic image exploiting these spectra and the reference map in order to test the proposed procedure and/or to validate endmember extraction [14,31,32,33] and/or fractional abundance estimation (Figure 1b) [31,32,34,35]. Among these authors, only Dobigeon et al. [32] and Jia and Qian [31] exploited synthetic images to validate spectral unmixing results both spectrally and spatially.

The lack of detailed knowledge of the ground truth was identified as the main reason for the many shortcomings in spatial validation of spectral unmixing results [4,18,24]. The authors detected and addressed five requirements for an accurate reference map: (i) these data should cover most of the study area rather than just a few pixels to characterize the overall accuracy of the results [6]; (ii) the spatial resolutions of these maps should be higher than those of fractional abundance maps that need to be validated [18,36,37]; (iii) these maps should be validated using other ground truth data to properly become ground truth data [36,38,39]. Because no map can fully and completely map the ground truth (i.e., territory [40]), it is more correct to call them reference maps than ground truth maps [41]; (iv) these maps should include accurate detail of fractional abundances to validate the full range of their values that were obtained by spectral unmixing [12,33,37,39]; (v) these data should minimize errors in co-localization and spatial resampling due to the comparison of different data at different spatial resolutions (i.e., remote data and reference maps) in the spatial validation procedures [33,42]. In Figure 2, the flowchart of the spatial validation model highlights the five requirements identified for accurate reference maps, highlighting that previous papers did not meet all requirements at the same time, but most papers met two or three requirements.

The first and second requirements were shared by most authors. Some created reference maps of the study area by exploiting high spatial resolution aerial imagery in order to validate spectral unmixing results that were obtained from coarse spatial resolution imagery (e.g., [18,37]). Dobigeon et al. [32] extracted information of the ground cover fraction distributions from digital photographs. Other authors exploited the data that were acquired during the field surveys in order to photo-interpret IKONOS image [23,33] or aerial photos [43] and created reference maps of the study area. In order to validate the fractional abundances obtained from MODIS images, Kim et al. [44] utilized a Landsat image classified with three different approaches. Kruse [45] performed the same procedures on AVIRIS (at spatial resolution equal to 20 m) and Hyperion (at spatial resolution equal to 30 m) images and exploited the results of the first data to evaluate the accuracy of the second. In order to validate the fractional abundances obtained from NOAA-AVHRR data, Puyou-Lascassies et al. [46] utilized SPOT-HRV images classified and aerial photographs. Williams et al. [36,39] spatially resampled a NEON D17 image at 0.25 m to validate the fractional abundances obtained from AVIRIS imagery. It is interesting to note that most of these reference maps represent urban contexts [18,23,31,33,37,38].

The third requirement was only met by Walton [38] and Williams et al. [36,39] which validated the reference map using airborne images and associated the error with it. The four requirements were met by few authors [12,33,37,39]. The reference fractional abundance maps ranking from 100 to 0% were produced by [12,33,37,39] in order to compare the results of the spectral unmixing proposed and by [43] in order to validate the results of three machine learning regression techniques. However, the authors started from knowledge of only a few abundance values and interpolated the rest of the range (e.g., [33]). The fifth requirement was only fulfilled by Cavalli [33] who compared the histograms of fractional abundance that were obtained from reference maps (FAMs) and fractional abundance maps.

Table 1. Requirements for accurate reference maps.

Requirements for Accurate Reference Maps		Proposed by	Met by
I° requirement	Covering most of the study area	[6]	Most of the authors
II° requirement	Spatial resolution higher than those of remote data	[18,37]	Most of the authors
III° requirement	Validation with other ground truths	[36,38,39]	[36,38,39]
IV° requirement	Sampling the full range of abundances	[12,33,37,39]	[12,33,37,39]
V° requirement	Minimizing co-localization and spatial resampling errors	[33]	[33]

summarizes the requirements and the authors that proposed and met them. In conclusion, most papers met two or three requirements of the reference maps and not all five (Figure 2).

Therefore, this work created reference fractional abundance maps, meeting all requirements, in order to analyze their advantages in improving the spatial validation of spectral unmixing results. Moreover, in order to gain more information about result accuracy, it employed not only FAMs that met five out of five requirements, but also other data: reference spectra, reference fractional abundance maps that meet four out of five requirements, synthetic images, and fractional abundance maps retrieved from synthetic images. Therefore, this paper not only fulfilled all the requirements for an accurate reference map that no other previous paper satisfied, but also, it utilized other data that no previous work exploited: reference fractional abundance maps that meet four out of five requirements, and fractional abundance maps retrieved from synthetic images. For this purpose, ground truth data of the historical center of Venice and a MIVIS image which was acquired simultaneously with an in situ campaign were exploited [47]. The results of this work emphasize that reference maps that meet all requirements and their integrated use with different types of reference data increase the spatial accuracy, improve the knowledge of accuracies, and upgrade the validation procedure.

2. Materials and Methods

2.1. Reference Map and Reference Spectra

2.1.1. In Situ and Remote Data

An urban area was chosen as the study area for two reasons: firstly, the spectral unmixing is the methodology most often employed to map urban cover materials (e.g., [20]); secondly, the urban pattern facilitates the creation of fractional reference abundance maps (e.g., [37]). Among many urban areas, the historical center of Venice was preferred because several field campaigns were carried out to spatially and spectrally characterize urban surface materials [23,47]. On 26 July 2001 at 10:45 a.m. (GMT), in conjunction with an in situ campaign, MIVIS (Multispectral Infrared and Visible Imaging Spectrometer) airborne data were acquired with a scan rate of 25 scans/sec at an altitude of 4000 m, corresponding to a ground resolution of 8 m (

Table 2. Characteristics of the MIVIS image.

KERRYPNX	Spatial Resolution (m)	Bands	Spectral Region	Spectral Resolution (μm)	Spectral Range (μm)
MIVIS	8	20	VIS	0.02	0.4–0.83
		8	NIR	0.05	1.15–1.55
		64	SWIR	0.09	2–2.5
		10	TIR	0.34–0.54	8.2–12.7

).

Moreover, in a previous work, different classifiers that were also applied to these remote data and a reference map that met only the first two requirements was utilized to validate their results [23]. Their comparison pointed out that the most consistent results were obtained from the spectral unmixing followed by the results obtained from a spectral angle mapper (SAM) supervised classification algorithm [23].

