1. Introduction
The recent technological progress in hyperspectral imaging has led to the emergence of a gradually increasing source of spectral information showing the characteristics of hyperspectral image acquisition via numerous available spectral bands. However, many difficulties appear while extracting valuable data for the final user of different applications. The most important problem is the presence of mixed pixel [
1] occurring when the size of two or more classes of land cover classes may be larger than the pixel. This difficulty appears in case where the size of two land cover classes may be larger than the pixel size, but some parts of their boundaries are localized within a single pixel [
2]. As the traditional hard classification algorithms cannot be applied to solve mixed pixels problem [
3,
4], the best solution consists in using spectral unmixing [
5,
6] and fuzzy classification [
7] in order to specify the endmembers (classes) and their presence ratio, while their precise spatial localization cannot be determined. In 1997, sub-pixel mapping methods were first proposed by Atkinson [
8] to approximate the spatial locations at a sub-pixel scale from coarse spatial resolution hyperspectral data. These techniques are based on either the original hyperspectral image or use the findings obtained by soft classification (i.e., spectral unmixing) as input. They have been utilized in several fields, such as forestry [
9], water mapping [
10], burned areas [
11], target detection [
12], and rural land cover objects [
13].
In fact, SPM (Sub-Pixel Mapping) aims essentially at improving the images spatial resolution relying on spatial dependence assumption between and within pixels [
14]. Despite the importance of this method, a great effort has to be made to construct more reliable and more efficient approach in order to satisfy the requirements of this large number of applications. SPM methods presented in the literature can be classified into three categories [
15].
The first one includes techniques relying on spatial dependence assumption and utilizing, as input, a hyperspectral image. As instance of these methods, we can mention linear optimization [
16], pixel swapping algorithm [
17], techniques relying on spatial attraction at a sub-pixel scale [
18,
19], Cokriging indicator [
20], and Markov random field [
21,
22,
23], as well as artificial-intelligence based on some techniques, such as differential evolution [
24], genetic algorithm [
25], and methods relying on hopfield neural network [
26]. Indeed, technique belonging to the first class are generally applied to exploit the spatial correlation of the elements between and within a mixed pixel, based on the assumption that close elements have higher correlation than distant ones. In fact, they do not provide a single solution, but they depend on the initialization step. In Reference [
27], Arun et al. suggested a technique relying on pixel-affinity and semi-variogram. Besides, authors, in Reference [
20], presented a method using semi-variogram approximated from fine spatial resolution training images.
The second class involves techniques used to inject a priori knowledge in the form of additional data to improve SPM accuracy. To remedy the problem of insufficient information, several methods that use an additional source of information have been proposed in the literature. Authors in References [
28,
29] have presented methods that add additional data by considering some parameters, such as spatial resolution (through the integration of panchromatic images [
30] or fused images) and spatial shift [
14,
28,
31,
32].
The third class contains techniques relying on spatial prior model. These methods allow transforming sub-pixel mapping into an inverse well-posed problem with a unique solution to form a fine spatial resolution map. Researchers, in Reference [
33], developed a subpixel mapping technique relying on MAP (Maximum A Posteriori) model, and winner takes all as strategy to choose the adequate class that should be considered in the classification problem. Fend et al. [
34,
35] constructed also the DCT (Discret Cosine Transform) dictionary to obtain more sub-pixel configurations.
On the other hand, authors in Reference [
36] classified sub-pixel mapping techniques into three classes according to the size and shape of the object [
37], which can be zonal [
26,
38,
39], linear [
13], or a point [
40].
The previously-mentioned methods (first category), relying on the spatial dependence assumption, are based on the assumption that the near sub-pixels are more similar than the distant one, which is not always the case. Another limitation of these techniques is the problem of insufficient information at the sub-pixel level, which affects negatively the mapping accuracy. It is worth noting that the total variation, used as a regularization, cannot provide an optimal distribution of the classes at the sub-pixel level. In addition, the discrete cosine transform dictionary does not depend on the input data. Thus, its atoms cannot be adapted to the image and cannot be employed in all possible output configurations. Unlike this standard dictionary that can be employed with all input images, that proposed in our work is a spatial one that can be adapted to each input image in order to ensure a spatial modeling that is very close to reality and can provide more possible configurations at sub-pixel scale.
To solve this problem and to adapt the atoms of the dictionary to the utilized data set, we propose, in this study, the K-SVD (K-singular value decomposition) algorithm applied in image compression [
41] or feature extraction [
42]. We also depict the mixed pixel with a linear combination of atoms produced by the K-SVD algorithm without taking into account any relation between classes through various mixed pixels which representing the isotropic TV (Total Variation) [
43] used as a regularization constraint for the characterization of the relation among neighboring sub-pixels in all abundance cards. Our proposal based on sparse representation [
44,
45] allows also the transformation of the ill-posed sub-pixel mapping problem into a well-posed one, which makes possible the convergence of the algorithm into a unique minimum of the cost function having unique optimal solution.
The remaining of this manuscript is organized as follows:
Section 2 presents some mathematical notations. In
Section 3, we define the sub- mapping model.
Section 4 shows the features of the proposed K-SVD learning process.
Section 5 describes the introduced sub-pixel mapping technique relying on the learned dictionary and isotropic total variation minimization. Then, in
Section 5, we discuss the experimental results.
Section 6 is a short conclusion where we show our future works.
5. Discussion
The line graph (
Figure 13) depicts the change of the overall accuracy rate depending on the variable
. Four representative curves were gathered in the same graph, showing a comparative study of our approach with the three other existing methods (AMCDSM-TV, AMCDSM-L, ASSM-TV). The red curve represents the accuracy rates obtained applying the SMKSVD-TV, while the others show those provided by the three other techniques. It is clear that the overall accuracy increases sharply until it reaches a particular value. The optimal overall accuracy rates are equal to
,
,
, and
for AMCDSM-TV, AMCDSM-L, ASSM-TV, SMKSVD-TV, respectively. The mentioned value was achieved at an optimum lambda value. After that, each line shows a decreasing overall accuracy as the value of
rises.
Despite the fact that the different methods gave curves with similar trends, the curve obtained by applying our proposed SMKSVD-TV stands out among the other ones thanks to a very high optimum overall accuracy. Besides, the overall accuracy in the SMKSVD-TV method is always the highest whatever the value of lambda is.
Table 13 illustrates the average of the overall accuracy obtained using different sets of data and applying various methods. The experimental study uses three data sets (Jasper ridge, Pavia university, and urban hyperspectral images) to show the overall accuracy for each of the already-mentioned techniques. A comparative study of the different employed methods and data sets proves that the SMKSVD-TV technique provided the highest overall accuracy, compared to the three existing method, even when changing the used data.
This rate is , for Jasper ridge hyperspectral image; , for Urban image; and for Pavia university. With regard to the average of the overall accuracy equal to for the different data sets, the proposed SMKSVD-TV method gave the best value (). This average is , , and for AMCDSM-TV, AMCDSM-TV, and ASSM-TV, respectively. The above results attest that the SMKSVD-TV is very efficient in terms of overall accuracy.