The city of Venice, known worldwide, is located inside a large lagoon in the northwestern Adriatic Sea which is named after the city (Figure 3a). One hundred and eighteen small islands, on which Venice is built, and three hundred and fifty-five bridges, that connect them together, are two of the special characteristics of Venice (Figure 3b). A bridge also connects the mainland to the western part of the city, which is the only portion accessible by wheeled vehicles. In the rest of the city, the movement of people and things is by pedestrian way or by boats. In the islands, the buildings arise close to one another and are separated by narrow streets (called “calli”), squares with different shapes (called “campi”), streams, and canals.

2.1.2. Reference Map Meeting All Requirements

In order to validate the fractional abundance maps retrieved from the MIVIS image, the reference map of Venice’s historical center, previously created by photo-interpreting a panchromatic IKONOS image [23,33], was exploited (i.e., the first and second requirements of reference maps, e.g., [36]). In order to meet the third requirement (e.g., [38]), this reference map was validated using other ground truth data. The city and lagoon portals of Venice supplied these additional ground truth data to validate it (i.e., the edges of pavement and canals, the ground limits of buildings and bridges, the map of houses of worship and public and private buildings, and limits and features of public green space [48,49]). These data were obtained integrating data that were derived from different methodologies (i.e., traditional topography, the GPS satellite system, and laser scanner [49]) and their characteristics of precision and spatial distribution generated ground truth maps at a spatial resolution of 0.02 m with a maximum error of 10%.

These data allowed us to validate four types of land covers: vegetation, pavement, and public and private roofs. Photointerpretation and knowledge of the area allowed us to differentiate between pavement materials and to eliminate all those areas whose cover material was uncertain. These steps led to the creation of six validated reference maps (Figure 4): (i) Asphalt, that paves the western part of the city and harbor areas; (ii) Lateritic tiles, that cover most buildings; (iii) Lead plates, that cover the public buildings and domes; (iv) Limestone, that came from the quarries of Pietra d’Istria (Italy), that decorates the pavement of the historical center, but only the decorations of the squares facing St. Mark’s Basilica and the train station were considered; (v) Trachyte rock, that came from quarries in the Euganean Hills (Italy), that paves pedestrian streets; (vi) Vegetation. Many uncertain areas were not included in the reference map to minimize the error as much as possible; however, it is impossible for the maps to totally match the ground truth [40].

In order to meet the fourth requirement (i.e., the characterization of the entire range of fractional abundances, e.g., [37]), the previous works started from the assumptions that when spatially resampling a mask, where all pixels are characterized by 100% abundance, at lower resolution, only some defined pixels will be characterized by 100% abundance; the abundances of the other pixels of the starting mask will decrease linearly with increasing distance from pixels having 100% abundance; and the distance between pixels with 100% abundance and those with 0% abundance will be equal to the spatial resolution of the remote image [12,33,37]. By knowing only a few abundance values, the previous papers interpolated the rest of the range, whereas the present work aims to best sample the entire range using more known values. For this purpose, the choice of spatial resolution of the starting reference maps was very important. The spatial resolution of 0.20 m was chosen to validate the results obtained from MIVIS image because the starting maps at this resolution allowed us to sample the range from 100 to 0% with 40 known values (i.e., 40 pixels with spatial resolution of 0.20 m is the same as 8 m). The ground truth data that were exploited to validate the previous reference map allowed us to achieve this spatial resolution [48,49]. Therefore, the reference map at spatial resolution of 0.20 m was divided into 6 masks corresponding to the 6 selected endmembers, and these data were used as input for the procedure of fractional abundance determining. Figure 5a shows a portion of previous reference map (white) [33] and validated reference map (green) of the trachyte endmember at the spatial resolution of 0.20 m. In Figure 3b, the area examined in Figure 5 is highlighted with a yellow square.

The proposed procedure, starting from the one previously presented, upgrades and complements it, since the previous procedure started from only three known values and interpolated the rest of the range considering a linear trend [33]. The new one involves two steps: the first step identifies the pixels that should have 100% abundance of each endmember. For this purpose, firstly, the starting reference maps were spatially resampled to the spatial resolution of the remote image by applying the nearest neighbor method; secondly, these resampled maps were spatially resampled to the resolution of the starting reference map by applying the bilinear convolution method; finally, the pixels remaining after the two spatial resamples were selected (red pixels in Figure 5b).

The literature highlights that nearest-neighbor resampling is the method most commonly employed in the ortho-rectification, resampling to a north–south grid process and aligning coarse-scale images on fine-scale ones (e.g., [50]). In any case, in order to identify the methods that best minimize error in spatial sampling, the results of the main image resampling methods (i.e., pixel aggregate, nearest neighbor, bilinear and cubic convolution methods) were compared (Figure 6 shows the same portion of the trachyte reference mask shown in Figure 5). In these figures, the starting mask (blue) is covered by a twice-resampled reference map which has two different colors: pixels are colored cyan when they fall within the boundaries of the source map, whereas they are colored red when they erroneously cross the boundaries of the source map. Figure 6a shows a smaller number of red pixels than the other figures, indicating that the application of nearest neighbor and bi-linear convolution methods, in this time order, minimizes errors due to image resampling.

The second step evaluates pixels that should have less than 100% abundance of the endmember. With regard to the portions of the mask surrounding the areas with 100% abundance, a set of areas equidistant from the 100% abundance pixels, within the higher resolution reference map, was created. A first area was created at a distance of one pixel, the second at a distance of two pixels, and so on until the distance was equal to the spatial resolution of the remote image (Figure 5c). These areas are associated with gradually decreasing abundance from 99 to 0%. With regard to the portions of the mask not surrounding the areas with 100% abundance, the directional filters at horizontal (0°) and vertical (90°) orientations were applied to this portion of the reference mask. A set of areas that were equidistant from the pixels identified with the filters was created within the higher-resolution reference map (Figure 5d). These areas are associated with gradually increasing abundance from 1 to 99%. In conclusion, all pixels associated with the 100 to 0% abundances of each endmember were identified and mapped to form the corresponding reference fractional abundance map (Figure 5d).

Lastly, to minimize co-localization and spatial resampling errors due to the comparison of different data (i.e., remote data and reference maps) at different spatial resolution (i.e., to meet the five requirements), the reference and retrieved fractional abundance maps were not compared, but the method proposed by Cavalli was exploited: the comparison of the abundance histograms of the reference fractional abundance maps and the fractional abundance maps obtained by spectral unmixing [33]. The histogram of reference fractional abundance maps was called the fractional abundance models (FAMs) [33].

2.1.3. Reference Spectra

Spectral characterization of the six identified materials was performed at several sites in the historical center to sample natural spectral variability and in the laboratory to better evaluate the spectral characteristics of the materials [23]. The field campaigns were conducted using a portable field spectrometer Analytical Spectral Devices FieldSpec Full-Range Pro [47]. The spectrometer sampled spectral range from 350 to 2500 nm using two detectors: one covered from 350 to 1050 nm with a spectral sampling interval of 1.4 nm, and the other covered from 1000 to 2500 nm with a spectral sampling interval of 2.0 nm. The field measurements were collected within 2 h of solar noon sets by acquiring 4–5 measures for each target from a height of 1 m using a field of view of 25° to fulfil the target dimension. A Spectralon reference standard (i.e., National Institute of Standards and Technology calibrated panel) was employed to derive absolute reflectance spectra [23,47]. Therefore, the created spectral library was spectral resampled according to the spectral characteristics of the MIVIS data to become the reference spectra.

2.2. Preprocessing and Spectral Unmixing Processing Chains

As regards radiometric calibration, MIVIS data were pre-processed employing an improved radiometric correction procedure [51]. Since the determination of the endmember was performed by exploiting reflectances that were acquired in situ and/or in the laboratory, radiance data were atmospherically corrected to obtain reflectance images. MIVIS radiance data were atmospherically corrected by means of the MODTRAN4 radiative transfer code [52,53]. Atmospheric correction data were validated using in situ spectral measurements [54]. After applying the spectral unmixing processing chain [55], the fractional abundance maps were geometrically corrected by exploiting an improved geometric correction procedure [56]. In order to compare the reference maps with the fractional abundance maps that were retrieved from MIVIS data, a set of ground control points (GCPs) was extracted from the IKONOS panchromatic image, and resampled to 0.20 m, to which the ground truth data used to validate the reference map [48,49] were superimposed. The MIVIS image yielded RMS errors of 0.064 pixels.

For most authors, the spectral unmixing processing chain applied to hyperspectral data includes not only endmember determination and inversion, but also dimension reduction (e.g., [12,13]). Therefore, minimum noise fraction (MNF) transformation [57] was applied to both datasets by using the forward MNF transform implemented in the ENVI 5.6 software package. In conclusion, MIVIS’s 92 bands were reduced to 47 bands.

Because multiple endmember spectral mixture analysis (MESMA) is the most widely used spectral unmixing methodology in the urban context (e.g., [17,18,37]), it was applied to the corrected MIVIS image exploiting an open-source software application ‘VIPER2-tools’ that is an ENVI add-on. MESMA, introduced by Roberts et al. [58], allows us to vary the number and types of endmembers for each pixel because spectral unmixing assumes that the set of endmembers present in a pixel is invariable. MESMA searches, among all possible combinations, the best combination of endmembers for each mixed pixel [58].

Since MESMA is based on LMM, mixed spectrum $S^{\text{'}}$ measured at pixel i can be denoted as a linear combination of endmember spectra:

$$S_i^{\text{'}}={{\displaystyle \sum }}_{k=1}^N{f}_{ki}*S_k+{\epsilon }_i$$

(1)

where N is the number of the endmembers, f_k is the fractional abundance of the endmember k, S_k is the spectrum of the endmember k, and ε is the residual error that takes into account the spectral and spatial characteristic of the sensor, noise and errors of the preprocessing and processing chains [4,12,13]. In the LMM, fractional abundances are subject to the following constraints:

$$f_k\ge 0, \mathrm{k}=1,\dots \dots ,\mathrm{N}$$

(2)

$${\displaystyle \sum }_{k=1}^N{f}_{ki}=1$$

(3)

2.3. Validation Methods of the Spectral Unmixing Results

The remote and in situ data of Venice historical center allowed us to spatially and spectrally validate the spectral unmixing results using five different reference data: reference spectra, synthetic image, reference fractional abundance maps that met four out of five requirements, fractional abundance maps retrieved from synthetic image, and FAMs that met all the requirements. A validation method was associated with each reference datum employed. Firstly, to validate the endmembers, the differences between the reference spectra and the spectra of the pixel constituents were measured for each pixel (called the first method, Figure 1a). This spectral validation method was employed by many authors (e.g., [20]). Secondly, to validate the endmembers, the similarity measures between the reference spectra and the pixel spectra of the real and synthetic image were compared (called the second method, Figure 7a). This spectral validation method was proposed by some authors (e.g., [31]); however, these authors created the synthetic images from reference maps that did not meet all the requirements (Figure 1b), whereas the reference fractional abundance maps that were employed to create the synthetic image met four out of five requirements (Figure 7a). Third, to validate the endmembers, the same spectral unmixing procedure was applied to both the true and synthetic images, and the differences between the reference spectra and the spectra of the pixel constituents, that were obtained from the true and synthetic images, were compared (called the third method, Figure 7b). Fourth, to spatially validate the results, the differences between the fractional abundance maps, that were obtained from the true and synthetic images, and the reference ones were calculated and compared (called the fourth method, Figure 7c). Lastly, to spatially validate the results, the differences between the fractional abundance maps, that were obtained from the true and synthetic images, and the FAMs were calculated and compared (called the fifth method, Figure 7d). In conclusion, the results of all these methods were compared. In the flowcharts of five validation methods shown in Figure 1a and Figure 7, the reference data are shown in green color and those utilized are highlighted with a yellow rectangle with a green border, whereas additional data utilized are highlighted with an orange rectangle.

Among these methods, four employ a synthetic image to spatially and spectrally validate the results (Figure 7a–d). In order to build the synthetic images, the six reference maps and spectral library were utilized and the procedure proposed by [33] was exploited in order to simulate the internal spectral variability of each endmember in the synthetic images. Each reference mask was multiplied by the corresponding endmember spectrum and its internal spectral variability [33] (i.e., imagining a chessboard of the same size and shape as the mask, its pixels that were associated with the black chessboard squares were multiplied by the mean spectrum plus the standard deviation, whereas those that were associated with the white chessboard squares were multiplied by the mean spectrum minus the standard deviation). The area previously mapped as a water body [33] was multiplied by the spectra of lagoon waters exploiting the same procedure again. In order to consider the spectral variability of all materials characterized in situ, the area remaining after removing the endmembers masks and that of the water body was multiplied by two average spectra utilizing the procedure proposed by [33]: imagining a checkerboard of the same size and shape as mask, one pixel as yes and one as no were multiplied by the average spectrum of the six identified endmembers plus the average spectrum of lagoon waters, and one pixel as no and one as yes were multiplied by the average spectrum of some ancillary materials (e.g., wood, cloth, plastic, and metal) plus the average spectrum of lagoon waters. Therefore, the starting hyperspectral image at a spatial resolution of 0.30 m with 47 bands was spatially resampled at 8 m to simulate the MIVIS image. This spatial resolution, different from that identified to create the reference maps and FAMs (i.e., 0.20 m), was chosen to maintain spectral variability in the synthetic image since the pixels at a resolution of 8 m did not contain an integer or an even number of pixels at a resolution of 0.30 m. Simulating the internal spectral variability of endmembers and preserving it during spatial resampling is important because in high-spatial-resolution images, spectral unmixing results are also driven by internal spectral variability [33].

Spectral and spatial accuracy were previously assessed not only using differences between the retrieved and reference data (i.e., bias [59], mean absolute error [19], mean of all [36], mean square error [32], Pearson correlation coefficient [15], reconstruction error [60], root mean square error [14], and systematic error [19]), but also using spectral similarity measures (i.e., Euclidean distance [31], maximum likelihood algorithm [36], spectral angular distance [14], and spectral information divergence [31]). Both approaches were also utilized in this work to spatially and spectrally validate the spectral unmixing results (

Table 3. The metrics performed to evaluate the spatial and spectral accuracy in the five methods.

Spectral Validation Methods	Reference Data Employed	Accuracy of the Pixel Constituent Spectra
		Error Measure	Similarity Measure
First (Figure 1a)	Reference spectra	RMSk	SADk—aSSMnk
Second (Figure 7a)	Spectra of synthetic image		SADk—aSSMnk
Third (Figure 7b)	Pixel constituent spectra obtained from synthetic image	RMSk
Spatial Validation Methods	Reference Data Employed	Accuracy of the Fractional Abundance Maps
		Error Measure	Similarity Measure
Fourth (Figure 7c)	Reference fractional abundance maps and fractional abundance maps obtained from synthetic image	MAEk
Fifth (Figure 7d)	FAMs and fractional abundance maps obtained from synthetic image	MAEk	KGE

).

Since the root mean square error (RMS) of the residual errors (e.g., [20]) and mean absolute error (MAE) (e.g., [19]) were the most commonly used, they were selected to validate the pixel constituent spectra and fractional abundance maps, respectively. The average value of the residual errors of all pixels i within endmember mask k (RMS_k) was given as:

$$RM{S}_{\mathrm{k}}=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^{ni}{\left({\epsilon }_i\right)}^2}{ni}}$$

(4)

where ni is the number of pixels i within endmember mask k; RMS_k values were calculated by applying “VIPER2-tools” which determines the best set of endmembers by minimizing the RMS value for each pixel.

The MAE calculated for all pixels within endmember mask k (MAE_k) was given as:

$$MA{E}_k= \left(\frac{{{\displaystyle \sum }}_{i=1}^{ni}\left|A_{classified}-A_{reference}\right|}{ni}\right)$$

(5)

where the A_classified and A_reference are fractional abundances of the same pixel that were retrieved from remote and reference data, respectively. MAE_k was calculated in the range of abundances from 100 to 0% (MAE_k-Totals), in the range from 100 to 50% (MAE_k-100–50%), and in the range from 49 to 0% (MAE_k-49–0%). This distinction of abundance range was performed since it divides pixels that are assigned to an endmember from those that are not assigned [61].

As regards the similarity measures performed for spectral validation, spectral angular distance (SAD) was chosen because it is the most widely employed (e.g., [14]). The average value of SAD calculated of all pixels i within endmember mask k (SAD_k) was given as [62]:

$$SA{D}_k=\frac{{{\displaystyle \sum }}_{i=1}^{ni}co{s}^{-1}\left(\frac{{{\displaystyle \sum }}_{j=1}^{nb}{S}_i^{\text{'}}*S_k}{{\left({{\displaystyle \sum }}_{j=1}^{nb}{\left(S_i^{\text{'}}\right)}^2\right)}^{0.5}{\left({{\displaystyle \sum }}_{j=1}^{nb}{\left(S_k\right)}^2\right)}^{0.5}}\right)}{ni}$$

(6)

where nb is the number of the bands j, ${\mathrm{S}}_i^{\text{'}}$ and S_k are spectra of the pixel and reference endmember. Therefore, the spectral similarity measurement is close to the value 0 when the pixel spectrum is spectrally similar to the reference spectra and it is close to the value 1 when the spectrum is not spectrally similar.

The metrics previously utilized to evaluate the similarity between the pixel spectrum and reference spectra (i.e., Euclidean distance [31], maximum likelihood algorithm [36], spectral angular distance [14], and spectral information divergence [31]) were also exploited by four supervised classifiers (i.e., minimum distance, maximum likelihood, spectral angular mapper, and spectral information divergence, respectively). Since every supervised classifier successfully uses a different algorithm to measure the similarity between the pixel spectrum and reference spectra, spectral similarity measurements achieved by the main supervised classifiers (i.e., binary encoding, minimum distance, maximum likelihood, spectral angular mapper, and spectral information divergence) were taken into consideration to better assess the spectral similarity [63]. Therefore, these supervised classifiers were applied to the input images utilizing the same reference spectra, that were exploited to determine the spectra of the pixel constituents; five different spectral similarity measurements within each endmember mask were evaluated and the average of their values were normalized. The normalized average of spectral similarity measures (aSSMn), that were calculated for all pixels within endmember mask k is expressed as

$$aSSM{n}_k=\frac{{{\displaystyle \sum }}_{i=1}^{ni}\frac{{{\displaystyle \sum }}_{C=1}^{nc} \left(\frac{\left(SS{M}_{Ci}-\left(SS{M}_{Cmin\_k}\right)\right)}{\left(\left(SS{M}_{Cmax\_k}\right)-\left(SS{M}_{Cmin\_k}\right)\right)}\right) }{nc}}{ni}$$

(7)

where nc is the number of the supervised classification C, SSM_Ci is spectral similarity measurement that was calculated at pixel i within endmember mask by the supervised classifier C, and SSM_{Cmin_k} and SSM_{Cmax_k} are the minimum and maximum values of the spectral similarity measurements that were calculated by the supervised classifier C within endmember mask k. Therefore, when the pixel spectrum is spectrally similar to the reference spectrum, the normalized average value of spectral similarity measurements is close to the value 1, and when the pixel spectrum is not spectrally similar, it is close to the value 0. In conclusion, for each endmember, mean and standard deviation of SAD_k and aSSMn_k values, that were evaluated within the corresponding mask, were calculated (i.e., SAD_k, aSSMn_k, σSAD_k, and σaSSMn_k).

Since the reference fractional abundance maps created meet four out of five requirements and the FAMs derived from these maps meet all requirements, also FAMs were exploited to spatially validate the fractional abundance maps. Kling–Gupta efficiency (KGE) was exploited to evaluate similarity between the FAM and the histogram of retrieved fractional abundances to the same endmember within its mask. KGE is given as [64]:

$$KGE=1-\sqrt{{\left(R-1\right)}^2+{\left(\frac{{\sigma }_{FAM}}{{\sigma }_{retrived fractional abundance histogram}}-1\right)}^2+{\left(\frac{{\mu }_{FAM}}{{\mu }_{retrieved fractional abundance histogram}}-1\right)}^2}$$

(8)

where R is linear correlation between the FAMs and retrieved fractional abundance histogram relative to the same endmember, σ is their standard deviation, and µ is their mean values. The KGE value is used to calibrate and evaluate not only the hydrological models, but also the bio-optical models [65]. The literature grouped the KGE values into three classes: values equal to 1, positive and negative values that indicate equal, approximately similar, and non-comparable distributions, respectively [64,65].

3. Results

3.1. Fractional Abundance Models (FAMs)

As mentioned above, the reference fractional abundance maps at a spatial resolution of 0.20 m allowed us to retrieve FAMs at a spatial resolution of 8 m: starting from the reference fractional abundance maps at spatial resolutions of 0.20 m, the range of the FAMs from 100 to 0% at the spatial resolution of 8 m was sampled with 40 known values in order to validate the results obtained from the MIVIS data. Figure 8 compares the FAMs of each selected endmember.

The analysis of the FAMs highlights that urban surface materials of Venice historical center which are characterized by large and non-jagged shapes (i.e., asphalt, lateritic tiles, and lead plates) show an initial linear trend and then tend to be zero confirming the results as reported by [12,33,37], whereas the other surface materials which are characterized by small and jagged shapes show a non-linear trend (i.e., limestone, trachyte rock, and vegetation).

3.2. Methods for Spectral Validating the Pixel Constituent Spectra

As mentioned in the previous section, the endmembers (i.e., pixel constituent spectra) were spectrally validated by measuring the difference (i.e., error) and similarity between the reference spectra and the spectrum of each image pixel (Table 3), which was placed within the reference mask of the corresponding endmember. The errors were calculated using RMS_k values (Equation (4)); the values obtained by applying the first and third methods (Figure 1a and Figure 7b) are shown in the first and second columns of

Table 4. In the first and third methods, spectral accuracy estimated with RMS_k values that were retrieved from true and synthetic images of MIVIS and the absolute differences between the RMS_k values obtained from true and synthetic image (ΔRMS_k).

	MIVIS Image	Synthetic MIVIS Image
Endmembers	RMSk	RMSk	ΔRMSk
Asphalt	0.023	0.011	0.012
Lateritic tiles	0.020	0.019	0.000
Lead plates	0.028	0.026	0.002
Limestone	0.029	0.018	0.011
Trachyte rock	0.028	0.013	0.015
Vegetation	0.023	0.021	0.002

, respectively. The absolute differences between the RMS_k (ΔRMS_k) values that were retrieved from the true and synthetic images are shown in the third column of Table 4.

Since the focus of spectral unmixing is the determination of endmembers (e.g., [13]), the residual error values (ε, Equation (1)) and, consequently, the RMS_k value (Equation (4)) attest to the accuracy of its results. With reference to the previous works (e.g., [66]), an RMS value smaller than 0.045 attests to good accuracy in determining endmembers. The RMS_k values obtained from the MIVIS image ranged from 0.029 to 0.020 and the values obtained from the synthetic image ranged from 0.026 to 0.011. Therefore, the RMS_k values obtained attest to good accuracy of the spectral unmixing procedure performed since they are below the previously proposed threshold (Figure 9).

The absolute differences between the RMS_k values obtained from true and synthetic images (ΔRMS_k) range from 0.015 to 0.002. It is important to mention that the spectral and spatial characteristics of the synthetic image are the same as those of the remote image and the procedure used to make the synthetic image is the same spectral unmixing procedure as that applied to the remote image. In other words, each spectrum of the synthetic image is the weighted sum of the reference spectra of the endmembers, whose weights are determined by the reference fractional abundance maps. Therefore, because differences between the RMS_k values obtained from the true and synthetic images are equal to about 30% of the RMS_k value obtained from the true image, the remaining 70% can be attributed to the spectral and spatial characteristics of the MIVIS image and confirms the choice of not only the set of endmembers, but also the linear mixture model.

In detail, the choices of the reference endmember set and the LMM are more accurate for the endmembers of building roofing materials and vegetation (i.e., more than 91% of the RMS_k values, Figure 10) than for the ones of street paving materials (i.e., less than 62% of the RMS_k values, respectively, Figure 10), where the effect of multiple scattering and the occurrence of other surface materials (i.e., the surface of street paving materials was mapped as totally empty of people and things) may be slightly greater.

As regards similarity measures between the reference spectra and the spectrum of each image pixel placed within the corresponding reference mask, they were evaluated using the SAD_k and aSSMn_k values (Equation (6) and Equation (7), respectively) and their standard deviation values. However, it is important to remember that, when the pixel spectrum is similar to the reference spectrum, the SAD value is close to 0 whereas the SSMn value is close to 1. Conversely, when the pixel spectrum is not similar to the reference spectrum, the value of SAD is close to 1, while the value of SSMn is close to 0. The SAD_k, σSAD_k, aSSMn_k, and σaSSMn_k values obtained from true and synthetic images are shown in the first and second four rows of

Table 5. In the second method, spectral accuracy estimated with SAD_k, σSAD_k, aSSMn_k, and σaSSMn_k values that were retrieved from true and synthetic images and the absolute differences between SAD_k, σSAD_k, aSSMn_k, and σaSSMn_k values obtained from true and synthetic images (ΔSAD_k, ΔσSAD_k, ΔaSSMn_k, and Δ σaSSMn_k).

	MIVIS Image
Endmembers	SADk	σSADk	aSSMnk	σaSSMnk
Asphalt	0.361	0.123	0.558	0.395
Lateritic tiles	0.299	0.069	0.659	0.255
Lead plates	0.282	0.079	0.693	0.247
Limestone	0.379	0.129	0.498	0.423
Trachyte rock	0.387	0.143	0.485	0.417
Vegetation	0.361	0.083	0.677	0.251
	Synthetic MIVIS Image
Endmembers	SADk	σSADk	aSSMnk	σaSSMnk
Asphalt	0.282	0.060	0.653	0.293
Lateritic tiles	0.271	0.042	0.772	0.217
Lead plates	0.253	0.065	0.714	0.245
Limestone	0.286	0.035	0.581	0.245
Trachyte rock	0.281	0.062	0.649	0.312
Vegetation	0.302	0.051	0.718	0.195
Endmembers	ΔSADk	ΔσSADk	ΔaSSMnk	ΔσaSSMnk
Asphalt	0.079	0.063	0.095	0.102
Lateritic tiles	0.028	0.027	0.113	0.038
Lead plates	0.029	0.014	0.021	0.002
Limestone	0.093	0.094	0.083	0.178
Trachyte rock	0.106	0.081	0.164	0.105
Vegetation	0.059	0.032	0.041	0.056

, respectively. The absolute differences between the values that were retrieved from true and synthetic images are shown in the third four rows of Table 5.

The spectra of the MIVIS image within the reference masks are slightly less similar to the corresponding reference spectra than those of the synthetic image (i.e., the average values of SAD_i and aSSMn_k obtained from the true image are equal to 0.345 and 0.595, respectively, and the average values of SAD_k and aSSMn_k obtained from the synthetic image are equal to 0.279 and 0.681, respectively). In the true and synthetic images’ processing, the endmembers showing the greatest values were lead plates, lateritic tiles and vegetation, whereas the ones showing the smallest values were the endmembers of street paving materials. As regards true image processing, the standard deviation of the SAD_k and aSSMn_k values calculated for endmembers of street paving materials (i.e., the average values of σSAD_k and σaSSMn_k were equal to 0.132 and 0.514, respectively) were greater than the ones that were calculated for building roofing materials (i.e., the average values of σSAD_k and saSSMn_k were equal to 0.077 and 0.251, respectively). Contrastingly, in the synthetic image processing, the standard deviation of the SAD_k and aSSMn_k values calculated for endmembers of street paving materials (i.e., the average values of σSAD_k and σaSSMn_k values were equal to 0.052 and 0.219, respectively) were quite similar to the ones that were calculated for building roofing materials (i.e., the average values of σSAD_k and σaSSMn_k values were equal to 0.053 and 0.283, respectively). In conclusion, the pixels within the reference masks of the street paving materials are less similar to the corresponding reference spectra than those inside the reference masks of the building roofing materials. Therefore, the similarity measures also confirmed what was evidenced by the error measures: the effect of multiple scattering and the presence of other surface materials affect the pixels of street paving materials more than the pixels of building roofing materials.

3.3. Methods for Spatial Validating the Fractional Abundance Maps

The error and similarity measures were also evaluated to spatially validate the spectral unmixing results. In the fourth method (Figure 7c), the spatial accuracy of fractional abundance maps was evaluated by measuring the differences between the fractional abundance retrieved from true and synthetic images and the fractional abundance of the corresponding reference map that met four out of five requirements. For each endmember, only the errors of pixels placed within the corresponding reference mask were considered. The errors were calculated with MAE_k-Totals, MAE_k-100–50%, and MAE_k-49–0% values (Equation (5)) and these values obtained from true and synthetic images are shown in Figure 11.

The MAE_k-Totals values obtained from the processing of the MIVIS image using reference fractional abundance maps were less than 3.0 (Figure 11). In detail, the endmembers of trachyte rock and asphalt showed the greatest errors, followed by limestone and then vegetation; lastly, the endmembers of building roofing materials showed the smallest values. Moreover, MAE_k-Totals values calculated from true images were about two times greater than ones calculated from synthetic images (i.e., the average value of MAE_k-Totals obtained from the true image was equal to 1.97, whereas the average value of MAE_k-Totals obtained from the synthetic image was equal to 0.99). Therefore, about 50% of MAE_k-Totals was due to sensor characteristics, the reference endmember set and LMM (Figure 12). In detail, the choices of the reference endmember set and the LMM were more accurate for the endmembers of building roofing materials and vegetation (i.e., more than 68% of the MAE_k-Totals values, Figure 12) than for ones of street paving materials (i.e., less than 48% of the MAE_k-Totals values, Figure 12).

As regards MAE_k-100–50% and MAE_k-49–0% values calculated from the MIVIS image, the MAE_k-49–0% values were about three times greater than the MAE_k-100–50% values (i.e., the average value of MAE_k-100–50% was equal to 0.48 whereas the average value of MAE_k-49–0% was equal to 1.48, Figure 11). As regards MAE_k-100–50% and MAE_k-49–0% values calculated from the synthetic image, MAE_k-49–0% values were about 2.5 times greater than MAE_k-100–50% values (i.e., the average value of MAE_k-100–50% obtained was equal to 0.29 whereas the average value of MAE_k-49–0% was equal to 0.69, Figure 11). These values show that fractional abundances from 49 to 0% are overestimated by maps retrieved from true and synthetic images [33], and this overestimation is more in maps retrieved from true images than in maps retrieved from synthetic images.

The fifth method (Figure 7d) aims to minimize errors in co-localization and spatial resampling due to the comparison of different data at different spatial resolutions in the spatial validation procedures. Therefore, the histograms of fractional abundances retrieved from true and synthetic images were compared with the FAMs in order to evaluate spatial accuracy using error and similarity measurements. Error measures calculated from true and synthetic images were measured with MAE_k-Totals, MAE_k-100–50%, and MAE_k-49–0% values and these values are shown in Figure 13.

In general, the MAE_k-Totals values obtained from the MIVIS image using FAMs were less than 1.9, whereas the values obtained from the synthetic images were less than 1.4 (Figure 13). In detail, as regards true image processing, the endmember of trachyte rock showed the greatest values, followed by asphalt and limestone, respectively; lastly, the endmembers of building roofing materials and vegetation showed the smallest values. As regards error values in the abundance ranges from 100 to 50% and from 49 to 0% that were calculated from true and synthetic images, MAE_k-49–0% values were about 2.5 times greater than MAE_k-100–50% values (i.e., the average values of ΔMAE_k-100–50% obtained from true and synthetic images were equal to 0.39 and 0.27, respectively, whereas the average values of ΔMAE_k-49–0% were equal to 0.93 and 0.72, respectively, Figure 13). In other words, the proportion by which fractional abundance maps recovered from the MIVIS image overestimating abundances from 49 to 0% was the same as that by which fractional abundance maps recovered from the synthetic image overestimating abundances from 49 to 0%.

The comparison of the results of the fourth and fifth methods shows that the MAE_k-Totals values obtained by exploiting reference fractional abundance maps ranged from 3.0 to 1.1 (Figure 11), whereas the MAE_k-Totals values obtained by exploiting FAMs ranges from 1.9 to 0.9 (Figure 13). Therefore, fractional abundance maps that were spatially validated using FAMs exhibit smaller error values than the same results that were spatially validated using reference fractional abundance maps. Since the reference fractional abundance maps meet four out of five requirements and the FAMs meet five out of five requirements, the error in co-localization and spatial resampling, due to the comparison of different data at different spatial resolutions in the spatial validation procedures, is equal to the differences between the MAE_k-Totals, MAE_k-100–50%, and MAE_k-49–0% values that were calculated with the reference maps and those that were assessed with the FAMs, and the percentages of these difference were called Error_Total, Error_100–50%, and Error_49–0% values (

Table 6. The error in co-localization and spatial resampling evaluated by MIVIS image processing with Error_Total, Error_100–50%, and Error_49–0% values.

	MIVIS Image
Endmembers	ErrorTotals	Error100–50%	Error49–0%
Asphalt	44.4%	33.3%	50.0%
Lateritic tiles	8.3%	16.7%	0.0%
Lead plates	18.2%	20.0%	16.7%
Limestone	39.1%	0.0%	39.3%
Trachyte rock	36.7%	0.0%	44.0%
Vegetation	26.7%	25.0%	27.3%

).

The average values of Error _Total, Error_100–50%, and Error_49–0% are equal to 29, 16, and 30%, respectively. In detail, the endmembers of street paving materials show greater errors in co-localization and spatial resampling (i.e., the average values of Error _Total, Error_100–50%, and Error_49–0% are equal to 40, 11, and 44%, respectively) than values those that were evaluated to retrieve the fractional abundance maps of building roofing material and vegetation endmembers (i.e., the average values are equal to 18, 21, and 15%, respectively). Moreover, the Error_49–0% values evaluated for the endmembers of street paving materials are two times greater than the Error_100–50% values evaluated for the other endmembers.

The similarity between the histogram obtained from each fractional abundance map and the FAM of the same endmember was measured using KGE values.

Table 7. In the fifth method, spatial accuracy is measured with KGE values that were obtained from true and synthetic images and the absolute difference between these values.

	MIVIS Image
	True	Synthetic	Difference
Endmembers	KGE	KGE	KGE
Asphalt	−0.010	0.050	0.060
Lateritic tiles	0.150	0.315	0.165
Lead plates	0.170	0.720	0.550
Limestone	−0.060	0.012	0.072
Trachyte rock	−0.080	−0.028	0.052
Vegetation	0.300	0.485	0.185

shows also the absolute differences between these values. However, it is important to remember that the literature highlights three KGE value ranges [64,65]: when the distributions are equal, the KGE value is equal to 1; when the distributions are comparable, the KGE value is positive or close to zero; when the distributions are not comparable, the KGE value is negative.

In general, the endmembers of building roofing materials and vegetation (i.e., KGE values are positive) show histograms more comparable with the FAMs than those of street paving materials (i.e., KGE values are negative). In detail, the vegetation endmember shows the greatest value, followed by lead plates and then lateritic tiles; the trachyte rock endmember shows the smallest value, followed by limestone and then asphalt.

4. Discussion and Conclusions

The literature shows that the most widely employed methodology for mapping urban land cover by remote sensing imagery is spectral unmixing (e.g., [20]). However, many authors pointed out the many shortcomings in the spatial validation of spectral unmixing results, identifying the lack of detailed knowledge of the ground truth as the primary reason (e.g., [24]). Keshava and Mustard [4] stated that, for this reason, the main focus is the determination of endmembers (i.e., the first procedure of spectral unmixing), rather than the retrieval of fractional abundance maps (i.e., the second procedure). Therefore, detailed knowledge of ground truth data is the main challenge to improve the spatial validation of spectral unmixing results.

A careful review of the literature allowed us to identify five requirements for providing detailed knowledge of the ground truth (e.g., [37]). However, most proposed reference maps met two or three requirements and none met all five (Table 1). The present work fulfilled all of them to determine an accurate reference map of Venice. This site was chosen because, on the one hand, the urban pattern (e.g., [20]) and, on the other hand, the extensive in situ surveys [23,47] provide favorable conditions for determining an accurate reference map. The first and second requirements of the reference map were met by most of the authors (e.g., [18]): this map should cover most of study area and its spatial resolution should be higher than that of the image to be analyzed. However, these authors did not choose the spatial resolution to better characterize the range of abundances (i.e., the fourth requirement), in contrast to the present work, which purposely chose the resolution of the reference map used for this reason. The third requirement was met by a few authors (e.g., [36]): the reference map should be validated with other ground truth data. In this work, the starting reference map at the spatial resolution of 0.50 m [33] was validated with other ground truth data that were characterized by the spatial resolutions of 0.02 m with a maximum error of 10% [48,49]. These characteristics allowed us to create the reference map at the spatial resolution of 0.20 m that were exploited to validate the results obtained from the MIVIS image. This resolution was chosen to fulfill the fourth requirement that was considered by some authors (e.g., [12]): since the pixel values of fractional abundance maps vary from 100 to 0%, the pixels of the reference map should represent the same range of abundances to properly validate them. The literature shows two shared assumptions (e.g., [33]): starting with pixels with 100% abundance that are derived from a mask (i.e., where each pixel has 100% abundance) at high resolution sampled at lower resolution, (i) the abundances of the remaining pixels within the starting mask decrease linearly with increasing distance from the pixel with 100% abundance and (ii) the distance between pixel with 100% and those with 0% is equal to the spatial resolution of the sensor. Therefore, the reference map that met three out of five requirements allowed us to create reference fractional abundance maps where the range from 100 to 0% was sampled with 40 known values. Finally, the fifth requirement aims to minimize errors in co-localization and spatial resampling due to the comparison of different data at different spatial resolutions (i.e., remote data and reference maps) in the procedures of the spatial validation [33,42]. For this purpose, the retrieved and reference fractional abundance maps were not compared, but their histograms of fractional abundances were compared. It is important to note that the analysis of the histograms obtained from reference fractional abundance maps (i.e., FAMs) shows that the number of pixels decreases linearly as the abundance values decrease only for the endmembers characterized by large and non-jagged shapes. Therefore, the number of known value abundances should increase and the number of interpolated abundance values should decrease, since the linear correlation between the number of pixels and their abundances cannot be taken for granted.

The sources of error in the application of spectral unmixing are many (e.g., [28]). As pointed out by Keshava and Mustard [4], most authors spectrally validated the results by obtaining an overall accuracy in determining endmembers. Similarly, to spatially validate the spectral unmixing results, previous authors utilized only one reference datum, achieving an overall accuracy in the validation of fractional abundance maps. In contrast, more reference data provide not only an overall value of the spectral and spatial accuracy, but also other information that can help minimize error in the validation procedure. This work employed five data: (i) FAMs that met all the requirements; (ii) reference spectra; (iii) reference fractional abundance maps that met four out of five requirements; (iv) a synthetic image; (v) fractional abundance maps retrieved from the synthetic image. It is important to note that no previous work has exploited reference fractional abundance maps that meet four of the five requirements, and fractional abundance maps retrieved from synthetic images in order to obtain more information about the validation procedure. A validation method was associated with each reference datum employed: three spectral validation methods and two spatial validation methods were performed. Therefore, performing these validation methods provided not only the spatial and spectral accuracies, but also the percentages of these values due to the errors in co-localization and spatial resampling and the percentages due the spatial and spectral characteristics of the image, the accuracies of reference endmember set employed, and the spectral unmixing model applied (i.e., LMM or NLMM). The comparison of results obtained with FAMs and reference fractional abundance maps quantifies the percentages of these values due to errors in co-localization and spatial resampling. As regards the other percentages, many works successfully utilized synthetic images to test and validate spectral unmixing procedures and their results (e.g., [14]): each spectrum of the synthetic image is the weighted sum of the reference spectra of the endmembers, whose weights are determined by the reference fractional abundance maps. However, no work applied the same spectral unmixing procedure to synthetic and true remote sensing images and compared their results. However, their comparison quantifies the percentage of accuracy related to spatial and spectral characteristics of the remote sensor, the reference endmember employed, and the spectral unmixing model applied. Analysis of the results, evaluated from the MIVIS image of Venice by exploiting the five reference data, shows the following details on accuracy values:

• The average spectral accuracy evaluated with RMS_k values is equal to 0.025 and the sensor characteristics and the accuracy of the reference endmember set and LMM affect about of 73% of these values;
• The average spectral accuracies evaluated with the SAD_k and aSSMn_k values are equal to 0.345 and 0.595 and the sensor characteristics and the accuracy of the reference endmember set and LMM affect more than 82% of these values;
• The average spatial accuracy evaluated with MAE_k-Totals values by using FAMs is equal to 1.32 and the sensor characteristics and the accuracy of the reference endmember set and LMM affect about of 78% of these values;
• The average spatial accuracy evaluated with MAE_k-Totals values by using reference fractional abundance maps is equal to 1.97 and the sensor characteristics and the accuracy of the reference endmember set and LMM affect about of 58% and the errors in co-localization and spatial resampling affect about of 29% of these values.

In conclusion, this work, on the one hand, researched and implemented all requirements for increasing ground truth knowledge and, on the other hand, researched and exploited various reference data; the integrated use of these two research tracks increased spatial accuracy, improved the knowledge of accuracies, and upgraded the validation procedure, emphasizing the importance of detailed knowledge of ground truth and minimization of co-localization and spatial resampling error.

One important aspect is that any proposed methodology should be replicable by others and that its results should be comparable with those of others. In order to measure the spatial and spectral accuracy of the spectral unmixing results, many metrics were proposed, but their outcomes are not easily comparable with those obtained from other data, because there are many different features (e.g., sensor and site characteristics) that drive the results. For this purpose, spatial and spectral accuracies of each endmember were calculated by considering only those values of the pixels that fall within the corresponding reference mask, because analysing only the area in which the presence of the endmember is verified reduces the variables related to that site and facilitates comparison of the results obtained with other data.

Incidentally, meeting all the requirements and exploiting different references are essential to improve the accuracy of the results and the quality of the validation procedure, but they are time-consuming.

Increasing the spatial accuracy of fractional abundance maps and upgrading the validation procedure of spectral unmixing results are the main challenges for monitoring urban sprawl using remote sensing techniques. In order to test this proposed procedure, it is planned to apply it to data with the same characteristics acquired on the same or different urban sites and compare the results obtained, while continuing to analyse the literature to seek new requirements for accurate reference fractional maps and for minimizing the errors.